Introduction of dual BEM/BIEM and its mathematical issues J T Chen ( 陳正宗特聘教授 )

1. Introduction of dual BEM/BIEM and its mathematical issuesJ T Chen (陳正宗特聘教授) National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 海洋大學河海工程學系 April 24, 13:30-14:30, 2007 (靜宜數學2007.ppt)

2. Outlines • Overview of BEM and dual BEM ( 中醫式的工程分析法) • Mathematical formulation Hypersingular BIE • Successful experiences • Nonuniqueness and its treatments Degenerate scale True and spurious eigensolution (interior prob.) Fictitious frequency (exterior acoustics) • Conclusions

3. Top ten countries of BEM and dual BEM • BEM USA (3569) , China (1372) , UK, Japan, Germany, France, Taiwan (580), Canada, South Korea, Italy (No.7) • Dual BEM (Made in Taiwan) UK (124), USA (98), Taiwan (70), China (46), Germany, France, Japan, Australia, Brazil, Slovenia (No.3) (ISI information Apr.20, 2007) 台灣加油 FEM Taiwan (No.9/1383)

4. Top ten countries of FEM, FDM and Meshless methods • FEM USA(9946), China(4107), Japan(3519), France, Germany, England, South Korea, Canada, Taiwan(1383), Italy • Meshless methods USA(297), China(179), Singapore(105), Japan(49), Spain, Germany, Slovakia, England, Portugal, Taiwan(28) • FDM USA(4041), Japan(1478), China(1411), England, France, Canada, Germany, Taiwan (559), South Korea, India (ISI information Apr.20, 2007)

5. Top three scholars on BEM and dual BEM • BEM Aliabadi M H (UK, Queen Mary College) Chen J T (Taiwan, Ocean Univ.) 103 篇 Mukherjee S (USA, Cornell Univ.) Leisnic D(UK, Univ. of Leeds) Tanaka M (Japan, Shinshu Univ.) • Dual BEM (Made in Taiwan) Aliabadi M H (UK, Queen Mary Univ. London) Chen J T (Taiwan, Ocean Univ.) 103 篇 Power H (UK, Univ. Nottingham) (ISI information Apr.20, 2007) NTOU/MSV 加油

6. Top 25 scholars on BEM/BIEM since 2001 NTOU/MSV Taiwan 北京清華姚振漢教授提供(Nov., 2006)

7. Overview of numerical methods Domain Boundary MFS,Trefftz method MLS, EFG IE DE PDE- variational 針 灸 把 脈 開刀 7

8. Number of Papers of FEM, BEM and FDM 6 2 1 (Data form Prof. Cheng A. H. D.)

9. Growth of BEM/BIEM papers (data from Prof. Cheng A.H.D.)

10. Advantages of BEM • Discretization dimension reduction • Infinite domain (half plane) • Interaction problem • Local concentration Disadvantages of BEM 北京清華 • Integral equations with singularity • Full matrix (nonsymmetric)

11. NTUCE Ó BEM and FEM • BEM and meshless methods can be seen as a supplement of FEM. • (2) BEM utilizes the discretization concept of FEM as well as the limitation. Whether the supplement is needed or not depends on its absolutely superior area than FEM. • Ｃrack & large scale problems

12. NTUCE Ó What Is Boundary Element Method ? the Nth constant or linear element geometry node N 4 5 1 2 6 3 1 2 西醫 郎中

13. NTUCE Ó Dual BEM Why hypersingular BIE is required (potential theory) Degenerate boundary 4 4 7 7 6 5 6 5 8 3 8 3 9 10 1 2 1 2 Artifical boundary introduced ! BEM Dual integral equations needed ! Dual BEM

14. NTUCE Ó Some researchers on Dual BEM Chen(1986) 436 citings in total Hong and Chen (1988） 71 citings ASCE Portela and Aliabadi (1992) 188 citings IJNME Mi and Aliabadi (1994) Wen and Aliabadi (1995) Chen and Chen (1995) 新竹清華 黎在良等---斷裂力學邊界數值方法(1996) 周慎杰(1999) Chen and Hong (1999) 76 citings ASME Niu and Wang (2001) Yu D H, Zhu J L, Chen Y Z, Tan R J … cite

15. NTUCE Ó Dual Integral Equations by Hong and Chen(1984-1986) Singular integral equation Hypersingular integral equation Cauchy principal value Hadamard principal value Boundary element method Dual boundary element method 1969 2006 1986 normal boundary degenerate boundary

16. Degenerate boundary geometry node (1,0.5) (-1,0.5) 4 7 the Nth constant or linear element N 6 5 8 3 (0,0) 1 2 (-1,-0.5) (1,-0.5) 5(+) 6(+) 5(+) 6(-) 5(+) 6(+) 5(+) 6(+) 5(+) 6(+) 5(+) 6(-) 5(+) 6(-) 5(+) 6(-)

17. How to get additional constraints The constraint equation is not enough to determine coefficients of p and q, Another constraint equation is required

18. BEM Cauchy kernel singular DBEM Hadamard kernel hypersingular crack 1888 Integral equation (1984) (2000) FMM Large scale Degenerate kernel Original data from Prof. Liu Y J Desktop computer fauilure

19. Fundamental solution • Field response due to source (space) • Green’s function • Casual effect (time) K(x,s;t,τ)

20. Green’s function, influence line and moment diagram Force Force s x s s=1/2 x=1/4 G(x,s) G(x,s) x s Moment diagram s:fixed x:observer Influence line s:moving x:observer(instrument)

21. Two systems u and U U(x,s) u(x) Domain(D) s source Boundary (B) Infinite domain

22. Dual integral equations for a domain point(Green’s third identity for two systems, u and U) Primary field Secondary field where U(s,x)=ln(r) is the fundamental solution.

23. Dual integral equations for a boundary point(x push to boundary) Singular integral equation Hypersingular integral equation where U(s,x) is the fundamental solution.

24. Potential theory • Single layer potential (U) • Double layer potential (T) • Normal derivative of single layer potential (L) • Normal derivative of double layer potential (M)

25. Physical examples for potentials Moment Force U:moment diagram T:moment diagram L:shear diagram M:shear diagram

26. Order of pseudo-differential operators • Single layer potential (U) --- (-1) • Double layer potential (T) --- (0) • Normal derivative of single layer potential (L) --- (0) • Normal derivative of double layer potential (M) --- (1) Pseudo differential operator Real differential operator

27. Calderon projector

28. How engineers avoid singularity BEM / BIEM Improper integral Singularity & hypersingularity Regularity Fictitious BEM Bump contour Limit process Fictitious boundary Achenbach et al. (1988) Null-field approach Guiggiani (1995) Gray and Manne (1993) Collocation point CPV and HPV Ill-posed Waterman (1965)

29. NTUCE Ó Definitions of R.P.V., C.P.V. and H.P.V.using bump approach • R.P.V. (Riemann principal value) • C.P.V.(Cauchy principal value) • H.P.V.(Hadamard principal value)

30. Principal value in who’s sense • Common sense • Riemann sense • Lebesgue sense • Cauchy sense • Hadamard sense (elasticity) • Mangler sense (aerodynamics) • Liggett and Liu’s sense The singularity that occur when the base point and field point coincide are not integrable. (1983)

31. Two approaches to understand HPV Differential first and then trace operator (Limit and integral operator can not be commuted) Trace first and then differential operator (Leibnitz rule should be considered)

32. Bump contribution (2-D) U T s s 0 x x 0 L s M s x x

33. Bump contribution (3-D) 0` s s x x 0 s s x x

34. Successful experiences since 1986

35. Solid rocket motor (Army工蜂火箭)

36. X-ray detection (三溫暖測試) Crack initiation crack growth Stress reliever

37. FEM simulation

38. Stress analysis

39. BEM simulation

40. Shong-Fon II missile (Navy)

41. IDF (Air Force)

42. Flow field

43. V-band structure (Tien-Gen missile)

44. FEM simulation

46. Seepage flow

47. Free surface seepage flow using hypersingular formulation FEM (iteration No.49) BEM(iteration No.13) Initial guess Initial guess After iteration After iteration Remesh area Remesh line

48. Meshes of FEM and BEM

49. a t=0 t=0 t=0 c t=0 t=0 e b Incomplete partition in room acoustics

50. oblique incident water wave Top view y Free water surface S x O breakwater z breakwater z O x S Water wave problem with breakwater