Weighted reproducing kernel collocation method based on error analysis for solving inverse elasticity problems

2019 ◽  
Vol 230 (10) ◽  
pp. 3477-3497 ◽  
Author(s):  
Judy P. Yang ◽  
Wen-Chims Hsin
Keyword(s):  
Error Analysis ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Elasticity Problems ◽  
Inverse Elasticity

Weighted Reproducing Kernel Collocation Method and Error Analysis for Inverse Cauchy Problems

2016 ◽  
Vol 08 (03) ◽  
pp. 1650030 ◽  
Author(s):  
Judy P. Yang ◽  
Pai-Chen Guan ◽  
Chia-Ming Fan
Keyword(s):  
Error Analysis ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Convergence Rates ◽  
Discrete Systems ◽  
Accurate Solution ◽  
Cauchy Problems ◽  
Linear Partial Differential Equations ◽  
Direct Collocation ◽  
Optimal Convergence Rates

In this work, the weighted reproducing kernel collocation method (weighted RKCM) is introduced to solve the inverse Cauchy problems governed by both homogeneous and inhomogeneous second-order linear partial differential equations. As the inverse Cauchy problem is known for the incomplete boundary conditions, how to numerically obtain an accurate solution to the problem is a challenging task. We first show that the weighted RKCM for solving the inverse Cauchy problems considered is formulated in the least-squares sense. Then, we provide the corresponding error analysis to show how the errors in the domain and on the boundary can be balanced with proper weights. The numerical examples demonstrate that the weighted discrete systems improve the accuracy of solutions and exhibit optimal convergence rates in comparison with those obtained by the traditional direct collocation method. It is shown that neither implementation of regularization nor implementation of iteration is needed to reach the desired accuracy. Further, the locality of reproducing kernel approximation gets rid of the ill-conditioned system.

scholarly journals Adjoint-weighted variational formulation for the direct solution of plane stress inverse elasticity problems

2008 ◽  
Vol 135 ◽  
pp. 012012 ◽  
Author(s):  
Paul E Barbone ◽  
Carlos E Rivas ◽  
Isaac Harari ◽  
Uri Albocher ◽  
Assad A Oberai ◽  
...  
Keyword(s):  
Plane Stress ◽  
Variational Formulation ◽  
Direct Solution ◽  
Elasticity Problems ◽  
Inverse Elasticity

scholarly journals Coefficient-Based Regression with Non-Identical Unbounded Sampling

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jia Cai
Keyword(s):  
Hilbert Space ◽  
Integral Operator ◽  
Error Analysis ◽  
Least Squares ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Least Squares Regression ◽  
Regression Problem ◽  
Novel Technique ◽  
Sample Error

We investigate a coefficient-based least squares regression problem with indefinite kernels from non-identical unbounded sampling processes. Here non-identical unbounded sampling means the samples are drawn independently but not identically from unbounded sampling processes. The kernel is not necessarily symmetric or positive semi-definite. This leads to additional difficulty in the error analysis. By introducing a suitable reproducing kernel Hilbert space (RKHS) and a suitable intermediate integral operator, elaborate analysis is presented by means of a novel technique for the sample error. This leads to satisfactory results.

scholarly journals Detecting Inverse Boundaries by Weighted High-Order Gradient Collocation Method

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1297 ◽  
Author(s):  
Judy P. Yang ◽  
Hon Fung Samuel Lam
Keyword(s):  
Collocation Method ◽  
Reproducing Kernel ◽  
Outer Boundary ◽  
Computational Cost ◽  
Weak Form ◽  
High Order ◽  
Unknown Boundary ◽  
Shape Functions ◽  
High Order Derivative ◽  
Special Cases

The weighted reproducing kernel collocation method exhibits high accuracy and efficiency in solving inverse problems as compared with traditional mesh-based methods. Nevertheless, it is known that computing higher order reproducing kernel (RK) shape functions is generally an expensive process. Computational cost may dramatically increase, especially when dealing with strong-from equations where high-order derivative operators are required as compared to weak-form approaches for obtaining results with promising levels of accuracy. Under the framework of gradient approximation, the derivatives of reproducing kernel shape functions can be constructed synchronically, thereby alleviating the complexity in computation. In view of this, the present work first introduces the weighted high-order gradient reproducing kernel collocation method in the inverse analysis. The convergence of the method is examined through the weights imposed on the boundary conditions. Then, several configurations of multiply connected domains are provided to numerically investigate the stability and efficiency of the method. To reach the desired accuracy in detecting the outer boundary for two special cases, special treatments including allocation of points and use of ghost points are adopted as the solution strategy. From four benchmark examples, the efficacy of the method in detecting the unknown boundary is demonstrated.

A gradient reproducing kernel collocation method for high order differential equations

2019 ◽  
Vol 64 (5) ◽  
pp. 1421-1454 ◽  
Author(s):  
Ashkan Mahdavi ◽  
Sheng-Wei Chi ◽  
Huiqing Zhu
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
High Order

scholarly journals A reproducing kernel Hilbert space approach in meshless collocation method

2019 ◽  
Vol 38 (2) ◽  
Author(s):  
Babak Azarnavid ◽  
Mahdi Emamjome ◽  
Mohammad Nabati ◽  
Saeid Abbasbandy
Keyword(s):  
Hilbert Space ◽  
Collocation Method ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Meshless Collocation ◽  
Space Approach
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