Proceedings of Design Engineering Technical Conference Design Automation Conference September 2004, Salt Lake, Utah DETC2004-57096 Automated Symmetry Exploitation in Engineering Analysis Dr. Krishnan Suresh Department of Mechanical Engineering University of Wisconsin, Madison suresh@engr.wisc.edu ABSTRACT It is well known that one can exploit symmetry to speed-up engineering analysis and improve accuracy, at the same time. Not surprisingly, most CAE systems have standard ‘provisions’ for exploiting symmetry. However, these provisions are inadequate in that they needlessly burden the design engineer with time consuming and error-prone tasks of symmetry detection, symmetry cell construction and reformulation. In this paper, we propose and discuss an automated methodology for symmetry exploitation. First, we briefly review the theory of point symmetry groups that symmetry exploitation rests on. We then address symmetry detection and ‘symmetry cell’ construction. We then address an important concept of boundary mapping of symmetry cells, and relate it to the irreducible representations of point symmetry groups. By formalizing these concepts, we show how automated symmetry exploitation can be achieved, and discuss an implementation of the proposed work within the FEMLAB CAE environment. Keywords: Symmetry, engineering analysis, eigen-value, group theory, irreducible representation, CAD/ CAE. 1. INTRODUCTION Geometric symmetry is fairly common in engineering. For example, Figure 1 illustrates a 3-teeth propeller and a 4-way connector hub. The former exhibits 3-fold symmetry, while the latter exhibits 16-fold symmetry. It is well known that one can exploit this symmetry both to speed-up improve accuracy of engineering analysis [1, 2]. Figure 1: Two symmetric components. Symmetry may be exploited either via block diagonal decomposition or via symmetry cell reformulation; both methods formally rests on group theory [3-6]. The decomposition approach has been successfully employed to analyze symmetric frame structures [7-9], and continuum problems [2, 5]. In this paper, we consider the symmetry cell approach since it neither requires meshing the entire solid nor creating the global matrix, often a distinct advantage over the decomposition method, while analyzing large systems [7]. The mathematics behind the symmetry cell approach is described in [3]. Most CAE systems have standard ‘provisions’ for exploiting symmetry via the symmetry cell approach. However, these provisions are inadequate in that they needlessly burden the design engineer with time consuming and error-prone tasks of symmetry detection, symmetry cell construction and reformulation. In this paper, we propose and discuss an automated methodology for symmetry exploitation. While the ensuing methodology is equally applicable to 2-D and 3-D solids, the illustrative examples are 2-D, and so is the pilot implementation of the proposed architecture within the FEMLAB™ CAD/ CAE environment [10]. The mathematical symbols used in this paper are defined below for easy reference.