Embedding of polytopes for topology optimization

• Published in: Journal of the Brazilian Society of Mechanical Sciences and Engineering 2018-02, Vol.40 (2), p.1-13, Article 57
• Main Authors: Thedin, Regis, Menezes, Ivan F. M., Pereira, Anderson, Carvalho, Marcio S.
• Format: Article
• Language: English
• Subjects: Embedding; Engineering; Filtration; Mapping; Matrix methods; Mechanical Engineering; Optimization algorithms; Polytopes; Regularization; Technical Paper; Topology optimization
• ISSN: 1678-5878; 1806-3691
• DOI: 10.1007/s40430-018-0981-3

A methodology for solving three-dimensional topology optimization problems through a two-level mesh representation approach is described and evaluated. Structural topology optimization problems are executed on a polytope-based mesh, which carries the design variable (and subsequently the density). Displacement field is determined using tetrahedron elements, embedded within the polytopes. The proposed mapping-based framework decouples the analysis routine and optimization algorithm from the specific choice of topology optimization formulation. The mapping-based formulation allows easy applicability of features such as regularization and density filters and symmetry through a mapping matrix. The embedding approach is demonstrated on minimum compliance problems and solid–void solution is obtained by employing continuation on the solid isotropic material with penalization penalty parameter. We show that the proposed approach is able to achieve solutions free of numerical anomalies (e.g., checkerboard pattern and one-node connections) without the application of any explicit regularization scheme nor filtering. Our solutions approach the theoretical solution as mesh size increases.