A ®nite point method for elasticity problems E. O~ nate * , F. Perazzo 1 , J. Miquel International Centre for Numerical Methods in Engineering, Universidad Polit ecnica de Catalu~ na, Gran Capit an s/n, 08034 Barcelona, Spain Abstract The basis of the ®nite point method FPM) for the fully meshless solution of elasticity problems in structural mechanics is described. A stabilization technique based on a ®nite calculus procedure is used to improve the quality of the numerical solution. The eciency and accuracy of the stabilized FPM in the meshless analysis of simple linear elastic structural problems is shown in some examples of applications. Ó 2001 Elsevier Science Ltd. All rights reserved. Keywords: Finite point method; Meshless method; Elasticity 1. Introduction Mesh free techniques have become quite popular in computational mechanics. A family of mesh free methods is based on smooth particle hydrodynamic procedures [1,2]. These techniques, also called free la- grangian methods, are typically used for problems in- volving large motions of solids and moving free surfaces in ¯uids. A second class of mesh free methods derive from generalized ®nite dierence GFD) techniques [3,4]. Here the approximation around each point is typically de®ned in terms of Taylor series expansions and the discrete equations are found by using point collocation. Among a third class of mesh free techniques we ®nd the so called diuse element DE) method [5], the element free Galerking EFG) method [6,7], the repro- ducing kernel particle RKP) method [8], the meshless local Petrov±Galerkin MLPG) method [9,10] and the method of ®nite spheres [11]. These methods use local interpolations for de®ning the approximate ®eld around a point in terms of values in adjacent points, whereas the discretized system of equations is typically obtained by integrating the Galerkin variational form over a suitable background grid [5±10]. The ®nite point method FPM) proposed in Refs. [12±16] is a truly meshless procedure. The approximation around each point is obtained by using standard moving least square techniques similarly as in DE and EFG methods. The discrete system of equations is obtained by sampling the governing dierential equations at each point as in GFD methods. The basis of the success of the FPM for solid and ¯uid mechanics applications is the stabilization of the discrete dierential equations. The stable form found by the ®nite calculus FIC) procedure presented in Refs. [18±23] corrects the errors introduced by the point collocation procedure, mainly next to boundary segments. In addi- tion, it introduces the necessary stabilization for treating high convection eects and it also allows equal order velocity±pressure interpolations in ¯uid ¯ow problems [22,23]. This paper extends preliminary successful work of the authors to derive a stabilized FPM for analysis of solid mechanics problems using the FIC approach [24]. The content of the paper is structured as follows. In Section 2 the basis of the FPM approximation is presented. The concept of the FIC procedure is detailed next. The discretization of the equilibrium equations in solid mechanics using a stabilized FPM via the FIC technique is described. The eciency of the stabilized FPM is veri®ed in several applications to the 2D and 3D analysis of simple linear elastic solids. Computers and Structures 79 2001) 2151±2163 www.elsevier.com/locate/compstruc * Corresponding author. Tel.: +34-93-205-7016; fax: +34-93- 401-6517. E-mail address: [email protected] E. O~ nate). URL: http://www.cimne.upc.es. 1 Professor at the Universidad T ecnica Federico Santamar õa, Valparaiso, Chile. Visiting Professor at CIMNE. 0045-7949/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII:S0045-794901)00067-0

A Finite Point Method for Elasticity Problems

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