2009 Method of Particular Solutions 1 A Study on Particular Solutions of Coupled PDE System Chia-Cheng Tasi 蔡加正 Department of Information Technology Toko.

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2009 Method of Particular Solutions 1 A Study on Particular Solutions of Coupled PDE System Chia-Cheng Tasi 蔡加正 Department of Information Technology Toko University, Chia-Yi County, Taiwan

2009 Method of Particular Solutions 2 Motivation and Introduction Method of Particular Solutions (MPS) Particular solutions of Chebyshev polynomials Numerical example I Particular solutions of spline Numerical example II Conclusions Overview

2009 Method of Particular Solutions 3 陳建宏柯永澤陳柏台辛敬業劉德源張建仁周宗仁林炤圭蕭松山岳景雲翁文凱臧效義郭世榮葉為忠曹登皓劉進賢陳正宗范佳銘系工河工機械河工機械林俊華尹彰尹彰電機河工 You are Welcome ! NTOU is a Kingdom of BEM

2009 Method of Particular Solutions 4 Motivation and Introduction 1.Boundary-type numerical method: BEM, Treffz method, MFS 2.Advantage: Reduction of dimensionalities 3.Disadvantage: Domain integration => the method of particular solutions (MPS) or the dual reciprocity method (DRM) 4.Active research fields of BEM: Singularity and Domain integration

2009 Method of Particular Solutions 5 Motivation and Introduction

2009 Method of Particular Solutions 6 Motivation and Introduction

2009 Method of Particular Solutions 7 Innovation Application DBEM MPSMRM Hypersingularity Motivation and Introduction How to prove? BIEM

2009 Method of Particular Solutions 8 RBF MFS MPS with Chebyshev Polynomials exponential convergence Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003) Motivation and Introduction Golberg (1995) Chebyshev

2009 Method of Particular Solutions 9 Motivation and Introduction

2009 Method of Particular Solutions 10 Method of particular solutions

2009 Method of Particular Solutions 11 Method of particular solutions Method of fundamental solutions, Trefftz method, boundary element method, et al.

2009 Method of Particular Solutions 12 Method of particular solutions (basis functions)

2009 Method of Particular Solutions 13 Coupled PDE system Product operator Polyharmonic operatorPoly-Helmholtz operator Helmholtz operatorLaplacian operator Hörmander operator decomposition technique Partial fraction decomposition Generating theorem ? Method of particular solutions

2009 Method of Particular Solutions 14 Particular solutions for the engineering problems Method of particular solutions (Hörmander Operator Decomposition technique)

2009 Method of Particular Solutions 15 Example

2009 Method of Particular Solutions 16 Example

2009 Method of Particular Solutions 17 Other examples Stokes flow Thermal Stokes flow

2009 Method of Particular Solutions 18 Other examples Thick plate Solid deformation

2009 Method of Particular Solutions 19 Remark Particular solutions for engineering problems Particular solutions for product operator H ö rmander operator decomposition technique

2009 Method of Particular Solutions 20 Particular solutions for Partial fraction decomposition Particular solutions for product operator Method of particular solutions (Partial fraction decomposition)

2009 Method of Particular Solutions 21 Partial fraction decomposition (Theorem)

2009 Method of Particular Solutions 22 Partial fraction decomposition (Proof 1)

2009 Method of Particular Solutions 23 Partial fraction decomposition (Proof 2)

2009 Method of Particular Solutions 24 Example (1)

2009 Method of Particular Solutions 25 Example (2)

2009 Method of Particular Solutions 26 Remark Partial fraction decomposition

2009 Method of Particular Solutions 27 Particular solutions of Chebyshev polynomials (why orthogonal polynomials) Fourier series: exponential convergence but Gibb’s phenomena Lagrange Polynomials: Runge phenomena Chebyshev Polynomials (one of the orthogonal polynomials): exponential convergence

2009 Method of Particular Solutions 28 Chebyshev interpolation (1)

2009 Method of Particular Solutions 29 Chebyshev interpolation (2)

2009 Method of Particular Solutions 30 Chebyshev interpolation (3)

2009 Method of Particular Solutions 31 Chebyshev interpolation (4)

2009 Method of Particular Solutions 32 Particular solutions of Chebyshev polynomials

2009 Method of Particular Solutions 33 Particular solutions of Chebyshev polynomials

2009 Method of Particular Solutions 34 Particular solutions of of Chebyshev polynomials (Generating Theorem)

2009 Method of Particular Solutions 35 Particular solutions of of Chebyshev polynomials (Generating Theorem)

2009 Method of Particular Solutions 36 Particular solutions of of Chebyshev polynomials (Generating Theorem)

2009 Method of Particular Solutions 37 Particular solutions of Chebyshev polynomials (poly-Helmholtz) Generating Theorem Golberg, M.A.; Muleshkov, A.S.; Chen, C.S.; Cheng, A.H.-D. (2003)

2009 Method of Particular Solutions 38 Particular solutions of Chebyshev polynomials (polyharmonic)

2009 Method of Particular Solutions 39 Particular solutions of Chebyshev polynomials (polyharmonic)

2009 Method of Particular Solutions 40 Method of fundamental solutions Fig. 2: Geometry configuration of the MFS.

2009 Method of Particular Solutions 41 Method of fundamental solutions (example)

2009 Method of Particular Solutions 42 Example (2D modified Helmholtz) Numerical example I

2009 Method of Particular Solutions 43 Example (2D Laplace) Numerical example I

2009 Method of Particular Solutions 44 Example (3D modified Helmholtz) Numerical example I

2009 Method of Particular Solutions 45 Example (3D Laplace) Numerical example I

2009 Method of Particular Solutions 46 Example (2D polyharmonic) Numerical example I

2009 Method of Particular Solutions 47 Example (2D product operator) Numerical example I

2009 Method of Particular Solutions 48 Example (Reissner plate) Numerical example I Fig. 1: Geometric configuration of the Ressiner plate model.

2009 Method of Particular Solutions 49 Example (Reissner plates: particular solutions)

2009 Method of Particular Solutions 50 Example (Reissner plates: particular solutions)

2009 Method of Particular Solutions 51 Example (Reissner plates: particular solutions)

2009 Method of Particular Solutions 52 Example (Reissner plates: particular solutions)

2009 Method of Particular Solutions 53 Example (Reissner plates: fundamental solutions)

2009 Method of Particular Solutions 54 Example (Reissner plates: numerical results)

2009 Method of Particular Solutions 55 Example (Reissner plates: numerical results)

2009 Method of Particular Solutions MFS+Chebyshev => spectral convergence 2. H ö rmander operator decomposition technique 3. Partial fraction decomposition 4. polyHelmholtz & Polyharmonic particular solutions 5. MFS for the product operator Conclusion

2009 Method of Particular Solutions 57 Thank you

2009 Method of Particular Solutions 58 Particular solutions of spline (APS)

2009 Method of Particular Solutions 59 Particular solutions of spline (APS)

2009 Method of Particular Solutions 60 Particular solutions of spline (Definition)

2009 Method of Particular Solutions 61 Particular solutions of spline (2D Poly- Helmholtz Operator)

2009 Method of Particular Solutions 62 Particular solutions of spline (2D Poly- Helmholtz Operator) Generating Theorem proof

2009 Method of Particular Solutions 63 Particular solutions of spline (2D Poly- Helmholtz Operator)

2009 Method of Particular Solutions 64 Particular solutions of spline (2D Poly- Helmholtz Operator)

2009 Method of Particular Solutions 65 Particular solutions of spline (3D Poly- Helmholtz Operator) Generating Theorem proof

2009 Method of Particular Solutions 66 Particular solutions of spline (Limit Behavior)

2009 Method of Particular Solutions 67 Particular solutions of spline (Limit Behavior)

2009 Method of Particular Solutions 68 Particular solutions of spline (2D&3D Polyharmonic Operator) Cheng (2000)

2009 Method of Particular Solutions 69 Numerical example II

2009 Method of Particular Solutions 70 Numerical example II

2009 Method of Particular Solutions 71 Numerical example II (BC)

2009 Method of Particular Solutions 72 Numerical example II (BC)

2009 Method of Particular Solutions 73 Numerical example II (MFS)

2009 Method of Particular Solutions 74 Numerical example II (results)

2009 Method of Particular Solutions MFS+Chebyshev => spectral convergence 2. H ö rmander operator decomposition technique 3. Partial fraction decomposition 4. polyHelmholtz & Polyharmonic particular solutions 5. MFS for the product operator 6. MFS+APS => scattered data in right-hand sides Conclusion

2009 Method of Particular Solutions 76 Thank you