Topology Optimization Using Wachspress-Type Interpolation with Hexagonal Elements

Abstract

Traditionally, standard Lagrangian‐type finite elements, such as quads and triangles, have been the elements of choice in the field of topology optimization. However, finite element meshes with these elements exhibit the well‐known “checkerboard” pathology in the solution of topology optimization problems. A feasible alternative to eliminate this long‐standing problem consists of using hexagonal elements with Wachspress‐type shape functions. The features of the hexagonal mesh include 2‐node connections (i.e. 2 elements are either not connected or connected by 2 nodes), and 3 edge‐based symmetry lines per element. In contrast, quads can display 1‐node connection, which can lead to checkerboard; and only have 2 edge‐based symmetry lines. We explore the Wachspress‐type hexagonal elements and show their advantages in solving topology optimization problems. We also discuss extensions of the work to account for material gradient effects.