Abstract
This work is concerned with the shape optimal design of an obstacle immersed in the Stokes–Brinkman fluid, which is also coupled with a thermal model in the bounded domain. The shape optimal problem is formulated and analyzed based on the framework of the continuous adjoint method, with the advantage that the computing cost of the gradients and sensitivities is independent of the number of design variables. Then, the velocity method is utilized to describe the domain deformation, and the Eulerian derivative for the cost functional is established by applying the differentiability of a minimax problem based on the function space parametrization technique. Moreover, an iterative algorithm is proposed to optimize the boundary of the obstacle in order to reduce the total dissipation energy. Finally, numerical examples are presented to illustrate the feasibility and effectiveness of our method.
Highlights
Shape Optimal Control Problem in FluidsWe present the general structure to solve the optimal control problems, which will be applied to the particular case of shape optimal problem in Stokes–Brinkman flow with heat transfer
The efficient computation requires a shape calculus which differs from its analog in vector spaces. e traditional approaches always involve the computation of the state derivative with respect to the domain, but the state parameters belong to the function spaces depending on the variable domain
We present the general structure to solve the optimal control problems, which will be applied to the particular case of shape optimal problem in Stokes–Brinkman flow with heat transfer
Summary
We present the general structure to solve the optimal control problems, which will be applied to the particular case of shape optimal problem in Stokes–Brinkman flow with heat transfer . Our work is to minimize a cost functional J which consists of the solution of the state equations: min J J(w, φ),. We need to solve following problem to obtain the optimal solution: seek(w, λ, φ) ∈ W × W × V, such that ∇L(w, λ, φ) 0. We can apply an iterative method to solve the control problem by choosing an initial value for the variable φ0. We compute the state equations and evaluate the cost functional and solve the adjoint equations. We give a suitable stopping criterion and derive the cost functional derivative J′[21, 22]
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