time domain room acoustic simulations using the spectral ... · the context of room acoustics.29,30...

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Time domain room acoustic simulations using the spectral element method

Pind Finnur Engsig-Karup Allan P Jeong Cheol Ho Hesthaven Jan S Mejling Mikael S Stroslashmann-Andersen Jakob

Published inJournal of the Acoustical Society of America

Link to article DOI10112115109396

Publication date2019

Document VersionPeer reviewed version

Link back to DTU Orbit

Citation (APA)Pind F Engsig-Karup A P Jeong C H Hesthaven J S Mejling M S amp Stroslashmann-Andersen J (2019)Time domain room acoustic simulations using the spectral element method Journal of the Acoustical Society ofAmerica 145(6) 3299-3310 httpsdoiorg10112115109396

Time domain room acoustic simulations using thespectral element method

Finnur Pind1 a) Allan P Engsig-Karup2 Cheol-Ho Jeong3 Jan S Hesthaven4 Mikael S Mejling2 and JakobStroslashmann-Andersen1

1Henning Larsen Copenhagen Denmark2Scientific Computing Section Department of Applied Mathematics and Computer Science Techni-

cal University of Denmark Kongens Lyngby Denmark3Acoustic Technology Group Department of Electrical Engineering Technical University of Den-

mark Kongens Lyngby Denmark4Chair of Computational Mathematics and Simulation Science Ecole polytechnique federale de Lau-

sanne Lausanne Switzerland

This paper presents a wave-based numerical scheme based on a spectral element methodcoupled with an implicit-explicit Runge-Kutta time stepping method for simulating roomacoustics in the time domain The scheme has certain features which make it highly attrac-tive for room acoustic simulations namely a) its low dispersion and dissipation propertiesdue to a high-order spatio-temporal discretization b) a high degree of geometric flexibilitywhere adaptive unstructured meshes with curvilinear mesh elements are supported and c) itssuitability for parallel implementation on modern many-core computer hardware A methodfor modelling locally reacting frequency dependent impedance boundary conditions withinthe scheme is developed in which the boundary impedance is mapped to a multipole rationalfunction and formulated in differential form Various numerical experiments are presentedwhich reveal the accuracy and cost-efficiency of the proposed numerical scheme a

ccopy2019 Acoustical Society of America [httpdxdoiorg(DOI number)]

[XYZ] Pages 1ndash14

Keywords Room acoustic wave-based simulations spectral element method high-order nu-merical schemes frequency dependent impedance boundary conditions curvilinear meshing

I INTRODUCTION

Room acoustic simulations are used in manyfields for example in building design1 virtual reality2

entertainment3 automotive design4 music5 and hear-ing research6 Since their inception in the 1960rsquos78 roomacoustic simulations have primarily been carried out bymeans of geometrical acoustics methods9 such as theray tracing method10 the image source method11 or thebeam tracing method12 In these methods several simpli-fying approximations regarding sound propagation andreflection are made which make the computational taskmore manageable These approximations will howeverdeteriorate the accuracy of the simulation because vari-ous important wave phenomena such as diffraction in-terference and scattering are not accurately capturedWave phenomena will be prominent in rooms where theroom dimensions and sizes of obstacles are compara-ble to the wavelength of the acoustic wave Small tomedium sized rooms and low to mid frequencies arehere of primary concern13 However large rooms can

a)fpinhenninglarsencomaA part of the results of this paper was previously presented atEuronoise 2018 in Creta

also exhibit wave phenomena eg the seat-dip effect14

Another problem associated with geometrical acousticsmethods is that they require simplified 3D models madeup of coarse planar polygons Fine geometrical detailsare typically replaced by assigning scattering coefficientsto planar surfaces and these coefficients are often basedon crude visual inspection15 Instead of using simplified3D models it would be more accurate to model directlythe complex and detailed geometry typically found in ar-chitectural models16

Thanks to the continuous advances in computationpower and in scientific computing theory the wave-based methods are becoming a viable alternative for roomacoustic simulations In these methods the governingphysics equations are solved numerically and they aretherefore from a physical point of view more accuratethan their geometrical counterparts since all wave phe-nomena is inherently accounted for17 Wave-based meth-ods that have been applied to room acoustic simula-tions include the finite-difference time-domain method(FDTD)18 the boundary element method (BEM)19 thelinear finite element method (h-FEM)20 the equiva-lent source method (ESM)21 the finite volume method(FVM)22 and the pseudospectral time-domain method(PSTD)23 A major drawback of the wave-based meth-

J Acoust Soc Am 7 May 2019 Spectral element room acoustic simulations 1

ods is the large computational effort needed24 Variousstudies have been carried out to bring down computationtimes eg parallel implementations of algorithms utiliz-ing many-core hardware such as GPUrsquos25 hybridizationof different algorithm types42627 and recently the usageof high-order numerical methods which have the poten-tial of being cost-efficient28 has been investigated withinthe context of room acoustics2930

The primary purpose of this paper is to present anumerical scheme based on a spectral element method(SEM)31ndash33 adapted for time domain room acoustic sim-ulations and to assess the suitability of using the SEMfor this task The SEM is known to be well-suited forcost-effective simulations of large scale problems overlong simulation times due to a high-order polynomial ba-sis discretization which leads to small numerical disper-sion and dissipation errors34 Furthermore the SEM iscapable of operating on unstructured adaptive mesheswith curvilinear mesh elements making it highly suit-able for simulating complex geometries Finally theSEM has also been shown to be well-suited for parallelcomputing3536 In the scheme presented here time step-ping is done by means of an implicit-explicit high-orderRunge-Kutta solver37 ensuring computational efficiencyrobustness and the maintenance of global high-order ac-curacy A method for incorporating locally reacting fre-quency dependent impedance boundary conditions in thescheme is presented

The SEM has several advantages compared to theother wave-based methods found in the literature TheFDTD method and the PSTD method are ill-suited fordealing with complex geometries3839 The FVM over-comes this drawback of limited geometrical flexibility22

however another challenge with the FVM is a flux recon-struction procedure that despite recent progress is notstraightforward to extend to arbitrarily high-order accu-racy in two and three spatial dimensions4041 The BEMhas the benefit of needing only to discretize the bound-ary surface instead of the domain volume however inthe BEM operators are dense Typically the FEM whichhas sparse operators and where the domain volume is dis-cretized is considered faster than BEM unless the volumeto surface area becomes very large24 In addition thereare other challenges relating to uniqueness of solutions inthe BEM42 High-order FEM typically referred to as thehp-FEM is another option43 The hp-FEM and the SEMare based on the same underlying theoretical frameworkand possess similar properties while differing in imple-mentation The key distinction between the hp-FEM andthe SEM is whether the expansion is modal or nodal Inhp-FEM the expansion basis is normally modal ie thebasis functions are of increasing order (hierarchical) Ina modal expansion the expansion coefficients do not haveany particular physical meaning In contrast in the SEMthe expansion basis is a non-hierarchical Lagrange basiswhich consists of polynomials of arbitrary order withsupport on the element Importantly the nodal expan-sion coefficients are associated with the solution valuesat the nodal points hence these can be interpreted read-

ily The discontinuous Galerkin finite element method(DGFEM) is another method which stems from a sim-ilar theoretical framework as the SEM Its main draw-backs relative to the SEM is that it requires more degreesof freedom and a flux reconstruction between elementsmust be computed However it relies only on local weakformulations defined for elements rather than for the fulldomain as in the SEM which makes it possible to ex-ploit the resulting locality in parallelization hp-FEMSEM and DGFEM have similar geometric flexibility44

The paper is structured as follows In Sec II thegoverning acoustics equations and the boundary condi-tion formulation are presented In Sec III the proposednumerical scheme is described Certain numerical prop-erties of the scheme such as numerical dispersion anddissipation and its computational efficiency are analyzedin Sec IV Section V presents various simulation resultsmade using the proposed scheme and finally some con-cluding remarks are given in Sec VI

II GOVERNING EQUATIONS amp BOUNDARY CONDI-

TIONS

Acoustic wave propagation in a lossless medium ina d dimensional enclosure is governed by the followingsystem of two coupled linear first-order partial differentialequations

vt = minus1

ρnablap

in Ωtimes [0 t]

pt = minusρc2nabla middot v(1)

where p(x t) is the sound pressure v(x t) is the parti-cle velocity x is the position in space of the domain Ωt is time ρ is the density of the medium and c is thespeed of sound in air (ρ = 12 kgm3 and c = 343 ms inthis study) These equations correspond to the linearizedEuler and continuity equation without flow This systemis exactly equivalent to the more commonly used secondorder wave equation

Sufficient boundary conditions must be supplied withthe system in Eq (1) and in room acoustics it is nat-ural to define the boundary conditions in terms of thecomplex frequency dependent surface impedance Z(ω)which can be estimated from material models or frommeasurements4546 The pressure and the particle ve-locity at the boundary are related through the surfaceimpedance in the frequency domain via

vn(ω) =p(ω)

Z(ω)= p(ω)Y (ω) (2)

where ω is the angular frequency p and vn = v middot n arethe Fourier transforms of the pressure and particle veloc-ity at the boundary respectively n is the surface normalunit vector and Y (ω) is the boundary admittance whichis convenient to use when implementing frequency depen-dent boundary conditions into the linearized Euler equa-tions The boundary admittance can be approximated

2 J Acoust Soc Am 7 May 2019 Spectral element room acoustic simulations

as a rational function on the form

Y (ω) =a0 + middot middot middot+ aN (minusjω)N

1 + middot middot middot+ bN (minusjω)N (3)

which can be rewritten by using partial fraction decom-position as47

Y (ω) =Yinfin +

Qsumk=1

Akλk minus jω

+

Ssumk=1

(Bk + jCk

αk + jβk minus jω+

Bk minus jCkαk minus jβk minus jω

)

(4)

where Q is the number of real poles λk and S is the num-ber of complex conjugate pole pairs αkplusmnjβk used in therational function approximation Yinfin Ak Bk Ck are nu-merical coefficients Any number of poles can be chosenone strategy being to choose enough poles such that theerror in the multipole approximation of the boundaryadmittance is below a predefined threshold

Equation (2) can be transformed to the time domainby means of an inverse Fourier transform

vn(t) =

int t

minusinfinp(tprime)y(tminus tprime) dtprime (5)

Then by applying an inverse Fourier transform onEq (4) and inserting it into Eq (5) the expression forthe velocity at the boundary becomes

vn(t) =Yinfinp(t) +

Qsumk=1

Akφk(t)

+

Ssumk=1

2[Bkψ

(1)k (t) + Ckψ

(2)k (t)

]

(6)

where φk ψ(1)k and ψ

(2)k are so-called accumulators They

are determined by the following set of ordinary differen-tial equations

dφkdt

+ λkφk(t) = p(t)

dψ(1)k

dt+ αkψ

(1)k (t) + βkψ

(2)k (t) = p(t)

dψ(2)k

dt+ αkψ

(2)k (t)minus βkψ(1)

k (t) = 0

(7)

This approach is often called the auxiliary differentialequations (ADE) method in the literature47ndash49 and hasthe benefit of being computationally efficient becausesolving a small set of linear ODErsquos requires only relativelyminor computations Furthermore this approach has lowmemory requirements because only one time step historyof accumulator values must be stored

III NUMERICAL DISCRETIZATION

In this section a high-order numerical scheme for thesolution of Eq (1) in two and three spatial dimensions is

derived High-order methods are methods which have aglobal error convergence rate O(hP ) of at least third or-der (P gt 2) where h is the mesh element side length Inthis study triangular mesh elements are used in 2D andhexahedral elements are used in 3D although elementsof different shapes can be used

A Spatial discretization

The domain Ω is partitioned into a set of non-overlapping elements Ωn n = 1 Nel A set of nodesis chosen and mapped into each element making upa total of K nodes across the mesh and having coor-dinates xi i = 1 K A finite element approxima-tion space V of globally continuous piece-wise poly-nomial functions of degree at most P is introducedV = φ isin C0(Ω)foralln isin 1 Nel φ(n) isin PP Assuch the global basis functions φ are defined by patch-ing together local polynomial nodal basis functions φ(n)which are defined locally on each element and in thisstudy taken to be Lagrange polynomials of order P Tosupport order P basis functions each element must con-tain KP = (P+1)(P+2)2 nodes in 2D for the triangularelements and KP = (P + 1)3 nodes in 3D for the hex-ahedral elements44 Figure 1 shows an example of a 2Dmesh of a rectangular domain supporting P = 4 orderbasis functions

0 Lx

0

Ly

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

17 18 19 20

21 22 23 24

FIG 1 An example of a mesh of a 2D rectangular domain

using triangular elements and having nodes for supporting

P = 4 basis functions The mesh is made up of Nel = 24

elements and contains 221 DOFrsquos (nodes)

The weak formulation of the governing equationsthrough the use of the Divergence theorem on the pres-sure equation in Eq (1) takes the following form

intΩ

vtφ dΩ = minus1

ρ

intΩ

nablap φ dΩintΩ

ptφ dΩ = minusρc2[int

Γ

φn middot v dΓminusint

Ω

v middot nablaφ dΩ

]

(8)

where Γ denotes the boundary of Ω Now introduce atruncated series expansion for the unknown variables v


and p in Eq (8)

v(x t) asympKsumi=1

vi(t)Ni(x)

p(x t) asympKsumi=1

pi(t)Ni(x)

(9)

where Ni(x) isin V is the set of global finite element basisfunctions possessing the cardinal property Ni(xj) = δij Substituting the approximations in Eq (9) for v andp into Eq (8) and choosing φ isin Ni(x)Ki=1 to definea nodal Galerkin scheme results in the following semi-discrete system

Mvprimex = minus1

ρSxp Mvprimey = minus1

ρSyp Mvprimez = minus1

ρSzp

Mpprime = ρc2(STx vx + STy vy + STz vz minus vnB

)

(10)

where vx vy vz represent the x y z components of theparticle velocity vn is computed using Eq (6) and wherethe following global matrices have been introduced

Mij =

intΩ

Nj Ni dΩ Sx(ij) =

intΩ

(Nj)x Ni dΩ

Sy(ij) =

intΩ

(Nj)y Ni dΩ Sz(ij) =

intΩ

(Nj)z Ni dΩ

Bij =

intΓ

Nj Ni dΓ

(11)

where the x y z subscripts in the integrals denote dif-ferentiation In Eq (10) M is typically called the massmatrix and S is called the stiffness matrix To determinethese matrices it is convenient to introduce the conceptof a local element matrix

Due to the nature of the global piece-wise basis func-tions the integrals in Eq (11) are only non-zero when thenodes i j belong to the same element32 This means thattwo basis functions Ni and Nj only contribute towardsentries Mij when xi xj isin Ωn due to the local supportof the basis functions This leads to the definition of thelocal element matrices as

M(n)ij =

intΩn

N(n)i N

(n)j dΩn

S(n)x(ij) =

intΩn

N(n)i (N

(n)j )x dΩn

S(n)y(ij) =

intΩn

N(n)i (N

(n)j )y dΩn

S(n)z(ij) =

intΩn

N(n)i (N

(n)j )z dΩn

i j = 1 KP

(12)

From the local element matrices it is possible to assem-ble the global matrices in Eq (11) by iterating over the

elements and summing the element contributions relyingon the property of domain decomposition eg

Mij =

intΩ

NiNj dΩ =

Nelsumn=1

intΩn

N(n)i N

(n)j dΩn (13)

where the integrals may be zero The element matri-ces are therefore dense whereas the global matrices aresparse

B Spatial integration and nodalmodal duality

To compute the element matrices in Eq (12) it isconvenient to introduce a special element called the ref-erence element Ωr In 2D it is a triangle given by

I2 =r = (r s)|(r s) ge minus1 r + s le 0

(14)

and in 3D it is a hexahedron given by

I3 =r = (r s t)| minus 1 le (r s t) le 1

(15)

On these elements one can define a hierarchical modal ba-sis as opposed to the nodal basis discussed above Thisimplies a possible modalnodal duality in the represen-tation of the local solutions that can be exploited forexact integration relying on the orthogonal properties ofthe local modal basis functions without resorting to nu-merical quadrature rules When using a modal basis anunknown function is represented as

u(r) =

Psumj=0

ujψj(r) r isin Id (16)

where ψj are the modal basis functions and the coef-ficients uj are weights On I2 a basis proposed byDubiner50 is chosen where the reference triangle elementis first mapped to a unit square quadrilateral element bythe mapping

T (r s)rarr (a b) T (r s) =

(2

1 + r

1minus sminus 1 s

) (17)

where (a b) are the coordinates in the quadrilateralelement This allows for defining a modal basis interms of tensor products from the 1D reference ele-ment I1 = [minus1 1] The intra-element nodal distribu-tion of the collocation points r of the 1D reference ele-ment used in this study is of the Legrende-Gauss-Lobatto(LGL) kind Using this nodal distribution avoids Rungersquosphenomenon44 Now the 2D modal basis is defined as

ψpq(r s) = φap(r) φbq(s) (18)

where

φap(r) = P00p (r) φbq(s) =

(1minus s

2

)2

P2p+10q (s) (19)

and where Pαβp (z) is the prsquoth order Jacobi polynomialwith parameters α β By constructing the basis func-tions ψpq in this manner they become orthonormal onI2


On I3 a similar orthonormal modal basis is con-structed using a tensor product of Jacobi polynomials

ψ(r s t) = P00i (r) P00

j (s) P00k (t)

i j k = 0 P(20)

The function values of the nodes u used in the nodalrepresentation and the weights u used in the modal rep-resentation of u relate to each other through

Vu = u (21)

where V is the generalized Vandermonde matrix with

Vij = ψj(ri) i j = 1 P + 1 (22)

Utilizing this the irsquoth local nodal basis function on thereference element can be expressed as44

Ni(r) =

P+1sumn=1

(VT)minus1

inψn(r) (23)

Inserting Eq (23) into the expression defining the ele-ment mass matrix M on the reference element yields

Mij =

P+1sumn=1

(VT)minus1

in

(VT)minus1

jn=(VVT

)minus1 (24)

using the orthonormality of the chosen modal basis andthus avoiding the use of numerical quadrature rules Theconnection to the mass matrix in Eq (12) is defined bythe coordinate mapping between reference element andany element in the physical space

M(n)ij =

intΩn

ψi(x)ψj(x) dΩn =

intΩr

J (n)ψi(r)ψj(r) dΩr

(25)where J (n) is the Jacobian of the coordinate mappingξ x(n) rarr r

Next write the derivative of the irsquoth local basis func-tion as

part

partrNi(r s t) =

P+1sumn=1

part

partrNi(rn sn tn) Nn(r s t) (26)

Inserting the above into the expression defining the ele-ment stiffness matrix in Eq (12) one finds that44

Sr =MDr (27)

whereDr = VrVminus1 (28)

is a differentiation matrix and

Vr(ij) =part

partrψj(ri si) (29)

The remaining element matrices Ss and St are definedsimilarly and again the Jacobian coordinate mapping isused to map between the reference element and an arbi-trary element in the mesh

C Time stepping and stability

In order to solve the ODE system in Eq (10) effi-ciently an explicit time stepping method is preferred51

Explicit time stepping comes with conditional stabilitywhich sets an upper bound on the time step size ∆t Inthe proposed numerical scheme there are two mechanismsat play which influence the maximum allowable time stepFirstly the usual global Courant-Friedrichs-Lewy (CFL)condition where ∆t le C1max |λi| where λi representsthe eigenvalues of the spatial discretization34 and C1

is a constant depending on the size of the stability re-gion of the time stepping method Secondly the stiffnessof the ADE equations (Eq (7)) For certain boundaryconditions the ADE equations can become stiff whichputs an excessively strict restriction on the time stepThis motivates the usage of an implicit-explicit time step-ping method where the main SEM semi-discrete system(Eq (10)) is integrated explicitly in time whereas theADErsquos which are trivial to solve are integrated implic-itly in time This way the time step size is dictatedsolely by the global CFL condition not by the boundaryADErsquos

A six-stage fourth-order implicit-explicit Runge-Kutta time stepping method is used Let F ex(u t) bea spatial discretization operator representing the righthand side of the main semi-discrete system ie ut =F ex(u t) which is to be solved explicitly Similarly letF im(w t) represent the right hand side of the ADE equa-tions ie wt = F im(w t) which are to be solved implic-itly Intermediate stages are calculated with

Ti = tn + ci∆t

Ui = un + ∆t

6sumj=1

aexij F

ex(Uj Tj)

Wi = wn + ∆t

6sumj=1

aimij F

im(Wj Tj)

(30)

and the next iterative step of the solution is given by

tn+1 = tn + ∆t

un+1 = un + ∆t

6sumj=1

bexij F

ex(Uj Tj)

wn+1 = wn + ∆t

6sumj=1

bimij Fim(Wj Tj)

(31)

The coefficients aex aim bex bim c of the Butcher tableauof the Runge-Kutta method can be found in37

For the SEM the eigenvalues λi scale with polyno-mial order P in the following way34

max |λi| sim C2P2γ (32)

where γ is the highest order of differentiation in the gov-erning equations (γ = 1 here) and the constant C2 is


dependent on the minimum element size in the meshThis means that using a very high polynomial order P results only in marginal benefits in cost-efficiency due toa severe restriction on the time step size

The temporal step size in 2D used in this work isgiven by44

∆t = CCFL min(∆ri) minrDc (33)

where ∆ri is the grid spacing between the LGL nodes inthe reference 1D element I1 = [minus1 1] and rD = A

s is theradius of the triangular elementsrsquo inscribed circle wheres is half the triangle perimeter and A is the area of the tri-angle Here min(∆ri) prop 1P 2 and min rD

c correspondsthe smallest element on the mesh thus the expressionscales in accordance with the conditional stability crite-rion described above The constant CCFL is on the orderof O(1)

In the 3D case the temporal step size is given by

∆t = CCFLmin (∆x ∆y ∆z)

c (34)

where ∆x∆y and ∆z are the grid spacings betweennodes on the mesh in each dimension Because the intra-element nodal distribution within each hexahedral ele-ment is based on LGL nodes this expression also scalesinversely with basis order P 2 and with element size thusscaling proportionally to the stability criterion Againthe constant CCFL is on the order of O(1)

IV NUMERICAL PROPERTIES OF THE SCHEME

A Numerical errors

Numerical errors will arise both due to the spatialdiscretization and the temporal discretization These er-rors will be a mixture of dispersion errors and dissipationerrors An error convergence test is presented using a 3Dcube domain of size (1 times 1 times 1)λ where λ representswavelength The domain has periodic boundaries and ismeshed uniformly with hexahedral elements The erroris defined as ε = 〈||pa minus pSEM||L2

〉 The L2 integrationis carried out numerically by using the global mass ma-trix M as an integrator and 〈〉 indicates time averagingsuch that the mean of the L2 error across all time stepsis taken The analytic solution is given by

pa(x y z t) = sin(2π(xminus ct)) + sin(2π(x+ ct))

+ sin(2π(y minus ct)) + sin(2π(y + ct))

+ sin(2π(z minus ct)) + sin(2π(z + ct))

(35)

The domain is excited by an initial pressure condition bysetting t = 0 in the equation above Mass lumping is usedin the simulation to improve computational efficiency seediscussion on mass lumping in Sec IV B Figure 2a showsthe results of the convergence test for various polynomialorders P Here the time step is set to be small enough(CCFL = 001 in Eq (34)) such that spatial truncationerrors dominate The results show how fast the numerical

errors decrease for different orders P as the mesh elementside length h is refined For a given mesh element sizeit is evident how the high-order basis functions result insignificantly lower numerical errors

In order to give insights into the effects of the tempo-ral errors another convergence test is carried out using alarger time step having CCFL = 075 in Eq (34) The re-sults are shown in Fig 2b The global error convergenceis unaffected for basis functions orders up to P = 4 butfor P gt 4 a loss of convergence rates is seen as expectedsince the time stepping method is only fourth-order ac-curate

100

101

102

10-8

10-6

10-4

10-2

100

a) CCFL = 001

100

101

102

10-8

10-6

10-4

10-2

100

b) CCFL = 075

FIG 2 (Color online) Convergence test for the 3D periodic

domain problem

The dispersive and dissipative properties of the SEMfor wave problems have been widely studied52ndash59 A com-monly used approach for analyzing these properties in fi-nite element methods for wave problems uses eigenvalueanalysis The eigenvalue analysis has been used to provethat the SEM is non-dissipative for wave problems53

However numerical dissipation can be introduced via thetime stepping method which is coupled with the SEMThe numerical dissipation of the complete scheme can bequantified by measuring the energy in the system underrigid boundary conditions given by

E(t) =

intΩ

1

2ρc2p(tx)2 +

ρ

2|v(tx)|2 dx (36)


and a discrete measure of the energy can be computed in3D by

E(tn) =1

2ρc2pTMp

+ρ

2

(vTxMvx + vTyMvy + vTzMvz

)

(37)

where the sparse global mass matrix M is employed as aquadrature free integrator The dissipative properties ofthe proposed scheme are tested numerically in Sec V B

Using the eigenvalue analysis to analyze the disper-sive properties of the SEM results in some ambiguity dueto multiple solutions of the eigenvalue problem A morecomplete approach is a so-called multi-modal analysiswhere all of the numerical modes are regarded as relevantmodes of wave propagation relying on the representationof the numerical solutions in terms of a weighted combi-nation of all the various numerical modes5760 In thisstudy a multi-modal analysis method is devised basedon the 1D advection equation which is representative ofthe single modes in the Euler equations

ut + cux = 0 (38)

Exact solutions of the 1D advection equation can bestated on the general form

u(x t) = f(kxminus ωt) = f((ωc)xminus ωt) (39)

where f(s) is any smooth function describing the initialcondition waveform Thus the initial condition takes theform

u0 = u(x 0) = f((ωc)x) (40)

By assuming a solution ansatz f(s) = ejs for a singlewave the exact solution after N time steps will havea phase shift corresponding to eminusωN∆t Knowing thisa relation between the numerical solution at time stepN uN and the initial condition u0 can be establishedthrough

u0 = uNeminusωN∆t (41)

where ω is the numerical frequency which will differ fromthe exact frequency ω due to the dispersion of the numer-ical scheme This non-linear equation can be solved nu-merically for ω and in this study a Levenberg-Marquardtalgorithm is used for this task By comparing the nu-merical frequency against the exact one the dispersionrelationship can be established since cdc = ww wherecd is the numerical wave speed This analysis comeswith the advantage that any numerical simulator thatsolves the problem to evaluate uN can be used and inthis way all dispersive properties spatial and temporalof the given numerical scheme are taken into accountFigure 3 shows a resulting dispersion relation for a givenspatio-temporal resolution Clearly the high-order dis-cretization results in reduced dispersion errors In 3Dfundamentally the same dispersion behavior will occuras in 1D although here the dispersion relations will bedependent on the wave propagation direction59

0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6

FIG 3 (Color online) Numerical dispersion relations in 1D

by means of a multi-modal analysis h = 01 ∆t = 005 and

c = 1

B Computational work effort amp mass lumping

As has been shown above the usage of high-orderbasis functions results in lower numerical errors for agiven mesh resolution meaning that coarser spatial reso-lutions can be employed in simulations thereby reducingcomputational cost significantly However when usingexplicit time stepping the temporal step size must bemade smaller when using high-order basis functions asdescribed in Sec III C This counterbalances the bene-fits of the coarser spatial mesh to a degree The relevantquestion then becomes for a given problem which orderof basis functions results in the most cost-effective simu-lation The optimal order will primarily depend on thedesired numerical accuracy the simulation time (roomimpulse response length) and the highest frequency ofinterest28

A simple measure which can give an indication ofthe computational cost is applied in this study Thecost is defined as

WP = Ntimestep middotNDOF3D (42)

This model assumes serial computations and does notconsider details such as matrix operator densities com-puter architecture and implementation details TheNDOF3D is evaluated from 1D numerical experiments inwhich a 1D periodic domain of length 8λ a lumped massmatrix and a time step size ∆t = 3

4min ∆x

c are employedunder the assumption that the same spatial resolution isneeded in 1D as in the axial directions in 3D This wayNDOF3D = N3

DOF1DFigure 4 shows the estimated relative computational

cost required by the different orders to propagate a wavein a 3D periodic domain with ε = 2 numerical accu-racy as a function of the simulation time measured inwave periods Nw The choice of ε = 2 is ascribed tothe audibility threshold for dispersion error61 The num-ber of wave periods Nw in a periodic domain can be re-lated to the impulse response length tIR and the highestfrequency of interest fh through Nw = fhtIR The fig-ure highlights a number of important properties of thescheme For P = 1 which corresponds to the classic lin-ear h-FEM the computational cost is vastly larger com-


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015

FIG 4 (Color online) Predicted relative computational cost

required to propagate a 3D wave on a periodic cube domain

while maintaining an error tolerance ε = 002

pared to the other basis function orders As the basisorder P is increased the efficiency improvement followsa trend of diminishing returns Furthermore the ben-efits of using high-order basis functions increases withsimulation time Comparing the computational cost forthis particular test case for the P gt 1 cases against theP = 1 case shows that speed-up factors in the range of104 to 109 can be expected depending on which value ofP is used and what simulation time is used ComparingP gt 2 against P = 2 shows speed-up factors in the rangeof 20 to 1000 However it should be emphasised thatthis is based on a simplified measure of the computa-tional cost and in reality other factors besides the spatialresolution and the number of time steps taken eg thosementioned above will influence the cost as well

Mass lumping can be used to improve the efficiencyof the scheme62 The global mass matrix M is made diag-onal rendering matrix multiplication trivial Mass lump-ing will reduce accuracy slightly but global convergencerates are maintained63 The SEM when used in conjunc-tion with quadrilateral elements in 2D and hexahedralelements in 3D allows for the usage of mass lumpingtechniques in a straightforward way namely

Mii = diagsumj

Mij (43)

Applying mass lumping for meshes based on triangularelements in 2D and on tetrahedrons in 3D is more chal-lenging although one can take inspiration from previousstudies64 In this study mass lumping is employed forall 3D simulations whereas all 2D simulations are donewithout the use of mass lumping

A simple test case is presented to demonstrate thetrade-offs in accuracy and efficiency when using masslumping In this test case P = 4 Table I shows a com-parison of numerical errors ε and measured CPU timeswhen simulating 100 wave periods on the 3D periodicdomain The CPU times are measured using a sequen-tial non-optimized proof-of-concept implementation ofthe numerical scheme on an Intel Xeon E5-2650v4 CPU

The results show that the numerical error is slightlyincreased when mass lumping is used The computation

Nel per dim DOF εNon-ML tNon-ML εML tML

2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s

TABLE I Numerical error ε and CPU times t with and with-

out mass lumping for P = 4 while simulating 100 wave peri-

ods on a periodic 3D cube domain

time however is significantly shorter when using masslumping It becomes more beneficial to use mass lump-ing on larger problems In another test case where anerror bound of ε = 1 is used simulating 100 wave pe-riods with four mesh elements in each spatial dimensionthe ML version is about 8 times faster than the non-MLversion

V SIMULATION RESULTS

A 2D circular domain

Consider a 2D circular domain with radius a = 05m centered at (0 0) m and having perfectly rigid bound-aries This test case is chosen to illustrate the geometricflexibility of the SEM The impulse response of a givensource-receiver pair is simulated for two cases one us-ing typical straight-sided triangular mesh elements andthe other where the boundary elements have been trans-formed to be curvilinear Figure 5 shows the straight-sided mesh When using straight-sided mesh elementsas is typically done in FEM simulations a curved do-main boundary will be poorly represented unless an ex-tremely fine mesh is used which leads to an undesirablyhigh computational cost The main benefit of using high-order numerical schemes is the ability to use a coarsermesh with large mesh elements without a reduction inaccuracy By utilizing curvilinear mesh elements it be-comes possible to use large mesh elements with high basisorders while at the same time capturing important geo-metrical details

In both cases P = 4 basis functions are used and arelatively fine spatial resolution is employed roughly 9points per wavelength (PPW) for the highest frequencyof interest (1 kHz) This means that only minimal dis-persion should occur The initial condition is a Gaus-sian pulse with spatial variance σ = 005 m2 the simula-tion time is 3 s and the time step size is computed usingEq (33) with CCFL = 075

Figure 6 shows the simulated frequency responsesobtained via Fourier transforms of the simulated impulseresponses The curvilinear approach results in a betterprediction of the analytic modes66 For the straight-sidedelements case there is an apparent mistuning of the sim-ulated modal frequencies and this mistuning increaseswith frequency Figure 7 shows the difference in modal


-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]

FIG 5 (Color online) Mesh of the 2D circular domain

made using distmesh65 The mesh consists of 60 elements 521

DOFrsquos The circumference error for the straight-sided mesh is

041 and the interior surface area error is 164 The source

location is shown with a red cross ((sx sy) = (03 01)) and

the receiver location is shown with a black star ((rx ry) =

(minus02minus01))

frequencies when comparing simulated versus analyticmodal frequencies

100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30

FIG 6 (Color online) Simulated frequency responses in the

2D circular domain with and without curvilinear boundary

elements Analytic modes are dashed vertical lines calculated

using Greenrsquos function66

B 3D cube room with rigid boundaries

Consider now a 3D 1 times 1 times 1 m cube shaped roomwith perfectly rigid boundaries The rigid cube is a testcase of interest because an analytic solution exists whichsimulations can be compared against67 The room im-pulse response is simulated for a given source-receiverpair using basis orders P = 1 2 4 6 In all cases thespatial resolution is made to be the same ie the num-ber of DOFrsquos on the mesh are fixed to 15625 such thatNel = 24 12 6 4 per dimension in a uniform hexahedralmesh for P = 1 2 4 6 respectively The spatial reso-lution in all cases corresponds to roughly 86 PPW at1 kHz The initial condition is a Gaussian pulse with

1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8

FIG 7 (Color online) Error in simulated modal frequencies

for the 2D cirular domain case

spatial variance σ = 02 m2 the simulation time is 3s and the time step size is determined using Eq (34)with CCFL = 02 The simulated frequency responses areshown in Fig 8 The figure shows how the usage of high-order polynomial basis functions results in a closer matchto the analytic solution for the given fixed spatial res-olution As the polynomial order is increased the validfrequency range of the simulation is effectively extendedThe numerical error manifests itself both via mistuningsof the exact modes due to dispersion mismatch of modalfrequency amplitudes and as noise in the valleys betweenmodal frequencies

The dispersion error is analyzed further in Fig 9where the numerical modal frequencies are comparedagainst the analytic modal frequencies The difference isconstant and smaller than 04 Hz for the first 35 modesfor P = 4 and P = 6 but increases fast with frequencyfor P = 1 being 07 Hz for the 1st mode to 390 Hz forthe 15th mode The numerical dissipation in the schemefor this test case calculated using Eq (37) is shown inFig 10 The dissipation is found to be very low less than003 in all cases

C Single 3D reflection from an impedance boundary

In order to assess how accurately the proposedscheme represents locally reacting frequency dependentimpedance boundary conditions a single reflection of aspherical wave hitting such a boundary is studied Forthis case an analytic solution exists68 The wave reflec-tion is studied under two different boundary conditionsIn both cases the boundary is modelled as a porous ma-terial having flow resistivity of σmat = 10000 Nsmminus4 buthaving thickness of either dmat = 002 m or dmat = 005m The surface impedance of these materials are esti-mated using Mikirsquos model46 and mapped to a six polerational function using a vector fitting algorithm69 Fig-ure 11 shows the surface admittance of these two mate-rials and the resulting rational function approximationUsing six poles is sufficient to perfectly capture the realand imaginary part of the admittance curves Figure12 shows the corresponding absorption coefficients of thetwo materials


200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20

FIG 8 (Color online) Simulated frequency responses of a cube shaped room with rigid boundaries for basis orders P = 1 2 4 6

while using a fixed spatial resolution (15625 DOFrsquos) The analytic solution is the dashed curve The source location is

(sx sy sz) = (025 075 060) and the receiver location is (rx ry rz) = (085 030 080) The responses have been offset by 40

dB to aid visibility

5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102


for the cube shaped room with rigid boundaries case

0 05 1 15 2 25 3

09997

09998

09999

1

FIG 10 (Color online) Numerical dissipation for the cube

shaped room with rigid boundaries case

A large 3D domain is used for the simulation and theresulting impulse response is windowed such that no par-asitic reflections from other surfaces influence the sim-ulated response The source is located 2 m from theimpedance boundary and the receiver is located 1 m fromthe boundary at the midpoint between the source andthe boundary A basis order of P = 4 is used and a highspatial resolution is employed roughly 14 PPW at 1 kHzensuring minimal numerical errors in the frequency range

102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m

FIG 11 (Color online) Rational function fitting of the normal

incidence admittance of the two porous materials used in the

single reflection test case

of interest The initial condition is again a Gaussian pulsespatial variance σ = 02 m2

The resulting complex pressure is shown in the fre-quency domain in Fig 13 The simulated pressurematches the analytic solution perfectly both in termsof amplitude and phase for both boundary conditiontested thus illustrating the high precision of the im-plementation of locally reacting frequency dependentboundary conditions in the numerical scheme


102

103

0

02

04

06

08

1

FIG 12 (Color online) Normal incidence absorption coeffi-

cient of the two porous materials used in the single reflection

test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase

FIG 13 (Color online) Simulated complex pressure of a sin-

gle reflection from a locally reacting frequency dependent

impedance boundary compared with the analytic solution

D 3D room with frequency dependent boundary conditions

As a final test case an impulse response in the 1 times1times 1 m cube shaped room is simulated under frequencydependent boundary conditions where the ceiling of theroom is made to be covered with a porous material Thesame two materials considered in Sec V C are used againin this test case The basis order used in the simulationis P = 4 the source and receiver positions the initialcondition and CCFL are the same as in the tests in Fig 8but the spatial resolution has been increased to Nel =10 elements per dimension corresponding to roughly 14PPW at 1 kHz

The resulting frequency responses are shown inFig 14 The figure shows how in the presence of theporous material the modal frequency peaks have both

decreased in amplitude due to sound absorption at theboundary and shifted in frequency due to a phase shiftat the boundary when compared to the perfectly rigidboundary case The frequency dependent behavior ofthe porous material is evident in the frequency responseAt lower frequencies the modal peaks are less damp-ened compared to the higher frequencies and clearly thedmat = 005 m material adds more damping than thedmat = 002 m material

100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]

Rigiddmat = 002 mdmat = 005 m

FIG 14 (Color online) Simulated frequency responses of a

cube shaped room with five rigid surfaces and the ceiling

covered with a porous material The case where all surfaces

are rigid is also shown Basis order P = 4 is used in the

simulation

VI CONCLUSION

In this study a time domain numerical schemeadapted for room acoustic simulations based on a spec-tral element method in space and an implicit-explicitRunge-Kutta method in time has been developed andevaluated The main benefits of this scheme are its high-order accuracy combined with its geometrical flexibil-ity allowing for accurate and cost-effective room acousticsimulations of complex geometries

The results presented in this study show that there isa significant improvement in cost-efficiency and accuracywhen high-order basis functions are used This has beenshown both via a multi-modal spatio-temporal disper-sion analysis and via various three dimensional numeri-cal experiments Furthermore it has been shown how thehigh geometric flexibility of the SEM makes it possible tosimulate domains with curved geometries with very highaccuracy Errors in estimating modal frequencies dueto poor representation of curved geometries when usingstraight-sided mesh elements are effectively mitigated byusing curvilinear boundary elements

The presented method for representing locally re-acting frequency dependent impedance boundary con-ditions is found to be highly accurate with an excellentmatch seen between analytic solutions and simulationsin the case of a normal incidence spherical wave being re-flected from a impedance boundary The solution of theboundary ADErsquos comes with minimal additional compu-tational cost and is carried out implicitly thus the solu-


tion of these equations has no influence over the stabilityconditions of the scheme

The fact that room acoustic simulations involvebroad frequency ranges tight error tolerances long sim-ulation times and large complex 3D domains makes theproposed scheme particularly suitable where high preci-sion is important As the simulation time gets longerthe frequency range gets broader and the desired accu-racy gets higher the benefits of using high-order methodsrelative to low-order methods become greater

ACKNOWLEDGMENTS

This research has been partially funded by the Inno-vation Fund in Denmark Benchmarking has been doneusing the infrastructure at the DTU Computing Center

1S Pelzer L Aspock D Schroder and M Vorlander Integrat-ing real-time room acoustics simulation into a CAD modelingsoftware to enhance the architectural design process Buildings4(2)113ndash138 2014

2R Mehra A Rungta A Golas M Lin and D ManochaWAVE Interactive wave-based sound propagation for virtual en-vironments IEEE Trans Vis Comp Graph 21(4)434ndash4422015

3N Raghuvanshi A Allen and J Snyder Numerical wave sim-ulation for interactive audio-visual applications J Acoust SocAm 139(4)2008ndash2009 2016

4M Aretz and M Vorlander Combined wave and ray based roomacoustic simulations of audio systems in car passenger compart-ments part i Boundary and source data Appl Acoust 7682ndash99 2014

5V Valimaki J D Parker L Savioja J O Smith and J SAbel Fifty years of artificial reverberation IEEE Trans AudioSpeech Lang Proc 20(5)1421ndash1448 2012

6J Xia B Xu S Pentony J Xu and J Swaminathan Effectsof reverberation and noise on speech intelligibility in normal-hearing and aided hearing-impaired listeners J Acoust SocAm 143(3)1523ndash1533 2018

7M R Schroeder and K H Kuttruff On frequency responsecurves in rooms Comparison of experimental theoretical andMonte Carlo results for the average frequency spacing betweenmaxima J Acoust Soc Am 34(1)76ndash80 1962

8A Krokstad S Strom and S Soersdal Calculating the acousti-cal room response by the use of a ray tracing technique J SoundVib 8(1)118ndash125 1968

9L Savioja and U P Svensson Overview of geometrical roomacoustic modeling techniques J Acoust Soc Am 138(2)708ndash730 2015

10A Kulowski Algorithmic representation of the ray tracing tech-nique Appl Acoust 18(6)449ndash469 1985

11H Lee and B-H Lee An efficient algorithm for the image modeltechnique Appl Acoust 24(2)87ndash115 1988

12S Laine S Siltanen T Lokki and L Savioja Accelerated beamtracing algorithm Appl Acoust 70(1)172ndash181 2009

13Y W Lam Issues for computer modelling of room acousticsin non-concert hall settings Acoust Sci Tech 26(2)145ndash1552005

14J LoVetri D Mardare and G Soulodre Modeling of the seatdip effect using the finite-difference time-domain method JAcoust Soc Am 100(4)2204ndash2212 1996

15T J Cox and P DrsquoAntonio Acoustic absorbers and diffuserstheory design and application Routledge Taylor amp Francis 3rdedition 2016 Ch 13

16M L S Vercammen Sound concentration caused by curvedsurfaces PhD thesis Eindhoven University of Technology TheNetherlands 2011

17M Vorlander Computer simulations in room acoustics Con-cepts and uncertainties J Acoust Soc Am 133(3)1203ndash12132013

18D Botteldooren Finite-difference time-domain simulation oflow-frequency room acoustic problems J Acoust Soc Am98(6)3302ndash3308 1995

19J A Hargreaves and T J Cox A transient boundary elementmethod model of Schroeder diffuser scattering using well mouthimpedance J Acoust Soc Am 124(5)2942ndash2951 2008

20T Okuzono T Otsuru R Tomiku and N Okamoto A finite-element method using dispersion reduced spline elements forroom acoustics simulation Appl Acoust 791ndash8 2014

21R Mehra N Raghuvanshi L Antani A Chandak S Cur-tis and D Manocha Wave-based sound propagation in largeopen scenes using an equivalent source formulation ACM TransGraph 32(2)191ndash1913 2013

22S Bilbao Modeling of complex geometries and boundary condi-tions in finite differencefinite volume time domain room acous-tics simulation IEEE Trans Audio Speech Lang Proc21(7)1524ndash1533 2013

23M Hornikx T Krijnen and L van Harten openPSTD Theopen source pseudospectral time-domain method for acousticpropagation Comp Phys Comm 203298ndash308 2016

24M Vorlander Auralization Fundamentals of Acoustics Mod-elling Simulation Algorithms and Acoustic Virtual RealitySpringer 2008 Ch 10

25L Savioja Real-time 3D finite-difference time-domain simulationof low-and mid-frequency room acoustics In 13th InternationalConference on Digital Audio Effects volume 1 2010

26A Southern S Siltanen D T Murphy and L Savioja Roomimpulse response synthesis and validation using a hybrid acousticmodel IEEE Trans Audio Speech Lang Proc 21(9)1940ndash1952 2013

27R P Munoz and M Hornikx Hybrid Fourier pseudospec-traldiscontinuous Galerkin time-domain method for wave prop-agation J Comp Phys 348416ndash432 2017

28H-O Kreiss and J Oliger Comparison of accurate methods forthe integration of hyperbolic equations Tellus 24(3)199ndash2151972

29J van Mourik and D Murphy Explicit higher-order FDTDschemes for 3D room acoustic simulation IEEE Trans AudioSpeech Lang Proc 22(12)2003ndash2011 2014

30B Hamilton and S Bilbao FDTD methods for 3-D room acous-tics simulation with high-order accuracy in space and time IEEETrans Audio Speech Lang Proc 25(11)2112ndash2124 2017

31A T Patera A spectral element method for fluid dynamicsLaminar flow in a channel expansion J Comp Phys 54(3)468ndash488 1984

32GE Karniadakis and SJ Sherwin Spectralhp Element Meth-ods for Computational Fluid Dynamics Oxford University Press2nd edition 2005

33D Kopriva Implementing Spectral Methods for Partial Differ-ential Equations Springer 2009

34AP Engsig-Karup C Eskilsson and D Bigoni A stabilisednodal spectral element method for fully nonlinear water wavesJ Comp Phys 3181ndash21 2016

35G Seriani A parallel spectral element method for acoustic wavemodeling J Comp Acoust 05(01)53ndash69 1997

36S Airiau M Azaiez FB Belgacem and R Guivarch Paral-lelization of spectral element methods In J M L M PalmaA A Sousa J Dongarra and V Hernandez editors High Per-formance Computing for Computational Science mdash VECPAR2002 pages 392ndash403 Springer 2003

37C A Kennedy and M H Carpenter Additive Runge-Kuttaschemes for convection-diffusion-reaction equations Appl Num


Math 44(1)139ndash181 200338M Hornikx Ten questions concerning computational urban

acoustics Build Enviro 106409ndash421 201639S Bilbao B Hamilton J Botts and L Savioja Finite volume

time domain room acoustics simulation under general impedanceboundary conditions IEEE Trans Audio Speech Lang Proc24(1)161ndash173 2016

40C-W Shu Essentially non-oscillatory and weighted essentiallynon-oscillatory schemes for hyperbolic conservation laws InA Quarteroni editor Advanced Numerical Approximation ofNonlinear Hyperbolic Equations pages 325ndash432 Springer 1998

41C-W Shu High-order finite difference and finite volume WENOschemes and discontinuous Galerkin methods for CFD Int JComp Fluid Dynamics 17(2)107ndash118 2003

42N Atalla and F Sgard Finite Element and Boundary Methodsin Structural Acoustics and Vibration CRC Press 1st edition2015 Ch 7

43I Babuska and BQ Guo The h p and h-p version of the finiteelement method basis theory and applications Adv Eng Softw15(3)159ndash174 1992

44J S Hesthaven and T Warburton Nodal DiscontinuousGalerkin MethodsmdashAlgorithms Analysis and ApplicationsSpringer 2008 Ch 13469 and 10

45A Richard E Fernandez-Grande J Brunskog and C-H JeongEstimation of surface impedance at oblique incidence based onsparse array processing J Acoust Soc Am 141(6)4115ndash41252017

46Y Miki Acoustical properties of porous materials - modificationsof Delany-Bazley models J Acoust Soc Jap 11(1)19ndash24 1990

47R Troian D Dragna C Bailly and M-A Galland Broadbandliner impedance eduction for multimodal acoustic propagation inthe presence of a mean flow J Sound Vib 392200ndash216 2017

48P Cazeaux and J S Hesthaven Multiscale modelling of soundpropagation through the lung parenchyma ESAIM M2AN48(1)27ndash52 2014

49A Taflove and S C Hagness Computational Electrodynam-ics The Finite-Difference Time-Domain Method Artech HouseInc 3 edition 2013 Ch 9

50M Dubiner Spectral methods on triangles and other domainsJ Sci Comp 6(4)345ndash390 1991

51T Okuzono T Yoshida K Sakagami and T Otsuru An ex-plicit time-domain finite element method for room acoustics sim-ulations Comparison of the performance with implicit methodsAppl Acoust 10476ndash84 2016

52M Ainsworth and H Wajid Dispersive and dissipative be-havior of the spectral element method SIAM J Num Anal47(5)3910ndash3937 2009

53S Sherwin Dispersion analysis of the continuous and discontin-uous Galerkin formulations In B Cockburn G E Karniadakisand C-W Shu editors Discontinuous Galerkin Methods pages425ndash431 Springer 2000

54G Gassner and D Kopriva A comparison of the dispersionand dissipation errors of Gauss and Gauss-Lobatto discontinu-ous Galerkin spectral element methods SIAM J Sci Comp33(5)2560ndash2579 2011

55G Seriani and SP Oliveira DFT modal analysis of spectral ele-ment methods for acoustic wave propagation J Comp Acoust16(04)531ndash561 2008

56Fang Q Hu MY Hussaini and P Rasetarinera An analysis ofthe discontinuous Galerkin method for wave propagation prob-lems J Comp Phys 151(2)921ndash946 1999

57S P Oliveira On multiple modes of propagation of high-orderfinite element methods for the acoustic wave equation In M LBittencourt NA Dumont and J S Hesthaven editors Spec-tral and High Order Methods for Partial Differential EquationsICOSAHOM 2016 pages 509ndash518 Springer 2017

58G Seriani and S P Oliveira Optimal blended spectral-elementoperators for acoustic wave modeling Geophysics 72(5)SM95ndash

SM106 200759Y Geng G Qin J Zhang W He Z Bao and Y Wang Space-

time spectral element method solution for the acoustic waveequation and its dispersion analysis Acoust Sci and Tech38(6)303ndash313 2017

60J Yu C Yan and Z Jiang Effects of artificial viscosity andupwinding on spectral properties of the discontinuous Galerkinmethod Comp Fluids 175276ndash292 2018

61J Saarelma J Botts B Hamilton and L Savioja Audibilityof dispersion error in room acoustic finite-difference time-domainsimulation as a function of simulation distance J Acoust SocAm 139(4)1822ndash1832 2016

62C A Felippa Q Guo and KC Park Mass matrix templatesGeneral description and 1D examples Arch Comp Meth Eng22(1)1ndash65 2015

63I Fried and D S Malkus Finite element mass matrix lumpingby numerical integration with no convergence rate loss Int JSol Struct 11(4)461ndash466 1975

64S Jund and S Salmon Arbitrary high-order finite elementschemes and high-order mass lumping Int J Appl Math CompSci 17(3)375ndash393 2007

65P-O Persson and G Strang A simple mesh generator in MatlabSIAM Review 46(2)329ndash345 2004

66F Jacobsen and P Juhl Fundamentals of General Linear Acous-tics Wiley 2013 Ch 7

67S Sakamoto Phase-error analysis of high-order finite differ-ence time domain scheme and its influence on calculation resultsof impulse response in closed sound field Acoust Sci Tech28(5)295ndash309 2007

68S-I Thomasson Reflection of waves from a point source by animpedance boundary J Acoust Soc Am 59(4)780ndash785 1976

69B Gustavsen and A Semlyen Rational approximation of fre-quency domain responses by vector fitting IEEE Trans PowDel 14(3)1052ndash1061 1999



Time domain room acoustic simulations using thespectral element method

Finnur Pind1 a) Allan P Engsig-Karup2 Cheol-Ho Jeong3 Jan S Hesthaven4 Mikael S Mejling2 and JakobStroslashmann-Andersen1

1Henning Larsen Copenhagen Denmark2Scientific Computing Section Department of Applied Mathematics and Computer Science Techni-

cal University of Denmark Kongens Lyngby Denmark3Acoustic Technology Group Department of Electrical Engineering Technical University of Den-

mark Kongens Lyngby Denmark4Chair of Computational Mathematics and Simulation Science Ecole polytechnique federale de Lau-

sanne Lausanne Switzerland

This paper presents a wave-based numerical scheme based on a spectral element methodcoupled with an implicit-explicit Runge-Kutta time stepping method for simulating roomacoustics in the time domain The scheme has certain features which make it highly attrac-tive for room acoustic simulations namely a) its low dispersion and dissipation propertiesdue to a high-order spatio-temporal discretization b) a high degree of geometric flexibilitywhere adaptive unstructured meshes with curvilinear mesh elements are supported and c) itssuitability for parallel implementation on modern many-core computer hardware A methodfor modelling locally reacting frequency dependent impedance boundary conditions withinthe scheme is developed in which the boundary impedance is mapped to a multipole rationalfunction and formulated in differential form Various numerical experiments are presentedwhich reveal the accuracy and cost-efficiency of the proposed numerical scheme a

ccopy2019 Acoustical Society of America [httpdxdoiorg(DOI number)]

[XYZ] Pages 1ndash14

Keywords Room acoustic wave-based simulations spectral element method high-order nu-merical schemes frequency dependent impedance boundary conditions curvilinear meshing

I INTRODUCTION

Room acoustic simulations are used in manyfields for example in building design1 virtual reality2

entertainment3 automotive design4 music5 and hear-ing research6 Since their inception in the 1960rsquos78 roomacoustic simulations have primarily been carried out bymeans of geometrical acoustics methods9 such as theray tracing method10 the image source method11 or thebeam tracing method12 In these methods several simpli-fying approximations regarding sound propagation andreflection are made which make the computational taskmore manageable These approximations will howeverdeteriorate the accuracy of the simulation because vari-ous important wave phenomena such as diffraction in-terference and scattering are not accurately capturedWave phenomena will be prominent in rooms where theroom dimensions and sizes of obstacles are compara-ble to the wavelength of the acoustic wave Small tomedium sized rooms and low to mid frequencies arehere of primary concern13 However large rooms can

a)fpinhenninglarsencomaA part of the results of this paper was previously presented atEuronoise 2018 in Creta

also exhibit wave phenomena eg the seat-dip effect14

Another problem associated with geometrical acousticsmethods is that they require simplified 3D models madeup of coarse planar polygons Fine geometrical detailsare typically replaced by assigning scattering coefficientsto planar surfaces and these coefficients are often basedon crude visual inspection15 Instead of using simplified3D models it would be more accurate to model directlythe complex and detailed geometry typically found in ar-chitectural models16

Thanks to the continuous advances in computationpower and in scientific computing theory the wave-based methods are becoming a viable alternative for roomacoustic simulations In these methods the governingphysics equations are solved numerically and they aretherefore from a physical point of view more accuratethan their geometrical counterparts since all wave phe-nomena is inherently accounted for17 Wave-based meth-ods that have been applied to room acoustic simula-tions include the finite-difference time-domain method(FDTD)18 the boundary element method (BEM)19 thelinear finite element method (h-FEM)20 the equiva-lent source method (ESM)21 the finite volume method(FVM)22 and the pseudospectral time-domain method(PSTD)23 A major drawback of the wave-based meth-









TIONS


vt = minus1

ρnablap

in Ωtimes [0 t]




vn(ω) =p(ω)

Z(ω)= p(ω)Y (ω) (2)







Y (ω) =Yinfin +

Qsumk=1

Akλk minus jω

+

Ssumk=1

(Bk + jCk



)

(4)



vn(t) =

int t



vn(t) =Yinfinp(t) +

Qsumk=1

Akφk(t)

+

Ssumk=1

2[Bkψ

(1)k (t) + Ckψ

(2)k (t)

]

(6)




dφkdt

+ λkφk(t) = p(t)

dψ(1)k

dt+ αkψ

(1)k (t) + βkψ

(2)k (t) = p(t)

dψ(2)k

dt+ αkψ


k (t) = 0

(7)







0 Lx

0

Ly

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

17 18 19 20

21 22 23 24






intΩ

vtφ dΩ = minus1

ρ

intΩ

nablap φ dΩintΩ


Γ


Ω


]

(8)



and p in Eq (8)

v(x t) asympKsumi=1

vi(t)Ni(x)

p(x t) asympKsumi=1

pi(t)Ni(x)

(9)


Mvprimex = minus1



ρSzp


)

(10)


Mij =

intΩ

Nj Ni dΩ Sx(ij) =

intΩ

(Nj)x Ni dΩ

Sy(ij) =

intΩ


intΩ

(Nj)z Ni dΩ

Bij =

intΓ

Nj Ni dΓ

(11)



M(n)ij =

intΩn

N(n)i N

(n)j dΩn

S(n)x(ij) =

intΩn

N(n)i (N

(n)j )x dΩn

S(n)y(ij) =

intΩn

N(n)i (N

(n)j )y dΩn

S(n)z(ij) =

intΩn

N(n)i (N

(n)j )z dΩn

i j = 1 KP

(12)



Mij =

intΩ

NiNj dΩ =

Nelsumn=1

intΩn

N(n)i N

(n)j dΩn (13)





(14)



(15)


u(r) =

Psumj=0




(2

1 + r

1minus sminus 1 s

) (17)



where


(1minus s

2

)2

P2p+10q (s) (19)




ψ(r s t) = P00i (r) P00

j (s) P00k (t)

i j k = 0 P(20)


Vu = u (21)


Vij = ψj(ri) i j = 1 P + 1 (22)


Ni(r) =

P+1sumn=1

(VT)minus1

inψn(r) (23)


Mij =

P+1sumn=1

(VT)minus1

in

(VT)minus1

jn=(VVT

)minus1 (24)


M(n)ij =

intΩn

ψi(x)ψj(x) dΩn =

intΩr




part

partrNi(r s t) =

P+1sumn=1

part



Sr =MDr (27)



Vr(ij) =part








Ti = tn + ci∆t

Ui = un + ∆t

6sumj=1

aexij F

ex(Uj Tj)

Wi = wn + ∆t

6sumj=1

aimij F

im(Wj Tj)

(30)


tn+1 = tn + ∆t

un+1 = un + ∆t

6sumj=1

bexij F

ex(Uj Tj)

wn+1 = wn + ∆t

6sumj=1

bimij Fim(Wj Tj)

(31)














c (34)



A Numerical errors






(35)




100

101

102

10-8

10-6

10-4

10-2

100

a) CCFL = 001

100

101

102

10-8

10-6

10-4

10-2

100

b) CCFL = 075


domain problem



E(t) =

intΩ

1

2ρc2p(tx)2 +

ρ

2|v(tx)|2 dx (36)



E(tn) =1

2ρc2pTMp

+ρ

2


)

(37)



ut + cux = 0 (38)




u0 = u(x 0) = f((ωc)x) (40)




0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6



c = 1






4min ∆x





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS



















































































TIONS


vt = minus1

ρnablap

in Ωtimes [0 t]




vn(ω) =p(ω)

Z(ω)= p(ω)Y (ω) (2)







Y (ω) =Yinfin +

Qsumk=1

Akλk minus jω

+

Ssumk=1

(Bk + jCk



)

(4)



vn(t) =

int t



vn(t) =Yinfinp(t) +

Qsumk=1

Akφk(t)

+

Ssumk=1

2[Bkψ

(1)k (t) + Ckψ

(2)k (t)

]

(6)




dφkdt

+ λkφk(t) = p(t)

dψ(1)k

dt+ αkψ

(1)k (t) + βkψ

(2)k (t) = p(t)

dψ(2)k

dt+ αkψ


k (t) = 0

(7)







0 Lx

0

Ly

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

17 18 19 20

21 22 23 24






intΩ

vtφ dΩ = minus1

ρ

intΩ

nablap φ dΩintΩ


Γ


Ω


]

(8)



and p in Eq (8)

v(x t) asympKsumi=1

vi(t)Ni(x)

p(x t) asympKsumi=1

pi(t)Ni(x)

(9)


Mvprimex = minus1



ρSzp


)

(10)


Mij =

intΩ

Nj Ni dΩ Sx(ij) =

intΩ

(Nj)x Ni dΩ

Sy(ij) =

intΩ


intΩ

(Nj)z Ni dΩ

Bij =

intΓ

Nj Ni dΓ

(11)



M(n)ij =

intΩn

N(n)i N

(n)j dΩn

S(n)x(ij) =

intΩn

N(n)i (N

(n)j )x dΩn

S(n)y(ij) =

intΩn

N(n)i (N

(n)j )y dΩn

S(n)z(ij) =

intΩn

N(n)i (N

(n)j )z dΩn

i j = 1 KP

(12)



Mij =

intΩ

NiNj dΩ =

Nelsumn=1

intΩn

N(n)i N

(n)j dΩn (13)





(14)



(15)


u(r) =

Psumj=0




(2

1 + r

1minus sminus 1 s

) (17)



where


(1minus s

2

)2

P2p+10q (s) (19)




ψ(r s t) = P00i (r) P00

j (s) P00k (t)

i j k = 0 P(20)


Vu = u (21)


Vij = ψj(ri) i j = 1 P + 1 (22)


Ni(r) =

P+1sumn=1

(VT)minus1

inψn(r) (23)


Mij =

P+1sumn=1

(VT)minus1

in

(VT)minus1

jn=(VVT

)minus1 (24)


M(n)ij =

intΩn

ψi(x)ψj(x) dΩn =

intΩr




part

partrNi(r s t) =

P+1sumn=1

part



Sr =MDr (27)



Vr(ij) =part








Ti = tn + ci∆t

Ui = un + ∆t

6sumj=1

aexij F

ex(Uj Tj)

Wi = wn + ∆t

6sumj=1

aimij F

im(Wj Tj)

(30)


tn+1 = tn + ∆t

un+1 = un + ∆t

6sumj=1

bexij F

ex(Uj Tj)

wn+1 = wn + ∆t

6sumj=1

bimij Fim(Wj Tj)

(31)














c (34)



A Numerical errors






(35)




100

101

102

10-8

10-6

10-4

10-2

100

a) CCFL = 001

100

101

102

10-8

10-6

10-4

10-2

100

b) CCFL = 075


domain problem



E(t) =

intΩ

1

2ρc2p(tx)2 +

ρ

2|v(tx)|2 dx (36)



E(tn) =1

2ρc2pTMp

+ρ

2


)

(37)



ut + cux = 0 (38)




u0 = u(x 0) = f((ωc)x) (40)




0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6



c = 1






4min ∆x





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS
















































































Y (ω) =Yinfin +

Qsumk=1

Akλk minus jω

+

Ssumk=1

(Bk + jCk



)

(4)



vn(t) =

int t



vn(t) =Yinfinp(t) +

Qsumk=1

Akφk(t)

+

Ssumk=1

2[Bkψ

(1)k (t) + Ckψ

(2)k (t)

]

(6)




dφkdt

+ λkφk(t) = p(t)

dψ(1)k

dt+ αkψ

(1)k (t) + βkψ

(2)k (t) = p(t)

dψ(2)k

dt+ αkψ


k (t) = 0

(7)







0 Lx

0

Ly

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

17 18 19 20

21 22 23 24






intΩ

vtφ dΩ = minus1

ρ

intΩ

nablap φ dΩintΩ


Γ


Ω


]

(8)



and p in Eq (8)

v(x t) asympKsumi=1

vi(t)Ni(x)

p(x t) asympKsumi=1

pi(t)Ni(x)

(9)


Mvprimex = minus1



ρSzp


)

(10)


Mij =

intΩ

Nj Ni dΩ Sx(ij) =

intΩ

(Nj)x Ni dΩ

Sy(ij) =

intΩ


intΩ

(Nj)z Ni dΩ

Bij =

intΓ

Nj Ni dΓ

(11)



M(n)ij =

intΩn

N(n)i N

(n)j dΩn

S(n)x(ij) =

intΩn

N(n)i (N

(n)j )x dΩn

S(n)y(ij) =

intΩn

N(n)i (N

(n)j )y dΩn

S(n)z(ij) =

intΩn

N(n)i (N

(n)j )z dΩn

i j = 1 KP

(12)



Mij =

intΩ

NiNj dΩ =

Nelsumn=1

intΩn

N(n)i N

(n)j dΩn (13)





(14)



(15)


u(r) =

Psumj=0




(2

1 + r

1minus sminus 1 s

) (17)



where


(1minus s

2

)2

P2p+10q (s) (19)




ψ(r s t) = P00i (r) P00

j (s) P00k (t)

i j k = 0 P(20)


Vu = u (21)


Vij = ψj(ri) i j = 1 P + 1 (22)


Ni(r) =

P+1sumn=1

(VT)minus1

inψn(r) (23)


Mij =

P+1sumn=1

(VT)minus1

in

(VT)minus1

jn=(VVT

)minus1 (24)


M(n)ij =

intΩn

ψi(x)ψj(x) dΩn =

intΩr




part

partrNi(r s t) =

P+1sumn=1

part



Sr =MDr (27)



Vr(ij) =part








Ti = tn + ci∆t

Ui = un + ∆t

6sumj=1

aexij F

ex(Uj Tj)

Wi = wn + ∆t

6sumj=1

aimij F

im(Wj Tj)

(30)


tn+1 = tn + ∆t

un+1 = un + ∆t

6sumj=1

bexij F

ex(Uj Tj)

wn+1 = wn + ∆t

6sumj=1

bimij Fim(Wj Tj)

(31)














c (34)



A Numerical errors






(35)




100

101

102

10-8

10-6

10-4

10-2

100

a) CCFL = 001

100

101

102

10-8

10-6

10-4

10-2

100

b) CCFL = 075


domain problem



E(t) =

intΩ

1

2ρc2p(tx)2 +

ρ

2|v(tx)|2 dx (36)



E(tn) =1

2ρc2pTMp

+ρ

2


)

(37)



ut + cux = 0 (38)




u0 = u(x 0) = f((ωc)x) (40)




0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6



c = 1






4min ∆x





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS












































































and p in Eq (8)

v(x t) asympKsumi=1

vi(t)Ni(x)

p(x t) asympKsumi=1

pi(t)Ni(x)

(9)


Mvprimex = minus1



ρSzp


)

(10)


Mij =

intΩ

Nj Ni dΩ Sx(ij) =

intΩ

(Nj)x Ni dΩ

Sy(ij) =

intΩ


intΩ

(Nj)z Ni dΩ

Bij =

intΓ

Nj Ni dΓ

(11)



M(n)ij =

intΩn

N(n)i N

(n)j dΩn

S(n)x(ij) =

intΩn

N(n)i (N

(n)j )x dΩn

S(n)y(ij) =

intΩn

N(n)i (N

(n)j )y dΩn

S(n)z(ij) =

intΩn

N(n)i (N

(n)j )z dΩn

i j = 1 KP

(12)



Mij =

intΩ

NiNj dΩ =

Nelsumn=1

intΩn

N(n)i N

(n)j dΩn (13)





(14)



(15)


u(r) =

Psumj=0




(2

1 + r

1minus sminus 1 s

) (17)



where


(1minus s

2

)2

P2p+10q (s) (19)




ψ(r s t) = P00i (r) P00

j (s) P00k (t)

i j k = 0 P(20)


Vu = u (21)


Vij = ψj(ri) i j = 1 P + 1 (22)


Ni(r) =

P+1sumn=1

(VT)minus1

inψn(r) (23)


Mij =

P+1sumn=1

(VT)minus1

in

(VT)minus1

jn=(VVT

)minus1 (24)


M(n)ij =

intΩn

ψi(x)ψj(x) dΩn =

intΩr




part

partrNi(r s t) =

P+1sumn=1

part



Sr =MDr (27)



Vr(ij) =part








Ti = tn + ci∆t

Ui = un + ∆t

6sumj=1

aexij F

ex(Uj Tj)

Wi = wn + ∆t

6sumj=1

aimij F

im(Wj Tj)

(30)


tn+1 = tn + ∆t

un+1 = un + ∆t

6sumj=1

bexij F

ex(Uj Tj)

wn+1 = wn + ∆t

6sumj=1

bimij Fim(Wj Tj)

(31)














c (34)



A Numerical errors






(35)




100

101

102

10-8

10-6

10-4

10-2

100

a) CCFL = 001

100

101

102

10-8

10-6

10-4

10-2

100

b) CCFL = 075


domain problem



E(t) =

intΩ

1

2ρc2p(tx)2 +

ρ

2|v(tx)|2 dx (36)



E(tn) =1

2ρc2pTMp

+ρ

2


)

(37)



ut + cux = 0 (38)




u0 = u(x 0) = f((ωc)x) (40)




0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6



c = 1






4min ∆x





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS













































































ψ(r s t) = P00i (r) P00

j (s) P00k (t)

i j k = 0 P(20)


Vu = u (21)


Vij = ψj(ri) i j = 1 P + 1 (22)


Ni(r) =

P+1sumn=1

(VT)minus1

inψn(r) (23)


Mij =

P+1sumn=1

(VT)minus1

in

(VT)minus1

jn=(VVT

)minus1 (24)


M(n)ij =

intΩn

ψi(x)ψj(x) dΩn =

intΩr




part

partrNi(r s t) =

P+1sumn=1

part



Sr =MDr (27)



Vr(ij) =part








Ti = tn + ci∆t

Ui = un + ∆t

6sumj=1

aexij F

ex(Uj Tj)

Wi = wn + ∆t

6sumj=1

aimij F

im(Wj Tj)

(30)


tn+1 = tn + ∆t

un+1 = un + ∆t

6sumj=1

bexij F

ex(Uj Tj)

wn+1 = wn + ∆t

6sumj=1

bimij Fim(Wj Tj)

(31)














c (34)



A Numerical errors






(35)




100

101

102

10-8

10-6

10-4

10-2

100

a) CCFL = 001

100

101

102

10-8

10-6

10-4

10-2

100

b) CCFL = 075


domain problem



E(t) =

intΩ

1

2ρc2p(tx)2 +

ρ

2|v(tx)|2 dx (36)



E(tn) =1

2ρc2pTMp

+ρ

2


)

(37)



ut + cux = 0 (38)




u0 = u(x 0) = f((ωc)x) (40)




0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6



c = 1






4min ∆x





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS




















































































c (34)



A Numerical errors






(35)




100

101

102

10-8

10-6

10-4

10-2

100

a) CCFL = 001

100

101

102

10-8

10-6

10-4

10-2

100

b) CCFL = 075


domain problem



E(t) =

intΩ

1

2ρc2p(tx)2 +

ρ

2|v(tx)|2 dx (36)



E(tn) =1

2ρc2pTMp

+ρ

2


)

(37)



ut + cux = 0 (38)




u0 = u(x 0) = f((ωc)x) (40)




0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6



c = 1






4min ∆x





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS













































































E(tn) =1

2ρc2pTMp

+ρ

2


)

(37)



ut + cux = 0 (38)




u0 = u(x 0) = f((ωc)x) (40)




0 005 01 015 02 025 03 035 04 045 05

ω∆t

095

1

105

c dc

P = 1

P = 2

P = 4

P = 6



c = 1






4min ∆x





0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS












































































0 1000 2000 3000 4000 5000 6000 7000 8000 9000 1000010

0

105

1010

1015






Mii = diagsumj

Mij (43)





2 512 01065 17 s 02815 9 s

3 1728 00217 269 s 00283 56 s

4 4096 00070 1617 s 00077 192 s

5 8000 00029 7314 s 00030 579 s











-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS












































































-05 0 05

x [m]

-05

-04

-03

-02

-01

0

01

02

03

04

05

y[m

]







(minus02minus01))


100 200 300 400 500 600 700 800 900 1000

-20

-10

0

10

20

30







1 2 3 4 5 6 7 8 9 10 11 12

0

1

2

3

4

5

6

7

8








200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS












































































200 400 600 800 1000 1200 1400

-140

-120

-100

-80

-60

-40

-20

0

20





5 10 15 20 25 30 35 40 45 50 5510

-1

100

101

102



0 05 1 15 2 25 3

09997

09998

09999

1




102

103

-2

0

2

4

610

-3

a) dmat = 002 m

102

103

-1

0

1

2

310

-3

b) dmat = 005 m







102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS












































































102

103

0

02

04

06

08

1



test case

102

103

0

02

04

06

08

1

1210

-3

a) Amplitude

102

103

-200

-150

-100

-50

0

50

100

150

200

b) Phase








100 200 300 400 500 600 700 800 900 1000

Frequency [Hz]

-70

-60

-50

-40

-30

-20

-10

Pout[dB]






simulation

VI CONCLUSION







ACKNOWLEDGMENTS














































































ACKNOWLEDGMENTS












































































time domain room acoustic simulations using the spectral ... · the context of room acoustics.29,30...

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