The numerical solutions of Green's functions for transversely isotropic elastic strata

2000 ◽  
Vol 21 (1) ◽  
pp. 49-58 ◽  
Author(s):  
Chen Rong ◽  
Xue Songtao ◽  
Chen Zhuchang ◽  
Chen Jun
Keyword(s):  
Green's Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Transversely Isotropic

Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions

2018 ◽  
Vol 39 (2) ◽  
pp. 607-625 ◽  
Author(s):  
Qiang Du ◽  
Yunzhe Tao ◽  
Xiaochuan Tian ◽  
Jiang Yang
Keyword(s):  
Green's Functions ◽  
Numerical Solutions ◽  
Green’S Functions ◽  
Diffusion Equations ◽  
Nonlocal Diffusion ◽  
Optimal Order ◽  
Numerical Approximations ◽  
Splitting Technique ◽  
Singular Measures ◽  
Nonlocal Models

AbstractNonlocal diffusion equations and their numerical approximations have attracted much attention in the literature as nonlocal modeling becomes popular in various applications. This paper continues the study of robust discretization schemes for the numerical solution of nonlocal models. In particular, we present quadrature-based finite difference approximations of some linear nonlocal diffusion equations in multidimensions. These approximations are able to preserve various nice properties of the nonlocal continuum models such as the maximum principle and they are shown to be asymptotically compatible in the sense that as the nonlocality vanishes, the numerical solutions can give consistent local limits. The approximation errors are proved to be of optimal order in both nonlocal and asymptotically local settings. The numerical schemes involve a unique design of quadrature weights that reflect the multidimensional nature and require technical estimates on nonconventional divided differences for their numerical analysis. We also study numerical approximations of nonlocal Green’s functions associated with nonlocal models. Unlike their local counterparts, nonlocal Green’s functions might become singular measures that are not well defined pointwise. We demonstrate how to combine a splitting technique with the asymptotically compatible schemes to provide effective numerical approximations of these singular measures.

scholarly journals The Fundamental Solutions for the Stress Intensity Factors of Modes I, II And III. The Axially Symmetric Problem

2015 ◽  
Vol 20 (2) ◽  
pp. 345-372
Author(s):  
B. Rogowski
Keyword(s):  
Stress Intensity ◽  
Stress Intensity Factors ◽  
Green's Functions ◽  
Green’S Functions ◽  
Transversely Isotropic ◽  
Fundamental Solutions ◽  
Engineering Practice ◽  
Crack Plane ◽  
Axially Symmetric ◽  
Intensity Factors

Abstract The subject of the paper are Green’s functions for the stress intensity factors of modes I, II and III. Green’s functions are defined as a solution to the problem of an elastic, transversely isotropic solid with a penny-shaped or an external crack under general axisymmetric loadings acting along a circumference on the plane parallel to the crack plane. Exact solutions are presented in a closed form for the stress intensity factors under each type of axisymmetric ring forces as fundamental solutions. Numerical examples are employed and conclusions which can be utilized in engineering practice are formulated.

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