Taylor-Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with
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ORIGINAL PAPER
Taylor‑Series Expansion Based Numerical Methods: A Primer, Performance Benchmarking and New Approaches for Problems with Non‑smooth Solutions Thibault Jacquemin1 · Satyendra Tomar1 · Konstantinos Agathos1 · Shoya Mohseni‑Mofidi2 · Stéphane P. A. Bordas1,3,4 Received: 27 April 2019 / Accepted: 29 July 2019 © CIMNE, Barcelona, Spain 2019
Abstract 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. Keywords Collocation method · Non-smooth problems · Singularities · Discontinuities · Generalized finite difference · Discretization-corrected particle strength exchange · Linear elasticity · Voronoi diagrams · Stabilization · Visibility criterion · Diffraction criterion · L-shape · Fichera’s corner · Comparison and performance study · Verification · Benchmarking Electronic supplementary material The online version of this article (https://doi.org/10.1007/s11831-019-09357-5) contains supplementary material, which is available to authorized users. * Stéphane P. A. Bordas [email protected] 1
Institute of Computational Engineering, University of Luxembourg, Maison du Nombre, 6 Avenue de la Fonte, 4364 Esch‑sur‑Alzette, Luxembourg
2
Fraunhofer-Institute for Mechanics of Materials IWM, Freiburg, Germany
3
Department of Medical Research, China Medical Unive
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