Jeng-Tzong Chen 海洋大學河工系與機械系 陳正宗終身特聘教授 Vice president, TwSIAM

1. Improvement of the indirect BEM and the MFS for solving fictitious-frequency problems in 2D exterior acoustics Jeng-Tzong Chen 海洋大學河工系與機械系 陳正宗終身特聘教授 Vice president, TwSIAM 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 (iwmm2018-MFS.ppt)

2. Outline • Paradise and parasite of BEM and MFS • Physical and umerical resonance • Focus on MFS and IBEM • Numerical results • Conclusions

3. Paradise parasite 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)?? In the sky, there is a paradise In the earth, there is Suchou and Hangzhou 上有天堂 下有蘇杭 Fundamental solution BEM Hangzhou 2015 Suzhou 2008 Green function Parasite in the BEM Paradise in the BEM

4. Paradiseof BEM (1984-2000) Infinite domain Paradise 刪繁就簡三秋樹 鄭板橋 3D→2D 領異標新二月花 Fracture and fatigue Stress concentration

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

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

7. Physical andnumericalresonance Fictitious frequency (BEM/BIEM) Physical resonance t(a,0)

8. 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)

9. Four regularization techniques No null-field equation Null-field equation Self-regularization method + CHIEF Extra point

10. Problem statement Domain Governing equation: Boundary condition: Exact solution:

11. Indirect BEM Indirect BIE Domain Fundamental solution Boundary Matching the boundary condition Discretization

12. Spectrum of minimum singular value double root single root

13. Operator Range base Domain base Rank deficiency 0 of How to represent ?? for

14. 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

15. Self-regularization technique (math. & physics) SVD has no component If Self-regularization technique by using the SVD

16. 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, Computers and Structures, Vol.145, pp. 12-22, 2014.

17. 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

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

19. Self-regularization technique Range deficiency Discrete system How to determine the field Self-regularization ??? Boundary data

20. Self-regularization+CHIEF idea Range deficiency Continuous system Extra source point

21. Analytical derivation of the presentindirect BIE Double root Degenerate kernel Fourier series

22. Analytical derivation of the presentindirect BIE Double root The failure positions are the same with the CHIEF method in the direct BEM

23. Present idea for the MFS Ordinary MFS: Present MFS: MFS source point Extra source point Constraint equations Single root Double root

24. Three methods of regularization.

25. Outline • Introduction • Problem statement • Present approach • Numerical results • Conclusions

26. Numerical examples Case 1 (no solution) Case 2 (infinite solution) Case 4 (no solution) Case 3 (infinite solution)

27. Bar chart of unitary vectors CHIEF

28. Contour plot Case 1 (no solution) Fail Fail Exact

29. Contour plot Case 1 (no solution) OK (0.5, 0.5) OK (0,0) Exact No treatment

30. Case 2 (infinite solution) Contour plot Exact OK OK

31. Case 2 (infinite solution) Contour plot Exact OK OK

32. Bar chart of unitary vectors CHIEF

33. Contour plot Case 3 (infinite solution) OK OK Exact

34. Case 4 (no solution) Contour plot Fail Fail Exact

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

36. Numerical examples (MFS) Case 4 Fail Exact solution Ordinary MFS OK Fail The present MFS with valid points The present MFS with invalid points

37. Conclusions • The bordered matrix obtained by the self-regularization technique is invertible for any wavenumber. • The problem of fictitious frequencies can be alleviated by adding extra fundamental solutions in the direct BEM and MFS. • The present approach can fill in the gap that there is no CHIEFconstraint in the indirect BEMand MFS. • The criterion of failure points of the present idea is the same with the CHIEF method.

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

39. New direction 八卦邊界元 Trigram (Bagua) boundary element method Real variable BEM Complex variable BEM 八卦 BEM Quaternion valued BEM Clifford algebra valued BEM

40. Thanks for your kind attentions 請多多指教 感謝楊建軍教授邀請 Welcome to visit the web site of MSVLAB/NTOU http://msvlab.hre.ntou.edu.tw/

41. Contour plot Case 3 Failure point 2017/07/11

42. 目前需要克服的問題 COLY(NELM+1)=0.9*DCOS(Pi/(14.)) COLX(NELM+1)=0.9*DSIN(Pi/(14.)) COLX(NELM+2)=0.9 COLY(NELM+2)=0 2017/07/11

43. Physical andnumericalresonance Fictitious frequency (BEM/BIEM) Physical resonance Present t(a,0) 2017/07/11

44. Self-regularization+CHIEF method 怎麼進場 自救 無法表現 可表現

45. Analytical derivation of 魯蛋版 indirect BIE