National Taiwan Ocean University MSVLAB Department of Harbor

National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering INTRODUCTION OF DUAL BEM/BIEM AND ITS ENGINEERING APPLICATIONS J T Chen, Distinguished Prof. Taiwan Ocean University June 07, 11: 00 -12: 00, 2007 (city-London 2007. ppt) 1

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

Top ten countries of BEM and dual BEM n n 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) 3

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

Top three scholars on BEM and dual BEM n n BEM Aliabadi M H (UK, Queen Mary College) Chen J T (Taiwan, Ocean Univ. ) 106 SCI papers 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. ) Power H (UK, Univ. Nottingham) NTOU/MSV 加油 (ISI information Apr. 20, 2007) 5

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

Overview of numerical methods Domain Boundary DE 開刀 7 IE PDE- variational 把脈 MFS, Trefftz method MLS, EFG 針灸 7

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

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

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) 10

BEM and FEM (1) 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 NTUCE Ó 11

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

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 13

Some researchers on Dual BEM Chen(1986) 436 citings in total cite Hong and Chen (1988） 74 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 AMR Niu and Wang (2001) Yu D H, Zhu J L, Chen Y Z, Tan R J … NTUCE Ó 14

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 15

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(+) dependency 5(+) 6(+) 5(+) 6(-) 16

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

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 18

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

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

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

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. 22

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. 23

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) 24

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

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 26

Calderon projector 27

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) 28

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 29

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) 30

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) 31

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

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

Successful experiences since 1986 34

Solid rocket motor (Army 蜂火箭) 35

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

FEM simulation 37

Stress analysis 38

BEM simulation 39

Shong-Fon II missile (Navy) 40

IDF (Air Force) 41

Flow field 42

V-band structure (Tien-Gen missile) 43

FEM simulation 44

45

Seepage flow 46

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

Meshes of FEM and BEM 48

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

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 50

Reflection and Transmission 51

Cracked torsion bar 52

IEEE J MEMS 53

54

55

Radiation and scattering problems Nonuniform radiaton scattering 56 2

BEM Adaptive Mesh FEM Dt. N interface 57 5

Strategy of adaptive BEM Singular Equation Error estimator Hypersingular Equation Error estimator u, t Solution 58 21

Nonuniform radiation： Dirichlet problem Numerical solution: BEM 64 ELEMENTS Numerical solution: FEM 2791 ELEMENTS 59 9

Is it possible ! No hypersingularity ! No subdomain ! 60

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 61

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 62

UT BEM + SVD (Present method) n versus k n Sub domain Determinant versus k n Dual BEM Determinant versus k 63

UT BEM+SVD k=3. 09 k=3. 84 k=4. 50 FEM (ABAQUS) k=3. 14 k=3. 82 k=4. 48 64

Disclaimer mmercial code) n The concepts, methods, and examples using our software for illustrative and educational purposes only. Our cooperation assumes no liability or responsibility to any person or company for direct or indirect damages resulting from the use of any information contained here. inherent weakness ? misinterpretation ? User 當自強 65

BEM trap ? Why engineers should learn mathematics ? n n n Well-posed ? Existence ? Unique ? Mathematics versus Computation equivalent ? Some examples 66

Numerical and physical resonance Physical resonance Numerical resonance radiation incident wave 67

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 68

Numerical phenomena (Fictitious frequency) t(a, 0) A story of NTU Ph. D. students 69

Numerical phenomena (Spurious eigensolution) 70

Treatments SVD updating term Burton & Miller method CHIEF method Mathematical analysis and numerical study for free vibration of plate using BEM-71 71

Conclusions • Introduction of dual BEM • The role of hypersingular BIE was examined • Successful experiences in the engineering applications using BEM were demonstrated • The traps of BIEM and BEM were shown 72