On the energy of a flow arising in shape optimization

Cardaliaguet, Pierre; Ley, Olivier

doi:10.4171/ifb/187

Cited by 8 publications

(12 citation statements)

References 16 publications

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“…We remark that 13) where the constant C is a bound on the perimeter of {u(·, t) = 0}, uniform for u ∈ E and t ∈ [0, t 0 ].…”

Section: Lemma 75 (Estimate On Characteristic Functions) There Exismentioning

confidence: 99%

Minimizing movements for dislocation dynamics with a mean curvature term

Forcadel¹,

Monteillet²

2009

ESAIM: COCV

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Abstract. We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.Mathematics Subject Classification. 53C44, 49Q15, 49L25, 28A75, 58A25.

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“…We remark that 13) where the constant C is a bound on the perimeter of {u(·, t) = 0}, uniform for u ∈ E and t ∈ [0, t 0 ].…”

Section: Lemma 75 (Estimate On Characteristic Functions) There Exismentioning

confidence: 99%

Minimizing movements for dislocation dynamics with a mean curvature term

Forcadel¹,

Monteillet²

2009

ESAIM: COCV

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“…The flow of the shape gradient then formally leads to an Hamilton-Jacobi equation for the evolution of the domain coupled with the state function. Recently, Cardaliaguet and Ley have performed in [7,8] a first step in the theoretical justification of this approach on a specific example. However, the level set grid has no particular reason to be useful for the computation of the state function.…”

Section: Introductionmentioning

confidence: 99%

On the ersatz material approximation in level-set methods

Dambrine

Kateb

2009

ESAIM: COCV

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Abstract. The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a ersatz material approximation [Allaire et al., J. Comput. Phys. 194 (2004) 363-393], a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the ersatz material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.Mathematics Subject Classification. 49Q10, 34A55, 49Q12.

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“…Such appears in the study of viscosity solutions when there are possibilities of nonuniqueness/fattening of the zero set (see for example [2]). …”

Section: The Continuum Limit and Existence Of Weak Solutionsmentioning

confidence: 99%

“…In [3] it was shown that a particular selection of the discrete scheme in [1] converges to viscosity solution of the mean curvature flow in the sense of [6]. Also see [2] where one studies a free boundary problem similar to (P), but satisfies the comparison principle. (In [2] the goal was to obtain an energy bound for the viscosity solution of the corresponding problem.…”

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A variational approach to a quasi-static droplet model

Grunewald

Kim²

2010

Calc. Var.

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We consider a quasi-static droplet motion based on contact angle dynamics on a planar surface. We derive a natural time-discretization and prove the existence of a weak global-in-time solution in the continuum limit. The time discrete interface motion is described in comparison with barrier functions, which are classical sub-and super-solutions in a local neighborhood. This barrier property is different from standard viscosity solutions since there is no comparison principle for our problem. In the continuum limit the barrier properties still hold in a modified sense.

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On the energy of a flow arising in shape optimization

Abstract: In [8] we have defined a viscosity solution for the gradient flow of the exterior Bernoulli free boundary problem. We prove here that the associated energy is non-increasing along the flow. For this we build a discrete gradient flow in the flavour of Almgren, Taylor and Wang [2].

Cited by 8 publications

References 16 publications

Minimizing movements for dislocation dynamics with a mean curvature term

Minimizing movements for dislocation dynamics with a mean curvature term

On the ersatz material approximation in level-set methods

A variational approach to a quasi-static droplet model

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