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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 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
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;
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:
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
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
Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work
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
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
Polygonal Finite Element • Laplace shape function Non-negative Linear completeness
Polygonal Finite Element • Laplace shape function for regular polygons • Closed-form expressions can be obtained by employing a symbolic program such as Maple.
Polygonal Finite Element • Numerical Integration
Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work
Topology optimization formulation • The discrete form of the problem is mathematically given by: • minimum compliance • compliant mechanism
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)
Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work
Numerical Results (Compliance Minimization) • Cantilever beam
Cantilever Beam Compliance Minimization (b) (a) (d) (c)
Numerical Results (Compliant Mechanism) • Force inverter
Higher Order Finite Element • Michell cantilever problem with circular support
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
Outline • Polygonal Finite Element • Topology optimization formulation • Numerical Results • Concluding remarks • Ongoing work
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.
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