infinite elements in acoustics

8/11/2019 Infinite Elements in Acoustics

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Infinite Elements in Acoustics

Peter Rucz

January 8, 2010
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Problem definition

n

n

f

=n f

Governing equations Helmholtz equation

2p(x) + k2p(x) = 0 x (1)

Boundary conditions

p(x) n =i0vn(x) x (2)

Sommerfeld radiation condition

limrrpr + ikp

= 0 (3)

We seek p(x) for the near and the far field, n and f for a givencircular frequency (). There are several methods for the solution.
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Numerical methods for the solution

BEM (Coupled FEM/BEM)

+ Non-local boundary conditions+ Can be applied for non-convex geometries Full, frequency depedent system matrices

PML (Absorbing layers)+ FE extension fast performance

+ Can be applied for non-convex geometries Truncation of the computational domain To obtain far field solution postprocessing is needed

Infinite elements (FEM+IEM)

+ Extension of the FE method+ Far field results obtained straightforwardly+ Frequency independent system matrices Can only be applied for convex geometries

(?) Application for CAA (Computational AeroAcoustics)
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Asymptotic form of a radiating solution

General expression of the solution of the 3D Helmholtz equation:

p(x, ) =n=0

nm=0

h(2)n (kr)Pmn (cos) {Anmsin(m) + Bnm(cos(m))}

+

n=0

n

m=0h

(1)

n (kr)Pm

n (cos) {Cnmsin(m) + Dnm(cos(m))} ,

where h(1)n , h

(2)n and are Pmn Hankel and Legendre functions.

Inwardly propagating waves can be omitted and by expanding

Hankel functions we get:

p(x, ) = eikrn=1

Gn(,,)

rn (4)
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Meshing with infinite elements

To fulfill the conditions of theAtkinsonWilcox theorem, theenclosing sphere of the radiating

object must be built by finiteelements. This ensures theconvexity but costs additionalDOFs for the computation.Infinite elements are joined to the

boundary of the surface.

f

n
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Element mappingA mapping transformation projects the infinite element into a standard,

finite parent space. Points at are mapped to the s= 1 line in the

parent element.

x =4

i=1

Mi(s, t)xi

M1 =(1 t)s

s 1

M2 =(1 + t)s

s 1

M3 =(1 + t)(1 +s)

2(s 1)

M4 =(1 t)(1 +s)

2(s

1)

Mapping nodes element geometry

Variable nodes pressure assigned and computed

Radial order number of nodes, order of approximation
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Formulation I. Solution scheme

The weak form in the FE scheme

A()q() = f() A =e

(w k2

w)d (5)

Conjugated AstleyLeis formulation

Shape functions: N(s, t)

Test functions: (s, t) =N(s, t)eik(s,t)

Weight functions: w(s, t) =D(s, t)N(s, t)e+ik(s,t)

where AstleyLeis weighting: D(s, t) = (1 s)2/4

Phase functions: (s, t) =a(s, t)(1 + s)/(1 s)

AstleyLeis weights ensure the finiteness of the integral (5). Phaseterms cancel each other out and the resulting system matrices (K,M and C) are frequency independent, non-hermitian and sparse.When the matrices are assembled the solution can be carried out

by direct or iterative formulas.


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Formulation II. Shape functions Lagrangian shape functions

Nl(s, t) =1 s

2 Lp

l(s)S(t),

where Lpl(s) is the standard Lagrange interpolating polynomial andS(t) is the standard shape function for a finite element.Hence the solution is sought (corresponding to (4)) in the form

l =

pi=1

iri

eik(ra)

Alternatively Jacobian polynomials can be used as shape functions,as they have the ortogonality property:

+11

(1 s)(1 + s)J,i (s)J,j (s)ds= iij , (6)

which results in a lower condition number of the system matrix. Inthis case a transformation matrix must be introduced to return to

the Lagrangian space.
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Demo application A transparent geoemtry (I.)

In this demo application a point

source radiates in free space. Atransparent FE geometry is builtaround the source. The solutionis carried out by a coupledFEM/BEM and a FEM/IEM

method and their accuracy iscompared to the analyticsolution. In the IEM case, infiniteelements with radial order of 5are attached to the outer

boundary of the FE geometry. Inthe FEM/BEM case animpedance boundary condition iscomputed for that region.
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Demo application The system matrices (II.)

The IEM/FEM system matrix

Assembling time: 29.14 s.

The FEM/BEM system matrix

Assembling time: 434.19 s.


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Demo application IEM solution (III.)

Relative L2 error norm of IE/FE solution: 0.0108


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Demo application BEM solution (IV.)

Relative L2 norm of FE/BE solution: 0.0172
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Real application Organ pipe modeling
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Comparison with previous results

Pipe: 4/16 Measurement Indirect BEM Coupled FE/BE IEM/FEM

Harmonic F [Hz] Stretch F [Hz] Stretch F [Hz] Stretch F [Hz] Stretch1. (Fund.) 129.87 1.000 131 1.000 128 1.000 126 1.0002. (Octave) 261.76 2.016 263 2.008 253 1.977 255 2.0243. 396.45 3.053 397 3.031 388 3.031 387 3.0714. 536.98 4.135 531 4.053 522 4.078 512 4.1355. 677.62 5.218 667 5.092 660 5.156 658 5.222

Pipe: 4/18 Measurement Indirect BEM Coupled FE/BE IEM/FEMHarmonic F [Hz] Stretch F [Hz] Stretch F [Hz] Stretch F [Hz] Stretch

1. (Fund.) 131.22 1.000 130 1.000 128 1.000 125 1.0002. (Octave) 262.44 2.000 262 2.008 252 1.969 253 2.0243. 400.38 3.051 394 3.025 387 3.023 384 3.0724. 547.08 4.169 529 4.056 521 4.070 519 4.1525. 680.99 5.190 664 5.095 660 5.156 655 5.240

Pipe: 4/18 Measurement Indirect BEM Coupled FE/BE IEM/FEMHarmonic F [Hz] Stretch F [Hz] Stretch F [Hz] Stretch F [Hz] Stretch1. (Fund.) 131.22 1.000 130 1.000 126 1.000 125 1.0002. (Octave) 265.12 2.020 262 2.007 255 2.024 253 2.0243. 401.73 3.061 395 3.024 388 3.079 384 3.0724. 543.71 4.143 529 4.053 524 4.159 519 4.1525. 679.64 5.190 665 5.095 662 5.254 656 5.248
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infinite elements in acoustics

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