M S V 1 Recent development of BEM/BIEM in vibration and acoustics 陳正宗 海洋大學 特聘教授 河海工程學系 Nov. 19, 2004, NSYSU, 14:10~16:00

M S V 2 Outlines Introduction Exterior acoustics - adaptive BEM Interior acoustics - multiply-connected eigenproblems Conclusions

M S V 3 Growth of BEM papers

M S V 4 Introduction Finite difference method (FDM) Finite element method (FEM) Boundary element method (BEM) Meshless method (MM) Boundary integral equation method (BIEM) FDM FEM Domain discretization BEM Boundary discretization MM BIEM No mesh No discretization for circular boundaries No mesh No node

M S V 5 Adaptive BEM for exterior radiation and scattering problems

M S V 6 Problem statement Non-uniform radiator problem Scattering problem

M S V 7 Adaptive scheme Solver Error indicator R.P.V. is Riemann Principal Value H.P.V. is Hadamard Principal Value C.P.V. is Cauchy Principal Value Singular formulation Hypersingular formulation

M S V 8 Adaptive mesh Uniform mesh Adaptive mesh

M S V 9 Refinement scheme h-versionp-versionr-version 1. Element number increasing 2. Interpolation function order increasing 3. Optimum nodal collocation 123

M S V 10 Mesh BEM FEM(DtN) Taiwan, NTOUUS Navy. Stanford Univ.

M S V 11 Non-uniform radiation ： Dirichlet problem Numerical solution: BEM Numerical solution: FEM(DtN) 64 ELEMENTS 2791 ELEMENTS Taiwan, NTOUUS Navy. Stanford Univ.

M S V 12 Non-uniform radiation ： Dirichlet problem Analytical solution: n=20

M S V 13 Superposition principle + ∥

M S V 14 Scattering ： Neumann problem Numerical solution: BEM Numerical solution: FEM( DtN) 63 ELEMENTS7816 ELEMENTS Taiwan, NTOUUS Navy. Stanford Univ.

M S V 15 Scattering ： Neumann problem Analytical solution: n=20

M S V 16 Fictitious frequency ： Non- uniform radiation problem ka u(a,0)

M S V 17 Fictitious frequency ： The scattering Dirichlet problem t(a, 0) ka

M S V 18 Summary Fictitious frequency depends on the formulation (singular or hypersingular) instead of B.C. (Dirichlet or Neumann). Burton & Miller method and CHIEEF method can overcome the problem of fictitious frequency. Fictitious frequency happens to be the true eigenvalues of the interior problem (Singular  Dirichlet, Hypersingular  Neumann).

M S V 19 Spurious eigenvalues for multiply-connected problems

M S V 20 Problem domain Doubly-connected domainMultiply-connected domain Simply-connected domain

M S V 21 BEM&BIEM BEM BIEM Boundary discretization Fourier series 0 0 u(θ) or t(θ) θ θ

M S V 22 The flowchart to determine the eigenvalues and mode shape by BEM Given G.E. and B.C. Solve the fundamental solution BIE for domain point Moving to the boundary BIE for boundary point Boundary element discretization Linear algebraic system Solve boundary data Potential SVD

M S V 23 The flowchart to determine the eigenvalues and mode shape by BIEM Degenerate kernelFourier series Null-field equation Algebraic equation Fourier Coefficients Potential Analytical Numerical SVD

M S V 24 Integral Formulation Null-field integral equations:

M S V 25 s x O R x ： source point ； s ： field point s r x Degenerate kernels

M S V 26 Degenerate kernels Degenerate kernels:

M S V 27 Degenerate kernels

M S V 28 Fourier series for boundary densities Fourier series:

M S V 29 Collocation points By choosing M terms of Fourier series, we select 2M+1 collocation points on the circle. 2M+1 terms

M S V 30 Integral representation Integral equation formulation:

M S V 31 Numerical examples Example 1

M S V 32 The eigenfrequenies by using singular equation More accurate ( ) Numerical [ ] exact Contaminated by spurious eigenvalues

M S V 33 Relation of spurious eigenvalue and true eigenvalue 0.5 True Spurious eigenvalue using singular formulation happens to be the true eigenvalue of the associated interior Dirichlet problem.

M S V 34 The eigenfrequenies by using hyper-singular equation ( ) Numerical [ ] exact More accurate Contaminated by spurious eigenvalues

M S V 35 Relation of spurious eigenvalue and true eigenvalue 0.5 True Spurious eigenvalue using hypersingular formulation happens to be the true eigenvalue of the associated interior Neumann problem.

M S V 36 The spurious eigenvalues are filtered by Burton&Miller method Only true eigenvalues appear

M S V 37 The former five eigenvalues of Helmholtz eigenproblem with an eccentric domain FEM [Chen et. al.] BEM [Chen and Zhou] BEM [Chen et. al.] Present method

M S V 38 The former five eigenmodes for eccentric case using present method, FEM and BEM

M S V 39 Numerical examples Example 2 R=1 c 1=0.3 c 2=0.4 e=0.5

M S V 40 Extraction of the spurious eigenvalues by using SVD updating document More accurate

M S V 41 The former five eigenvalues for a multiply- connected problem with two unequal holes using different approaches Method k i k1k1 k2k2 k3k3 k4k4 k5k5 Burton & Miller method Direct BEM + SVD Updating Null-field BEM + SVD Updating Fictitious BEM + SVD Updating Direct BEM + CHIEF method Null-field BEM + CHIEF method Fictitious BEM + CHIEF method FEM Present method

M S V 42 The former five modes for a circle domain with two unequal holes using present method, BEM and FEM

M S V 43 Summary Spurious eigenvalues depend on formulation (singular or hyper-singular). Spurious eigenvalues are independent of B.C. (Dirichlet or Neumann). Spurious eigenvalues happens to be the true eigenvalues of the interior problem (Dirichlet  singular, Neumann  hypersingular). To overcome the spurious eigenvalues  Burton&Miller, SVD updating term, SVD updating document …….

M S V 44 Conclusions Exterior acoustic problems (radiation and scattering) were solved by using adaptive BEM. Good accuracy and efficiency of the present method were obtained in comparison with those with FEM. Spurious eigenvalues embedded in the BIEM/BEM were examined and filtered out in this study. Both the fictitious frequency and spurious eigenvalue depend on the formulation instead of B.C..

M S V 45 歡迎參觀海洋大學力學聲響振動實驗室 烘培雞及捎來伊妹兒 URL:

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