Inﬁnite Elements in Acoustics

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Infinite Elements in Acoustics

Peter Rucz

January 8, 2010

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Problem definition

Γ

n Ωn

Γ_∞ Ωf

Ω=Ω_n∪Ω_f

Governing equations

I Helmholtz equation

∇²p(x) +k²p(x) = 0 x∈Ω (1)

I Boundary conditions

∇p(x)·n=−iωρ0vn(x) x∈Γ (2)

I Sommerfeld radiation condition

r→∞lim r^α ∂p

∂r+ ikp

= 0 (3)

We seekp(x) for the near and the far field,Ω_nand Ω_f for a given circular frequency (ω). There are several methods for the solution.

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Numerical methods for the solution

I BEM (Coupled FEM/BEM)

+ Non-local boundary conditions

+ Can be applied for non-convex geometries – Full, frequency depedent system matrices

I 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

I 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, ω) =

∞

X

n=0 n

X

m=0

h⁽²⁾_n (kr)P_n^m(cosθ){Anmsin(mϕ) +Bnm(cos(mϕ))}

+

∞

X

n=0 n

X

m=0

h⁽¹⁾_n (kr)P_n^m(cosθ){C_nmsin(mϕ) +D_nm(cos(mϕ))},

whereh⁽¹⁾n ,h⁽²⁾n and are P_n^m Hankel and Legendre functions.

Inwardly propagating waves can be omitted and by expanding Hankel functions we get:

p(x, ω) = e^−ikr

∞

X

n=1

Gn(θ, ϕ, ω)

rⁿ (4)

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The Atkinson-Wilcox theorem

TheAtkinson–Wilcox theorem states that the sound field at any point which lies outside a circumscribing sphere, which itself encloses all radiating and scattering sources, can be written as a multipole expansion of type (4).

(PointA satisfies this condition while pointB does not.)

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Meshing with infinite elements

To fulfill the conditions of the Atkinson–Wilcox theorem, the enclosing sphere of the radiating object must be built by finite elements. This ensures the convexity but costs additional DOFs for the computation.

Infinite elements are joined to the boundary of the surface.

Γ

Γ_∞ Ωf

Ω_n

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Element mapping

A mapping transformation projects the infinite element into a standard, finite parent space. Points at ”∞” are mapped to thes= 1line in the parent element.

x= 4 X

i=1

Mi(s, t)x_i

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)

I Mapping nodes– element geometry

I Variable nodes– pressure assigned and computed

I Radial order– number of nodes, order of approximation

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Formulation I. – Solution scheme

The weak form in the FE scheme A(ω)q(ω) =f(ω) Aαβ=

Z

Ω_e

(∇wα· ∇φβ−k²wαφβ)dΩ (5) Conjugated Astley–Leis formulation

I Shape functions: N(s, t)

I Test functions: φ(s, t) =N(s, t)e^−ikµ(s,t)

I Weight functions: w(s, t) =D(s, t)N(s, t)e^+ikµ(s,t) where

I Astley–Leis weighting: D(s, t) = (1−s)²/4

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

Astley–Leis weights ensure the finiteness of the integral (5). Phase terms cancel each other out and the resulting system matrices (K, M andC) 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

I Lagrangian shape functions

N_l(s, t) =1−s

2 L^p_l(s)S(t),

where L^p_l(s)is the standard Lagrange interpolating polynomial and S(t)is the standard shape function for a finite element.

Hence the solution is sought (corresponding to (4)) in the form φl=

p

X

i=1

αi

rⁱe^−ik(r−a)

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

+1

Z

−1

(1−s)^α(1 +s)^βJ_i^α,β(s)J_j^α,β(s)ds=µiδij, (6) which results in a lower condition number of the system matrix. In this 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. A transparent FE geometry is built around the source. The solution is carried out by a coupled FEM/BEM and a FEM/IEM method and their accuracy is compared to the analytic

solution. In the IEM case, infinite elements with radial order of 5 are attached to the outer

boundary of the FE geometry. In the FEM/BEM case an

impedance boundary condition is computed 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.)

RelativeL2 error norm of IE/FE solution: 0.0108

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Demo application – BEM solution (IV.)

RelativeL2 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] Stretch

1. (Fund.) 129.87 1.000 131 1.000 128 1.000 126 1.000

2. (Octave) 261.76 2.016 263 2.008 253 1.977 255 2.024

3. 396.45 3.053 397 3.031 388 3.031 387 3.071

4. 536.98 4.135 531 4.053 522 4.078 512 4.135

5. 677.62 5.218 667 5.092 660 5.156 658 5.222

1. (Fund.) 131.22 1.000 130 1.000 128 1.000 125 1.000

2. (Octave) 262.44 2.000 262 2.008 252 1.969 253 2.024

3. 400.38 3.051 394 3.025 387 3.023 384 3.072

4. 547.08 4.169 529 4.056 521 4.070 519 4.152

5. 680.99 5.190 664 5.095 660 5.156 655 5.240

1. (Fund.) 131.22 1.000 130 1.000 126 1.000 125 1.000

2. (Octave) 265.12 2.020 262 2.007 255 2.024 253 2.024

3. 401.73 3.061 395 3.024 388 3.079 384 3.072

4. 543.71 4.143 529 4.053 524 4.159 519 4.152

5. 679.64 5.190 665 5.095 662 5.254 656 5.248