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International Journal for Multiscale Computational Engineering

Published 6 issues per year

ISSN Print: 1543-1649

ISSN Online: 1940-4352

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HIGHER ORDER MULTIPOINT MESHLESS FINITE DIFFERENCE METHOD FOR TWO-SCALE ANALYSIS OF HETEROGENEOUS MATERIALS

Volume 17, Issue 3, 2019, pp. 239-260
DOI: 10.1615/IntJMultCompEng.2019028866
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ABSTRACT

This article is an introduction to the numerical homogenization of the heterogeneous material with periodic structure by the new Multipoint solution approach-a higher order extension of the Meshless Finite Difference Method (MFDM). The recently developed Multipoint method follows the original Collatz higher order concept and the essential idea of the MFDM-the moving weighted least squares approximation, using the arbitrarily irregular cloud of nodes as well as various formulations of boundary value problems. The method improves the former procedure without the necessity of providing additional unknowns to both the mesh and the MFD operator. The Multipoint meshless method, like the MFDM, may be used at the macro and the micro levels in the two-scale analysis of heterogeneous materials based on the single Representative Volume Element (RVE). The analysis of the convergence of the effective material parameters for the set of meshes was conducted and compared with the FEM. The error analysis at the micro and macro level confirm the high quality of the Multipoint solution, which may also be used as the improved reference solution instead of the true analytical one for the a posteriori error estimation.

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CITED BY
  1. Jaworska Irena, Generalization of the Multipoint meshless FDM application to the nonlinear analysis, Computers & Mathematics with Applications, 87, 2021. Crossref

  2. Klimczak Marek, Cecot Witold, Higher Order Multiscale Finite Element Method for Heat Transfer Modeling, Materials, 14, 14, 2021. Crossref

  3. Jaworska Irena, Multipoint Meshless FD Schemes Applied to Nonlinear and Multiscale Analysis, in Computational Science – ICCS 2022, 13353, 2022. Crossref

  4. Albuquerque‐Ferreira Augusto César, Ureña Miguel, Ramos Higinio, A technique for generating adapted discretizations to solve partial differential equations with the generalized finite difference method, Mathematical Methods in the Applied Sciences, 2022. Crossref

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