Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier-Stokes equations with mixed-boundary conditions in a channel

Differentiability results and sensitivity calculation for optimal control of incompressible two-phase Navier-Stokes equations with surface tension

We analyze optimal control problems for two-phase Navier-Stokes equations with surface tension. Based on $L_p$-maximal regularity of the underlying linear problem and recent well-posedness results of the problem for sufficiently small data we show the differentiability of the solution with respect to initial and distributed controls for appropriate spaces resulting form the $L_p$-maximal regularity setting. We consider first a formulation where the interface is transformed to a hyperplane. Then we deduce differentiability results for the solution in the physical coordinates. Finally, we state …

Navier-Stokes流的形状Optimization

Shape Optimization for Navier-Stokes Flow

本文研究了定常非齐次Navier-斯托克斯方程驱动的牛顿粘性不可压缩流体的最优形状设计问题。我们使用三种方法来获得一些给定的成本泛函的形状梯度的结构。第一个是使用Piola变换，并推导出状态导数及其相关的伴随状态;第二个是使用一个包含拉格朗日泛函的极小极大公式的可微性与函数空间参数化技术;最后一个是使用一个极小极大公式的可微性与函数空间嵌入技术。最后，我们将梯度型算法应用到我们的问题中，数值例子表明我们的理论是有用的，所提出的算法是可行的。

二维约束 Navier-Stokes 方程的最优控制

Optimal control of the 2D constrained Navier-Stokes equations

本文研究能量守恒约束下的二维 Navier-Stokes 方程及其最优控制，利用 Galerkin 逼近和形式 Lagrange 方法证明解的适定性、解映射性质与一阶必要最优性条件。

Shape sensitivity of a 2D Fluid-Structure Interaction problem between a viscous incompressible fluid and an incompressible elastic structure

We study the shape differentiability of a general functional depending on the solution of a bidimensional stationary Stokes-Elasticity system, with respect to the reference domain of the elastic structure immersed in a viscous fluid. The differentiability with respect to reference elastic domain variations are considered under shape perturbations with diffeomorphisms. The shape-derivative is then calculated with the use of the velocity method. This derivative involves the material derivatives of the solution of this Fluid-Structure Interaction problem. The adjoint method is used to obtain a si…

Topological optimization and minimal compliance in linear elasticity

We investigate a fixed domain approach in shape optimization, using a regularization of the Heaviside function both in the cost functional and in the state system. We consider the compliance minimization problem in linear elasticity, a well known application in this area of research. The optimal design problem is approached by an optimal control problem defined in a prescribed domain including all the admissible unknown domains. This approximating optimization problem has good differentiability properties and a gradient algorithm can be applied. Moreover, the paper also includes several numeri…

点wise跟踪最优控制问题对于Cahn-Hilliard Navier-Stokes系统

Pointwise Tracking Optimal Control Problem for Cahn Hilliard Navier Stokes system

本文研究了Cahn-Hilliard Navier-Stokes系统的点wise跟踪最优控制问题，分析了控制量对相变量的影响，并推导了伴随系统的一阶必要最优性条件。