Shape optimization for Stokes flow

Numer Methods Biomed Eng

2010

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SUMMARYThis paper presents the problem of shape optimization of two-dimensional viscous flow governed by the time-dependent Navier-Stokes equations. The minimization problem of the viscous dissipated energy was established in the fluid domain. The discretization of Navier-Stokes equations is accomplished using a new stabilized finite element method in space and finite difference in time. This new method does not need a stabilization parameter or calculation of high-order derivatives. We derive the structures of the discrete Eulerian derivative of the cost functional by a discrete adjoint method with a function space parametrization technique. A gradient-type optimization algorithm with a mesh adaptation technique and a mesh moving strategy is effectively implemented.

Section: Introductionmentioning

confidence: 97%

Unsteady flow optimization by a stabilized finite element method

Numer Methods Biomed Eng

2010

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“…There has been a significant amount of research on optimal control and optimal shape design for problems governed by the steady and unsteady Euler and Navier-Stokes equations. For instance, articles [1][2][3][4] deal with the shape optimization for the incompressible flow, references [5][6][7][8] address the optimal control for the incompressible flow.…”

Section: Introductionmentioning

confidence: 99%

A new stabilized finite element method for optimal control for a Ladyzhenskaya model for unsteady flows

Numerical Methods Partial

Kong

2011

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This article considers the time-dependent optimal control problem of tracking the velocity for the viscous incompressible flows which is governed by a Ladyzhenskaya equations with distributed control. The existence of the optimal solution is shown and the first-order optimality condition is established. The semidiscrete-in-time approximation of the optimal control problem is also given. The spatial discretization of the optimal control problem is accomplished by using a new stabilized finite element method which does not need a stabilization parameter or calculation of high order derivatives. Finally a gradient algorithm for the fully discrete optimal control problem is effectively proposed and implemented with some numerical examples.

“…], we use the state derivative approach to solve a shape optimization problem governed by a Robin problem, and in [11,12], we derive the expression of shape gradients for Stokes and Navier-Stokes optimization problem by this approach, respectively. In this paper, we use this approach and a weak implicit function theorem to derive the structures of shape gradients with respect to the shape of the variable domain for some given cost functionals in shape optimization problems for time-dependent Navier-Stokes flow with small regularity data.…”

Section: Introductionmentioning

confidence: 99%

Optimal shape design for the time‐dependent Navier–Stokes flow

Numerical Methods in Fluids

Zhuang

2007

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SUMMARYThis paper is concerned with the problem of shape optimization of two-dimensional flows governed by the time-dependent Navier-Stokes equations. We derive the structures of shape gradients for time-dependent cost functionals by using the state derivative and its associated adjoint state. Finally, we apply a gradienttype algorithm to our problem, and numerical examples show that our theory is useful for practical purposes and the proposed algorithm is feasible in low Reynolds number flows.