2016/08/08 by Guiqin Liu, G. R. Liu
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Artificial intelligence #Artificial neural network #Computational mechanics #Computer science #Diffuse element method #Electromagnetic Simulation and Numerical Methods #Finite element method #Function approximation #Interpolation (computer graphics) #Mathematical analysis #Mathematical optimization #Mathematics #Meshfree methods #Mixed finite element method #Moving least squares #Numerical methods in engineering #Partial differential equation #Physics #Smoothing #Solid mechanics
paper · doi:10.1142/s0219876216300014
openalex publication_date 2016/08/08 · crossref created 2016/08/08 · crossref issued 2016/08/31 · crossref published 2016/08/31 · crossref published-online 2016/08/31 · crossref published-print 2016/10/01 · crossref deposited 2025/06/04 · openalex created_date 2025/10/10 · crossref indexed 2026/08/01 · openalex updated_date 2026/08/02
This review paper presents a methodological study on possible and existing meshfree methods for solving the partial differential equations (PDEs) governing solid mechanics problems, based mainly on the research work in the past two decades at the authors group. We start with a discussion on the general steps in a meshfree method based on nodes, with the displacements as the primary variables. We then examine the major techniques used in each of these steps: (1) techniques for displacement function approximations using nodes, (2) approximation of the gradient of the displacements or strains based on nodes and a background T-cells that can be automatically generated and refined, and (3) formulation techniques for producing algebraic equations. The function approximation techniques include node-based interpolation methods, cell-based interpolation methods, function smoothing techniques, and moving least squares approximation techniques. The gradient approximation includes direct differentiation, gradient smoothing, and special strain construction. Formulation techniques include strong-form, weakform, local weakform, weak-strong-form, and weakened weakform (W2). In theory, a meshfree method can be developed using a combination of function approximation, gradient approximation, and formulation techniques, which can lead to matrix of a large number of possible methods. This review attempts to provide an overall methodological review, rather than a usual review of comparing different methods. We hope to show readers the differences between the forests, and just between the trees.