Research ArticleGeneralized Finite Difference Time Domain Methodand Its Application to Acoustics

Jianguo Wei1 Song Wang1 Qingzhi Hou2 and Jianwu Dang23

1School of Computer Software Tianjin University Tianjin 300072 China2Tianjin Key Laboratory of Cognitive Computing and Application Tianjin University Tianjin 300072 China3Japan Advanced Institute of Science and Technology Ishikawa 923-1292 Japan

Correspondence should be addressed to Qingzhi Hou qhoutjueducn

Received 28 August 2014 Revised 11 January 2015 Accepted 11 January 2015

Academic Editor Shaofan Li

Copyright copy 2015 Jianguo Wei et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Ameshless generalized finite difference time domain (GFDTD)method is proposed and applied to transient acoustics to overcomedifficulties due to use of grids or mesh Inspired by the derivation of meshless particle methods the generalized finite differencemethod (GFDM) is reformulated utilizing Taylor series expansion It is in a way different from the conventional derivation ofGFDM in which a weighted energy norm was minimized The similarity and difference between GFDM and particle methodsare hence conveniently examined It is shown that GFDM has better performance than the modified smoothed particle method inapproximating the first- and second-order derivatives of 1D and 2D functions To solve acoustic wave propagation problems GFDMis used to approximate the spatial derivatives and the leap-frog scheme is used for time integration By analog with FDTD thewhole algorithm is referred to as GFDTD Examples in one- and two-dimensional domain with reflection and absorbing boundaryconditions are solved and good agreements with the FDTD reference solutions are observed even with irregular point distributionThe developed GFDTD method has advantages in solving wave propagation in domain with irregular and moving boundaries

1 Introduction

Partial differential equations (PDEs) modeling problems inscience and engineering such as electromagnetics acousticsand hydrodynamics are usually solved by numericalmethodsthat discretize the computational domain with mesh or gridsGrid-basedmethods such as finite differencemethod (FDM)finite elementmethod (FEM) and boundary elementmethod(BEM) [1 2] have had much achievements and still domi-nate the field of scientific computing However numericaldifficulties originating from usage of grids often emergeFor complicated and irregular geometry implementation ofboundary conditions could be a big challenge for FDMGeneration of grids with high quality is not an easy task inFEM and BEM Moreover when free surface and movingboundaryinterface have to be treated the transformation ofgrids will turn the conventional grid-based methods into adifficult time-consuming process Numerical accuracy oftendegenerates and divergence problem occurs

In recent 20 years to overcome numerical difficulties dueto use of grids or mesh meshless methods (MMs) based on

different techniques have been proposed and widely usedin many fields such as hydrodynamics [3] astrophysics [4]and solid mechanics [3 5] Among the MMs generalizedfinite difference method (GFDM) is the one that evolvedfrom traditional FDM [6 7] and many different formshave been developed [8] Benito and his coauthors madegreat contribution to its recent development [9ndash11] For heatconduction problem it has been compared with the element-free Garlerkin (EFG) method (one of the most used MMs insolid mechanics) and better performance has been observed[10] Recently GFDM was used to solve the wave equations[11] and Burgersrsquo equations [12] and simulate seismic wavepropagation problems in heterogeneousmedia [13] An appli-cation to the detonation shock dynamics [14] was also carriedout Nevertheless few work on computational acoustics hasbeen reported

For acoustic wave propagation problems the concentra-tion is on the ones in confined domain for which grid-based methods like FDTD and TDFEM (time-domain finite-element methods) [15] are mostly used However moving

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 640305 13 pageshttpdxdoiorg1011552015640305

2 Mathematical Problems in Engineering

boundary exists in many acoustic problems like sound wavepropagation inside a deforming vocal tract This problem ishardly solved by conventional grid-based methods andMMsprovide a possibility As one of the MMs GFDM is extendedto transient acoustics in this paper which is helpful to solvewave propagation problems with moving boundary in thefuture

Inspired by the derivation of meshless particle methodswe firstly formulated the GFDM in a way different from theoriginal one that minimizes an energy norm Such that therelationship between GFDM and meshless particle meth-ods like smoothed particle hydrodynamics (SPH) and itsimprovements can be conveniently examined Compari-son with the modified dmoothed particle hydrodynamics(MSPH) method which has better performances than SPHand its corrections [16] shows higher approximation accu-racy of the GFDM especially at the boundary region Byanalog with FDTD a method referred to as generalized finitedifference time domain (GFDTD) is proposed in whichGFDM is used to discretize the spatial operators and the leap-frog algorithm is used for time integration To show its goodperformance and efficiency the GFDTD method is appliedto transient acoustics Comparison with conventional FDTDsolutions is presented and discussed

2 Generalized Finite DifferenceMethod (GFDM)

Other than conventional derivation of GFDM byminimizingan energy norm [10] a different derivation of GFDM is pre-sented in this section Taylor series expansion of 119891(119909 119910)around point (119909

0 1199100) remaining up to second-order terms

yields

119891 asymp 1198910+ ℎ1205971198910

120597119909+ 1198961205971198910

120597119910+ℎ2

2

12059721198910

1205971199092+1198962

2

12059721198910

1205971199102+ ℎ119896

12059721198910

120597119909120597119910 (1)

where 119891 = 119891(119909 119910) 1198910= 119891(119909

0 1199100) ℎ = 119909 minus 119909

0 and 119896 =

119910 minus 1199100

By multiplying both sides of (1) with 1199082ℎ and 1199082119896 (119908 isa weighting function with compact support) and integrating

the resulted equations over the support domainΩ we get twoequations and the following as an example is the result for1199082ℎ

intΩ

1198911199082ℎ dV asymp intΩ

11989101199082ℎ dV + int

Ω

1205971198910

1205971199091199082ℎ2dV

+ intΩ

1205971198910

1205971199101199082ℎ119896 dV + 1

2intΩ

12059721198910

12059711990921199082ℎ3dV

+1

2intΩ

12059721198910

12059711991021199082ℎ1198962dV + int

Ω

12059721198910

1205971199091205971199101199082ℎ2119896 dV

(2)

where dV is a volume measureRepeating the same procedure with 1199082ℎ22 119908211989622 and

1199082ℎ119896 instead of 1199082ℎ and 1199082119896 we get other three equationsand the following is the result for 1199082ℎ22

intΩ

1198911199082ℎ2

2dV asymp int

Ω

1198910

1199082ℎ2

2dV + int

Ω

1205971198910

120597119909

1199082ℎ3

2dV

+ intΩ

1205971198910

120597119910

1199082ℎ2119896

2dV + 1

2intΩ

12059721198910

12059711990921199082ℎ4

2dV

+1

2intΩ

12059721198910

12059711991021199082ℎ21198962

2dV + int

Ω

12059721198910

120597119909120597119910

1199082ℎ3119896

2dV

(3)

To approximate the integrations byRiemann sum the volumeof the support domain Ω is divided into 119873 points withassociated volumes dV

119894 (119894 = 1 2 119873) Equations (2)

(3) and the other three constitute a system of five equationswritten in matrix form as

APDfP = bP (4)

with

AP =

(((((((((((((((((

(

119873

sum119894=1

1199082119894ℎ2119894dV119894

119873

sum119894=1

1199082119894ℎ119894119896119894dV119894

119873

sum119894=1

1199082119894

ℎ3119894

2dV119894

119873

sum119894=1

1199082119894

ℎ1198941198962119894

2dV119894

119873

sum119894=1

1199082119894ℎ2119894119896119894dV119894

119873

sum119894=1

1199082119894ℎ119894119896119894dV119894

119873

sum119894=1

11990821198941198962119894dV119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2dV119894

119873

sum119894=1

1199082119894

1198963119894

2dV119894

119873

sum119894=1

1199082119894ℎ1198941198962119894dV119894

119873

sum119894=1

1199082119894

ℎ3119894

2dV119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2dV119894

119873

sum119894=1

1199082119894119895

ℎ4119894

4dV119894

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4dV119894

119873

sum119894=1

1199082119894

ℎ3119894ℎ119894

2dV119894

119873

sum119894=1

1199082119894119895

ℎ1198941198962119894

2dV119894

119873

sum119894=1

1199082119894

1198963119894

2dV119894

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4dV119894

119873

sum119894=1

1199082119894

1198964119894

4dV119894

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2dV119894

119873

sum119894=1

1199082119894ℎ2119894119896119894dV119894

119873

sum119894=1

1199082119894ℎ1198941198962119894dV119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2dV119894

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2dV119894

119873

sum119894=1

1199082119894ℎ21198941198962119894dV119894

)))))))))))))))))

)

Mathematical Problems in Engineering 3

DfP =

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

bP =

(((((((((((((((

(

minus1198910

119873

sum119894=1

1199082119894ℎ119894dV119894+119873

sum119894=1

1198911198941199082119894ℎ119894dV119894

minus1198910

119873

sum119894=1

1199082119894119896119894dV119894+119873

sum119894=1

1198911198941199082119894119896119894dV119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2dV119894+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2dV119894

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2dV119894+119873

sum119894=1

1198911198941199082119894

1198962119894

2dV119894

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894dV119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894dV119894

)))))))))))))))

)

(5)

where119908119894= 119908(119889

119894 119889119898)with 119889

119894= radic(119909

119894minus 1199090)2 + (119910

119894minus 1199100)2 ℎ119894=

119909119894minus 1199090 and 119896

119894= 119910119894minus 1199100 and 119889

119898is a measure of the support

sizeThe conventional derivation of GFDM is presented in

appendix It is clear that the difference between the conven-tional and the current derivation is not only the procedure butalso the final form The conventional derivation loses termdV119894(see (A5)) If all the points in the domain have the same

volume dV119894at both sides of (4) will be cancelled and the two

final forms will be the same However dV119894can hardly be the

same when points are irregularly spaced From this point ofview our derived final form is more general and takes pointirregularity into account

3 Modified Smoothed ParticleHydrodynamics (MSPH)

As a modification to SPH the MSPH method improves theaccuracy of the approximations especially at points near theboundary of the domain [16] It uses Taylor series expansionof function 119891(119909 119910) as in (1) Similar to the derivations of (2)and (3) but with different weight functions 120597119908120597119909 120597119908120597119910

12059711990821205971199092 12059711990821205971199102 and 1205972119908120597119909120597119910 the following equationsas examples for 120597119908120597119909 and 12059711990821205971199092 are obtained

intΩ

119891120597119908

120597119909dV = int

Ω

1198910

120597119908

120597119909ℎ dV +

1205971198910

120597119909intΩ

120597119908

120597119909ℎ dV

+1205971198910

120597119910intΩ

120597119908

120597119909119896 dV + 1

2

12059721198910

1205971199092intΩ

120597119908

120597119909ℎ2dV

+1

2

12059721198910

1205971199102intΩ

120597119908

1205971199091198962dV +

12059721198910

120597119909120597119910intΩ

120597119908

120597119909ℎ119896 dV

intΩ

1198911205971199082

1205971199092dV = int

Ω

1198910

1205971199082

1205971199092dV +

120597119891

120597119909intΩ

1205971199082

1205971199092ℎ dV

+120597119891

120597119910intΩ

1205971199082

1205971199092119896 dV + 1

2

12059721198910

1205971199092intΩ

1205971199082

1205971199092ℎ2dV

+1

2

12059721198910

1205971199102intΩ

1205971199082

12059711990921198962dV +

12059721198910

120597119909120597119910intΩ

1205971199082

1205971199092ℎ119896 dV

(6)

Again the Riemann sum over the support domain Ω is usedto approximate the integrations and a system of five equationsis obtained as

(((((((((((((((

(

119873

sum119894=1

120597119908119894

120597119909ℎ119894dV119894

119873

sum119894=1

120597119908119894

120597119909119896119894dV119894

119873

sum119894=1

120597119908119894

120597119909

ℎ2119894

2dV119894

119873

sum119894=1

120597119908119894

120597119909

1198962119894

2dV119894

119873

sum119894=1

120597119908119894

120597119909ℎ119894119896119894dV119894

119873

sum119894=1

120597119908119894

120597119910ℎ119894dV119894

119873

sum119894=1

120597119908119894

120597119910119896119894dV119894

119873

sum119894=1

120597119908119894

120597119910

ℎ2119894

2dV119894

119873

sum119894=1

120597119908119894

120597119910

1198962119894

2dV119894

119873

sum119894=1

120597119908119894

120597119910ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

12059711990921198962119894

2dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

12059711991021198962119894

2dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910ℎ119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910119896119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910

ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910

1198962119894

2dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910ℎ119894119896119894dV119894

)))))))))))))))

)

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

4 Mathematical Problems in Engineering

=

(((((((((((((((

(

minus1198910

119873

sum119894=1

120597119908119894

120597119909dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119909dV119894

minus1198910

119873

sum119894=1

120597119908119894

120597119910dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119910dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199092dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199092dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199102dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199102dV119894

minus1198910

119873

sum119894=1

1205971199082119894

120597119909120597119910dV119894+119873

sum119894=1

119891119894

1205971199082119894

120597119909120597119910dV119894

)))))))))))))))

)

(7)

Compared with formula (4) the only difference is theterms multiplied to both sides of (1) In GFDM 1199082ℎ 11990821198961199082ℎ22 119908211989622 and 1199082ℎ119896 are used instead of 120597119908120597119909120597119908120597119910 12059711990821205971199092 12059711990821205971199102 and 1205972119908120597119909120597119910 in MSPH As aresult GFDM avoids computing the derivatives of the weightfunction and hence saves computational efforts and leads tomore choice of the weight function

4 Numerical Tests forApproximation of Derivatives

In previous sections the deviation of GFDM and MSPH ispresented In this section to compare the performance of thetwo methods they are used to approximate the derivativesof certain 1D and 2D functions For the convenience ofevaluation a global error measure is defined as follows

Error119906=

1

|119906|maxradic1

119873

119873

sum119894=1

(119906(119890)119894minus 119906(119899)119894)2

(8)

where 119906 can be 120597119891120597119909 120597119891120597119910 12059721198911205971199092 and 12059721198911205971199102 andthe superscripts (119890) and (119899) refer to the exact and numericalsolutions respectively

The quartic spline function is used as the weight function119908119894

119908119894(119889) =

1 minus 6(119889

119889119898

)2

+ 8(119889

119889119898

)3

minus 3(119889

119889119898

)4

119889 le 119889119898

0 119889 gt 119889119898

(9)

where 119889119898is the kernel radius taken as 21Δ119909 (Δ119909 is the space

interval) which is usually used in meshless methods

41 One-Dimensional Case Consider the following function

119891 (119909) = (119909 minus 05)4 119909 isin [0 1] (10)

Figure 1 shows the first- and second-order derivatives esti-mated by GFDM and MSPH and the exact results when

the domain is discretized into 21 equally spaced points It isseen that GFDM has better performance in both derivativesespecially for the points near boundaries When the numberof points increases to 51 the results are similar as exhibited inFigure 2 Error analysis shown in Table 1 indicates that GFDMhas higher accuracy With increasing number of points theglobal error decreases

42 Two-Dimensional Case For the function

119891 (119909 119910) = sin120587119909 sin120587119910 119909 119910 isin [0 1] times [0 1] (11)

its first- and second-order derivatives together with estima-tions by GFDM and MSPH are shown in Figure 3 In eachdirection 21 points are employed As expected GFDM hashigher approximation accuracy than MSPH for both first-and second-order derivatives as shown in Table 2

5 Generalized Finite Difference Time DomainMethod for Computational Acoustics

For computational acoustics the mostly used approach isthe FDTD method which was originally designed for thesimulation of electromagnetics [1 2] As a finite differencescheme its applicability to complex problems suffers fromaforementioned difficulties for which the generalized finitedifference can be a good alternative In this section togetherwith the basics of computational acoustics a meshlessmethod is proposed in which GFDM is used to discretizethe spatial derivatives and the leap-frog algorithm is used todiscretize the temporal derivatives By analog with FDTDit is referred to as generalized finite difference time domain(GFDTD) method and is expected to have advantages due toits meshless property

The governing equations for acoustic wave propagationproblems are

1205880

120597k120597119905= minusnabla119901

1

11988820

120597119901

120597119905= minus1205880nabla sdot k

(12)

Mathematical Problems in Engineering 5

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 1 Estimates of the first derivative (a) and second derivative (b) of function 119891(119909) = (119909 minus 05)4 with 21 equally spaced points on [0 1]

Table 1 Approximation errors in the derivatives of the function119891(119909) = (119909 minus 05)4

Error () 120597119891120597119909 12059721198911205971199092

GFDM with 21 points 077 581MSPH with 21 points 077 602GFDM with 51 points 011 155MSPH with 51 points 011 161

Table 2 Approximation errors in the derivatives of the function119891(119909 119910) = sin120587119909 sin120587119910

Error () 120597119891120597119909 120597119891120597119910 12059721198911205971199092 12059721198911205971199102

GFDM 029 029 287 287MSPH 047 047 471 471

where 119901 is pressure k is particle velocity 1205880is the density of

the medium and 1198880is the speed of sound

51 Spatial Derivative Approximations by GFDM The spatialderivatives on the right-hand side of (12) are approximatedby GFDM By solving (4) we get the approximations of 120597119891120597119909and 120597119891120597119910 That is the derivatives of variable 119891 at point(1199090 1199100) can be approximated by function values at points

inside the support domain centered at (1199090 1199100) as

1205971198910

120597119909= minus11989801198910+119873

sum119894=1

119891119894119898119894

1205971198910

120597119910= minus12057801198910+119873

sum119894=1

119891119894120578119894

(13)

where 1198980and 120578

0are the difference coefficients for center

point and 119898119894and 120578

119894are coefficients for other points in the

support domain AsGFDMcan reproduce constant functions[4] we have

1198980=119873

sum119894=1

119898119894 120578

0=119873

sum119894=1

120578119894 (14)

By taking both 119901 and k in (12) as 119891 the approximated spatialoperators in (12) are accordingly obtained

52 Explicit Leap-Frog Scheme in GFDTD Generally thetemporal derivatives on the left-hand side of (12) can beintegrated by any time marching algorithms Inspired bythe conventional FDTD method the second-order accurateexplicit leap-frog scheme is used herein in which twovariables 119901 and k are alternatively calculated The velocity iscomputed at the half time step and the pressure is calculatedat the integer time step [2] After temporal approximationsthe semi-discretization of (12) becomes

1205880

k119899+12 minus k119899minus12

Δ119905= minusnabla119901119899

1

11988820

119901119899+1 minus 119901119899

Δ119905= minus1205880nabla sdot k119899+12

(15)

where superscript 119899 represents the time step The leap-frogscheme is conditionally stable and the time step Δ119905 shouldsatisfy the Courant-Friedrichs-Lewy (CFL) condition that isΔ119905 le Δ119909(radicdim sdot 119888

0) where dim is the dimension of the

problem

6 Mathematical Problems in Engineering

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 2 Estimates of the first derivative (a) and second derivative (b) of function119891(119909) = (119909 minus 05)4 with 51 equally spaced points on [0 1]

Substituting (13) into the right-hand side of (15) as theapproximation of the first-order spatial derivatives the fulldiscretized system of equations becomes

k119899+120

= k119899minus120

+Δ119905

1205880

sdot [(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894) (120578

01199011198990minus119873

sum119894=1

120578119894119901119899119894)]

119879

119901119899+10= 1199011198990+ Δ119905120588011988820

sdot [(1198980119906119899+120

minus119873

sum119894=1

119898119894119906119899+12119894

)

+(1205780V119899+120

minus119873

sum119894=1

120578119894V119899+12119894

)]

(16)

where the particle velocity is a 2D vector k = [119906 V]119879By analog with FDTD the full discretization scheme

is referred to as generalized finite difference time domain(GFDTD) method

6 Numerical Results

To validate the proposed GFDTD method it is appliedto one- and two-dimensional wave propagation problemsThree cases are presented The first two examine the acousticwave propagation in one- and two-dimensional domainrespectively and the third one is a real case with differenttypes of boundary conditions All the cases use (9) as theweight function The other parameters are 120588

0= 1 kgm3 and

1198880= 3464ms and the time interval Δ119905 is set to be 1 120583s to

satisfy the CFL condition FDTD solutions are chosen as thereference

61 One-Dimensional Case In this case 501 points are equallyspaced in the domain [0 1] In themiddle of it there is a wavesource in Gaussian pulse form

gp (119909) = 119890minus25|119909minus05| (17)

As shown in Figure 4 the simulated results at two timelevels 119905 = 250 120583s and 500120583s have good agreement with theFDTD solutions Compared with FDTD the relative errorsconcerning the pressure 119875FDTD and 119875GFDTD are

Error250=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0267 sdot 10minus2

Error500=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0519 sdot 10minus2

(18)

62 Two-Dimensional Case The Gaussian wave propagationin two-dimensional domain is simulated in this section Thelength of the square domain is 01m and 101 points areuniformly distributed in each direction The wave sourcestarts from the middle of the domain Figure 5 compares thesolutions of FDTDandGFDTDat 119905 = 20 120583sThe results along119910 = 005m are shown in Figure 5(c) Again good agreementis observed and the relative error is less than 2

To show the advantage of the proposed GFDTD over theconventional FDTD irregular point distribution is examinedAll the points used above are allowed to have plusmn10 per-turbation around their original locations to make the distri-bution irregular part of which is shown in Figure 6(a) and

Mathematical Problems in Engineering 7

GFDM

MSPH

Exact

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

(a)

GFDM

MSPH

Exact

x

y

z

2

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

2

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

(b)

Figure 3 Exact (top) and estimated (a) 120597119891120597119909 and (b) 12059721198911205971199092 of function 119891(119909 119910) = sin120587119909 sin120587119910 by GFDM (middle) and MSPH (bottom)

the result is shown in Figure 6(b) The comparison of theresult along 119910 = 005m shown in Figure 6(c) indicatesthat with irregular distribution of computational points theGaussian wave propagates as well as before

In the GFDTD simulation of wave propagation withirregular point distribution the volume associated to each

point had better to be considered as analyzed at the end ofSection 2 In 2D case the volume associated with a givenpoint is the area that the point dominates Here we useDelaunay triangulation andVoronoi diagram [17] to calculatethe area and the results are shown in Figure 7 The volume ofeach point is shown in Figure 7(a) Due to the designed plusmn10

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

boundary exists in many acoustic problems like sound wavepropagation inside a deforming vocal tract This problem ishardly solved by conventional grid-based methods andMMsprovide a possibility As one of the MMs GFDM is extendedto transient acoustics in this paper which is helpful to solvewave propagation problems with moving boundary in thefuture

Inspired by the derivation of meshless particle methodswe firstly formulated the GFDM in a way different from theoriginal one that minimizes an energy norm Such that therelationship between GFDM and meshless particle meth-ods like smoothed particle hydrodynamics (SPH) and itsimprovements can be conveniently examined Compari-son with the modified dmoothed particle hydrodynamics(MSPH) method which has better performances than SPHand its corrections [16] shows higher approximation accu-racy of the GFDM especially at the boundary region Byanalog with FDTD a method referred to as generalized finitedifference time domain (GFDTD) is proposed in whichGFDM is used to discretize the spatial operators and the leap-frog algorithm is used for time integration To show its goodperformance and efficiency the GFDTD method is appliedto transient acoustics Comparison with conventional FDTDsolutions is presented and discussed

2 Generalized Finite DifferenceMethod (GFDM)

Other than conventional derivation of GFDM byminimizingan energy norm [10] a different derivation of GFDM is pre-sented in this section Taylor series expansion of 119891(119909 119910)around point (119909

0 1199100) remaining up to second-order terms

yields

119891 asymp 1198910+ ℎ1205971198910

120597119909+ 1198961205971198910

120597119910+ℎ2

2

12059721198910

1205971199092+1198962

2

12059721198910

1205971199102+ ℎ119896

12059721198910

120597119909120597119910 (1)

where 119891 = 119891(119909 119910) 1198910= 119891(119909

0 1199100) ℎ = 119909 minus 119909

0 and 119896 =

119910 minus 1199100

By multiplying both sides of (1) with 1199082ℎ and 1199082119896 (119908 isa weighting function with compact support) and integrating

the resulted equations over the support domainΩ we get twoequations and the following as an example is the result for1199082ℎ

intΩ

1198911199082ℎ dV asymp intΩ

11989101199082ℎ dV + int

Ω

1205971198910

1205971199091199082ℎ2dV

+ intΩ

1205971198910

1205971199101199082ℎ119896 dV + 1

2intΩ

12059721198910

12059711990921199082ℎ3dV

+1

2intΩ

12059721198910

12059711991021199082ℎ1198962dV + int

Ω

12059721198910

1205971199091205971199101199082ℎ2119896 dV

(2)

where dV is a volume measureRepeating the same procedure with 1199082ℎ22 119908211989622 and

1199082ℎ119896 instead of 1199082ℎ and 1199082119896 we get other three equationsand the following is the result for 1199082ℎ22

intΩ

1198911199082ℎ2

2dV asymp int

Ω

1198910

1199082ℎ2

2dV + int

Ω

1205971198910

120597119909

1199082ℎ3

2dV

+ intΩ

1205971198910

120597119910

1199082ℎ2119896

2dV + 1

2intΩ

12059721198910

12059711990921199082ℎ4

2dV

+1

2intΩ

12059721198910

12059711991021199082ℎ21198962

2dV + int

Ω

12059721198910

120597119909120597119910

1199082ℎ3119896

2dV

(3)

To approximate the integrations byRiemann sum the volumeof the support domain Ω is divided into 119873 points withassociated volumes dV

119894 (119894 = 1 2 119873) Equations (2)

(3) and the other three constitute a system of five equationswritten in matrix form as

APDfP = bP (4)

with

AP =

(((((((((((((((((

(

119873

sum119894=1

1199082119894ℎ2119894dV119894

119873

sum119894=1

1199082119894ℎ119894119896119894dV119894

119873

sum119894=1

1199082119894

ℎ3119894

2dV119894

119873

sum119894=1

1199082119894

ℎ1198941198962119894

2dV119894

119873

sum119894=1

1199082119894ℎ2119894119896119894dV119894

119873

sum119894=1

1199082119894ℎ119894119896119894dV119894

119873

sum119894=1

11990821198941198962119894dV119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2dV119894

119873

sum119894=1

1199082119894

1198963119894

2dV119894

119873

sum119894=1

1199082119894ℎ1198941198962119894dV119894

119873

sum119894=1

1199082119894

ℎ3119894

2dV119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2dV119894

119873

sum119894=1

1199082119894119895

ℎ4119894

4dV119894

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4dV119894

119873

sum119894=1

1199082119894

ℎ3119894ℎ119894

2dV119894

119873

sum119894=1

1199082119894119895

ℎ1198941198962119894

2dV119894

119873

sum119894=1

1199082119894

1198963119894

2dV119894

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4dV119894

119873

sum119894=1

1199082119894

1198964119894

4dV119894

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2dV119894

119873

sum119894=1

1199082119894ℎ2119894119896119894dV119894

119873

sum119894=1

1199082119894ℎ1198941198962119894dV119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2dV119894

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2dV119894

119873

sum119894=1

1199082119894ℎ21198941198962119894dV119894

)))))))))))))))))

)

Mathematical Problems in Engineering 3

DfP =

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

bP =

(((((((((((((((

(

minus1198910

119873

sum119894=1

1199082119894ℎ119894dV119894+119873

sum119894=1

1198911198941199082119894ℎ119894dV119894

minus1198910

119873

sum119894=1

1199082119894119896119894dV119894+119873

sum119894=1

1198911198941199082119894119896119894dV119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2dV119894+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2dV119894

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2dV119894+119873

sum119894=1

1198911198941199082119894

1198962119894

2dV119894

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894dV119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894dV119894

)))))))))))))))

)

(5)

where119908119894= 119908(119889

119894 119889119898)with 119889

119894= radic(119909

119894minus 1199090)2 + (119910

119894minus 1199100)2 ℎ119894=

119909119894minus 1199090 and 119896

119894= 119910119894minus 1199100 and 119889

119898is a measure of the support

sizeThe conventional derivation of GFDM is presented in

appendix It is clear that the difference between the conven-tional and the current derivation is not only the procedure butalso the final form The conventional derivation loses termdV119894(see (A5)) If all the points in the domain have the same

volume dV119894at both sides of (4) will be cancelled and the two

final forms will be the same However dV119894can hardly be the

same when points are irregularly spaced From this point ofview our derived final form is more general and takes pointirregularity into account

3 Modified Smoothed ParticleHydrodynamics (MSPH)

As a modification to SPH the MSPH method improves theaccuracy of the approximations especially at points near theboundary of the domain [16] It uses Taylor series expansionof function 119891(119909 119910) as in (1) Similar to the derivations of (2)and (3) but with different weight functions 120597119908120597119909 120597119908120597119910

12059711990821205971199092 12059711990821205971199102 and 1205972119908120597119909120597119910 the following equationsas examples for 120597119908120597119909 and 12059711990821205971199092 are obtained

intΩ

119891120597119908

120597119909dV = int

Ω

1198910

120597119908

120597119909ℎ dV +

1205971198910

120597119909intΩ

120597119908

120597119909ℎ dV

+1205971198910

120597119910intΩ

120597119908

120597119909119896 dV + 1

2

12059721198910

1205971199092intΩ

120597119908

120597119909ℎ2dV

+1

2

12059721198910

1205971199102intΩ

120597119908

1205971199091198962dV +

12059721198910

120597119909120597119910intΩ

120597119908

120597119909ℎ119896 dV

intΩ

1198911205971199082

1205971199092dV = int

Ω

1198910

1205971199082

1205971199092dV +

120597119891

120597119909intΩ

1205971199082

1205971199092ℎ dV

+120597119891

120597119910intΩ

1205971199082

1205971199092119896 dV + 1

2

12059721198910

1205971199092intΩ

1205971199082

1205971199092ℎ2dV

+1

2

12059721198910

1205971199102intΩ

1205971199082

12059711990921198962dV +

12059721198910

120597119909120597119910intΩ

1205971199082

1205971199092ℎ119896 dV

(6)

Again the Riemann sum over the support domain Ω is usedto approximate the integrations and a system of five equationsis obtained as

(((((((((((((((

(

119873

sum119894=1

120597119908119894

120597119909ℎ119894dV119894

119873

sum119894=1

120597119908119894

120597119909119896119894dV119894

119873

sum119894=1

120597119908119894

120597119909

ℎ2119894

2dV119894

119873

sum119894=1

120597119908119894

120597119909

1198962119894

2dV119894

119873

sum119894=1

120597119908119894

120597119909ℎ119894119896119894dV119894

119873

sum119894=1

120597119908119894

120597119910ℎ119894dV119894

119873

sum119894=1

120597119908119894

120597119910119896119894dV119894

119873

sum119894=1

120597119908119894

120597119910

ℎ2119894

2dV119894

119873

sum119894=1

120597119908119894

120597119910

1198962119894

2dV119894

119873

sum119894=1

120597119908119894

120597119910ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

12059711990921198962119894

2dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

12059711991021198962119894

2dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910ℎ119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910119896119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910

ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910

1198962119894

2dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910ℎ119894119896119894dV119894

)))))))))))))))

)

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

4 Mathematical Problems in Engineering

=

(((((((((((((((

(

minus1198910

119873

sum119894=1

120597119908119894

120597119909dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119909dV119894

minus1198910

119873

sum119894=1

120597119908119894

120597119910dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119910dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199092dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199092dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199102dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199102dV119894

minus1198910

119873

sum119894=1

1205971199082119894

120597119909120597119910dV119894+119873

sum119894=1

119891119894

1205971199082119894

120597119909120597119910dV119894

)))))))))))))))

)

(7)

Compared with formula (4) the only difference is theterms multiplied to both sides of (1) In GFDM 1199082ℎ 11990821198961199082ℎ22 119908211989622 and 1199082ℎ119896 are used instead of 120597119908120597119909120597119908120597119910 12059711990821205971199092 12059711990821205971199102 and 1205972119908120597119909120597119910 in MSPH As aresult GFDM avoids computing the derivatives of the weightfunction and hence saves computational efforts and leads tomore choice of the weight function

4 Numerical Tests forApproximation of Derivatives

In previous sections the deviation of GFDM and MSPH ispresented In this section to compare the performance of thetwo methods they are used to approximate the derivativesof certain 1D and 2D functions For the convenience ofevaluation a global error measure is defined as follows

Error119906=

1

|119906|maxradic1

119873

119873

sum119894=1

(119906(119890)119894minus 119906(119899)119894)2

(8)

where 119906 can be 120597119891120597119909 120597119891120597119910 12059721198911205971199092 and 12059721198911205971199102 andthe superscripts (119890) and (119899) refer to the exact and numericalsolutions respectively

The quartic spline function is used as the weight function119908119894

119908119894(119889) =

1 minus 6(119889

119889119898

)2

+ 8(119889

119889119898

)3

minus 3(119889

119889119898

)4

119889 le 119889119898

0 119889 gt 119889119898

(9)

where 119889119898is the kernel radius taken as 21Δ119909 (Δ119909 is the space

interval) which is usually used in meshless methods

41 One-Dimensional Case Consider the following function

119891 (119909) = (119909 minus 05)4 119909 isin [0 1] (10)

Figure 1 shows the first- and second-order derivatives esti-mated by GFDM and MSPH and the exact results when

the domain is discretized into 21 equally spaced points It isseen that GFDM has better performance in both derivativesespecially for the points near boundaries When the numberof points increases to 51 the results are similar as exhibited inFigure 2 Error analysis shown in Table 1 indicates that GFDMhas higher accuracy With increasing number of points theglobal error decreases

42 Two-Dimensional Case For the function

119891 (119909 119910) = sin120587119909 sin120587119910 119909 119910 isin [0 1] times [0 1] (11)

its first- and second-order derivatives together with estima-tions by GFDM and MSPH are shown in Figure 3 In eachdirection 21 points are employed As expected GFDM hashigher approximation accuracy than MSPH for both first-and second-order derivatives as shown in Table 2

5 Generalized Finite Difference Time DomainMethod for Computational Acoustics

For computational acoustics the mostly used approach isthe FDTD method which was originally designed for thesimulation of electromagnetics [1 2] As a finite differencescheme its applicability to complex problems suffers fromaforementioned difficulties for which the generalized finitedifference can be a good alternative In this section togetherwith the basics of computational acoustics a meshlessmethod is proposed in which GFDM is used to discretizethe spatial derivatives and the leap-frog algorithm is used todiscretize the temporal derivatives By analog with FDTDit is referred to as generalized finite difference time domain(GFDTD) method and is expected to have advantages due toits meshless property

The governing equations for acoustic wave propagationproblems are

1205880

120597k120597119905= minusnabla119901

1

11988820

120597119901

120597119905= minus1205880nabla sdot k

(12)

Mathematical Problems in Engineering 5

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 1 Estimates of the first derivative (a) and second derivative (b) of function 119891(119909) = (119909 minus 05)4 with 21 equally spaced points on [0 1]

Table 1 Approximation errors in the derivatives of the function119891(119909) = (119909 minus 05)4

Error () 120597119891120597119909 12059721198911205971199092

GFDM with 21 points 077 581MSPH with 21 points 077 602GFDM with 51 points 011 155MSPH with 51 points 011 161

Table 2 Approximation errors in the derivatives of the function119891(119909 119910) = sin120587119909 sin120587119910

Error () 120597119891120597119909 120597119891120597119910 12059721198911205971199092 12059721198911205971199102

GFDM 029 029 287 287MSPH 047 047 471 471

where 119901 is pressure k is particle velocity 1205880is the density of

the medium and 1198880is the speed of sound

51 Spatial Derivative Approximations by GFDM The spatialderivatives on the right-hand side of (12) are approximatedby GFDM By solving (4) we get the approximations of 120597119891120597119909and 120597119891120597119910 That is the derivatives of variable 119891 at point(1199090 1199100) can be approximated by function values at points

inside the support domain centered at (1199090 1199100) as

1205971198910

120597119909= minus11989801198910+119873

sum119894=1

119891119894119898119894

1205971198910

120597119910= minus12057801198910+119873

sum119894=1

119891119894120578119894

(13)

where 1198980and 120578

0are the difference coefficients for center

point and 119898119894and 120578

119894are coefficients for other points in the

support domain AsGFDMcan reproduce constant functions[4] we have

1198980=119873

sum119894=1

119898119894 120578

0=119873

sum119894=1

120578119894 (14)

By taking both 119901 and k in (12) as 119891 the approximated spatialoperators in (12) are accordingly obtained

52 Explicit Leap-Frog Scheme in GFDTD Generally thetemporal derivatives on the left-hand side of (12) can beintegrated by any time marching algorithms Inspired bythe conventional FDTD method the second-order accurateexplicit leap-frog scheme is used herein in which twovariables 119901 and k are alternatively calculated The velocity iscomputed at the half time step and the pressure is calculatedat the integer time step [2] After temporal approximationsthe semi-discretization of (12) becomes

1205880

k119899+12 minus k119899minus12

Δ119905= minusnabla119901119899

1

11988820

119901119899+1 minus 119901119899

Δ119905= minus1205880nabla sdot k119899+12

(15)

where superscript 119899 represents the time step The leap-frogscheme is conditionally stable and the time step Δ119905 shouldsatisfy the Courant-Friedrichs-Lewy (CFL) condition that isΔ119905 le Δ119909(radicdim sdot 119888

0) where dim is the dimension of the

problem

6 Mathematical Problems in Engineering

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 2 Estimates of the first derivative (a) and second derivative (b) of function119891(119909) = (119909 minus 05)4 with 51 equally spaced points on [0 1]

Substituting (13) into the right-hand side of (15) as theapproximation of the first-order spatial derivatives the fulldiscretized system of equations becomes

k119899+120

= k119899minus120

+Δ119905

1205880

sdot [(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894) (120578

01199011198990minus119873

sum119894=1

120578119894119901119899119894)]

119879

119901119899+10= 1199011198990+ Δ119905120588011988820

sdot [(1198980119906119899+120

minus119873

sum119894=1

119898119894119906119899+12119894

)

+(1205780V119899+120

minus119873

sum119894=1

120578119894V119899+12119894

)]

(16)

where the particle velocity is a 2D vector k = [119906 V]119879By analog with FDTD the full discretization scheme

is referred to as generalized finite difference time domain(GFDTD) method

6 Numerical Results

To validate the proposed GFDTD method it is appliedto one- and two-dimensional wave propagation problemsThree cases are presented The first two examine the acousticwave propagation in one- and two-dimensional domainrespectively and the third one is a real case with differenttypes of boundary conditions All the cases use (9) as theweight function The other parameters are 120588

0= 1 kgm3 and

1198880= 3464ms and the time interval Δ119905 is set to be 1 120583s to

satisfy the CFL condition FDTD solutions are chosen as thereference

61 One-Dimensional Case In this case 501 points are equallyspaced in the domain [0 1] In themiddle of it there is a wavesource in Gaussian pulse form

gp (119909) = 119890minus25|119909minus05| (17)

As shown in Figure 4 the simulated results at two timelevels 119905 = 250 120583s and 500120583s have good agreement with theFDTD solutions Compared with FDTD the relative errorsconcerning the pressure 119875FDTD and 119875GFDTD are

Error250=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0267 sdot 10minus2

Error500=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0519 sdot 10minus2

(18)

62 Two-Dimensional Case The Gaussian wave propagationin two-dimensional domain is simulated in this section Thelength of the square domain is 01m and 101 points areuniformly distributed in each direction The wave sourcestarts from the middle of the domain Figure 5 compares thesolutions of FDTDandGFDTDat 119905 = 20 120583sThe results along119910 = 005m are shown in Figure 5(c) Again good agreementis observed and the relative error is less than 2

To show the advantage of the proposed GFDTD over theconventional FDTD irregular point distribution is examinedAll the points used above are allowed to have plusmn10 per-turbation around their original locations to make the distri-bution irregular part of which is shown in Figure 6(a) and

Mathematical Problems in Engineering 7

GFDM

MSPH

Exact

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

(a)

GFDM

MSPH

Exact

x

y

z

2

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

2

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

(b)

Figure 3 Exact (top) and estimated (a) 120597119891120597119909 and (b) 12059721198911205971199092 of function 119891(119909 119910) = sin120587119909 sin120587119910 by GFDM (middle) and MSPH (bottom)

the result is shown in Figure 6(b) The comparison of theresult along 119910 = 005m shown in Figure 6(c) indicatesthat with irregular distribution of computational points theGaussian wave propagates as well as before

In the GFDTD simulation of wave propagation withirregular point distribution the volume associated to each

point had better to be considered as analyzed at the end ofSection 2 In 2D case the volume associated with a givenpoint is the area that the point dominates Here we useDelaunay triangulation andVoronoi diagram [17] to calculatethe area and the results are shown in Figure 7 The volume ofeach point is shown in Figure 7(a) Due to the designed plusmn10

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

DfP =

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

bP =

(((((((((((((((

(

minus1198910

119873

sum119894=1

1199082119894ℎ119894dV119894+119873

sum119894=1

1198911198941199082119894ℎ119894dV119894

minus1198910

119873

sum119894=1

1199082119894119896119894dV119894+119873

sum119894=1

1198911198941199082119894119896119894dV119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2dV119894+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2dV119894

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2dV119894+119873

sum119894=1

1198911198941199082119894

1198962119894

2dV119894

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894dV119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894dV119894

)))))))))))))))

)

(5)

where119908119894= 119908(119889

119894 119889119898)with 119889

119894= radic(119909

119894minus 1199090)2 + (119910

119894minus 1199100)2 ℎ119894=

119909119894minus 1199090 and 119896

119894= 119910119894minus 1199100 and 119889

119898is a measure of the support

sizeThe conventional derivation of GFDM is presented in

appendix It is clear that the difference between the conven-tional and the current derivation is not only the procedure butalso the final form The conventional derivation loses termdV119894(see (A5)) If all the points in the domain have the same

volume dV119894at both sides of (4) will be cancelled and the two

final forms will be the same However dV119894can hardly be the

same when points are irregularly spaced From this point ofview our derived final form is more general and takes pointirregularity into account

3 Modified Smoothed ParticleHydrodynamics (MSPH)

As a modification to SPH the MSPH method improves theaccuracy of the approximations especially at points near theboundary of the domain [16] It uses Taylor series expansionof function 119891(119909 119910) as in (1) Similar to the derivations of (2)and (3) but with different weight functions 120597119908120597119909 120597119908120597119910

12059711990821205971199092 12059711990821205971199102 and 1205972119908120597119909120597119910 the following equationsas examples for 120597119908120597119909 and 12059711990821205971199092 are obtained

intΩ

119891120597119908

120597119909dV = int

Ω

1198910

120597119908

120597119909ℎ dV +

1205971198910

120597119909intΩ

120597119908

120597119909ℎ dV

+1205971198910

120597119910intΩ

120597119908

120597119909119896 dV + 1

2

12059721198910

1205971199092intΩ

120597119908

120597119909ℎ2dV

+1

2

12059721198910

1205971199102intΩ

120597119908

1205971199091198962dV +

12059721198910

120597119909120597119910intΩ

120597119908

120597119909ℎ119896 dV

intΩ

1198911205971199082

1205971199092dV = int

Ω

1198910

1205971199082

1205971199092dV +

120597119891

120597119909intΩ

1205971199082

1205971199092ℎ dV

+120597119891

120597119910intΩ

1205971199082

1205971199092119896 dV + 1

2

12059721198910

1205971199092intΩ

1205971199082

1205971199092ℎ2dV

+1

2

12059721198910

1205971199102intΩ

1205971199082

12059711990921198962dV +

12059721198910

120597119909120597119910intΩ

1205971199082

1205971199092ℎ119896 dV

(6)

Again the Riemann sum over the support domain Ω is usedto approximate the integrations and a system of five equationsis obtained as

(((((((((((((((

(

119873

sum119894=1

120597119908119894

120597119909ℎ119894dV119894

119873

sum119894=1

120597119908119894

120597119909119896119894dV119894

119873

sum119894=1

120597119908119894

120597119909

ℎ2119894

2dV119894

119873

sum119894=1

120597119908119894

120597119909

1198962119894

2dV119894

119873

sum119894=1

120597119908119894

120597119909ℎ119894119896119894dV119894

119873

sum119894=1

120597119908119894

120597119910ℎ119894dV119894

119873

sum119894=1

120597119908119894

120597119910119896119894dV119894

119873

sum119894=1

120597119908119894

120597119910

ℎ2119894

2dV119894

119873

sum119894=1

120597119908119894

120597119910

1198962119894

2dV119894

119873

sum119894=1

120597119908119894

120597119910ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

12059711990921198962119894

2dV119894

119873

sum119894=1

1205971199082119894

1205971199092ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102119896119894dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

12059711991021198962119894

2dV119894

119873

sum119894=1

1205971199082119894

1205971199102ℎ119894119896119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910ℎ119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910119896119894dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910

ℎ2119894

2dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910

1198962119894

2dV119894

119873

sum119894=1

1205971199082119894

120597119909120597119910ℎ119894119896119894dV119894

)))))))))))))))

)

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

4 Mathematical Problems in Engineering

=

(((((((((((((((

(

minus1198910

119873

sum119894=1

120597119908119894

120597119909dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119909dV119894

minus1198910

119873

sum119894=1

120597119908119894

120597119910dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119910dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199092dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199092dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199102dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199102dV119894

minus1198910

119873

sum119894=1

1205971199082119894

120597119909120597119910dV119894+119873

sum119894=1

119891119894

1205971199082119894

120597119909120597119910dV119894

)))))))))))))))

)

(7)

Compared with formula (4) the only difference is theterms multiplied to both sides of (1) In GFDM 1199082ℎ 11990821198961199082ℎ22 119908211989622 and 1199082ℎ119896 are used instead of 120597119908120597119909120597119908120597119910 12059711990821205971199092 12059711990821205971199102 and 1205972119908120597119909120597119910 in MSPH As aresult GFDM avoids computing the derivatives of the weightfunction and hence saves computational efforts and leads tomore choice of the weight function

4 Numerical Tests forApproximation of Derivatives

In previous sections the deviation of GFDM and MSPH ispresented In this section to compare the performance of thetwo methods they are used to approximate the derivativesof certain 1D and 2D functions For the convenience ofevaluation a global error measure is defined as follows

Error119906=

1

|119906|maxradic1

119873

119873

sum119894=1

(119906(119890)119894minus 119906(119899)119894)2

(8)

where 119906 can be 120597119891120597119909 120597119891120597119910 12059721198911205971199092 and 12059721198911205971199102 andthe superscripts (119890) and (119899) refer to the exact and numericalsolutions respectively

The quartic spline function is used as the weight function119908119894

119908119894(119889) =

1 minus 6(119889

119889119898

)2

+ 8(119889

119889119898

)3

minus 3(119889

119889119898

)4

119889 le 119889119898

0 119889 gt 119889119898

(9)

where 119889119898is the kernel radius taken as 21Δ119909 (Δ119909 is the space

interval) which is usually used in meshless methods

41 One-Dimensional Case Consider the following function

119891 (119909) = (119909 minus 05)4 119909 isin [0 1] (10)

Figure 1 shows the first- and second-order derivatives esti-mated by GFDM and MSPH and the exact results when

the domain is discretized into 21 equally spaced points It isseen that GFDM has better performance in both derivativesespecially for the points near boundaries When the numberof points increases to 51 the results are similar as exhibited inFigure 2 Error analysis shown in Table 1 indicates that GFDMhas higher accuracy With increasing number of points theglobal error decreases

42 Two-Dimensional Case For the function

119891 (119909 119910) = sin120587119909 sin120587119910 119909 119910 isin [0 1] times [0 1] (11)

its first- and second-order derivatives together with estima-tions by GFDM and MSPH are shown in Figure 3 In eachdirection 21 points are employed As expected GFDM hashigher approximation accuracy than MSPH for both first-and second-order derivatives as shown in Table 2

5 Generalized Finite Difference Time DomainMethod for Computational Acoustics

For computational acoustics the mostly used approach isthe FDTD method which was originally designed for thesimulation of electromagnetics [1 2] As a finite differencescheme its applicability to complex problems suffers fromaforementioned difficulties for which the generalized finitedifference can be a good alternative In this section togetherwith the basics of computational acoustics a meshlessmethod is proposed in which GFDM is used to discretizethe spatial derivatives and the leap-frog algorithm is used todiscretize the temporal derivatives By analog with FDTDit is referred to as generalized finite difference time domain(GFDTD) method and is expected to have advantages due toits meshless property

The governing equations for acoustic wave propagationproblems are

1205880

120597k120597119905= minusnabla119901

1

11988820

120597119901

120597119905= minus1205880nabla sdot k

(12)

Mathematical Problems in Engineering 5

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 1 Estimates of the first derivative (a) and second derivative (b) of function 119891(119909) = (119909 minus 05)4 with 21 equally spaced points on [0 1]

Table 1 Approximation errors in the derivatives of the function119891(119909) = (119909 minus 05)4

Error () 120597119891120597119909 12059721198911205971199092

GFDM with 21 points 077 581MSPH with 21 points 077 602GFDM with 51 points 011 155MSPH with 51 points 011 161

Table 2 Approximation errors in the derivatives of the function119891(119909 119910) = sin120587119909 sin120587119910

Error () 120597119891120597119909 120597119891120597119910 12059721198911205971199092 12059721198911205971199102

GFDM 029 029 287 287MSPH 047 047 471 471

where 119901 is pressure k is particle velocity 1205880is the density of

the medium and 1198880is the speed of sound

51 Spatial Derivative Approximations by GFDM The spatialderivatives on the right-hand side of (12) are approximatedby GFDM By solving (4) we get the approximations of 120597119891120597119909and 120597119891120597119910 That is the derivatives of variable 119891 at point(1199090 1199100) can be approximated by function values at points

inside the support domain centered at (1199090 1199100) as

1205971198910

120597119909= minus11989801198910+119873

sum119894=1

119891119894119898119894

1205971198910

120597119910= minus12057801198910+119873

sum119894=1

119891119894120578119894

(13)

where 1198980and 120578

0are the difference coefficients for center

point and 119898119894and 120578

119894are coefficients for other points in the

support domain AsGFDMcan reproduce constant functions[4] we have

1198980=119873

sum119894=1

119898119894 120578

0=119873

sum119894=1

120578119894 (14)

By taking both 119901 and k in (12) as 119891 the approximated spatialoperators in (12) are accordingly obtained

52 Explicit Leap-Frog Scheme in GFDTD Generally thetemporal derivatives on the left-hand side of (12) can beintegrated by any time marching algorithms Inspired bythe conventional FDTD method the second-order accurateexplicit leap-frog scheme is used herein in which twovariables 119901 and k are alternatively calculated The velocity iscomputed at the half time step and the pressure is calculatedat the integer time step [2] After temporal approximationsthe semi-discretization of (12) becomes

1205880

k119899+12 minus k119899minus12

Δ119905= minusnabla119901119899

1

11988820

119901119899+1 minus 119901119899

Δ119905= minus1205880nabla sdot k119899+12

(15)

where superscript 119899 represents the time step The leap-frogscheme is conditionally stable and the time step Δ119905 shouldsatisfy the Courant-Friedrichs-Lewy (CFL) condition that isΔ119905 le Δ119909(radicdim sdot 119888

0) where dim is the dimension of the

problem

6 Mathematical Problems in Engineering

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 2 Estimates of the first derivative (a) and second derivative (b) of function119891(119909) = (119909 minus 05)4 with 51 equally spaced points on [0 1]

Substituting (13) into the right-hand side of (15) as theapproximation of the first-order spatial derivatives the fulldiscretized system of equations becomes

k119899+120

= k119899minus120

+Δ119905

1205880

sdot [(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894) (120578

01199011198990minus119873

sum119894=1

120578119894119901119899119894)]

119879

119901119899+10= 1199011198990+ Δ119905120588011988820

sdot [(1198980119906119899+120

minus119873

sum119894=1

119898119894119906119899+12119894

)

+(1205780V119899+120

minus119873

sum119894=1

120578119894V119899+12119894

)]

(16)

where the particle velocity is a 2D vector k = [119906 V]119879By analog with FDTD the full discretization scheme

is referred to as generalized finite difference time domain(GFDTD) method

6 Numerical Results

To validate the proposed GFDTD method it is appliedto one- and two-dimensional wave propagation problemsThree cases are presented The first two examine the acousticwave propagation in one- and two-dimensional domainrespectively and the third one is a real case with differenttypes of boundary conditions All the cases use (9) as theweight function The other parameters are 120588

0= 1 kgm3 and

1198880= 3464ms and the time interval Δ119905 is set to be 1 120583s to

satisfy the CFL condition FDTD solutions are chosen as thereference

61 One-Dimensional Case In this case 501 points are equallyspaced in the domain [0 1] In themiddle of it there is a wavesource in Gaussian pulse form

gp (119909) = 119890minus25|119909minus05| (17)

As shown in Figure 4 the simulated results at two timelevels 119905 = 250 120583s and 500120583s have good agreement with theFDTD solutions Compared with FDTD the relative errorsconcerning the pressure 119875FDTD and 119875GFDTD are

Error250=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0267 sdot 10minus2

Error500=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0519 sdot 10minus2

(18)

62 Two-Dimensional Case The Gaussian wave propagationin two-dimensional domain is simulated in this section Thelength of the square domain is 01m and 101 points areuniformly distributed in each direction The wave sourcestarts from the middle of the domain Figure 5 compares thesolutions of FDTDandGFDTDat 119905 = 20 120583sThe results along119910 = 005m are shown in Figure 5(c) Again good agreementis observed and the relative error is less than 2

To show the advantage of the proposed GFDTD over theconventional FDTD irregular point distribution is examinedAll the points used above are allowed to have plusmn10 per-turbation around their original locations to make the distri-bution irregular part of which is shown in Figure 6(a) and

Mathematical Problems in Engineering 7

GFDM

MSPH

Exact

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

(a)

GFDM

MSPH

Exact

x

y

z

2

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

2

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

(b)

Figure 3 Exact (top) and estimated (a) 120597119891120597119909 and (b) 12059721198911205971199092 of function 119891(119909 119910) = sin120587119909 sin120587119910 by GFDM (middle) and MSPH (bottom)

the result is shown in Figure 6(b) The comparison of theresult along 119910 = 005m shown in Figure 6(c) indicatesthat with irregular distribution of computational points theGaussian wave propagates as well as before

In the GFDTD simulation of wave propagation withirregular point distribution the volume associated to each

point had better to be considered as analyzed at the end ofSection 2 In 2D case the volume associated with a givenpoint is the area that the point dominates Here we useDelaunay triangulation andVoronoi diagram [17] to calculatethe area and the results are shown in Figure 7 The volume ofeach point is shown in Figure 7(a) Due to the designed plusmn10

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

=

(((((((((((((((

(

minus1198910

119873

sum119894=1

120597119908119894

120597119909dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119909dV119894

minus1198910

119873

sum119894=1

120597119908119894

120597119910dV119894+119873

sum119894=1

119891119894

120597119908119894

120597119910dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199092dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199092dV119894

minus1198910

119873

sum119894=1

1205971199082119894

1205971199102dV119894+119873

sum119894=1

119891119894

1205971199082119894

1205971199102dV119894

minus1198910

119873

sum119894=1

1205971199082119894

120597119909120597119910dV119894+119873

sum119894=1

119891119894

1205971199082119894

120597119909120597119910dV119894

)))))))))))))))

)

(7)

Compared with formula (4) the only difference is theterms multiplied to both sides of (1) In GFDM 1199082ℎ 11990821198961199082ℎ22 119908211989622 and 1199082ℎ119896 are used instead of 120597119908120597119909120597119908120597119910 12059711990821205971199092 12059711990821205971199102 and 1205972119908120597119909120597119910 in MSPH As aresult GFDM avoids computing the derivatives of the weightfunction and hence saves computational efforts and leads tomore choice of the weight function

4 Numerical Tests forApproximation of Derivatives

In previous sections the deviation of GFDM and MSPH ispresented In this section to compare the performance of thetwo methods they are used to approximate the derivativesof certain 1D and 2D functions For the convenience ofevaluation a global error measure is defined as follows

Error119906=

1

|119906|maxradic1

119873

119873

sum119894=1

(119906(119890)119894minus 119906(119899)119894)2

(8)

where 119906 can be 120597119891120597119909 120597119891120597119910 12059721198911205971199092 and 12059721198911205971199102 andthe superscripts (119890) and (119899) refer to the exact and numericalsolutions respectively

The quartic spline function is used as the weight function119908119894

119908119894(119889) =

1 minus 6(119889

119889119898

)2

+ 8(119889

119889119898

)3

minus 3(119889

119889119898

)4

119889 le 119889119898

0 119889 gt 119889119898

(9)

where 119889119898is the kernel radius taken as 21Δ119909 (Δ119909 is the space

interval) which is usually used in meshless methods

41 One-Dimensional Case Consider the following function

119891 (119909) = (119909 minus 05)4 119909 isin [0 1] (10)

Figure 1 shows the first- and second-order derivatives esti-mated by GFDM and MSPH and the exact results when

the domain is discretized into 21 equally spaced points It isseen that GFDM has better performance in both derivativesespecially for the points near boundaries When the numberof points increases to 51 the results are similar as exhibited inFigure 2 Error analysis shown in Table 1 indicates that GFDMhas higher accuracy With increasing number of points theglobal error decreases

42 Two-Dimensional Case For the function

119891 (119909 119910) = sin120587119909 sin120587119910 119909 119910 isin [0 1] times [0 1] (11)

its first- and second-order derivatives together with estima-tions by GFDM and MSPH are shown in Figure 3 In eachdirection 21 points are employed As expected GFDM hashigher approximation accuracy than MSPH for both first-and second-order derivatives as shown in Table 2

5 Generalized Finite Difference Time DomainMethod for Computational Acoustics

For computational acoustics the mostly used approach isthe FDTD method which was originally designed for thesimulation of electromagnetics [1 2] As a finite differencescheme its applicability to complex problems suffers fromaforementioned difficulties for which the generalized finitedifference can be a good alternative In this section togetherwith the basics of computational acoustics a meshlessmethod is proposed in which GFDM is used to discretizethe spatial derivatives and the leap-frog algorithm is used todiscretize the temporal derivatives By analog with FDTDit is referred to as generalized finite difference time domain(GFDTD) method and is expected to have advantages due toits meshless property

The governing equations for acoustic wave propagationproblems are

1205880

120597k120597119905= minusnabla119901

1

11988820

120597119901

120597119905= minus1205880nabla sdot k

(12)

Mathematical Problems in Engineering 5

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 1 Estimates of the first derivative (a) and second derivative (b) of function 119891(119909) = (119909 minus 05)4 with 21 equally spaced points on [0 1]

Table 1 Approximation errors in the derivatives of the function119891(119909) = (119909 minus 05)4

Error () 120597119891120597119909 12059721198911205971199092

GFDM with 21 points 077 581MSPH with 21 points 077 602GFDM with 51 points 011 155MSPH with 51 points 011 161

Table 2 Approximation errors in the derivatives of the function119891(119909 119910) = sin120587119909 sin120587119910

Error () 120597119891120597119909 120597119891120597119910 12059721198911205971199092 12059721198911205971199102

GFDM 029 029 287 287MSPH 047 047 471 471

where 119901 is pressure k is particle velocity 1205880is the density of

the medium and 1198880is the speed of sound

51 Spatial Derivative Approximations by GFDM The spatialderivatives on the right-hand side of (12) are approximatedby GFDM By solving (4) we get the approximations of 120597119891120597119909and 120597119891120597119910 That is the derivatives of variable 119891 at point(1199090 1199100) can be approximated by function values at points

inside the support domain centered at (1199090 1199100) as

1205971198910

120597119909= minus11989801198910+119873

sum119894=1

119891119894119898119894

1205971198910

120597119910= minus12057801198910+119873

sum119894=1

119891119894120578119894

(13)

where 1198980and 120578

0are the difference coefficients for center

point and 119898119894and 120578

119894are coefficients for other points in the

support domain AsGFDMcan reproduce constant functions[4] we have

1198980=119873

sum119894=1

119898119894 120578

0=119873

sum119894=1

120578119894 (14)

By taking both 119901 and k in (12) as 119891 the approximated spatialoperators in (12) are accordingly obtained

52 Explicit Leap-Frog Scheme in GFDTD Generally thetemporal derivatives on the left-hand side of (12) can beintegrated by any time marching algorithms Inspired bythe conventional FDTD method the second-order accurateexplicit leap-frog scheme is used herein in which twovariables 119901 and k are alternatively calculated The velocity iscomputed at the half time step and the pressure is calculatedat the integer time step [2] After temporal approximationsthe semi-discretization of (12) becomes

1205880

k119899+12 minus k119899minus12

Δ119905= minusnabla119901119899

1

11988820

119901119899+1 minus 119901119899

Δ119905= minus1205880nabla sdot k119899+12

(15)

where superscript 119899 represents the time step The leap-frogscheme is conditionally stable and the time step Δ119905 shouldsatisfy the Courant-Friedrichs-Lewy (CFL) condition that isΔ119905 le Δ119909(radicdim sdot 119888

0) where dim is the dimension of the

problem

6 Mathematical Problems in Engineering

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 2 Estimates of the first derivative (a) and second derivative (b) of function119891(119909) = (119909 minus 05)4 with 51 equally spaced points on [0 1]

Substituting (13) into the right-hand side of (15) as theapproximation of the first-order spatial derivatives the fulldiscretized system of equations becomes

k119899+120

= k119899minus120

+Δ119905

1205880

sdot [(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894) (120578

01199011198990minus119873

sum119894=1

120578119894119901119899119894)]

119879

119901119899+10= 1199011198990+ Δ119905120588011988820

sdot [(1198980119906119899+120

minus119873

sum119894=1

119898119894119906119899+12119894

)

+(1205780V119899+120

minus119873

sum119894=1

120578119894V119899+12119894

)]

(16)

where the particle velocity is a 2D vector k = [119906 V]119879By analog with FDTD the full discretization scheme

is referred to as generalized finite difference time domain(GFDTD) method

6 Numerical Results

To validate the proposed GFDTD method it is appliedto one- and two-dimensional wave propagation problemsThree cases are presented The first two examine the acousticwave propagation in one- and two-dimensional domainrespectively and the third one is a real case with differenttypes of boundary conditions All the cases use (9) as theweight function The other parameters are 120588

0= 1 kgm3 and

1198880= 3464ms and the time interval Δ119905 is set to be 1 120583s to

satisfy the CFL condition FDTD solutions are chosen as thereference

61 One-Dimensional Case In this case 501 points are equallyspaced in the domain [0 1] In themiddle of it there is a wavesource in Gaussian pulse form

gp (119909) = 119890minus25|119909minus05| (17)

As shown in Figure 4 the simulated results at two timelevels 119905 = 250 120583s and 500120583s have good agreement with theFDTD solutions Compared with FDTD the relative errorsconcerning the pressure 119875FDTD and 119875GFDTD are

Error250=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0267 sdot 10minus2

Error500=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0519 sdot 10minus2

(18)

62 Two-Dimensional Case The Gaussian wave propagationin two-dimensional domain is simulated in this section Thelength of the square domain is 01m and 101 points areuniformly distributed in each direction The wave sourcestarts from the middle of the domain Figure 5 compares thesolutions of FDTDandGFDTDat 119905 = 20 120583sThe results along119910 = 005m are shown in Figure 5(c) Again good agreementis observed and the relative error is less than 2

To show the advantage of the proposed GFDTD over theconventional FDTD irregular point distribution is examinedAll the points used above are allowed to have plusmn10 per-turbation around their original locations to make the distri-bution irregular part of which is shown in Figure 6(a) and

Mathematical Problems in Engineering 7

GFDM

MSPH

Exact

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

(a)

GFDM

MSPH

Exact

x

y

z

2

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

2

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

(b)

Figure 3 Exact (top) and estimated (a) 120597119891120597119909 and (b) 12059721198911205971199092 of function 119891(119909 119910) = sin120587119909 sin120587119910 by GFDM (middle) and MSPH (bottom)

the result is shown in Figure 6(b) The comparison of theresult along 119910 = 005m shown in Figure 6(c) indicatesthat with irregular distribution of computational points theGaussian wave propagates as well as before

In the GFDTD simulation of wave propagation withirregular point distribution the volume associated to each

point had better to be considered as analyzed at the end ofSection 2 In 2D case the volume associated with a givenpoint is the area that the point dominates Here we useDelaunay triangulation andVoronoi diagram [17] to calculatethe area and the results are shown in Figure 7 The volume ofeach point is shown in Figure 7(a) Due to the designed plusmn10

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 1 Estimates of the first derivative (a) and second derivative (b) of function 119891(119909) = (119909 minus 05)4 with 21 equally spaced points on [0 1]

Table 1 Approximation errors in the derivatives of the function119891(119909) = (119909 minus 05)4

Error () 120597119891120597119909 12059721198911205971199092

GFDM with 21 points 077 581MSPH with 21 points 077 602GFDM with 51 points 011 155MSPH with 51 points 011 161

Table 2 Approximation errors in the derivatives of the function119891(119909 119910) = sin120587119909 sin120587119910

Error () 120597119891120597119909 120597119891120597119910 12059721198911205971199092 12059721198911205971199102

GFDM 029 029 287 287MSPH 047 047 471 471

where 119901 is pressure k is particle velocity 1205880is the density of

the medium and 1198880is the speed of sound

51 Spatial Derivative Approximations by GFDM The spatialderivatives on the right-hand side of (12) are approximatedby GFDM By solving (4) we get the approximations of 120597119891120597119909and 120597119891120597119910 That is the derivatives of variable 119891 at point(1199090 1199100) can be approximated by function values at points

inside the support domain centered at (1199090 1199100) as

1205971198910

120597119909= minus11989801198910+119873

sum119894=1

119891119894119898119894

1205971198910

120597119910= minus12057801198910+119873

sum119894=1

119891119894120578119894

(13)

where 1198980and 120578

0are the difference coefficients for center

point and 119898119894and 120578

119894are coefficients for other points in the

support domain AsGFDMcan reproduce constant functions[4] we have

1198980=119873

sum119894=1

119898119894 120578

0=119873

sum119894=1

120578119894 (14)

By taking both 119901 and k in (12) as 119891 the approximated spatialoperators in (12) are accordingly obtained

52 Explicit Leap-Frog Scheme in GFDTD Generally thetemporal derivatives on the left-hand side of (12) can beintegrated by any time marching algorithms Inspired bythe conventional FDTD method the second-order accurateexplicit leap-frog scheme is used herein in which twovariables 119901 and k are alternatively calculated The velocity iscomputed at the half time step and the pressure is calculatedat the integer time step [2] After temporal approximationsthe semi-discretization of (12) becomes

1205880

k119899+12 minus k119899minus12

Δ119905= minusnabla119901119899

1

11988820

119901119899+1 minus 119901119899

Δ119905= minus1205880nabla sdot k119899+12

(15)

where superscript 119899 represents the time step The leap-frogscheme is conditionally stable and the time step Δ119905 shouldsatisfy the Courant-Friedrichs-Lewy (CFL) condition that isΔ119905 le Δ119909(radicdim sdot 119888

0) where dim is the dimension of the

problem

6 Mathematical Problems in Engineering

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 2 Estimates of the first derivative (a) and second derivative (b) of function119891(119909) = (119909 minus 05)4 with 51 equally spaced points on [0 1]

Substituting (13) into the right-hand side of (15) as theapproximation of the first-order spatial derivatives the fulldiscretized system of equations becomes

k119899+120

= k119899minus120

+Δ119905

1205880

sdot [(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894) (120578

01199011198990minus119873

sum119894=1

120578119894119901119899119894)]

119879

119901119899+10= 1199011198990+ Δ119905120588011988820

sdot [(1198980119906119899+120

minus119873

sum119894=1

119898119894119906119899+12119894

)

+(1205780V119899+120

minus119873

sum119894=1

120578119894V119899+12119894

)]

(16)

where the particle velocity is a 2D vector k = [119906 V]119879By analog with FDTD the full discretization scheme

is referred to as generalized finite difference time domain(GFDTD) method

6 Numerical Results

To validate the proposed GFDTD method it is appliedto one- and two-dimensional wave propagation problemsThree cases are presented The first two examine the acousticwave propagation in one- and two-dimensional domainrespectively and the third one is a real case with differenttypes of boundary conditions All the cases use (9) as theweight function The other parameters are 120588

0= 1 kgm3 and

1198880= 3464ms and the time interval Δ119905 is set to be 1 120583s to

satisfy the CFL condition FDTD solutions are chosen as thereference

61 One-Dimensional Case In this case 501 points are equallyspaced in the domain [0 1] In themiddle of it there is a wavesource in Gaussian pulse form

gp (119909) = 119890minus25|119909minus05| (17)

As shown in Figure 4 the simulated results at two timelevels 119905 = 250 120583s and 500120583s have good agreement with theFDTD solutions Compared with FDTD the relative errorsconcerning the pressure 119875FDTD and 119875GFDTD are

Error250=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0267 sdot 10minus2

Error500=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0519 sdot 10minus2

(18)

62 Two-Dimensional Case The Gaussian wave propagationin two-dimensional domain is simulated in this section Thelength of the square domain is 01m and 101 points areuniformly distributed in each direction The wave sourcestarts from the middle of the domain Figure 5 compares thesolutions of FDTDandGFDTDat 119905 = 20 120583sThe results along119910 = 005m are shown in Figure 5(c) Again good agreementis observed and the relative error is less than 2

To show the advantage of the proposed GFDTD over theconventional FDTD irregular point distribution is examinedAll the points used above are allowed to have plusmn10 per-turbation around their original locations to make the distri-bution irregular part of which is shown in Figure 6(a) and

Mathematical Problems in Engineering 7

GFDM

MSPH

Exact

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

(a)

GFDM

MSPH

Exact

x

y

z

2

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

2

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

(b)

Figure 3 Exact (top) and estimated (a) 120597119891120597119909 and (b) 12059721198911205971199092 of function 119891(119909 119910) = sin120587119909 sin120587119910 by GFDM (middle) and MSPH (bottom)

the result is shown in Figure 6(b) The comparison of theresult along 119910 = 005m shown in Figure 6(c) indicatesthat with irregular distribution of computational points theGaussian wave propagates as well as before

In the GFDTD simulation of wave propagation withirregular point distribution the volume associated to each

point had better to be considered as analyzed at the end ofSection 2 In 2D case the volume associated with a givenpoint is the area that the point dominates Here we useDelaunay triangulation andVoronoi diagram [17] to calculatethe area and the results are shown in Figure 7 The volume ofeach point is shown in Figure 7(a) Due to the designed plusmn10

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

GFDMMSPHExact

Firs

t der

ivat

ive e

stim

ate

Location

05

04

03

02

01

0

minus01

minus02

minus03

minus04

minus050 02 04 06 08 1

(a)

Location0 02 04 06 08 1

Seco

nd d

eriv

ativ

e esti

mat

e

3

25

2

15

1

05

0

GFDMMSPHExact

(b)

Figure 2 Estimates of the first derivative (a) and second derivative (b) of function119891(119909) = (119909 minus 05)4 with 51 equally spaced points on [0 1]

Substituting (13) into the right-hand side of (15) as theapproximation of the first-order spatial derivatives the fulldiscretized system of equations becomes

k119899+120

= k119899minus120

+Δ119905

1205880

sdot [(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894) (120578

01199011198990minus119873

sum119894=1

120578119894119901119899119894)]

119879

119901119899+10= 1199011198990+ Δ119905120588011988820

sdot [(1198980119906119899+120

minus119873

sum119894=1

119898119894119906119899+12119894

)

+(1205780V119899+120

minus119873

sum119894=1

120578119894V119899+12119894

)]

(16)

where the particle velocity is a 2D vector k = [119906 V]119879By analog with FDTD the full discretization scheme

is referred to as generalized finite difference time domain(GFDTD) method

6 Numerical Results

To validate the proposed GFDTD method it is appliedto one- and two-dimensional wave propagation problemsThree cases are presented The first two examine the acousticwave propagation in one- and two-dimensional domainrespectively and the third one is a real case with differenttypes of boundary conditions All the cases use (9) as theweight function The other parameters are 120588

0= 1 kgm3 and

1198880= 3464ms and the time interval Δ119905 is set to be 1 120583s to

satisfy the CFL condition FDTD solutions are chosen as thereference

61 One-Dimensional Case In this case 501 points are equallyspaced in the domain [0 1] In themiddle of it there is a wavesource in Gaussian pulse form

gp (119909) = 119890minus25|119909minus05| (17)

As shown in Figure 4 the simulated results at two timelevels 119905 = 250 120583s and 500120583s have good agreement with theFDTD solutions Compared with FDTD the relative errorsconcerning the pressure 119875FDTD and 119875GFDTD are

Error250=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0267 sdot 10minus2

Error500=

10038171003817100381710038171003817119875FDTD minus 119875GFDTD

1003817100381710038171003817100381721003817100381710038171003817119875

FDTD10038171003817100381710038172= 0519 sdot 10minus2

(18)

62 Two-Dimensional Case The Gaussian wave propagationin two-dimensional domain is simulated in this section Thelength of the square domain is 01m and 101 points areuniformly distributed in each direction The wave sourcestarts from the middle of the domain Figure 5 compares thesolutions of FDTDandGFDTDat 119905 = 20 120583sThe results along119910 = 005m are shown in Figure 5(c) Again good agreementis observed and the relative error is less than 2

To show the advantage of the proposed GFDTD over theconventional FDTD irregular point distribution is examinedAll the points used above are allowed to have plusmn10 per-turbation around their original locations to make the distri-bution irregular part of which is shown in Figure 6(a) and

Mathematical Problems in Engineering 7

GFDM

MSPH

Exact

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

(a)

GFDM

MSPH

Exact

x

y

z

2

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

2

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

(b)

Figure 3 Exact (top) and estimated (a) 120597119891120597119909 and (b) 12059721198911205971199092 of function 119891(119909 119910) = sin120587119909 sin120587119910 by GFDM (middle) and MSPH (bottom)

the result is shown in Figure 6(b) The comparison of theresult along 119910 = 005m shown in Figure 6(c) indicatesthat with irregular distribution of computational points theGaussian wave propagates as well as before

In the GFDTD simulation of wave propagation withirregular point distribution the volume associated to each

point had better to be considered as analyzed at the end ofSection 2 In 2D case the volume associated with a givenpoint is the area that the point dominates Here we useDelaunay triangulation andVoronoi diagram [17] to calculatethe area and the results are shown in Figure 7 The volume ofeach point is shown in Figure 7(a) Due to the designed plusmn10

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

GFDM

MSPH

Exact

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

4

2

0

minus2

minus4

x

y

z

1

05

0 002

0406

081

(a)

GFDM

MSPH

Exact

x

y

z

2

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

2

x

y

z

0

minus2

minus4

minus6

minus8

minus101

05

0 002

0406

081

(b)

Figure 3 Exact (top) and estimated (a) 120597119891120597119909 and (b) 12059721198911205971199092 of function 119891(119909 119910) = sin120587119909 sin120587119910 by GFDM (middle) and MSPH (bottom)

the result is shown in Figure 6(b) The comparison of theresult along 119910 = 005m shown in Figure 6(c) indicatesthat with irregular distribution of computational points theGaussian wave propagates as well as before

In the GFDTD simulation of wave propagation withirregular point distribution the volume associated to each

point had better to be considered as analyzed at the end ofSection 2 In 2D case the volume associated with a givenpoint is the area that the point dominates Here we useDelaunay triangulation andVoronoi diagram [17] to calculatethe area and the results are shown in Figure 7 The volume ofeach point is shown in Figure 7(a) Due to the designed plusmn10

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Am

plitu

de (P

a)

Location (m)

t = 250120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(a)A

mpl

itude

(Pa)

Location (m)

t = 500120583s06

05

04

03

02

01

0

minus010 02 04 06 08 1

GFDTDFDTD

(b)

Figure 4 Gaussian wave propagation at two time points (a) 119905 = 250 120583s and (b) 500 120583s

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

FDTD

(a)

GFDTD

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

(b)

The results along y = 005m030250201501005

0minus005minus01minus015

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

GFDTDFDTD

(c)

Figure 5Gaussianwave propagation in a square domain at time 119905 = 20 120583s (a)TheFDTDresult (b) theGFDTDresult and (c) the comparisonalong 119910 = 005m

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

x (m)

y(m

)

0052

005

0048

0046

0044

0042

0052 0054 0056 0058 006 0062

(a)

03

02

01

0

minus01

minus02

01

005

0 0002

004006

00801

Am

plitu

de (P

a)

y (m)x (m)

Irregular points distribution

(b)

030250201501005

0minus005minus01

0 001 002 003 004 005 006 007 008 009 01

Am

plitu

de (P

a)

Location (m)

The result along y = 005m

RegularIrregular

(c)

Figure 6 Gaussian wave propagation in a square domain with irregular point distribution (a) Part of the irregular distribution (b) simulatedresults and (c) comparison with regular point distribution after 20 120583s

Area of the irregular points

12

11

1

09

0801

008006

004002

0

01008

006004

0020

times10minus6

Are

a(m

2)

x (m)

y (m)

(a)

Area of the irregular points0044

0043

0042

0041

004

0039

0038

0043 0044 0045 0046 0047 0048 0049 005

x (m)

y(m

)

(b)

Figure 7 (a) Area associated with irregular distributed points and (b) Voronoi diagram

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

perturbation the volume fluctuates around 1minus6m2 Based onDelaunay triangulation and Voronoi diagram the area asso-ciated with a red point is indicated by the area surrounded bythe blue lines as shown in Figure 7(b) To compare the resultswith regular and irregular point distributions the values onregularly distributed points have to be firstly interpolatedby the calculated results with irregularly spaced points Thethird-order accurate cubic interpolation method [17] is usedherein The relative error is

Error =10038171003817100381710038171003817119875

iriegular minus 119875regular100381710038171003817100381710038172

1003817100381710038171003817119875regular10038171003817100381710038172

= 158 sdot 10minus2 (19)

The relatively small error is due to the gentle irregular pointdistributionThat is if the point irregularity is small the effectof dV119894is negligible This is consistent with Benitorsquos results [9ndash

11]

63 Two-Dimensional Case with Effect of Boundary Con-ditions In Sections 61 and 62 wave propagation insidea domain is simulated In this section to show the effectof boundaries which is of high importance in transientacoustics sound wave propagation in a rectangular tube isstudied Two different kinds of boundary conditions includ-ing reflection and absorbing boundary are considered At theleft edge of the computational domain there is a Gaussianpulse as the source term The upper and bottom boundariesare reflection layers and the second-order Murrsquos absorbingboundary condition [18] exists at the right side boundary Inthis case the same 120588

0and 1198880are used as before and the time

interval Δ119905 is 1 120583s Inside the computational domain 64 times 100points with spatial interval Δ119909 = Δ119910 = 1mm are evenlyspaced

631 Source Term The left is a wave source with pressuregiven by the Gaussian pulse

gp (119905) = 119890minus(119905minus119879)0291198792

(20)

where 119879 = 06461198910and 119891

0= 10KHz

632 Reflection Boundary Condition To simulate reflectionsat the upper and bottomwall boundaries themodel proposedby Yokota et al [19 20] and widely used in room acousticsis employed herein In this model the normal componentof particle velocity and the pressure of the points on theboundary are supposed to satisfy the following condition

knorm =119901

119885norm (21)

where 119885norm is the normal acoustic impedance on theboundary given by

119885norm = 120588011988801 + radic1 minus 120572norm

1 minus radic1 minus 120572norm (22)

Here the normal sound absorption coefficient 120572norm is takenas 02 as in [19]

633 Absorbing Boundary Condition At the right boundarysecond-order Murrsquos absorbing boundary condition [18] isapplied

1

1198880

1205972119901

120597119909120597119905+1

11988820

1205972119901

1205971199052+1

2

1205972119901

1205971199102= 0 (23)

By applying (12) to (23) and performing time integration (23)degenerates to

120597119901

120597119909+1

1198880

120597119901

120597119905minus11988801205880

2

120597119906

120597119910= 0 (24)

When GFDTD is used the discrete form of (24) is obtainedas

119901119899+10minus 1199011198990

Δ119905=119888201205880

2(1205780119906119899+120

minus119873

sum119894=1

120578119894119906119899+12119894

)

minus 1198880(11989801199011198990minus119873

sum119894=1

119898119894119901119899119894)

(25)

634 Results After applying the source term and the twoboundary conditions into our case the wave is consideredto be propagating from left to right inside a tube and getsabsorbed at the end of it Figure 8 shows the simulated resultsafter 200120583s with a color map image that clearly depicts thepressure distribution In Figure 9 the results after 350 120583s aredepicted and the absorbing boundary at the right edge leadsto no reflection

7 Conclusion

A new derivation of the generalized finite difference method(GFDM)with Taylor series expansion generates the same for-mulation as its conventional derivation and clearly demon-strates its relationship with meshless particle methodsGFDM has better performance in derivative approxima-tions than the particle methods The proposed generalizedfinite difference time domain (GFDTD) method has beensuccessfully applied to one- and two-dimensional acousticwave propagation problems with reflection and absorbingboundary conditions The numerical results are in line withthe FDTD reference solutions even with irregular pointdistribution The GFDTD method has high potentials insolving transient acoustic problems with moving boundarieswhich deserves further studies

Appendix

Conventional Derivation of GFDM

Considering the 2D case for the sameTaylor expansion in (1)we consider an energy norm 119861

119861 =119873

sum119894=1

[[1198910minus 119891119894+ ℎ119894

1205971198910

120597119909+ 119896119894

1205971198910

120597119910+ ℎ2119894

12059721198910

1205971199092+ 1198962119894

12059721198910

1205971199102

+ ℎ119894119896119894

12059721198910

120597119909120597119910]119908119894]

2

(A1)

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11y

(mm

)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 8 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 200 120583s

y(m

m)

x (mm)

FDTD

10

20

30

40

50

60

20 40 60 80 100

(Pa)

08

06

04

02

0

(a)

y(m

m)

x (mm)

(Pa)

10

20

30

40

50

60

20 40 60 80 100

08

06

04

02

0

GFDTD

(b)

Am

plitu

de (P

a)

Location (m)

The result along y = 0034

080604020

minus02

0 001 002 003 004 005 006 007 008 009 01

FDTDGFDTD

(c)

Figure 9 Soundwave propagation in a two-dimension tube (a) FDTD results (b) GFDTD results and (c) the comparison along 119910 = 0034mat 119905 = 350 120583s

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

where 119891119894= 119891(119909

119894 119910119894) 1198910= 119891(119909

0 1199100) ℎ119894= 119909119894minus1199090 119896119894= 119910119894minus1199100

and 119908119894is weighing function with compact support

The solution of the derivatives is obtained by minimizingthe norm 119861 that is

120597119861

120597 119863119891= 0 (A2)

with

119863119891119879=

1205971198910

1205971199091205971198910

12059711991012059721198910

120597119909212059721198910

120597119910212059721198910

120597119909120597119910 (A3)

For example the first equation is

1198910

119873

sum119894=1

1199082119894ℎ119894minus119873

sum119894=1

1198911198941199082119894ℎ119894+1205971198910

120597119909

119873

sum119894=1

1199082119894ℎ2119894+1205971198910

120597119910

119873

sum119894=1

1199082119894ℎ119894119896119894

+12059721198910

1205971199092

119873

sum119894=1

1199082119894

ℎ3119894

2+12059721198910

1205971199102

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

+12059721198910

120597119909120597119910

119873

sum119894=1

1199082119894ℎ2119894119896119894= 0

(A4)

Equation (A4) and the other four give the followingsystem

119873

sum119894=1

1199082119894ℎ2119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ119894119896119894

119873

sum119894=1

11990821198941198962119894

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894

2

119873

sum119894=1

1199082119894

ℎ2119894119896119894

2

119873

sum119894=1

1199082119894

ℎ4119894

4

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2119873

sum119894=1

1199082119894

1198962119894ℎ119894

2

119873

sum119894=1

1199082119894

1198963119894

2

119873

sum119894=1

1199082119894

ℎ21198941198962119894

4

119873

sum119894=1

1199082119894

1198964119894

4

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2119873

sum119894=1

1199082119894ℎ2119894119896119894

119873

sum119894=1

1199082119894ℎ1198941198962119894

119873

sum119894=1

1199082119894

ℎ3119894119896119894

2

119873

sum119894=1

1199082119894

ℎ1198941198963119894

2

119873

sum119894=1

1199082119894ℎ21198941198962119894

1205971198910

120597119909

1205971198910

120597119910

12059721198910

1205971199092

12059721198910

1205971199102

12059721198910

120597119909120597119910

=

minus1198910

119873

sum119894=1

1199082119894ℎ119894+119873

sum119894=1

1198911198941199082119894ℎ119894

minus1198910

119873

sum119894=1

1199082119894119896119894+119873

sum119894=1

1198911198941199082119894119896119894

minus1198910

119873

sum119894=1

1199082119894

ℎ2119894

2+119873

sum119894=1

1198911198941199082119894

ℎ2119894

2

minus1198910

119873

sum119894=1

1199082119894

1198962119894

2+119873

sum119894=1

1198911198941199082119894

1198962119894

2

minus1198910

119873

sum119894=1

1199082119894ℎ119894119896119894+119873

sum119894=1

1198911198941199082119894ℎ119894119896119894

(A5)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported in part by the National NaturalScience Foundation of China (nos 51478305 and 61175016)and Key Program (no 61233009) The authors thank theanonymous reviewers for their most useful suggestions

References

[1] M N O Sadiku Numerical Techniques in ElectromagneticsCRC Press 2000

[2] D M Sullivan Electromagnetic Simulation Using the FDTDMethod John Wiley amp Sons 2013

[3] J W Swegle and S W Attaway ldquoOn the feasibility of usingSmoothed Particle Hydrodynamics for underwater explosioncalculationsrdquo Computational Mechanics vol 17 no 3 pp 151ndash168 1995

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

[4] T Belytschko Y Krongauz D Organ M Fleming and P KryslldquoMeshless methods an overview and recent developmentsrdquoComputer Methods in Applied Mechanics and Engineering vol139 no 1ndash4 pp 3ndash47 1996

[5] G R Johnson R A Stryk and S R Beissel ldquoSPH for highvelocity impact computationsrdquo Computer Methods in AppliedMechanics and Engineering vol 139 no 1-4 pp 347ndash373 1996

[6] N Perrone and R Kao ldquoA general finite difference method forarbitrary meshesrdquo Computers amp Structures vol 5 no 1 pp 45ndash57 1975

[7] P S Jensen ldquoFinite difference techniques for variable gridsrdquoComputers amp Structures vol 2 no 1-2 pp 17ndash29 1972

[8] T Liszka and J Orkisz ldquoThe finite difference method at arbi-trary irregular grids and its application in applied mechanicsrdquoComputers and Structures vol 11 no 1-2 pp 83ndash95 1980

[9] J J Benito FUrea and LGavete ldquoInfluence of several factors inthe generalized finite differencemethodrdquoAppliedMathematicalModelling vol 25 no 12 pp 1039ndash1053 2001

[10] L Gavete M L Gavete and J J Benito ldquoImprovements ofgeneralized finite differencemethod and comparisonwith othermeshless methodrdquo Applied Mathematical Modelling vol 27 no10 pp 831ndash847 2003

[11] J J Benito F Urena and L Gavete ldquoSolving parabolicand hyperbolic equations by the generalized finite differencemethodrdquo Journal of Computational and Applied Mathematicsvol 209 no 2 pp 208ndash233 2007

[12] C-M Fan and P-W Li ldquoGeneralized finite difference methodfor solving two-dimensional Burgersrsquo equationsrdquo Procedia Engi-neering vol 79 pp 55ndash60 2014

[13] J J Benito F Urena L Gavete E Salete and A Muelas ldquoAGFDM with PML for seismic wave equations in heterogeneousmediardquo Journal of Computational and AppliedMathematics vol252 pp 40ndash51 2013

[14] Y L Chen K B Huang and X Yu ldquoNumerical study ofdetonation shock dynamics using generalized finite differencemethodrdquo Science China Physics Mechanics and Astronomy vol54 no 10 pp 1883ndash1888 2011

[15] J F Lee R Lee and A Cangellaris ldquoTime-domain finite-element methodsrdquo IEEE Transactions on Antennas and Propa-gation vol 45 no 3 pp 430ndash442 1997

[16] G M Zhang and R C Batra ldquoModified smoothed particlehydrodynamics method and its application to transient prob-lemsrdquo Computational Mechanics vol 34 no 2 pp 137ndash1462004

[17] M U Guide The mathworks Inc Natick Mass USA 2013[18] G Mur ldquoAbsorbing boundary conditions for the finite-

difference approximation of the time-domain electromagnetic-field equationsrdquo IEEE Transactions on Electromagnetic Compat-ibility vol 23 no 4 pp 377ndash382 1981

[19] T Yokota S Sakamoto and H Tachibana ldquoVisualization ofsound propagation and scattering in roomsrdquo Acoustical Scienceand Technology vol 23 no 1 pp 40ndash46 2002

[20] S Sakamoto H Nagatomo A Ushiyama and H TachibanaldquoCalculation of impulse responses and acoustic parameters ina hall by the finite-difference time-domain methodrdquo AcousticalScience and Technology vol 29 no 4 pp 256ndash265 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of