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See detailPoint Collocation Methods for Linear Elasticity Problems
Jacquemin, Thibault Augustin Marie UL

Presentation (2020, June 19)

Point collocation is the oldest way to solve partial differential equations. Methods based on collocation have been studied since decades and many variations have been proposed over the years. More ... [more ▼]

Point collocation is the oldest way to solve partial differential equations. Methods based on collocation have been studied since decades and many variations have been proposed over the years. More recently, those methods have shown a greater interest thanks to the advances in computing hardware. The collocation methods offer a great flexibility with regards to the discretization of a defined domain and the approximation of the field derivatives. This presentation will introduce the bases of the collocation methods and of the generalized finite difference method. The importance of the selection of the nodes involved in the approximation of the field derivatives will then be presented. Finally two aspects for which the method is particularly attractive will be detailed: the solution of a PDE from a given geometry with minimum discretization effort and the adaptivity of a model based on a posteriori error estimation. [less ▲]

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See detailTaylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non-smooth Solutions
Jacquemin, Thibault Augustin Marie UL; Tomar, Satyendra UL; Agathos, Konstantinos UL et al

in Archives of Computational Methods in Engineering (2019)

We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these ... [more ▼]

We provide a primer to numerical methods based on Taylor series expansions such as generalized finite difference methods and collocation methods. We provide a detailed benchmarking strategy for these methods as well as all data files including input files, boundary conditions, point distribution and solution fields, so as to facilitate future benchmarking of new methods. We review traditional methods and recent ones which appeared in the last decade. We aim to help newcomers to the field understand the main characteristics of these methods and to provide sufficient information to both simplify implementation and benchmarking of new methods. Some of the examples are chosen within a subset of problems where collocation is traditionally known to perform sub-par, namely when the solution sought is non-smooth, i.e. contains discontinuities, singularities or sharp gradients. For such problems and other simpler ones with smooth solutions, we study in depth the influence of the weight function, correction function, and the number of nodes in a given support. We also propose new stabilization approaches to improve the accuracy of the numerical methods. In particular, we experiment with the use of a Voronoi diagram for weight computation, collocation method stabilization approaches, and support node selection for problems with singular solutions. With an appropriate selection of the above-mentioned parameters, the resulting collocation methods are compared to the moving least-squares method (and variations thereof), the radial basis function finite difference method and the finite element method. Extensive tests involving two and three dimensional problems indicate that the methods perform well in terms of efficiency (accuracy versus computational time), even for non-smooth solutions. [less ▲]

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See detailWeak and strong from meshless methods for linear elastic problem under fretting contact conditions
Kosec, Gregor; Slak, Jure; Depolli, Matja et al

in Tribology International (2019), 138

We present numerical computation of stresses under fretting fatigue conditions derived from closed form expressions. The Navier-Cauchy equations, that govern the problem, are solved with strong and weak ... [more ▼]

We present numerical computation of stresses under fretting fatigue conditions derived from closed form expressions. The Navier-Cauchy equations, that govern the problem, are solved with strong and weak form meshless numerical methods. The results are compared to the solution obtained from well-established commercial package ABAQUS, which is based on finite element method (FEM). The results show that the weak form meshless solution exhibits similar behavior as the FEM solution, while, in this particular case, strong form meshless solution performs better in capturing the peak in the surface stress. This is of particular interest in fretting fatigue, since it directly influences crack initiation. The results are presented in terms of von Mises stress contour plots, surface stress profiles, and the convergence plots for all three methods involved in the study. [less ▲]

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