## 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 problem 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 can 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 checkerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.

Original language | English (US) |
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Pages (from-to) | 569-583 |

Number of pages | 15 |

Journal | Structural and Multidisciplinary Optimization |

Volume | 37 |

Issue number | 6 |

DOIs | |

State | Published - Feb 2009 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Software
- Computer Science Applications
- Computer Graphics and Computer-Aided Design
- Control and Optimization

## Keywords

- Checkerboard
- Continuous approximation of material distribution
- Projection functions
- Topology optimization
- Wachspress interpolation functions