Perturbation and stability analysis of strong form collocation with reproducing kernel approximation

2011 ◽  
Vol 88 (2) ◽  
pp. 157-179 ◽  
Author(s):  
Hsin-Yun Hu ◽  
Jiun-Shyan Chen ◽  
Sheng-Wei Chi
Keyword(s):  
Stability Analysis ◽  
Reproducing Kernel ◽  
Strong Form ◽  
Kernel Approximation

NONLINEAR THIN-PLATE BENDING ANALYSES USING THE HERMITE REPRODUCING KERNEL APPROXIMATION

2012 ◽  
Vol 09 (01) ◽  
pp. 1240012 ◽  
Author(s):  
SATOYUKI TANAKA ◽  
SHOTA SADAMOTO ◽  
SHIGENOBU OKAZAWA
Keyword(s):  
Thin Plate ◽  
Reproducing Kernel ◽  
Geometrical Nonlinearity ◽  
Mathematical Formulation ◽  
Thin Plates ◽  
Plate Bending ◽  
Kernel Approximation ◽  
Hermite Reproducing Kernel Approximation ◽  
Lagrange Strain ◽  
Thin Plate Bending

This study analyzed thin-plate bending problems with a geometrical nonlinearity using the Hermite reproducing kernel approximation and sub-domain-stabilized conforming integration. In thin-plate bending analyses, the deflections and rotations satisfy so-called Kirchhoff mode reproducing conditions. It is then possible to solve large deflection analyses of thin plates, such as elastic bucking problems, with high accuracy and efficiency. Total Lagrangian method is applied to solve the geometrical nonlinearity of the thin plates' deflections and rotations. The Green–Lagrange strain and second Piola–Kirchhoff stress forms are adopted to represent the strains and stresses in the thin plates. Mathematical formulation and some numerical examples are also demonstrated.

Solving Inverse Laplace Equation with Singularity by Weighted Reproducing Kernel Collocation Method

2017 ◽  
Vol 09 (05) ◽  
pp. 1750065 ◽  
Author(s):  
Judy P. Yang ◽  
Pai-Chen Guan ◽  
Chia-Ming Fan
Keyword(s):  
Inverse Problems ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Cauchy Problems ◽  
Overdetermined System ◽  
Numerical Examples ◽  
Kernel Approximation ◽  
Laplace Equations ◽  
Laplace Type ◽  
Least Squares Minimization

This work introduces the weighted collocation method with reproducing kernel approximation to solve the inverse Laplace equations. As the inverse problems in consideration are equipped with over-specified boundary conditions, the resulting equations yield an overdetermined system. Following our previous work, the weighted collocation method using a least-squares minimization has shown to solve the inverse Cauchy problems efficiently without using techniques such as iteration and regularization. In this work, we further consider solving the inverse problems of Laplace type and introduce the Shepard functions to deal with singularity. Numerical examples are provided to demonstrate the validity of the method.

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