National Taiwan Ocean University MSVLAB Department of Harbor
National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Dual BEM since 1986 J T Chen (陳正宗特聘教授) Department of Harbor and River Engineering National Taiwan Ocean University, Keelung, Taiwan 8: 30 -9: 50, Nov. 19, 2006 1
Outlines n n Overview of BEM and dual BEM Mathematical formulation Hypersingular BIE Nonuniqueness and its treatments Degenerate scale True and spurious eigensolution (interior prob. ) Fictitious frequency (exterior acoustics) Conclusions 2
Top ten countries of BEM and dual BEM n n BEM USA (3423) , China (1288) , UK, Japan, Germany, France, Taiwan (551), Canada, South Korea, Italy (No. 7) Dual BEM (Made in Taiwan) UK (119), USA (90), Taiwan (69), China (46), Germany, France, Japan, Australia, Brazil, Slovenia (No. 3) (ISI information Nov. 06, 2006) 台灣加油 FEM Taiwan (No. 9/1311) 3
ten countries of FEM, FDM and Meshless metho n n n FEM USA, China, Japan, France, Germany, England, South Korea, Canada, Taiwan, Italy Meshless methods USA, China, Singapore, Japan, Spain, Germany, Slovakia, England, France, Taiwan FDM USA, Japan, China, England, France, Germany, Canada, Taiwan, South Korea, Italy (ISI information Nov. 06, 2006) 4
Top three scholars on BEM and dual BEM n n BEM Aliabadi M H (UK, Queen Mary College) Mukherjee S (USA, Cornell Univ. ) Chen J T (Taiwan, Ocean Univ. ) 56 篇 Tanaka M (Japan, Shinshu Univ. ) Dual BEM (Made in Taiwan) Aliabadi M H (UK, Queen Mary Univ. London) Chen J T (Taiwan, Ocean Univ. ) 43 篇 Power H (UK, Univ Nottingham) (ISI information Nov. 06, 2006) NTOU/MSV 加油 5
Overview of numerical methods Domain DE 6 PDEvariational IE Boundary MFS Trefftz method MLS, EFG 6
Number of Papers of FEM, BEM and FDM 6 2 1 (Data form Prof. Cheng A. H. D. ) 7
Growth of BEM/BIEM papers (data from Prof. Cheng A. H. D. ) 8
Advantages of BEM n n Discretization dimension reduction Infinite domain (half plane) Interaction problem Local concentration Disadvantages of BEM n n Integral equations with singularity 北京清華 Full matrix (nonsymmetric) 9
What Is Boundary Element Method ? • Finite element method 1 Boundary element method 5 2 4 6 3 1 geometry node N 2 the Nth constant or linear element NTUCE Ó 11
Dual BEM Why hypersingular BIE is required (potential theory) Degenerate boundary 4 7 6 5 8 3 10 1 4 7 8 3 9 2 Artifical boundary introduced ! BEM 1 2 Dual integral equations needed ! Dual BEM NTUCE Ó 12
Some researchers on Dual BEM Chen(1986) 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 … NTUCE Ó 13
Dual Integral Equations by Hong and Chen(1984 -1986) Singular integral equation Cauchy principal value Boundary element method 1969 normal boundary Hypersingular integral equation Hadamard principal value Dual boundary element method 1986 2006 degenerate boundary NTUCE Ó 14
Degenerate boundary (-1, 0. 5) geometry node (1, 0. 5) 4 7 N 6 5 8 the Nth constant or linear element 3 (0, 0) 1 2 (-1, -0. 5) (1, -0. 5) 5(+) 6(+) 5(+) 6(-) 5(+) 6(+) 5(+) 6(-) 15
How to get additional constraints The constraint equation is not enough to determine coefficients of p and q, Another constraint equation is required 16
BEM DBEM Cauchy Hadamard kernel singular hypersingular crack 1888 Integral equation Original data from Prof. Liu Y J (1984) (2000) FMM Large scale Degenerate kernel Desktop computer fauilure 17
Fundamental solution n Field response due to source (space) Green’s function Casual effect (time) K(x, s; t, τ) 18
Green’s function, influence line and moment diagram Force s G(x, s) s=1/2 x Moment diagram s: fixed x: observer Force x s G(x, s) x=1/4 s Influence line s: moving x: observer(instrument) 19
Two systems u and U U(x, s) Domain(D) u(x) s source Boundary (B) Infinite domain 20
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. 21
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. 22
Potential theory n n Single layer potential (U) Double layer potential (T) Normal derivative of single layer potential (L) Normal derivative of double layer potential (M) 23
Physical examples for potentials Force Moment U: moment diagram T: moment diagram L: shear diagram M: shear diagram 24
Order of pseudo-differential operators n n 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 25
Calderon projector 26
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 HPV(1993) CPV and HPV Ill-posed Collocation point Waterman (1965) 27
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) NTUCE Ó 28
Principal value in who’s sense n n n n 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) 29
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) 30
Bump contribution (2 -D) U s 0 T s x x 0 L s x M s x 31
Bump contribution (3 -D) s x 0 ` s x 0 s s x x 32
Successful experiences since 1986 33
Solid rocket motor ( 蜂火箭) 34
X-ray detection (三溫暖測試) 35
FEM simulation 36
Stress analysis 37
BEM simulation 38
Shong-Fon II missile 39
IDF 40
Flow field 41
V-band structure (Tien-Gen missile) 42
FEM simulation 43
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Seepage flow 45
Meshes of FEM and BEM 46
Screen in acoustics a t=0 b t=0 c t=0 e t=0 47
Water wave problem with breakwater oblique incident water wave Top view y Free water surface S O breakwater z breakwater x z O S x 48
Reflection and Transmission 49
Cracked torsion bar 50
IEEE J MEMS 51
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Is it possible ! No hypersingularity ! No subdomain ! 54
Degenerate boundary problems n Multi-domain BEM Subdomain 1 r=1 u=0 Subdomain 1 interface Subdomain 2 r=1 u=0 n r=1 Dual BEM u=0 55
Conventional BEM in conjunction with SVD Singular Value Decomposition Rank deficiency originates from two sources: (1). Degenerate boundary (2). Nontrivial eigensolution Nd=5 Nd=4 Nd=5 56
UT BEM + SVD (Present method) n versus Determinant n k versus k Dual BEM Determinant versus k 57
UT BEM+SVD k=3. 09 k=3. 84 k=4. 50 FEM (ABAQUS) k=3. 14 k=3. 82 k=4. 48 58
BEM trap ? Why engineers should learn mathematics ? n Well-posed ? Existence ? Unique ? n Mathematics versus Computation n Some examples n n 59
Numerical phenomena (Degenerate scale) Commercial ode output ? Error (%) of 125 torsional rigidity 5 0 a Previous approach : Try and error on a Present approach : Only one trial 60
Numerical and physical resonance Physical resonance Numerical resonance radiation incident wave 61
Numerical phenomena (Fictitious frequency) t(a, 0) A story of NTU Ph. D. students 62
Numerical phenomena (Spurious eigensolution) 63
Some findings `Laplace Helmholtz Ling 1947 Analytical solution Caulk 1983 Bird & Steele 1991 Naghdi 1991 Analytical solution Present method Analytical solution Lee & Manoogian 1992 Tsaur et al. 2004 Analytical solution Present method 房營光1995 ? Tsaur et al. 64
Torsional rigidity ? 65
Stress concentration at point B Present method Steele & Bird Naghdi’s results The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified. --ASME Applied Mechanics Review 66
A half-plane problem with two alluvial valleys subject to the incident SH-wave Canyon Matrix SH-Wave 3 a 房[93]將正弦和餘弦函數的正交特性使用錯誤,以 至於推導出錯誤的聯立方程,求得錯誤的結果。 --亞太學報 67
Limiting case of two canyons Present method Tsaur et al. ’s results [103] 68
A half-plane problem with a circular inclusion subject to the incident SH-wave y h a Inclusion x Matrix SHWave 69
Surface displacements of a inclusion problem under the ground surface When I solved this problem I could find no published results for comparison. I also verified my results using the limiting cases. I did not have the benefit of published results for comparing the intermediate cases. I would note that due to precision limits in the Fortran compiler that I was using at the time. --Private communication Present method Tsaur et al. ’s results [102] Manoogian and Lee’s results [62] 70
Journals related to mechanics in Taiwan SCI EI 71
National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering 72
Conclusions • Introduction of dual BEM • The role of hypersingular BIE was examined. • Successful experiences in the engineering applications using BEM were demonstrated. • The trap of BIEM and BEM were shown • Previous errors by other investigators were identified 73
The End Thanks for your kind attention 74
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