checkerboard-free topology optimization using polygonal finite elements

Anderson Pereira

Tecgraf - Computer Graphics Technology Group

Department of Civil and Environmental EngineeringUniversity of Illinois at Urbana-Champaign

MECOM del Bicentenario

15 - 18 November 2010 - Buenos Aires, Argentina

Checkerboard-free topology optimization using polygonal finite elements

Cameron Talischi, Ivan Menezes and Glaucio H. Paulino

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;

Checkerboard:

One-node hinges:

4

Motivation

• In this work, we examine the use of polygonal meshes consisting of convex polygons in topology optimization to address the aforementioned issues

P

2.792

2.2

1.7921.0

0.788

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

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

Solution obtained with 9101 elements

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

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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

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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

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Cantilever Beam Compliance Minimization

(a) (b)

(c) (d)

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Numerical Results (Compliant Mechanism)

• Force inverter

19

Force Inverter Compliant Mechanism

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Higher Order Finite Element

• Michell cantilever problem with circular support

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Higher Order Finite Element

Solution based on a Voronoi meshSolution based on a T6 mesh

• Michell cantilever problem with circular support

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.

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Ongoing research

• Well-posed formulation of topology optimization problem based on level set (implicit function) description and extension to other objective functions.

P=1

80

50