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Checkerboard-free topology optimization using polygonal finite elements

Checkerboard-free topology optimization using polygonal finite elements. Anderson Pereira. Cameron Talischi , Ivan Menezes and Glaucio H. Paulino. MECOM del Bicentenario 15 - 18 November 2010 - Buenos Aires, Argentina.

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Checkerboard-free topology optimization using polygonal finite elements

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  1. Checkerboard-free topology optimization using polygonal finite elements Anderson Pereira Cameron Talischi, Ivan Menezes and Glaucio H. Paulino MECOM del Bicentenario15 - 18 November 2010 - Buenos Aires, Argentina • Tecgraf - Computer Graphics Technology GroupDepartment of Civil and Environmental Engineering • University of Illinois at Urbana-Champaign

  2. Motivation • In topology optimization, parameterization of shape and topology of the design has been traditionally carried out on uniform grids; • Conventional computational approaches use uniform meshes consisting of Lagrangian-type finite elements (e.g. linear quads) to simplify domain discretization and the analysis routine; • However, as a result of these choices, several numerical artifacts such as the well-known “checkerboard” pathology and one-node connections may appear;

  3. Motivation • In topology optimization, parameterization of shape and topology of the design has been traditionally carried out on uniform grids; • Conventional computational approaches use uniform meshes consisting of Lagrangian-type finite elements (e.g. linear quads) to simplify domain discretization and the analysis routine; One-node hinges: Checkerboard:

  4. Motivation • In this work, we examine the use of polygonal meshes consisting of convex polygons in topology optimization to address the aforementioned issues T. Lewinski and G. I. N. Rozvany. Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains. Struct Multidisc Optim (2008) 35:165–174

  5. Motivation • In this work, we examine the use of polygonal meshes consisting of convex polygons in topology optimization to address the abovementioned issues Solution obtained with 9101 elements T. Lewinski and G. I. N. Rozvany. Exact analytical solutions for some popular benchmark problems in topology optimization III: L-shaped domains. Struct Multidisc Optim (2008) 35:165–174

  6. Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work

  7. Polygonal Finite Element • Isoparametric finite element formulation constructed using Laplace shape function. Pentagon Hexagon Heptagon • The reference elements are regular n-gons inscribed by the unit circle. N. Sukumar and E. A. Malsch. Recent advances in the construction of polygonal finite element interpolants. 2006. Archives of Computational Methods in Engineering, 13(1):129--163

  8. Polygonal Finite Element • Isoparametric finite element formulation constructed using Laplace shape function. • Isoparametric mapping N. Sukumar and E. A. Malsch. Recent advances in the construction of polygonal finite element interpolants. 2006. Archives of Computational Methods in Engineering, 13(1):129--163

  9. Polygonal Finite Element • Laplace shape function Non-negative Linear completeness

  10. Polygonal Finite Element • Laplace shape function for regular polygons • Closed-form expressions can be obtained by employing a symbolic program such as Maple.

  11. Polygonal Finite Element • Numerical Integration

  12. Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work

  13. Topology optimization formulation • The discrete form of the problem is mathematically given by: • minimum compliance • compliant mechanism

  14. Relaxation • The Solid Isotropic Material with Penalization (SIMP) assumes the following power law relationship: • In compliance minimization, the intermediate densities have little stiffness compared to their contribution to volume for large values of p Sigmund, Bendsoe (1999)

  15. Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work

  16. Numerical Results (Compliance Minimization) • Cantilever beam

  17. Cantilever Beam Compliance Minimization (b) (a) (d) (c)

  18. Numerical Results (Compliant Mechanism) • Force inverter

  19. Force Inverter Compliant Mechanism

  20. Higher Order Finite Element • Michell cantilever problem with circular support

  21. Higher Order Finite Element • Michell cantilever problem with circular support Solution based on a T6 mesh Solution based on a Voronoi mesh Talischi C., Paulino G.H., Pereira A., and Menezes I.F.M. Polygonal finite elements for topology optimization: A unifying paradigm. International Journal for Numerical Methods in Engineering, 82(6):671–698, 2010

  22. Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work

  23. Concluding remarks • Solutions of discrete topology optimization problems may suffer from numerical instabilities depending on the choice of finite element approximation; • These solutions may also include a form of mesh-dependency that stems from the geometric features of the spatial discretization; • Unstructured polygonal meshes enjoy higher levels of directional isotropy and are less susceptible to numerical artifacts.

  24. Ongoing research • Well-posed formulation of topology optimization problem based on level set (implicit function) description and extension to other objective functions. 50 P =1 80

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