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

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

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

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

Top ten countries of BEM and dual 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)

Top ten countries of FEM, FDM and Meshless methods 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)

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

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

Research topics of NTOU / MSV LAB using null-field BIEs (2005-2007) (2008-2011) Null-field BIEM NUMPDE revision Navier Equation Laplace Equation Helmholtz Equation Biharmonic Equation BiHelmholtz Equation Elasticity (holes and/or inclusions) EABE MRC,CMES ASME JAM 2006 JSV (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) (Plate with circulr holes) Screw dislocation (Interior and exterior Acoustics) SH wave (exterior acoustics) (Inclusions) (Free vibration of plate) Indirect BIEM (Stokes flow) JCA CMAME 2007 ASME EABE JoM (Free vibration of plate) Direct BIEM (Inclusion) (Piezoleectricity) (Beam bending) Green function for an annular plate SDEE ICOME 2006 SH wave Impinging canyons Degenerate kernel for ellipse Torsion bar (Inclusion) Imperfect interface (Flexural wave of plate) Green function of`circular inclusion (special case:static) CMC Image method (Green function) Added mass SH wave Impinging hill Green function of half plane (Hole and inclusion) Previous work (done) Interested topic Present project Effective conductivity Water wave impinging circular cylinders URL: http://ind.ntou.edu.tw/~msvlab E-mail: jtchen@mail.ntou.edu.tw 海洋大學工學院河工所力學聲響振動實驗室 Frame2007-c.ppt

Research collaborators 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

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

哲人日已遠 典型在宿昔 C B Ling (1909-1993) 林致平院士(數學力學家) C B Ling (mathematician and expert in mechanics) 省立中興大學第一任校長 林致平校長 (民國五十年~民國五十二年) 林致平所長(中研院數學所) 林致平院士(中研院) PS: short visit (J T Chen) of Academia Sinica 2006 summer `

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

Motivation BEM / BIEM (mesh required) 除四舊 Numerical methods for engineering problems FDM / FEM / BEM / 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 ?

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

Present approach Advantages of degenerate kernel No principal value Fundamental solution No principal value CPV and HPV Advantages of degenerate kernel No principal value 2. Well-posed 3. No boundary-layer effect 4. Exponetial convergence

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

Motivation and literature review Analytical methods for solving Laplace problems with circular holes Conformal mapping Bipolar coordinate Special solution Chen and Weng, 2001, “Torsion of a circular compound bar with imperfect interface”, ASME Journal of Applied Mechanics 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

Fourier series approximation 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

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

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

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

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) Definitions of R.P.V., C.P.V. and H.P.V. using bump approach Ó NTUCE

Principal value in who’s sense 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 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)

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

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

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

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

How to separate the region

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

Separable form of fundamental solution (1D) Separable property continuous discontinuous

Separable form of fundamental solution (2D)

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

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

Adaptive observer system collocation point

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

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

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

Linear algebraic equation where Index of collocation circle Index of routing circle Column vector of Fourier coefficients (Nth routing circle)

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

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

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

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

Numerical examples Laplace equation (EABE 2006, EABE2006,EABE 2007) (CMES 2006, ASME 2007, JoM2007, 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)

Laplace equation 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

Steady state heat conduction problems Case 1 Case 2

Case 1: Isothermal line Exact solution FEM-ABAQUS (Carrier and Pearson) FEM-ABAQUS (1854 elements) BEM-BEPO2D (N=21) Present method (M=10)

Relative error of flux on the small circle

Convergence test - Parseval’s sum for Fourier coefficients

Laplace equation 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

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

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

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

Laplace equation 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

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

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

Laplace equation 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

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

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)

Torsional rigidity ?

Laplace equation 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

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

Shear stress σzq around the hole of radius a1 (x axis) Honein’s data (1992) Quarterly of Applied Mathematics Present method (M=20)

Shear stress σzq around the hole of radius a1 Stress approach Analytical Displacement approach Steele’s data (1992) Present method (M=20) Honein’s data (1992) Present method (M=20) 4.647 13.13% 5.349 0.02% 5.348 5.345 0.06%

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

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

Convergence test and boundary-layer effect analysis

Patterns of the electric field for e0=2, e1=9 and e2=5 Present method (L=20) Emets & Onofrichuk (1996)

Three identical inclusions forming an equilateral triangle

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

Laplace equation 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

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

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

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

Numerical examples Laplace equation Eigen problem Exterior acoustics Biharmonic equation

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

Example 1

The former five true eigenvalues by using different approaches k1 k2 k3 k4 k5 FEM (ABAQUS) 2.03 2.20 2.62 3.15 3.71 BEM (Burton & Miller) 2.06 2.23 2.67 3.22 3.81 (CHIEF) 2.05 (null-field) 2.04 2.65 3.21 3.80 (fictitious) 2.21 2.66 Present method 2.22 Analytical solution[19]

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

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

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

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

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

Fictitious frequencies

Soft-basin effect Present method 18 14 3

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

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

Numerical examples Laplace equation Eigen problem Exterior acoustics Biharmonic equation

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

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)

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

Comparison for Algebraic convergence Exponential convergence (160) BIE (Kelmanson) Present method Analytical solution (28) Algebraic convergence u1 (320) (640) (36) Exponential convergence (∞) (44) DOF of BIE (Kelmanson) DOF of present method

Contour plot of Streamline for -Q/90 Q/20 Q/5 -Q/30 Q/2 Q Present method (N=81) -Q/90 Q/20 Q/5 -Q/30 Q/2 Kelmanson (Q=0.0740, n=160) Q e Kamal (Q=0.0738)

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

Free vibration of plate

Comparisons with FEM

Disclaimer (commercial code) The concepts, methods, and examples using our software are 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

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

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

Numerical and physical resonance Numerical resonance radiation incident wave

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

Numerical phenomena (Spurious eigensolution)

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

Conclusions 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.

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

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