INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng (2012) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4264 Patch-averaged assumed strain finite elements for stress analysis ‡ G. Castellazzi 1, * ,† and P. Krysl 2 1 University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy 2 University of California, San Diego, 9500 Gilman Dr., MC 0085, La Jolla, CA 92093-0085, USA SUMMARY A finite element model for linear-elastic small deformation problems is presented. The formulation is based on a weighted residual that requires a priori the satisfaction of the kinematic equation. In this approach, an averaged strain-displacement matrix is constructed for each node of the mesh by defining an appropriate patch of elements, yielding a smooth representation of strain and stress fields. Connections with traditional and similar procedure are explored. Linear quadrilateral four-node and linear hexahedral eight-node elements are derived. Various numerical tests show the accuracy and convergence properties of the proposed elements in comparison with extant finite elements and analytic solutions. Specific examples are also included to illustrate the ability to resist numerical locking in the incompressible limit and insensitive response in the presence of shape distortion. Furthermore, the numerical inf-sup test is applied to a selection of problems to show the stability of the present formulation. Copyright © 2012 John Wiley & Sons, Ltd. Received 8 March 2011; Revised 25 November 2011; Accepted 30 November 2011 KEY WORDS: linear elasticity; patch averaging; assumed strain; finite element; volumetric locking; PASE 1. INTRODUCTION Finite element method is a widely common tool to solve engineering problems, and it is still the focus of intensive research today. Faster and more accurate procedure are often the target of this research. Finite elements in general can suffer from lack of robustness when particular conditions appear, such as geometry distortions and constrained deformations (almost incompressible materi- als). In particular, low-order elements result in poor performance when the element geometry is not a parallelogram in the first case [1], whereas they tend to a locking response in the second one. Several theoretical contributions to avoid these phenomena, for instance, for low-order quadrilat- eral elements, are nowadays available and adopted in many general purpose finite element codes. One simple and practically important way to avoid overly stiff response among these is to reduce the order of numerical quadrature employed [2]. Moreover, also the mixed-enhanced strain finite ele- ments exhibits an excellent behavior [3]. Another distinct approach can be recognized in the nodal quadrature approach. The method (see, e.g., [4]) consists of an average nodal pressure formulation proposed to alleviate volumetric locking with the justification that constraint counting is in favor of locking-free behavior for simplex meshes. Subsequently, an ad hoc technique was presented in [5] using an averaged B matrix as the strain-displacement operator on the basis of the well-known Flanagan–Belytschko formulas. Some further developments of these techniques can then be found in [6–8]. Recently, a displacement-based assumed strain finite element formulation with nodal inte- gration was proposed in [9,10] and was shown to provide an excellent behavior also for the almost *Correspondence to: G. Castellazzi, University of Bologna, Viale Risorgimento 2, 40136 Bologna, Italy. † E-mail: giovanni.castellazzi@unibo.it ‡ Please ensure that you use the most up to date class file available from the NME home page at http://www3.interscience. wiley.com/journal/1430/home Copyright © 2012 John Wiley & Sons, Ltd.