National Taiwan Ocean University MSVLAB Department of Harbor

National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering Null field integral approach for engineering problems with circular boundaries J. T. Chen Ph. D. 終身特聘教授 Taiwan Ocean University Keelung, Taiwan Dec. 26, 15: 30 -17: 00, 2007 交通大學 機械系 邀請演講 交大機械 2007. ppt 1

Overview of numerical methods Domain Boundary DE 開刀 2 IE PDE- variational 把 脈 MFS, Trefftz method MLS, EFG 針 灸 2

Number of Papers of FEM, BEM and FDM 6 2 1 (Data form Prof. Cheng A. H. D. ) 3

Growth of BEM/BIEM papers (data from Prof. Cheng A. H. D. ) 4

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) 5

Top ten countries of FEM, FDM and Meshless methods n n n 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 6

Active scholars on BEM and dual BEM n n BEM Aliabadi M H (UK, Queen Mary College) Chen J T (Taiwan, Ocean Univ. ) 113 SCI papers 479 citing 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) (ISI information Apr. 20, 2007) 7

Top 25 scholars on BEM/BIEM since 2001 NTOU/MSV Taiwan 北京清華姚振漢教授提供(Nov. , 2006) 8

Research topics of NTOU / MSV LAB using null-field BIEs (2005 -2007) (2008 -2011) Null-field BIEM NUMPDE revision Navier Equation Laplace Equation EABE Elasticity (holes and/or inclusions) Helmholtz Equation MRC, CMES (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) EABE (Inclusion) (Piezoleectricity) (Interior and exterior Acoustics) Screw dislocation CMAME 2007 Jo. M SH wave (exterior acoustics) (Inclusions) (Beam bending) SDEE ICOME 2006 Torsion bar (Inclusion) Imperfect interface Degenerate kernel for ellipse Added mass Green function of half plane (Hole and inclusion) Effective conductivity (Plate with circulr holes) SH wave Impinging hill JSV (Free vibration of plate) Indirect BIEM (Stokes flow) (Free vibration of plate) Direct BIEM Green function for an annular plate SH wave Impinging canyons CMC Image method (Green function) Bi. Helmholtz Equation ASME JAM 2006 JCA ASME Biharmonic Equation (Flexural wave of plate) Green function of`circular inclusion (special case: static) Previous work (done) Water wave impinging circular cylinders Interested topic Present project URL: http: //ind. ntou. edu. tw/~msvlab E-mail: jtchen@mail. ntou. edu. tw 海洋大學 學院河 所力學聲響振動實驗室 Frame 2007 -c. ppt 9

Research collaborators n n n n Dr. I. L. Chen (高海大) Dr. K. H. Chen (宜蘭大學) Dr. S. Y. Leu Dr. W. M. Lee (中華技術學院) Mr. S. K. Kao Mr. Y. Z. Lin Mr. S. Z. Hsieh Mr. H. Z. Liao Mr. G. N. Kehr (2007) Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao (2006) Mr. A. C. Wu Mr. P. Y. Chen (2005) Mr. Y. T. Lee Mr. S. K. Kao Mr. S. Z. Shieh Mr. Y. Z. Lin 10

Top 25 scholars on BEM/BIEM since 2001 北京清華姚振漢教授提供(Nov. , 2006) 11

Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 13

Motivation Numerical methods for engineering problems FDM / FEM / BIEM / Meshless method BEM / BIEM (mesh required) Treatment of singularity and hypersingularity Boundary-layer effect Convergence rate Ill-posed model 除四舊 Mesh free for circular boundaries ? 14

Motivation and literature review BEM/BIEM Singular and hypersingular Bump contour Improper integral Limit process Regular Fictitious BEM Null-field approach CPV and HPV Fictitiou s boundar y Collocation point Ill-posed 15

Present approach Degenerate kernel Fundamental solution No principal value CPV and HPV Advantages of degenerate kernel value 1. No principal 2. Well-posed 3. No boundary-layer effect 16

Engineering problem with arbitrary geometries Straight boundary (Legendre polynomial) (Fourier series) Degenerate boundary (Chebyshev polynomial) Circular boundary Elliptic boundary (Mathieu function) 17

Motivation and literature review Analytical methods for solving Laplace problems with circular holes Conformal mapping Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics Bipolar coordinate Special solution Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Honein, Honein and Hermann, 1992, “On two circular inclusions in harmonic problem”, Quarterly of Applied Mathematics Limited to doubly connected domain 18

Fourier series approximation n n Ling (1943) - torsion of a circular tube Caulk et al. (1983) - steady heat conduction with circular holes Bird and Steele (1992) - harmonic and biharmonic problems with circular holes Mogilevskaya et al. (2002) - elasticity problems with circular boundaries 19

Contribution and goal However, they didn’t employ the nullfield integral equation and degenerate kernels to fully capture the circular boundary, although they all employed Fourier series expansion. n To develop a systematic approach for solving Laplace problems with multiple holes is our goal. n 20

Outlines (Direct problem) n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 21

Boundary integral equation and null-field integral equation Interior case Exterior case Degenerate (separate) form 22

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 23

Principal value in who’s sense n n n n Riemann sense (Common 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 24 not integrable. (1983)

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) 25

Bump contribution (2 -D) U s 0 T s x x 0 L s x M s x 26

Bump contribution (3 -D) s x 0 ` s x 0 s s x x 27

Outlines (Direct problem) n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n Numerical examples Degenerate scale Conclusions 28

Gain of introducing the degenerate kernel Degenerate kernel Fundamental solution interior CPV and HPV exterior No principal value? 29

How to separate the region 30

Expansions of fundamental solution and boundary density n Degenerate kernel - fundamental solution n Fourier series expansions - boundary density 31

Separable form of fundamental solution (1 D) Separable property continuous discontinuo us 32

Separable form of fundamental solution (2 D) 33

Boundary density discretization Fourier series Ex. constant element Present method Conventional BEM 34

Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 35

Adaptive observer system collocation point 36

Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 37

Vector decomposition technique for potential gradient True normal direction Non-concentric case: Special case (concentric case) : 38

Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 39

Linear algebraic equation wher e Index of collocation circle Index of routing circle Column vector of Fourier coefficients (Nth routing circle) 40

Physical meaning of influence coefficient kth circular boundary mth collocation point xm on the jth circular boundary cosnθ, sinnθ boundary distributions Physical meaning of the influence coefficient 41

Flowchart of present method Potential gradient Degenerate kernel Analytical Fourier series Adaptive observer system Vector decompositio n Potential of domain point Collocation point and matching B. C. Linear algebraic Fourier coefficients equation Numerical 42

Comparisons of conventional BEM and present method Boundary density discretization Auxiliary system Formulation Observer system Singularity Convergence Boundary layer effect Conventional BEM Constant, linear, quadratic… elements Fundamental solution Boundary integral equation Fixed observer system CPV, RPV and HPV Linear Appear Present method Fourier series expansion Degenerate kernel Null-field integral equation Adaptive observer system Disappear Exponential Eliminate 43

Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 44

Numerical examples n n n Laplace equation (EABE 2006, EABE 2007) (CMES 2006, ASME 2007, Jo. M 2007, CMC) (MRC 2007, NUMPDE 2007, ASME 2007) Eigen problem (JCA) Exterior acoustics (CMAME 2007, SDEE 2008) Biharmonic equation (JAM, ASME 2006, IJNME) Plate vibration (JSV 2007) 45

Laplace equation n n n Steady state heat conduction problems Electrostatic potential of wires Flow of an ideal fluid pass cylinders A circular bar under torque An infinite medium under antiplane shear Half-plane problems 46

Steady state heat conduction problems Case 1 Case 2 47

Case 1: Isothermal line Exact solution (Carrier and Pearson) FEM-ABAQUS (1854 elements) BEM-BEPO 2 D (N=21) Present method (M=10) 48

Relative error of flux on the small circle 49

Convergence test - Parseval’s sum for Fourier coefficients Parseval’s sum 50

Laplace equation n n n Steady state heat conduction problems Electrostatic potential of wires Flow of an ideal fluid pass cylinders A circular bar under torque An infinite medium under antiplane shear Half-plane problems 51

Electrostatic potential of wires Two parallel cylinders held positive and negative potentials Hexagonal electrostatic potential 52

Contour plot of potential Exact solution (Lebedev et al. ) Present method (M=10) 53

Contour plot of potential Onishi’s data (1991) Present method (M=10) 54

Laplace equation n n n Steady state heat conduction problems Electrostatic potential of wires Flow of an ideal fluid pass cylinders A circular bar under torque An infinite medium under antiplane shear Half-plane problems 55

Flow of an ideal fluid pass two parallel cylinders is the velocity of flow far from the cylinders is the incident angle 56

Velocity field in different incident angle Present method (M=10) 57

Laplace equation n n n Steady state heat conduction problems Electrostatic potential of wires Flow of an ideal fluid pass cylinders A circular bar under torque An infinite medium under antiplane shear Half-plane problems 58

Torsion bar with circular holes removed Torque The warping function Boundary condition on where 59

Axial displacement with two circular holes Dashed line: exact solution Solid line: first-order solution Caulk’s data (1983) ASME Journal of Applied Mechanics Present method (M=10) 60

Torsional rigidity ? 61

Laplace equation n n n Steady state heat conduction problems Electrostatic potential of wires Flow of an ideal fluid pass cylinders A circular bar under torque An infinite medium under antiplane shear Half-plane problems 62

Infinite medium under antiplane shear The displacement Boundary condition on Total displacement 63

Shear stress σzq around the hole of radius a 1 (x axis) Honein’s data (1992) Quarterly of Applied Mathematics Present method (M=20) 64

Shear stress σzq around the hole of radius a 1 Stress approach Steele’s data (1992) Present method 5. 349 0. 02% 4. 647 13. 13% (M=20) Analytical Honein’s data (1992) 5. 348 Displacement approach Present method 0. 06% 5. 345 (M=20) 65

Extension to inclusion n Anti-plane piezoelectricity problems In-plane electrostatics problems Anti-plane elasticity problems 66

Two circular inclusions with centers on the y axis Honein et al. ’sdata (1992) Equilibrium of traction Present method (L=20) 67

Convergence test and boundary-layer effect analysis boundary-layer effect 68

Emets & Onofrichuk (1996) Present method (L=20) Patterns of the electric field for e 0=2, e 1=9 and e 2=5 69

Three identical inclusions forming an equilateral triangle 70

Tangential stress distribution around the inclusion located at the origin Present method (L=20), agrees well with Gong’s data (1995) 71

Laplace equation n n n Steady state heat conduction problems Electrostatic potential of wires Flow of an ideal fluid pass cylinders A circular bar under torque An infinite medium under antiplane shear Half-plane problems 72

Half-plane problems Dirichlet boundary condition (Lebedev et al. ) Mixed-type boundary condition (Lebedev et al. ) 73

Dirichlet problem Isothermal line Exact solution (Lebedev et al. ) Present method (M=10) 74

Mixed-type problem Isothermal line Exact solution (Lebedev et al. ) Present method (M=10) 75

Numerical examples n n Laplace equation Eigen problem Exterior acoustics Biharmonic equation 76

Problem statement Simply-connected domain Doubly-connected domain Multiply-connected domain 77

Example 1 78

The former five true eigenvalues by using different approaches k 1 k 2 k 3 k 4 k 5 FEM (ABAQUS) 2. 03 2. 20 2. 62 3. 15 3. 71 BEM 2. 06 2. 23 2. 67 3. 22 3. 81 BEM (CHIEF) 2. 05 2. 23 2. 67 3. 22 3. 81 BEM (null-field) 2. 04 2. 20 2. 65 3. 21 3. 80 BEM (fictitious) 2. 04 2. 21 2. 66 3. 21 3. 80 Present method 2. 05 2. 22 2. 66 3. 21 3. 80 Analytical solution[19] 2. 05 2. 23 2. 66 3. 21 3. 80 (Burton & Miller) 79

The former five eigenmodes by using present method, FEM and BEM 80

Numerical examples n n n Laplace equation Eigen problem Exterior acoustics Water wave Biharmonic equation 81

Sketch of the scattering problem (Dirichlet condition) for five cylinders . u=0 . y u=0 x . u=0 82

The contour plot of the real-part solutions of total field for (a) Present method (M=20) (b) Multiple Dt. N method (N=50) 83

The contour plot of the real-part solutions of total field for (a) Present method (M=20) (b) Multiple Dt. N method (N=50) 84

Fictitious frequencies 85

Soft-basin effect Present method 14 18 3 86

1. Introduction 2. Problem statement 3. Method of solution 4. Numerical examples 5. Concluding remarks Water wave impinging four cylinders 87

Maximum free-surface elevation Perrey-Debain et al. (JSV 2003 ) Present method (2007) 88

Numerical examples n n Laplace equation Eigen problem Exterior acoustics Biharmonic equation 89

Plate problems Geometric data: Essential boundary conditions: on and and on on (Bird & Steele, 1991) 90

Contour plot of displacement Present method (N=101) Bird and Steele (1991) (No. of nodes=3, 462, No. of elements=6, 606) FEM mesh FEM (ABAQUS) 91

Stokes flow problem Governing equation: Angular velocity: Boundary conditions: and on (Stationary) Eccentricity : 92

Comparison for (160) (28) BIE (Kelmanson) Present method Analytical solution Algebraic convergence u 1 (320) (640) (36) Exponential convergence (∞) (44) DOF of BIE (Kelmanson) DOF of present method 93

Contour plot of Streamline for 0 Q/20 Q/5 Q/2 -Q/90 -Q/30 Q Present method (N=81) 0 Q/20 Q/5 Q/2 -Q/90 -Q/30 Q Kelmanson (Q=0. 0740, n=160) e Kamal (Q=0. 0738) 94

Numerical examples n n n Laplace equation Eigen problem Exterior acoustics Biharmonic equation Plate vibration 95

Free vibration of plate 96

Comparisons with FEM 97

Disclaimer (commercial 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 98

BEM trap ? Why engineers should learn mathematics ? n n n Well-posed ? Existence ? Unique ? Mathematics versus Computation Some examples 99

Numerical phenomena (Degenerate scale) Commercial ode output ? Error (%) of 125 torsional rigidity 5 0 a 100

Numerical and physical resonance Physical resonance Numerical resonance radiation incident wave 101

Numerical phenomena (Fictitious frequency) t(a, 0) A story of NTU Ph. D. students 102

Numerical phenomena (Spurious eigensolution) 103

Outlines n n Motivation and literature review Mathematical formulation Expansions of fundamental solution and boundary density Adaptive observer system Vector decomposition technique Linear algebraic equation n n Numerical examples Conclusions 104

Conclusions n n A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve BVPs with arbitrary circular holes and/or inclusions. Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series. 105

Conclusions n n n Free of boundary-layer effect Free of singular integrals Well posed Exponetial convergence Mesh-free approach 106

The End Thanks for your kind attentions. Your comments will be highly appreciated. URL: http: //msvlab. hre. ntou. edu. tw/ 107

108

109