Improvement of the indirect BEM and the MFS

Improvement of the indirect BEM and the MFS for solving fictitious-frequency problems in 2 D exterior acoustics Jeng-Tzong Chen 海洋大學河系與機械系陳正宗終身特聘教授 Vice president, Tw. SIAM Convener of Civil & Hydraulic Engineering Program, MOST, Taiwan Fellow of STAM, ROC Life-time Distinguished Professor Dept. Harbor and River Engineering Dept. Mechanical and Mechatronic Engineering Taiwan Ocean University 13: 00 -13: 40, August 3, 2018 International Academic Communication Center of CSUST (iwmm 2018 -MFS. ppt) Page 1

Outline 2018/06/28 u Paradise and parasite of BEM and MFS u Physical and umerical resonance u Focus on MFS and IBEM u Numerical results u Conclusions Page 2

In the sky, there is a paradise In the earth, there is Suchou and Hangzhou 上有天堂下有蘇杭 BEM and MFS always needs fundamental solution or Green function 邊界元素法有格林函數與基本解天堂(paradise)? ? 寄生蟲(parasite)? ? Fundamental solution BEM Suzhou 2008 Hangzhou 2015 Green function Paradise in the BEM Parasite in the BEM Page 3

Paradise of BEM (1984 -2000) 3 D→ 2 D 鄭板橋刪繁就簡三秋樹領異標新二月花 Paradise Stress concentration 2020/11/10 Infinite domain Fracture and fatigue Page 4

Parasite of BEM (2000 -2018) Minimum singular value t(a, 0) Rank-deficient systems 0/0 Mechanics Research Communications Radius Degenerate scale Fictitious frequency Engineering Analysis with Boundary Elements Pitfall Parasite Dependency Spurious eigenvalue International Journal for Numerical Methods in Engineering Degenerate boundary Page 5 Acta Thermochimita Parasite of BEM. pptx made by York

(Hangzhou: paradise in the world) (U(x, s): parasite in the BEM) BEM/BIEM (Disadvantage: Failure) Engineering problems Test function PDE and IE are not equivalent PDE model Match maker (ask for help) (1) Fundamental sol. (2) Green function IE model(BEM) Existence unique solution (1) (2) (3) (4) 引狼入室 (set a fox to keep geese) Degenerate scale Degenerate boundary Spurious eigenvalues Fictitious frequency Range deficiency Rank deficiency Singular matrix Nonunique solution Fredholm alternative theorem Infinite solution (Not sufficient) No solution (Not necessary) 2015 Master Thesis Oral Defense , MSV, National Taiwan Ocean University, Keelung Page 6 6

Physical and numerical resonance Physical resonance Fictitious frequency (BEM/BIEM) t(a, 0) 2018/06/28 Page 7

Proof of existence for the fictitious frequency in MFS ─ circulants I-Lin Chen, Using the method of fundamental solutions in conjunction with the degenerate kernel in cylindrical acoustic problems, Journal of the Chinese Institute of Engineers, Vol. 29, No. 3, pp. 445 -457 (2006) Page 8

Four regularization techniques Burton & Miller method CHIEF method Null-field equation Extra point 2018/06/28 Direct BEM indirect BEM (MFS) or Mixed potential method No null-field Self-regularization method + equation CHIEF Present approach Page 9

Problem statement Domain Governing equation: Boundary condition: Exact solution: 2018/06/28 Page 10

Indirect BEM Domain Indirect BIE Fundamental solution Boundary Matching the boundary condition 2018/06/28 Discretization Page 11

Spectrum of minimum singular value single root 2018/06/28 double root Page 12

Rank deficiency 0 Range base Operator How to represent 2018/06/28 for of ? ? Page 13

Self-regularization technique Fichera’s method for the problem of degenerate scale constraint equation When the size is at a degenerate scale, the Fichera’s method can obtain the unique solution 2018/06/28 Page 14

Self-regularization technique (math. & physics) SVD has no If component Self-regularization technique by using the SVD 2018/06/28 Page 15

Application to free-free structure (physics)` Ordinary stiffness matrix SVD & Bordered matrix flexibility matrix regularization J. T. Chen, W. S. Huang, Y. C. Tu and J. W. Lee, 2014, A self-regularized approach for deriving the free-free stiffness and flexibility matrices, Page 16 Computers and Structures, Vol. 145, pp. 12 -22, 2014.

Spectrum of U by using the self-regularization method 2. 409 3. 838 5. 144 5. 529 6. 390 7. 027 7. 599 8. 431 8. 668 8. 783 9. 776 9. 947 2018/06/28 Page 17

A new point of view of CHIEF Exterior problem governed by the Helmholtz equation Direct BEM Indirect BEM Domain Fictitious frequency G. E. Null-field equation is a CHIEF equation Relation between CHIEF and self-reg. method Risk of failure points 2018/06/28 B. Cs. No null-field equation How to find CHIEF equation ? or A self-regularization method Constraint equation ? Failure points ? Page 18

Self-regularization technique Range deficiency Discrete system How to determine the field Self-regularization Boundary data 2018/06/28 ? ? ? Page 19

Self-regularization+CHIEF idea Range deficiency Continuous system Extra source point 2018/06/28 Page 20

Analytical derivation of the present indirect BIE Double root Degenerate kernel Fourier series 2018/06/28 Page 21

Analytical derivation of the present indirect BIE Double root The failure positions are the same with the CHIEF method in the direct BEM 2018/06/28 Page 22

Present idea for the MFS Ordinary MFS: Present MFS: MFS source point Extra source point Constraint equations Single root Double root 2018/06/28 Page 23

Three methods of regularization. The conventional method The self-regularization method The present method (Adding extra collocation points & extra source points) ( combined with the right singular vector and the left singular vector ) (Adding extra source points & the right singular vector ) The self-regularization method The present method The conventional method Page 24

Outline u Introduction u Problem statement u Present approach u Numerical results u Conclusions 2018/06/28 Page 25

Numerical examples Case 1 (no solution) Case 3 (infinite solution) 2018/06/28 Case 2 (infinite solution) Case 4 (no solution) Page 26

Bar chart of unitary vectors CHIEF 2018/06/28 Page 27

Case 1 (no solution) Exact 2018/06/28 Contour plot Fail Page 28

Case 1 (no solution) Contour plot Exact OK (0, 0) OK (0. 5, 0. 5) No treatment 2018/06/28 Page 29

Case 2 (infinite solution) Contour plot Exact 2018/06/28 OK OK Page 30

Case 2 (infinite solution) Contour plot Exact 2018/06/28 OK OK Page 31

Bar chart of unitary vectors CHIEF 2018/06/28 Page 32

Case 3 (infinite solution) Exact 2018/06/28 Contour plot OK OK Page 33

Case 4 (no solution) Contour plot Exact 2018/06/28 Fail Page 34

Case 4 (no solution) Contour plot Case 4 (no solution) Failure points Exact Fail OK No treatment 2018/06/28 2017/07/11 Page 35

Numerical examples (MFS) Case 4 Fail Exact solution Fail The present MFS with invalid points 2018/06/28 Ordinary MFS OK The present MFS with valid points Page 36

Conclusions l The bordered matrix obtained by the selfregularization technique is invertible for any wavenumber. l The problem of fictitious frequencies can be alleviated by adding extra fundamental solutions in the direct BEM and MFS. l The present approach can fill in the gap that there is no CHIEF constraint in the indirect BEM and MFS. l The criterion of failure points of the present idea is the same with the CHIEF method. 2018/06/28 Page 37

singular Nonsingular Fredholm alternative theorem (Singular matrix) Interior Laplace problem(Degenerate scale) Role of c Singular to nonsingular Discriminant A D. O. F. more SVD is so expensive Stiffness and flexibility(Freefree structure) Exterior Helmholtz problem (Fictitious) c=0 Infinite solutions c≠ 0 No solution c=0 c≠ 0 Forces and moments equilibrium Forces and moments unequilibrium c=0 c≠ 0 Participation factor Free of participation factor Degenerate boundary c=0 True and spurious eignevalue c=0 c≠ 0 Page c≠ 0 38

New direction 八卦邊界元 Trigram (Bagua) boundary element method P a g e Real variable BEM Complex variable BEM 八卦 3 9 BEM Clifford algebra valued BEM 2020/11/10 Quaternion valued BEM Page 39