INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 2008; 74:416–446 Published online 10 September 2007 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.2181 Meshfree and finite element nodal integration methods M. A. Puso 1, ∗, † , J. S. Chen 2 , E. Zywicz 1 and W. Elmer 2 1 Methods Development Group, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, U.S.A. 2 Department of Civil Engineering, UCLA, Los Angeles, CA 90095, U.S.A. SUMMARY Nodal integration can be applied to the Galerkin weak form to yield a particle-type method where stress and material history are located exclusively at the nodes and can be employed when using meshless or finite element shape functions. This particle feature of nodal integration is desirable for large deformation settings because it avoids the remapping or advection of the state variables required in other methods. To a lesser degree, nodal integration can be desirable because it relies on fewer stress point evaluations than most other methods. In this work, aspects regarding stability, consistency, efficiency and explicit time integration are explored within the context of nodal integration. Both small and large deformation numerical examples are provided. Copyright 2007 John Wiley & Sons, Ltd. Received 15 March 2007; Revised 5 June 2007; Accepted 23 July 2007 KEY WORDS: nodal integration; particle methods; meshfree; explicit time integration 1. INTRODUCTION The Galerkin method requires stress and strain evaluations at specific points in a discretization to integrate the weak form of equations. Nodal integration, strictly speaking, only evaluates stress at the nodes and is a form of particle method. These particle type methods offer a Lagrangian solution to very large deformation problems such as penetrator, earth moving and metal forming type analyses. Nodal integration has been applied to meshless shape functions in a number of previous works [1–6] and to finite element shape functions in [7–10]. Three classical nodal integration schemes, nodal strain method [4], stabilized conforming nodal integration (SCNI) [3] and nodal averaging [8], are considered here and their differences are based on how strain is calculated. ∗ Correspondence to: M. A. Puso, Methods Development Group, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, U.S.A. † E-mail: puso@llnl.gov Contract/grant sponsor: U.S. Department of Energy; contract/grant number: W-7405-Eng-48 Copyright 2007 John Wiley & Sons, Ltd.