Level Set Method for Shape and Topology Optimization of Contact Problems

Myśliński, Andrzej

doi:10.1007/978-3-642-04802-9_23

Cited by 8 publications

(20 citation statements)

References 25 publications

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“…In future studies, the method needs to be expanded onto problems with multiple phases, where one of them could be void. Finally, the method should be augmented by topological derivatives to mitigate the dependence of the optimization results on the initial design; see for example Sokolowski and Zochowski (2005) and Myśliński (2009).…”

Section: Discussionmentioning

confidence: 99%

“…Myślińsky pioneered level set methods for shape and topology optimization of unilateral contact problems where one of the bodies is rigid; see for example Myśliński (2009), Myśliński (2013, Myśliński (2014) and Myśliński (2015). The finite element method in combination with an Ersatz material approach is used to model contact problems with and without friction.…”

Section: Introductionmentioning

confidence: 99%

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Level set topology optimization of problems with sliding contact interfaces

Lawry

Maute

2015

Struct Multidisc Optim

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This paper introduces a topology optimization method for the design of two-component structures and two-phase material systems considering sliding contact and separation along interfaces. The geometry of the contact interface is described by an explicit level set method which allows for both shape and topological changes in the optimization process. The mechanical model assumes infinitesimal strains, a linear elastic material behavior, and a quasi-static response. The contact conditions are enforced by a stabilized Lagrange multiplier method and an active set strategy. The mechanical model is discretized by the extended finite element method which retains the crispness of the level set geometry description and allows for the convenient integration of the weak form of the contact conditions at the phase boundaries. The formulation of the optimization problem is regularized by introducing a perimeter penalty into the objective function. The optimization problem is solved by a nonlinear programming scheme computing the design sensitivities by the adjoint method. The main characteristics of the proposed method are studied by numerical examples in two dimensions. Consideration of contact leads to the formation of barb-type features that increase the interface stiffness. The numerical results further demonstrate the significant difference in the optimized geometries when assuming perfect bonding versus considering contact.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Level set topology optimization of problems with sliding contact interfaces

Lawry

Maute

2015

Struct Multidisc Optim

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“…The stress field sðuðxÞÞ satisfies the system of equations [1][2][3] Às ij ðuðxÞÞ ;j ¼ f i ðxÞ; x 2 O; i; j ¼ 1; 2,…”

Section: Problem Formulationmentioning

confidence: 99%

“…The material derivative method is employed in monograph [2] to calculate the sensitivity of solutions to contact problems as well as the derivatives of domain depending functionals with respect to variations of the boundary of the domain occupied by the body. Shape optimization of a dynamic contact problem with a given Coulomb friction and heat flow is considered in [3]. In this paper the material derivative method is employed to formulate a necessary optimality condition.…”

Section: Introductionmentioning

confidence: 99%

“…Level set methods are numerically efficient and robust procedures for the tracking of interfaces, which allow domain boundary shape changes in the course of iteration. Applications of the level set methods in structural optimization can be found, among others, in [3,7,[14][15][16][17]. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative of an appropriately defined cost functional with respect to the variations of the free boundary.…”

Section: Introductionmentioning

confidence: 99%

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Level set method for optimization of contact problems

Myśliński

2008

Engineering Analysis with Boundary Elements

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This paper deals with the numerical solution of topology and shape optimization problems of an elastic body in unilateral contact with a rigid foundation. The contact problem with the prescribed friction is described by an elliptic variational inequality of the second order governing a displacement field. The structural optimization problem consists in finding such shape of the boundary of the domain occupied by the body that the normal contact stress along the contact boundary of the body is minimized. The shape of this boundary and its evolution is described using the level set approach. Level set methods are numerically efficient and robust procedures for the tracking of interfaces. They allow domain boundary shape changes in the course of iteration. The evolution of the domain boundary and the corresponding level set function is governed by the Hamilton-Jacobi equation. The speed vector field driving the propagation of the level set function is given by the Eulerian derivative of an appropriately defined functional with respect to the free boundary. In this paper the necessary optimality condition is formulated. The level set method, based on the classical shape gradient, is coupled with the bubble or topological derivative method, which is precisely designed for introducing new holes in the optimization process. The holes are supposed to be filled by weak phase mimicking voids. Since both methods capture a shape on a fixed Eulerian mesh and rely on a notion of gradient computed through an adjoint analysis, the coupling of these two method yields an efficient algorithm. Moreover the finite element method is used as the discretization method. Numerical examples are provided and discussed.

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Level set shape and topology optimization of finite strain bilateral contact problems

Lawry

Maute

2017

Numerical Meth Engineering

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This paper presents a method for the optimization of multicomponent structures comprised of 2 and 3 materials considering large motion sliding contact and separation along interfaces. The structural geometry is defined by an explicit level set method, which allows for both shape and topology changes. The mechanical model assumes finite strains, a nonlinear elastic material behavior, and a quasi-static response. Identification of overlapping surface position is handled by a coupled parametric representation of contact surfaces. A stabilized Lagrange method and an active set strategy are used to model frictionless contact and separation. The mechanical model is discretized by the extended FEM, which maintains a clear definition of geometry. Face-oriented ghost penalization and dynamic relaxation are implemented to improve the stability of the physical response prediction. A nonlinear programming scheme is used to solve the optimization problem, which is regularized by introducing a perimeter penalty into the objective function. Design sensitivities are determined by the adjoint method. The main characteristics of the proposed method are studied by numerical examples in 2 dimensions. The numerical results demonstrate improved design performance when compared to models optimized with a small strain assumption. Additionally, examples with load path dependent objectives display nonintuitive designs. KEYWORDS extended finite element method, finite strain, level set methods, stabilized lagrange multiplier method, sliding contact, topology optimization [Correction added on 24 January 2018, after first online publication: absolute value symbol was missing in Equation 49, and have since been added.]

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Level Set Method for Shape and Topology Optimization of Contact Problems

Cited by 8 publications

References 25 publications

Level set topology optimization of problems with sliding contact interfaces

Level set topology optimization of problems with sliding contact interfaces

Level set method for optimization of contact problems

Level set shape and topology optimization of finite strain bilateral contact problems

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