Structural and Multidisciplinary Optimization
C. Talischi, G.H. Paulino, and C. H. Le
Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Newmark Laboratory, MC-250, 205 North Mathews Avenue, Urbana, IL 61801-2397, U.S.A.
Abstract
Traditionally, standard Lagrangian-type finite elements, such as linear quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these conventional elements exhibit the well-known "checkerboard" pathology in the iterative solution of topology optimization problems. A feasible alternative to eliminate such long-standing problems consists of using hexagonal (honeycomb) elements with Wachspress-type shape functions. The features of the hexagonal mesh include two-node connections (i.e. two elements are either not connected or connected by two nodes), and three edge-based symmetry lines per element. In contrast, quads display one-node connections, which can lead to checkerboard; and only have two edge-based symmetry lines. In addition, Wachspress rational shape functions satisfy the partition of unity condition and lead to conforming finite element approximations. We explore the Wachspress-type hexagonal elements and present their implementation using three approaches for topology optimization: element-based, continuous approximation of material distribution, and minimum length-scale through projection functions. Examples are presented that demonstrate the advantages of the proposed element in achieving the advantages of the proposed element in acheiveing checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.
KEY WORDS: Topology Optimization, Checkerboard, Wachspress interpolation functions, Continuous approximation of material distribution, Projection functions