Spurious Eigensolutions and Fictitious Frequencies for Acoustic Problems with the Mixed-type Boundary Conditions by using BEM 邊界元素法於混合型邊界條件問題之 假根及虛擬頻率探討 國立台灣海洋大學河工二館 307 教室 中華民國九十二年七月十五日 研 究 生： 林宗衛 指導老師： 陳正宗教授

海洋大學力學聲響振動實驗室 Outlines Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research 02 Techniques and treatments

海洋大學力學聲響振動實驗室 Outlines Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research 03 Techniques and treatments

海洋大學力學聲響振動實驗室 Motivation 04 DirichletNeumannMixed-type spurious Eigenvalue fictitious frequency spurious Eigenvalue fictitious frequency spurious Eigenvalue fictitious frequency Complex-valued BEM NYNY?? Real-part BEM Y-Y-?? Imaginary-part BEM Y-Y-?? MRM Y-Y-??

海洋大學力學聲響振動實驗室 Outlines 05 Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research 05 Techniques and treatments

海洋大學力學聲響振動實驗室 Governing equation where D : the domain of interest k : the wave number x : the domain point u(x): the acoustic potential 06

海洋大學力學聲響振動實驗室 Types of boundary conditions Mixed-type u=u NeumannDirichlet t= =u 07

海洋大學力學聲響振動實驗室 The null-field integral formulations D DcDc 8 D DcDc

海洋大學力學聲響振動實驗室 D c : the complementary domain of D, x,s: the field and source point, u(s): the potential on the boundary t(s): the normal derivative of potential on the boundary U (s,x): the kernel function, T(s,x) M(s,x) L (s,x) 9

海洋大學力學聲響振動實驗室 U (s,x) kernel in different methods Complex-valued BEM: Real-part BEM: Imaginary-part BEM: MRM: 10

海洋大學力學聲響振動實驗室 Discretization (singular formulation) 11

海洋大學力學聲響振動實驗室 Rearrangement of the influences matrices 12

海洋大學力學聲響振動實驗室 Discretization (hypersingular formulation) 13

海洋大學力學聲響振動實驗室 Outlines 14 Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research Techniques and treatments

海洋大學力學聲響振動實驗室 k t : the true eigenvalues 15 Extraction of the true eigensolutions by using SVD updating terms

海洋大學力學聲響振動實驗室 Detection of the spurious eigensolutions by using SVD updating documents by using the Fredhohm ’ s alternative theorem 16 T: transpose ks: the spurious eigenvalues

海洋大學力學聲響振動實驗室 Burton & Miller method [ikU(s, x)+L(s, x)]u(s) = [ikT(s, x)+M(s, x)]t(s) +) 17

海洋大學力學聲響振動實驗室 CHIEF method 18

海洋大學力學聲響振動實驗室 Outlines 19 Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research Techniques and treatments

海洋大學力學聲響振動實驗室 Interior problems 20

海洋大學力學聲響振動實驗室 A finite rod s 0 =0 x x 0 1 s 1 =1 Type I : u(0)=0; t(1)=0 Type II: t(0)=0; u(1)=0 21 B.C.:

海洋大學力學聲響振動實驗室 Comparison of the eigenfuctions Interior problem Dirichlet u(0)=0, u(1)=0 Neumann t(0)=0, t(1)=0 Mixed-type (u(0)=0, t(1)=0) TrueSpuriousTrueSpuriousTrueSpurious Complex – valued BEM UT Sink- -Cosk- LM Sink- -Cosk- Real-part BEM UT Sink CoskSink LM Sink CoskSink Imaginary-part BEM UT Sink CoskSink LM Sink CoskSink 22

海洋大學力學聲響振動實驗室 23 Real-part BEM Imaginary-part BEM Complex-valued BEM T T T T T T T T T T T T S S S S S S

海洋大學力學聲響振動實驗室 A circular cavity u 2 =0t 1 =0 R=1 m 24 Elements: N=30

海洋大學力學聲響振動實驗室 Detection the eigenvalues using the complex-valued BEM 25

海洋大學力學聲響振動實驗室 True Spurious Updating term Updating document Detection of true and spurious eigenvalues using the real-part BEM 26

海洋大學力學聲響振動實驗室 Detection of true and spurious eigenvalues using the real-part BEM True Spurious Updating term Updating document 27

海洋大學力學聲響振動實驗室 The comparison of the spurious eigenvalues with different B.C. Neumann or DirichletMixed-typed 28

海洋大學力學聲響振動實驗室 The spurious eigenvalues using different methods Complex- valued BEM Real-part BEM Imaginary-part BEM MRM Singular formulation - Hyperingular formulation - where 29

海洋大學力學聲響振動實驗室 The comparison for the former five eigenmodes Real –part BEM FEM 30

海洋大學力學聲響振動實驗室 The comparison for the former five eigenvalues k1k1 k2k2 k3k3 k4k4 k5k5 Real –part BEM FEM ( ABAQUS )

海洋大學力學聲響振動實驗室 Exterior problems 32

海洋大學力學聲響振動實驗室 A semi-infinite rod s 0 =1 x 0 1 s1=s1= 33 B.C.: mt(1)+u(1)=n

海洋大學力學聲響振動實驗室 Comparison of the fictitious frequencies Interior problem Dirichlet u(1)=0 Neumann t (1)=0 Mixed-type mt(1)+u(1)=n Fictitious eigenfunctions T(s,x 0 )=0 UT cosk LM sink U(s,x 0 )=0 UT sink LM cosk 34

海洋大學力學聲響振動實驗室 T(s,x)=0U(s,x)=0 35 The fictitious frequency of 1-D problem

海洋大學力學聲響振動實驗室 A circular radiator 36 R=1 m Elements: N=30

海洋大學力學聲響振動實驗室 The positions of irregular for u(1,0) and their treatments x 37

海洋大學力學聲響振動實驗室 The positions of irregular for t(1, p ) and their treatments x 38

海洋大學力學聲響振動實驗室 Comparison of the BEM and exact solutions for radiation problem BEM solutionAnalytical solution 39

海洋大學力學聲響振動實驗室 Outlines 40 Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research Techniques and treatments

海洋大學力學聲響振動實驗室 The null-field integral formulations 41

海洋大學力學聲響振動實驗室 Semi-analytical approach 42

海洋大學力學聲響振動實驗室 43 Real-part BEM Imaginary-part BEM Complex-valued BEM Interior problem

海洋大學力學聲響振動實驗室 44 Exterior problem Singular Formulation Hypersingular Formulation

海洋大學力學聲響振動實驗室 Outlines 45 Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research Techniques and treatments

海洋大學力學聲響振動實驗室 Conclusions The results in the numerical experiment match well with those in our semi-analytical results. The eigensolutions and fictitious frequencies were solved successfully by using semi-analytical approach and BEMs 46

海洋大學力學聲響振動實驗室 Conclusions The spurious eigenvalues and fictitious frequencies depend on the representation no matter what the given types of B.C. for the problem are specified. 47

海洋大學力學聲響振動實驗室 Outlines 48 Motivation Boundary integral equations Examples Conclusions Semi-analytical approach Further research Techniques and treatments

海洋大學力學聲響振動實驗室 Further research 49 It will be interesting to know that if spurious eigenvalues and fictitious frequencies locate at the zeros of the Mathieu functions in the shape of ellipse. The quantity of the fictitious frequencies depend on the modal participation factor.

海洋大學力學聲響振動實驗室 Further research 50 To derive the fictitious frequencies analytically by using the complex-valued BEM for mixed-type boundary condition problem. The extension to multiple radiators & scatters and half-plane problems using the similar algorithm can be conducted to examine the occurrence of the fictitious frequencies.

海洋大學力學聲響振動實驗室 Further research 51 The extension to 3-D problems deserves further study to reconfirm the conclusion proposed in the thesis.

海洋大學力學聲響振動實驗室 報告完畢 敬請指教 52