Engineering Mechanics Institute Conference 2013

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Structural Topology Optimization using Polytopes

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Arun Gain
University of Illinois at Urbana-Champaign
United States

Glaucio Paulino
University of Illinois at Urbana-Champaign
United States

Leonardo Duarte
Tecgraf, Pontifical Catholic University of Rio de Janeiro
Brazil

Ivan Menezes
Tecgraf, Pontifical Catholic University of Rio de Janeiro
Brazil

Abstract:
In the past few decades, topology optimization methods have been applied to a wide range of practical applications. In the literature, typically a uniform grid of linear quads/bricks is used for topology optimization problems. Numerical anomalies, such as checkerboard pattern and one-node connections arise out of such formulations. Constraints in the geometrical features of spatial discretization can result in mesh dependent designs [2]. Polygonal elements which do not suffer from such numerical anomalies have been investigated in the past in two-dimensional topology optimization [1, 2]. In the current work, we propose to use of polyhedral meshes to address the geometric features of the domain discretization. Polyhedral meshes provide a greater flexibility in discretizing complex domains. Moreover, techniques such as mesh refinement and coarsening produce elements which are inherently polyhedral. Typically, in order to solve the state equation on polyhedral mesh, the global stiffness matrix calculation would require dealing with each polyhedral element separately, in physical coordinates. Numerical integration to obtain element stiffness matrix on physical coordinates can be computationally expensive and inaccurate. In this work, we solve the state equation using an efficient hybrid mimetic approach. The mimetic hybrid approach reduces the volume integrals, encountered during element stiffness matrix calculations, into surface integrals, thus reducing the computational cost. The features of the current approach are demonstrated using various numerical examples for compliance minimization problem.


References
[1] Gain A.L., Paulino G.H. (2012) Phase-field based topology optimization with polygonal elements: a finite volume approach for the evolution equation. Journal of Structural and Multidisciplinary Optimization. 46(3): 327-342
[2] Talischi C, Paulino GH, Pereira A, Menezes IFM (2010) Polygonal finite elements for topology optimization: A unifying paradigm. International Journal for Numerical Methods in Engineering 82: 671 – 698

 

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