Particular solutions for some engineering problems

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Particular solutions for some engineering
problems
2008 NTOU

• Chia-Cheng Tasi
• ???
• Department of Information Technology
• Toko University, Chia-Yi County, Taiwan

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Overview
Motivation Method of Particular Solutions
(MPS) Particular solutions of polyharmonic
spline Numerical example I Particular solutions
of Chebyshev polynomials Numerical example
II Conclusions
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Motivation
BEM has evolved as a popular numerical technique
for solving linear, constant coefficient partial
differential equations. Other boundary type
numerical methods Treffz method, MFS Advantage
Reduction of dimensionalities (3D-gt2D,
2D-gt1D) Disadvantage domain integration for
nonhomogeneous problem For inhomogeneous
equations, the method of particular solution
(MPS) is needed. In BEM, it is called the dual
reciprocity boundary element method (DRBEM)
(Partridge, et al., 1992).
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Motivation and Literature review
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Motivation
RBF
Golberg (1995) Chebyshev
MPS with Chebyshev Polynomials
spectral convergence
MFS
Golberg, M.A. Muleshkov, A.S. Chen, C.S.
Cheng, A.H.-D. (2003)
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Motivation
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Motivation
We note that the polyharmonic and the
poly-Helmholtz equations are encountered in
certain engineering problems, such as high order
plate theory, and systems involving the coupling
of a set of second order elliptic equations, such
as a multilayered aquifer system, or a multiple
porosity system. These coupled systems can be
reduced to a single partial differential equation
by using the Hörmander operator decomposition
technique. The resultant partial differential
equations usually involve the polyharmonic or the
products of Helmholtz operators. Hence My study
is to fill an important gap in the application of
boundary methods to these engineering problems.
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Method of particular solutions
Method of particular solutions
Method of fundamental solutions, Trefftz method,
boundary element method, et al.
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Method of particular solutions
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Method of particular solutions (basis functions)
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Method of particular solutions (Hörmander
Operator Decomposition technique)
Particular solutions for the engineering problems
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Example
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Example
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Other examples
Stokes flow
Thermal Stokes flow
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Other examples
Thick plate
Solid deformation
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Remark
Particular solutions for product operator
Particular solutions for engineering problems
Hörmander operator decomposition technique
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Method of particular solutions (Partial fraction
decomposition)
Particular solutions for
Particular solutions for product operator
Partial fraction decomposition
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Partial fraction decomposition (Theorem)
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Partial fraction decomposition (Proof 1)
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Partial fraction decomposition (Proof 2)
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Example (1)
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Example (2)
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Remark
Partial fraction decomposition
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Particular solutions of polyharmonic spline (APS)
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Particular solutions of polyharmonic spline (APS)
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Particular solutions of polyharmonic spline
(Definition)
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Particular solutions of polyharmonic spline
(Generating Theorem)
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Particular solutions of polyharmonic spline
(Generating Theorem)
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Particular solutions of polyharmonic spline
(Generating Theorem)
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Particular solutions of polyharmonic spline (2D
Poly-Helmholtz Operator)
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Particular solutions of polyharmonic spline (2D
Poly-Helmholtz Operator)
proof
Generating Theorem
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Particular solutions of polyharmonic spline (2D
Poly-Helmholtz Operator)
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Particular solutions of polyharmonic spline (2D
Poly-Helmholtz Operator)
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Particular solutions of polyharmonic spline (3D
Poly-Helmholtz Operator)
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Particular solutions of polyharmonic spline (2D
Poly-Helmholtz Operator)
proof
Generating Theorem
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Particular solutions of polyharmonic spline
(Limit Behavior)
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Particular solutions of polyharmonic spline
(Limit Behavior)
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Numerical example I
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Numerical example I
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Numerical example I (BC)
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Numerical example I (BC)
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Numerical example I (MFS)
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Numerical example I (results)
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Particular solutions of Chebyshev polynomials
(why orthogonal polynomials)
Fourier series exponential convergence but
Gibbs phenomena Lagrange Polynomials Runge
phenomena Jacobi Polynomials (orthogonal
polynomials) exponential convergence
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Particular solutions of Chebyshev polynomials
(why Chebyshev)
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Chebyshev interpolation (1)
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Chebyshev interpolation (2)
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Particular solutions of Chebyshev polynomials
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Particular solutions of Chebyshev polynomials
(poly-Helmholtz)
Generating Theorem
Golberg, M.A. Muleshkov, A.S. Chen, C.S.
Cheng, A.H.-D. (2003)
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Particular solutions of Chebyshev polynomials
(polyharmonic)
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Particular solutions of Chebyshev polynomials
(polyharmonic)
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Method of fundamental solutions
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Method of fundamental solutions (example)
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Numerical example II
Example (2D modified Helmholtz)
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Numerical example II
Example (2D Laplace)
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Numerical example II
Example (3D modified Helmholtz)
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Numerical example II
Example (3D Laplace)
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Numerical example II
Example (2D polyharmonic)
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Numerical example II
Example (2D product operator)
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Conclusion
1. MFSAPS gt scattered data in right-hand
sides 2. MFSChebyshev gt spectral convergence 3.
Hörmander operator decomposition technique 4.
Partial fraction decomposition 5. polyHelmholtz
Polyharmonic particular solutions 6. MFS for the
product operator
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Thank you