Shape optimization in the Navier‐Stokes flow with thermal effects

Abstract

In this article, we consider the shape optimization problem of a body immersed in the incompressible fluid governed by Navier-Stokes equations coupling with a thermal model. Based on the continuous adjoint method, we formulate and analyze the shape optimization problem. Then we derive the structure of shape gradient for the cost functional by using the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique. Moreover, we present a gradient-type algorithm to the shape optimization problem. Finally, numerical examples demonstrate the feasibility and effectiveness of the proposed algorithm.

Analysis of a System of Hemivariational Inequalities Arising in Non-Stationary Stokes Equation with Thermal Effects

“…Therefore, numerous studies focus on the fluid problems with thermal effects -cf. [2,20,21,27]. In these papers, either Dirichlet or Neumann boundary conditions are considered, so that all the models lead to a system of equations.…”

Shape inverse problem of thermodynamic equations based on domain derivative method

“…Yan et al recovered the shape of a solid in the incompressible fluid driven by the Stokes flow [9]. Based on the continuous adjoint method, Yan et al solved the shape optimization problem in the incompressible fluid governed by Navier-Stokes equations coupling with a thermal model in [10]. They derived the structure of shape gradient for the cost functional by employing the differentiability of a minimax formulation involving a Lagrange functional with the function space parametrization technique and proposed a gradient-type algorithm to the optimal control problem.…”

Shape Identification for Stokes-Oseen Problem Based on Domain Derivative Method

“…Harbrecht and Tausch considered the numerical solution of a shape identification problem for the heat equation [7]. Yan et al recovered the shape of a solid in the incompressible fluid driven by the Stokes flow [8], and considered the shape optimization problem of a body immersed in the incompressible fluid governed by Navier-Stokes equations coupling with a thermal model in [9].…”