A topological derivative method for topology optimization

Norato, Julián A.; Bendsøe, Martin P.; Haber, Robert B.; Tortorelli, Daniel A.

doi:10.1007/s00158-007-0094-6

Cited by 124 publications

(68 citation statements)

References 23 publications

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Section: Topological Derivativementioning

confidence: 99%

“…The Topological Derivative describes the sensitivity of an objective functional for introducing an infinitesimal hole into the design space. This sensitivity can be used for topology optimization using a level set method [1].Regarding the mean compliance J σ = Ω ε 0 σ dε dΩ, the general form of the TD isWe denote the stresses σ ϕ and strains ε ϕ in the polar coordinate system, where ϕ indicates the circumferential direction on the hole boundary Γ ρ (x). This is one of the research results from Eschenauer, Kobelev and Schumacher [2] and named the Topological Derivative by Sokolowski and Zochowski [3].…”

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confidence: 99%

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On the calculation of Topological Derivatives considering an exemplary nonlinear material model

Weider

Schumacher

2016

Proc Appl Math and Mech

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In this contribution, a new approach for dealing with material nonlinearities in structural optimization is presented. The method combines the advantage of the calculation of the Topological Derivatives in elasticity problems with the use of metamodelling for capturing complex correlations. For an exemplary nonlinear material model, the calculation of the Topological Derivative and the resulting difference to the linear elastic material model is shown. An academic example shows the difference between the linear and the nonlinear material model. Topological DerivativeTopological Derivatives are already used for optimization subject to linear static problems. In this paper, a numerical approach for dealing with material nonlinearities is presented. We consider a functional J (Ω), whereis the domain for a topology optimization problem subject to a boundary value problem in mechanics. Assuming that the following limit existsthe functional TJ (x) is called the Topological Derivative (TD). Here, B ρ (x) denotes the ball with radius ρ around x ∈ Ω and Ω \ B ρ (x) denotes the domain with the cut-out. The Topological Derivative describes the sensitivity of an objective functional for introducing an infinitesimal hole into the design space. This sensitivity can be used for topology optimization using a level set method [1].Regarding the mean compliance J σ = Ω ε 0 σ dε dΩ, the general form of the TD isWe denote the stresses σ ϕ and strains ε ϕ in the polar coordinate system, where ϕ indicates the circumferential direction on the hole boundary Γ ρ (x). This is one of the research results from Eschenauer, Kobelev and Schumacher [2] and named the Topological Derivative by Sokolowski and Zochowski [3]. The determination of (2) is based on the adjoint state approach. The known analytical form for the stress distribution around circular cavities in linear elasticity leads to a closed analytical form for the sensitivity with respect to boundary value problems in linear elasticity. Numerical ApproachTo extend this approach to an exemplary nonlinear material model, we use numerical methods for the calculation of the TD in shell structures. The elasto-plastic material shown in Fig. 1a with a nearly constant tangent modulus is used. For numerical stability the tangent modulus has to be grater than zero. As an analytical stress distribution is not known, a finite element model with a circular hole in a planar shell is loaded under two in-plane outer stresses (Fig. 1b). This microcell represents a section with an infinitesimal hole transformed to the principal direction. The resulting stresses and strains in circumferential direction on the boundary of the hole are calculated with an implicit quasi-static finite element simulation. The sensitivity is then calculated with the general expression (2) for the compliance. Various combinations of stress states for the microcell provide the sample points for the meta-model of the TD.

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Section: Topological Derivativementioning

confidence: 99%

mentioning

confidence: 99%

On the calculation of Topological Derivatives considering an exemplary nonlinear material model

Weider

Schumacher

2016

Proc Appl Math and Mech

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“…In order to apply a topology optimization algorithm to problem (2.7) such as those proposed in [5,7,13,16,26], we need to know the expression of the topological derivative of the functional I γ Ω (u Ω ). We assume that the topological derivative of the objective functional I Ω (u Ω ) is known.…”

Section: Topology Perturbationsmentioning

confidence: 99%

A penalty method for topology optimization subject to a pointwise state constraint

Amstutz¹

2009

ESAIM: COCV

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Abstract. This paper deals with topology optimization of domains subject to a pointwise constraint on the gradient of the state. To realize this constraint, a class of penalty functionals is introduced and the expression of the corresponding topological derivative is obtained for the Laplace equation in two space dimensions. An algorithm based on these concepts is proposed. It is illustrated by some numerical applications.Mathematics Subject Classification. 49Q10, 49Q12, 49M30, 35J05.

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“…More recently, a new class of methodologies for structural topology optimization has emerged based on the use of the topological derivative of the relevant objective functionals [29,12,25,5,24,26]. The notion of topological derivative itself is a relatively new concept, introduced just over a decade ago [12,29].…”

Section: Introductionmentioning

confidence: 99%

Topological Derivative-Based Topology Structural Optimization Under Drucker-Prager-Type Stress Constraints

Amstutz¹,

Novotny²,

Neto³

2014

Proceedings of 10th World Congress on Computational Mechanics

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A topological derivative method for topology optimization

Cited by 124 publications

References 23 publications

On the calculation of Topological Derivatives considering an exemplary nonlinear material model

On the calculation of Topological Derivatives considering an exemplary nonlinear material model

A penalty method for topology optimization subject to a pointwise state constraint

Topological Derivative-Based Topology Structural Optimization Under Drucker-Prager-Type Stress Constraints

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