Abstract
This paper is concerned with the numerical simulation for shape optimization of the Stokes flow around a solid body. The shape gradient for the shape optimization problem in a viscous incompressible flow is evaluated by the velocity method. The flow is governed by the steady-state Stokes equations coupled with a thermal model. The structure of continuous shape gradient of the cost functional is derived by employing the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. A gradient-type algorithm is applied to the shape optimization problem. Numerical examples show that our theory is useful for practical purpose, and the proposed algorithm is feasible and effective.
Highlights
The problem of finding the shape design of a system described by the incompressible Stokes equations arises in many design problems such as aerospace, automotive, hydraulic, ocean, and structural and wind engineering
The shape optimization problem can be transformed into the minimization problem without constraint condition by the Lagrange multiplier and the adjoint equations using adjoint variables corresponding to the state equations
The optimal shape design of a body subjected to the minimum viscous dissipated energy has been a challenging task for a long time, and it has been investigated by several authors
Summary
The problem of finding the shape design of a system described by the incompressible Stokes equations arises in many design problems such as aerospace, automotive, hydraulic, ocean, and structural and wind engineering. The problem is to optimize the shape of the domain in order to minimize a cost functional that depends on the solutions. In our paper [7], we solved a shape reconstruction problem for the inverse Stokes problem and investigated the numerical simulation by the domain derivation and the regularized Gauss-Newton iterative method. We will study the energy minimization problem for Stokes flow with convective heat transfer in spite of its lack of rigorous mathematical justification in case the Lagrange formulation is not convex. For the numerical solution of the viscous energy minimization problem, we introduce a gradient-type algorithm with mesh adaptation technique, while the partial differential systems are discretized by means of the finite-element method. We propose a gradient-type algorithm for the shape optimization problem, and the examples verify that our method is efficient and useful in the numerical implementations
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