Computational aspects of nonlinear and multiscale analyses by the multipoint meshless FDM

Abstract

The paper focuses on the recently developed higher order extension of the MFDM – the multipoint meshless finite difference method and introduces the computational schemes and algorithms of the method application to the boundary value problems, especially to the nonlinear and multiscale analyses. There are two basic versions of the multipoint approach that allow accurate and effective solving of engineering problems. The main advantage of the multipoint general version is its generality – the basic relations of derivatives from the unknown function depend on the domain discretization only and are independent of the type of problem being solved. This feature allows to divide the multipoint computational strategy into two stages and is advantageous from the calculation efficiency point of view. The multipoint method algorithms, applied to such engineering problems as two-scale analysis of heterogeneous materials and nonlinear analysis, are developed and briefly presented. This manuscript is an extension of the paper Multipoint Meshless FD Schemes Applied to Nonlinear and Multiscale Analysis by I. Jaworska (2022) published in Lecture Notes in Computer Science [1]. In this extended version computational aspects of the algorithms are deepened, the subsections explaining specific features of the multipoint meshless approach application to the numerical homogenization and nonlinear analysis are included, and finally, the advantages and disadvantages of the method are discussed. The paper is illustrated with several examples (completed in the extended version) of the multipoint numerical analysis.