A shape optimization formulation of weld pool determination

Chakib, Abdelkrim; Ellabib, Abdellatif; Nachaoui, Abdeljalil; Nachaoui, Mourad

doi:10.1016/j.aml.2011.09.017

Cited by 10 publications

(10 citation statements)

References 3 publications

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“…El Yazidi and Ellabib 18 used the genetic algorithm (GA) with the finite element to estimate the configuration of depletion layer in semiconductor. Chakib et al 19 used the GA to solve a shape optimization problem in weld pool. El Yazidi and Ellabib 20 used an augmented Lagrangian approach to solve a bilateral free boundary.…”

Section: Introductionmentioning

confidence: 99%

An iterative method for optimal control of bilateral free boundaries problem

Yazidi

Ellabib

2021

Math Methods in App Sciences

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In this paper, a bilateral free boundaries problem is considered. This kind of inverse problems appears in the theory of semiconductors and multi‐phase problems. Using a shape functional and some regularization terms, an optimal control problem is formulated. In addition, we prove its solution existence. The first optimality conditions and the shape gradient are computed. With the finite element method, we write the discrete version of the optimal control problem. To design our proposed scheme, we based on the conjugate gradient, where we use the genetic algorithm to find the best initial guess for the gradient method. At each mesh regeneration, we perform a mesh refinement in order to avoid any domain singularities. Some numerical examples are shown to demonstrate the validity of the theoretical results and to prove the robustness and efficiency of the proposed scheme, especially to identify free boundaries with jump points.

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Section: Introductionmentioning

confidence: 99%

An iterative method for optimal control of bilateral free boundaries problem

Yazidi

Ellabib

2021

Math Methods in App Sciences

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“…Instability of the problem. To prove the instability result of the inverse problem (3), we adopt the method already used in [8,13,15,14]. Thus, a local regularity argument is used to show the compactness of the Riesz operator corresponding to the shape Hessian at a solution ω * ∈ Ω d0 of the inverse problem.…”

Section: 2mentioning

confidence: 99%

“…Note that the regularity H 2 (Ω d0 ) is due to a local regularity argument (as the one used in [8,13,15,14]), since the object ω * has a C 2 boundary and since the condition on ∂ω * is homogeneous (and then smooth), the solution of problem ( 28) is globally H 1 (Ω), but locally H 2 (Ω d0 ).…”

Section: 2mentioning

confidence: 99%

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Shape optimization method for an inverse geometric source problem and stability at critical shape

Afraites¹,

Masnaoui²,

Nachaoui³

2022

DCDS-S

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This work deals with a geometric inverse source problem. It consists in recovering inclusion in a fixed domain based on boundary measurements. The inverse problem is solved via a shape optimization formulation. Two cost functions are investigated, namely, the least squares fitting, and the Kohn-Vogelius function. In this case, the existence of the shape derivative is given via the first order material derivative of the state problems. Furthermore, using the adjoint approach, the shape gradient of both cost functions is characterized. Then, the stability is investigated by proving the compactness of the Hessian at the critical shape for both considered cases. Finally, based on the gradient method, a steepest descent algorithm is developed, and some numerical experiments for non-parametric shapes are discussed.

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“…For details concerning the differentiation with respect to the domain( see [30,28,29] and the book [23]). We can also use the techniques developed in [13,14,15] .…”

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confidence: 99%

A new coupled complex boundary method (CCBM) for an inverse obstacle problem

Afraites¹

2022

DCDS-S

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In the present work we introduce and study a new method for solving the inverse obstacle problem. It consists in identifying a perfectly conducting inclusion ω contained in a larger bounded domain Ω via boundary measurements on ∂Ω. In order to solve this problem, we use the coupled complex boundary method (CCBM), originaly proposed in [16]. The new method transforms our inverse problem to a complex boundary problem with a complex Robin boundary condition coupling the Dirichlet and Neumann boundary data. Then, we optimize the shape cost function constructed by the imaginary part of the solution in the whole domain in order to determine the inclusion ω. Thanks to the tools of shape optimization, we prove the existence of the shape derivative of the complex state with respect to the domain ω. We characterize the gradient of the cost functional in order to make a numerical resolution. We then investigate the stability of the optimization problem and explain why this inverse problem is severely ill-posed by proving compactness of the Hessian of cost functional at the critical shape. Finally, some numerical results are presented and compared with classical methods.

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A shape optimization formulation of weld pool determination

Cited by 10 publications

References 3 publications

An iterative method for optimal control of bilateral free boundaries problem

An iterative method for optimal control of bilateral free boundaries problem

Shape optimization method for an inverse geometric source problem and stability at critical shape

A new coupled complex boundary method (CCBM) for an inverse obstacle problem

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