Boundary element method applied to topology optimization using the level set method and an alternative velocity regularization


The topology optimization (TO) is a valuable tool in the early stages of structural engineering design. It enables the determination of the structural layout accounting for the required performance and utilizing less amount of material. In this study, an algorithm for TO is proposed, which is based on two computational procedures. On one hand the boundary element method (BEM), which is efficient for mechanical modelling and remeshing due to its mesh dimension reduction. On the other hand, the level set method (LSM) is an efficient approach to parameterize the design domain. Moreover, it handles complex topology changes without difficulties. The new feature presented here is showing a different formulation of the problem and explore its benefits. The idea is based on the augmented Lagrangian method in which shape sensitivity is used to drive the topology search. The shape derivative takes advantage of conformal and invertible mappings contributing for global stability. To reduce the susceptibility to local minima, a topology perturbation scheme based on local stresses is also adopted. The normal boundary velocity field may be locally singular. In this case the Peng regularization is utilized to maintain stability. These improvements make the algorithm convergent even on the presence of local instabilities. The LSM provides the structural geometry from its zero-level-set curve. Then, this curve is discretised through the BEM. The classical upwind fashion respecting strict CFL conditions is utilised for solving LSM equations. Local holes may be included at each time step, which enables topology changes based on local stress. Classical benchmark examples are used to illustrate the efficiency of the numerical procedure.

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Financial support for this research (Grant 2012/24944-5, 2015/07931-5) provided by São Paulo Research Foundation (FAPESP) is greatly appreciated.

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Correspondence to Hugo Luiz Oliveira.

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Clip 1 Cantilever beam with an inferior corner load immersed in an extended design domain (PNG 6 kb)

Clip 2 Michell-type structure with a central load immersed in an extended design domain (PNG 6 kb)

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Oliveira, H.L., Leonel, E.D. Boundary element method applied to topology optimization using the level set method and an alternative velocity regularization. Meccanica 54, 549–563 (2019).

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  • Boundary element method
  • Level set method
  • Shape derivative