Computer-Aided Design 42 (2010) 808–816 Contents lists available at ScienceDirect Computer-Aided Design journal homepage: www.elsevier.com/locate/cad Algebraic reduction of beams for CAD-integrated analysis Kavous Jorabchi, Joshua Danczyk, Krishnan Suresh ∗ 2050 Mechanical Engineering, 1513 University Ave, Madison, WI 53706, University of Wisconsin, Madison, United States article info Article history: Received 16 October 2007 Accepted 6 May 2010 Keywords: Beams Finite element analysis CAD-integrated analysis Dimensional reduction abstract Beams are high aspect ratio structural members that are used extensively in civil, automotive, aerospace, and MEMS applications. In all such applications, one must typically analyze and optimize the beams through computer simulations. Standard 3D finite element analysis (FEA) of beams can be used in such simulations; it is however prone to errors, and is computationally expensive for thin structures. Therefore, a common strategy is to carry out a dimensionally reduced 1D beam analysis. Unfortunately, 1D beam analysis is hard to automate and integrate with 3D CAD. In this paper, we propose an alternate ‘‘algebraic reduction’’ method that combines the generality of 3D FEA, and the computational efficiency of 1D beam analysis. This is achieved via a dual-representation framework where the geometry of the beam is captured via a 3D finite element mesh, while the physics is captured via a 1D beam model. The proposed method is formally established, and supported through numerical experiments. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Beams 1 are high aspect ratio structural members. Due to their high strength-to-weight ratio they are used extensively, for example, in civil structures, automotive body panels, aerospace structures, and MEMS applications. Fig. 1, for example, shows a high aspect ratio micro-cantilever beam used as an MEMS vibration sensor [1]. In theory, it is possible to analyze beams via standard 3D finite element analysis [2]. However, the recommended strategy is 1D beam analysis [3] for reasons discussed below (other less-common methods are also reviewed later in this section). Standard 3D finite element analysis (FEA) (see Fig. 2) has reached a high degree of reliability over the past few decades, making it the de facto analysis method today. However, high aspect ratio beams pose unique challenges to 3D FEA. Specifically, consider the beam problem in Fig. 3(a). If one uses a coarse finite element mesh (element size ≫ thickness) as in Fig. 3(b), the presence of poor quality elements leads to Poisson and shear locking [4]. On the other hand, if a high quality mesh (element size ∼ thickness) is used, the computational cost grows rapidly with the aspect ratio as illustrated in Fig. 4 (aspect ratio is the overall length divided by the thickness of the hollow beam). ∗ Corresponding author. Tel.: +1 608 262 3594; fax: +1 608 265 2316. E-mail address: suresh@engr.wisc.edu (K. Suresh). 1 In this paper the word ‘‘beam’’ is used in two contexts: ‘‘beam’’ without any prefix implies the actual 3D geometric model, whereas ‘‘1D beam’’ implies 1D (line) idealization of the 3D geometry. Thus, despite the generality and ease of 3D FEA, it is rarely used today to analyze high aspect ratio structures. Indeed, the recommended strategy for thin beam-like geome- tries is 1D beam analysis [5]. This method entails explicitly com- puting the 1D beam axis and extracting the cross-sectional prop- erties (see Fig. 5); 1D beam analysis does not suffer from locking and ill-conditioning problems, and is highly efficient. However, 1D beam analysis poses numerous automation chal- lenges. Specifically, as the beam becomes increasingly complex, computing the 1D beam geometry from a 3D CAD model can be cumbersome [6,7]. Further, coupling the 1D geometry to 3D struc- tural elements is non-trivial (see Fig. 5). Finally, post-processing and visualizing the 1D analysis results within the 3D environment defeats the very purpose of 3D modeling. Besides 3D FEA and 1D analysis, other methods have been proposed for analyzing thin structures [8]. A popular method is based on the concept of solid-shell elements that use an anisotropic space for the finite element basis functions, i.e., relatively low order shape functions are used across the thickness to overcome ill-conditioning, etc. However, solid-shell methods entail a priori orientation of the finite element mesh [9], which can pose difficulties for standard finite element mesh generators. Reduced integration techniques have also been proposed by several researchers to suppress the deficiencies of standard FEA [10,11]; however, under-integration causes generation of hourglass modes and needs stabilization. Yet another technique is to use hybrid or mixed variational principles for stresses and displacements [12]. Hybrid elements can be computationally expensive since construction of the element stiffness entails inverting a sizeable matrix [12]. 0010-4485/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cad.2010.05.001