1 / 109

Null field integral approach for engineering problems with circular boundaries

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.

griffin-watson
Download Presentation

Null field integral approach for engineering problems with circular boundaries

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

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

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

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

  5. Top ten countries of BEM and dual BEM • 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)

  6. Top ten countries of FEM, FDM and Meshless methods • 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 (ISI information Apr.20, 2007)

  7. Active scholars on BEM and dual BEM • 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) Chen J T (Taiwan, Ocean Univ.) Power H (UK, Univ. Nottingham) (ISI information Apr.20, 2007)

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

  9. (Interior and exterior Acoustics) SH wave (exterior acoustics) (Inclusions) 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 EABE MRC,CMES ASME JAM 2006 JSV Elasticity (holes and/or inclusions) (Plate with circulr holes) (Potential flow) (Torsion) (Anti-plane shear) (Degenerate scale) (Free vibration of plate) Indirect BIEM Screw dislocation (Stokes flow) JCA CMAME 2007 (Free vibration of plate) Direct BIEM JoM ASME EABE (Beam bending) (Inclusion) (Piezoleectricity) Green function for an annular plate SDEE ICOME 2006 SH wave Impinging canyons Degenerate kernel for ellipse (Flexural wave of plate) Torsion bar(Inclusion) Imperfect interface Green function of`circular inclusion (special case:static) CMC SH wave Impinging hill Added mass Image method (Green function) Green function of half plane (Hole and inclusion) Previous work (done) Interested topic Present project Water wave impinging circular cylinders Effective conductivity 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

  10. 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

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

  12. 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 `

  13. 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

  14. Motivation 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 ?

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

  16. Present approach Degenerate kernel 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

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

  18. Motivation and literature review Analytical methods for solving Laplace problems with circular holes Special solution Bipolar coordinate Conformal mapping 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 Lebedev, Skalskaya and Uyand, 1979, “Work problem in applied mathematics”, Dover Publications Limited to doubly connected domain

  19. 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

  20. Contribution and goal • However, they didn’t employ the null-field integral equationand 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.

  21. 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

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

  23. NTUCE Ó 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)

  24. 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)

  25. 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)

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

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

  28. 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

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

  30. How to separate the region

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

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

  33. Separable form of fundamental solution (2D)

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

  35. 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

  36. collocation point Adaptive observer system

  37. 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

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

  39. 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

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

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

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

  43. Comparisons of conventional BEM and present method

  44. 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

  45. 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)

  46. 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

  47. Steady state heat conduction problems Case 1 Case 2

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

  49. Relative error of flux on the small circle

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

More Related