Towards a procedure to automatically improve finite element solutions by interpolation covers Jaehyung Kim, Klaus-Jürgen Bathe ⇑ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA article info Article history: Received 12 August 2013 Accepted 26 September 2013 Available online 7 November 2013 Keywords: Finite elements 3-Node triangular element 4-Node tetrahedral element Improvement of displacements and stresses Interpolation covers Enrichment of interpolations abstract In a previous paper (Kim and Bathe, 2013) [1], we introduced a scheme to improve finite element displacement and stress solutions by the use of interpolation covers. In the present paper we show how the scheme can be used to automatically improve finite element solutions. As in Ref. (Kim and Bathe, 2013) [1], we focus on the use of the low-order finite elements for the analysis of solids, namely, the 3-node triangular and 4-node tetrahedral elements with the use of interpolation covers. An error indica- tor is employed to automatically establish which order cover to apply at the finite element mesh nodes to best improve the accuracy of the solution. Some two- and three-dimensional problems are solved to illustrate the procedure. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction In standard finite element analysis, the numerical solution is improved by changing the locations of the mesh nodes, increasing the mesh density, or as another option, using a more powerful ele- ment. A scheme to proceed differently has been discussed in Ref. [1]. This method uses interpolation covers over patches of ele- ments to enrich the displacement interpolations and increase the solution accuracy. The order of the interpolations can vary depend- ing on what improvement in accuracy is needed. The scheme is clo- sely related to the numerical manifold method proposed by Shi [2,3]. We refer to Ref. [1] for a detailed description of the scheme using interpolation covers and further references pertaining to its development. The scheme was established in detail to improve the stress solu- tions when using the 3-node triangular element in two-dimen- sional analyses and the 4-node tetrahedral element in three- dimensional analyses. The use of these classical elements is attrac- tive because these elements can be used to mesh very complex geometries, and they are robust and lead to relatively small band- widths, but almost always it would be of much value to have better stress predictions [4,5]. The objective in the present paper is to use the scheme of Ref. [1] and present a fully automatic procedure to adaptively choose the orders of the interpolation covers with the aim to increase the solution accuracy for meshes using the low-order elements. Since the interpolation covers are compatible in displacements, an arbitrary combination of covers and order of interpolations can be chosen. Of course, an ideal adaptive scheme should give more accuracy at a smaller computational cost than using the tra- ditional approach of using a finer mesh or higher-order finite elements. In the adaptive interpolation procedure, we shall use cover or- ders up to cubic, to provide a flexible adaptive range. We focus our discussion on the analysis of problems in solid mechanics, but similar ideas can directly be applied to the analysis of problems in heat transfer, fluid flow and multiphysics. In the following sections, we first briefly review the scheme using interpolation covers to improve the accuracy of solutions, we then present the adaptive scheme to automatically choose the covers, and finally we give example solutions to illustrate the performance of the method. 2. The finite element formulation enriched with covers In this section, we briefly review the finite element formulation enriched with covers for low-order elements, merely to provide the foundation for the sections to follow. A detailed description also referring to other related research works is given in Ref. [1]. Let us assume that a mesh of 3-node triangular (or 4-node tet- rahedral) elements has been used to obtain a displacement and stress solution of a two-dimensional (or three-dimensional) prob- lem in solid mechanics. Fig. 1(a) shows a node i, the two-dimen- sional elements connected to that node and the linear interpolation function h i used in the solution. We define C i to be 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.09.007 ⇑ Corresponding author. Tel.: +1 6179265199. E-mail address: kjb@mit.edu (K.J. Bathe). Computers and Structures 131 (2014) 81–97 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc