Solution of Shape Optimization Problem and Its Application to Product Design

Azegami, Hideyuki

doi:10.1007/978-981-10-2633-1_6

Cited by 6 publications

(4 citation statements)

References 11 publications

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“…Rabago et al [31] on the other hand considered a Neumann-type extension for solving the mentioned velocity field. In our context we shall consider two approaches based on H 1 -gradient method [1]. In particular, for sufficiently small parameter γ > 0, and λ ∈ {0, 1} we shall use the following H 1 -gradient method: Solve for…”

Section: 2mentioning

confidence: 99%

A shape optimization problem constrained with the Stokes equations to address maximization of vortices

Simon

Notsu

2022

EECT

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<p style='text-indent:20px;'>We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-norm of the curl and the <i>det-grad</i> measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.</p>

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Section: 2mentioning

confidence: 99%

A shape optimization problem constrained with the Stokes equations to address maximization of vortices

Simon

Notsu

2022

EECT

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“…Let Ω 0 = Ω(0) be an original domain with boundary Ω 0 = Γ M ∪Γ F , where Γ M is a moving boundary, that is, Γ M is deformed in the computational steps of the optimization process, and Γ F is a fixed boundary, that is, Γ F is fixed in the computational steps of the optimization process. By (1), Ω(t) is given as…”

Section: Shape Optimization Problemmentioning

confidence: 99%

“…In this case, the objective function is the mean compliance and the constraint function is the total volume. In order to solve this problem, a nonparametric boundary variation can be formulated by selecting a one‐parameter family of continuous one‐to‐one mappings from the original domain to variable domains . The shape gradients for the domain variations can be evaluated using the adjoint variable method.…”

Section: Introductionmentioning

confidence: 99%

“…In order to solve this problem, a nonparametric boundary variation can be formulated by selecting a one-parameter family of continuous one-to-one mappings from the original domain to variable domains. 1 The shape gradients 2 for the domain variations can be evaluated using the adjoint variable method. The solution to the shape optimization problem is obtained by an iterative calculation of the domain variations.…”

Section: Introductionmentioning

confidence: 99%

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Shape optimization using time evolution equations

Murai

Kawamoto

Kondoh

2018

Numerical Meth Engineering

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Summary In this study, we deal with a numerical solution based on time evolution equations to solve the optimization problem for finding the shape that minimizes the objective function under given constraints. The design variables of the shape optimization problem are defined on a given original domain on which the boundary value problems of partial differential equations are defined. The variations of the domain are obtained by the time integration of the solution to derive the time evolution equations defined in the original domain. The shape gradient with respect to the domain variations are given as the Neumann boundary condition defined on the original domain boundary. When the constraints are satisfied, the decreasing property of the objective function is guaranteed by the proposed method. Furthermore, the proposed method is used to minimize the heat resistance under a total volume constraint and to solve the minimization problem of mean compliance under a total volume constraint.

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Second Derivatives of Cost Functions and $$H^1$$ Newton Method in Shape Optimization Problems

Azegami

2017

Mathematical Analysis of Continuum Mechanics and Industrial Applications II

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Solution of Shape Optimization Problem and Its Application to Product Design

Cited by 6 publications

References 11 publications

A shape optimization problem constrained with the Stokes equations to address maximization of vortices

A shape optimization problem constrained with the Stokes equations to address maximization of vortices

Shape optimization using time evolution equations

Second Derivatives of Cost Functions and $$H^1$$ Newton Method in Shape Optimization Problems

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