1 Null integral equations and their applications J. T. Chen Ph.D. Taiwan Ocean University Keelung, Taiwan June 04-06, 2007 BEM 29 in WIT Bem talk.ppt National Taiwan Ocean University MSVLAB Department of Harbor and River Engineering

2 Research collaborators Dr. I. L. Chen Dr. K. H. Chen Dr. I. L. Chen Dr. K. H. Chen Dr. S. Y. Leu Dr. W. M. Lee Dr. S. Y. Leu Dr. W. M. Lee Mr. Y. T. Lee Mr. Y. T. Lee Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. W. C. Shen Mr. C. T. Chen Mr. G. C. Hsiao Mr. A. C. Wu Mr.P. Y. Chen Mr. A. C. Wu Mr.P. Y. Chen Mr. J. N. Ke Mr. H. Z. Liao Mr. J. N. Ke Mr. H. Z. Liao

3 URL: 海洋大學工學院河工所力學聲響振動實驗室 nullsystem2007.ppt` Elasticity & Crack Problem Laplace Equation Research topics of NTOU / MSV LAB on null-field BIE ( ) Navier Equation Null-field BIEM Biharmonic Equation Previous research and project Current work (Plate with circulr holes ) BiHelmholtz Equation Helmholtz Equation (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) (Inclusion) (Piezoleectricity) (Beam bending) Torsion bar (Inclusion) Imperfect interface Image method (Green function) Green function of half plane (Hole and inclusion) (Interior and exterior Acoustics) SH wave (exterior acoustics) (Inclusions) (Free vibration of plate) Indirect BIEM ASME JAM 2006 MRC,CMESEABE ASME JoM EABE CMAME 2007 SDEE JCA NUMPDE revision JSV SH wave Impinging canyons Degenerate kernel for ellipse ICOME 2006 Added mass 李應德 Water wave impinging circular cylinders Screw dislocation Green function for an annular plate SH wave Impinging hill Green function of`circular inclusion (special case:staic) Effective conductivity CMC (Stokes flow) (Free vibration of plate) Direct BIEM (Flexural wave of plate)

4 Prof. C B Ling ( ) Fellow of Academia Sinica He devoted himself to solve BVPs with holes. PS: short visit (J T Chen) of Academia Sinica 2006 summer ` C B Ling (mathematician and expert in mechanics)

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

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

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

8 Present approach 1.No principal value 2. Well-posed 2. Well-posed 3. No boundary-layer effect 3. No boundary-layer effect 4. Exponetial convergence 4. Exponetial convergence 5. Meshless 5. Meshless Advantages of degenerate kernel Degenerate kernel Fundamental solution CPV and HPV No principal value

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

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

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

12 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. 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. To develop a systematic approach for solving Laplace problems with multiple holes is our goal.

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

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

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

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

17 How to separate the region

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

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

20 Separable form of fundamental solution (2D)

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

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

23 Adaptive observer system collocation point

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

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

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

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

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

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

30 Comparisons of conventional BEM and present method BoundarydensitydiscretizationAuxiliarysystemFormulationObserversystemSingularityConvergenceBoundarylayereffect ConventionalBEMConstant,linear,quadratic…elementsFundamentalsolutionBoundaryintegralequationFixedobserversystem CPV, RPV and HPV LinearAppear PresentmethodFourierseriesexpansionDegeneratekernelNull-fieldintegralequationAdaptiveobserversystemDisappearExponentialEliminate

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

32 Numerical examples Laplace equation (EABE 2005, EABE 2007) Laplace equation (EABE 2005, EABE 2007) (CMES 2005, ASME 2007, JoM2007) (CMES 2005, ASME 2007, JoM2007) (MRC 2007, NUMPDE revision) (MRC 2007, NUMPDE revision) Membrane eigenproblem (JCA) Membrane eigenproblem (JCA) Exterior acoustics (CMAME, SDEE ) Exterior acoustics (CMAME, SDEE ) Biharmonic equation (JAM, ASME 2006 ) Biharmonic equation (JAM, ASME 2006 ) Plate eigenproblem (JSV ) Plate eigenproblem (JSV )

33 Laplace equation A circular bar under torque A circular bar under torque (free of mesh generation) (free of mesh generation)

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

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

36 Torsional rigidity ?

37 Extension to inclusion Extension to inclusion Anti-plane elasticity problems Anti-plane elasticity problems (free of boundary layer effect) (free of boundary layer effect)

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

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

40 Numerical examples Biharmonic equation Biharmonic equation (exponential convergence) (exponential convergence)

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

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

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

44 Comparison for DOF of BIE (Kelmanson) DOF of present method BIE (Kelmanson) Present method Analytical solution (160) (320) (640) u1u1 (28) (36) (44) (∞) Algebraic convergence Exponential convergence

45 Contour plot of Streamline for Present method (N=81) Kelmanson (Q=0.0740, n=160) Kamal (Q=0.0738) e Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0 Q/2 Q Q/5 Q/20 -Q/90 -Q/30 0

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

47 Some findings `LaplaceHelmholtz Ling 1947 Analytical solution Bird & Steele 1991 房營光 1995 Analytical solution Lee & Manoogian Caulk 1983 Naghdi 1991 Analytical solution Tsaur et al Analytical solution Present method Present method (semi-analytical) Tsaur et al. ?

48 Stress concentration at point B Present method Naghdi’s results Steele & Bird The two approaches disagree by as much 11%. The grounds for this discrepancy have not yet been identified. --ASME Applied Mechanics Review

49 A half-plane problem with two alluvial valleys subject to the incident SH-wave Canyon Matrix 3a SH-Wave Tsaur et al. pointed out that Fang made a mistake of misusing the orthogonal relation.

50 Limiting case of two canyons Present method Tsaur et al.’s results [103]

51 Inclusion Matrix h SH- Wave a x y A half-plane problem with a circular inclusion subject to the incident SH-wave

52 Surface displacements of a inclusion problem under the ground surface Present method Tsaur et al.’s results [102] Manoogian and Lee’s results [62] 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

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

54 Conclusions A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve Laplace Helmholtz and Biharminic problems with circular boundaries. A systematic approach using degenerate kernels, Fourier series and null-field integral equation has been successfully proposed to solve Laplace Helmholtz and Biharminic problems with circular boundaries. Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series. Numerical results agree well with available exact solutions, Caulk’s data, Onishi’s data and FEM (ABAQUS) for only few terms of Fourier series.

55 Conclusions Four previous results were examined. Four previous results were examined.. `Laplace Helmholtz` Ling 1947 Analytical solution Bird & Steele 1991 房營光 1995 Analytical solution Lee & Manoogian1992 ? ? ? ?

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

57 The End Thanks for your kind attentions. Your comments will be highly appreciated. URL:

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