1 The numerical methods for Helmholtz equation 報告人：陳義麟 國立高雄海洋技術學院造船系副教授 於海洋大學河海工程系 基隆 2003/10/23

Outline Helmholtz equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 2

Helmholtz equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 3

Helmholtz equation Time domain Wave equation Fourier transformation Frequency domain Helmholtz equation 4

Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 5

Engineering applications 1.Waveguide problem 2.Vibration of membranes 3.Water wave diffraction problem 4.Exterior acoustic problem 5.Elastic wave problem 6

Two-dimensional Helmholtz problem with a circular domain: G.E. : B D is the angle along the circular domain is the radius of the circular domain is the potential function denotes the Laplacian operator 7 D B Interior : Exterior : Problem statement k is the wave number

Wave equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 8

Numerical Methods Finite Element Method Finite Difference Method Boundary Element Method mesh Methods Meshless Methods 9

Helmholtz equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 10

11 Original system Reciprocal work theorem Auxiliary system UT (singular) formulation LM (hypersingular) formulation Kernel function Boundary integral equation method

12 Discrete the boundary integral equation (BEM) x2 x x3 x4 Influence matrix

13 Helmholtz equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks

Interior problem (Eigenproblem) For the Dirichlet B.C., u=0 To obtain the nontrivial solution 14

Field solution Field integral equation x s : Influence row vector 15

Exterior problem (radiation or scattering) For the Neumann B.C., Field solution u(a,0) 16

Fictitious frequency ka For the Neumann B.C.,, fail at the fictitious frequency 17

Helmholtz equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 18

Trefftz method Field solution : whereis complex type 19 where is the number of complete functions is the unknown coefficient is the T-complete function which satisfies the Helmholtz equation

T-complete set functions : 20 T-complete set The superscripts “I” and “E” denote the interior and exterior problems, respectively. Interior: Exterior:

For the Dirichlet B.C. u=0 By matching the boundary condition at 21 Derivation of unknown coefficients Field solution: Interior : Find the eigenvalue and eigenvector

22 Derivation of unknown coefficients Field solution: Exterior : For the Dirichlet B.C. By matching the boundary condition at

Helmholtz equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 23

Field solution : is the number of source points in the MFS is the unknown coefficient is the fundamental solution is the complementary domain is the source point is the collocation point Method of Fundamental Solutions (MFS) 24 B B’ R a D R B B’B’ D a Interior: Exterior: r r

Symmetry property for kernel : 25 Degenerate kernel : Separable property of kernel function

26 By matching the boundary condition for interior problem with Dirichlet B. C., u=0. Derivation of unknown coefficients Found the igenvalue and eigenvector

27 By matching the boundary condition for exterior problem with Dirichlet B. C. Derivation of unknown coefficients Failed at det|T|=0.

Numerical example (interior problem) B D BEMTrefftz 28

Numerical example (exterior problem) BEM MFS k t Fig.3 The contour plot for the real-part solutions. Radiator 29 ? Trefftz

Multiply domain ? Degenerate boundary ? ? Degenerate scale ? Further research 30

Helmholtz equation Engineering applications Numerical methods Boundary integral equation method BEM for interior and exterior problems The Trefftz method The MFS method Concluding remarks 31

Concluding Remarks 1.The three numerical methods have been demonstrated. 2.The BEM has the mesh concept, the others are meshless. 3.The BEM and MFS adopted the fundamental solution as a kernel function and basis function, respectively. 4.The Trefftz method adopted the T-complete set as a basis function. 5.The drawbacks of those numerical methods are the objective of research. 32

The End Thanks for your attention 33

Degenerate kernel (step1) 34 Step 1 S x r x: variable s: fixed

Degenerate kernel (step2, step3) 35 x s A B Step 2 RARA Step 3 x B A s RBRB