The application of adjoint method for shape optimization in Stokes–Oseen flow

This paper presents a numerical method for shape optimization of a body immersed in an incompressible viscous flow governed by Stokes–Oseen equations. The purpose of this work is to optimize the shape that minimizes a given cost functional. Based on the continuous adjoint method, the shape gradient of the cost functional is derived by involving a Lagrangian functional with the function space parametrization technique. Then, a gradient‐type algorithm is applied to the shape optimization problem. The numerical examples indicate the proposed algorithm is feasible and effective in low Reynolds number flow. Copyright © 2016 John Wiley & Sons, Ltd.     

Shape Hessian for generalized Oseen flow by differentiability of a minimax: A Lagrangian approach

The goal of this paper is to compute the shape Hessian for a generalized Oseen problem with nonhomogeneous Dirichlet boundary condition by the velocity method. The incompressibility will be treated by penalty approach. The structure of the shape gradient and shape Hessian with respect to the shape of the variable domain for a given cost functional are established by an application of the Lagrangian method with function space embedding technique. This work was supported by the National Natural Science Fund of China (No. 10371096) for ZM Gao and YC Ma.     

A numerical method for the viscous incompressible Oseen flow in shape reconstruction

This paper is concerned with the shape reconstruction of a bounded domain with a viscous incompressible fluid driven by the Oseen equations. For the approximate solution of the ill-posed and nonlinear problem we propose a regularized Gauss-Newton method. A theoretical foundation for the method is given by establishing the differentiability of the boundary value problem with respect to the boundary in the sense of the domain derivative. The results of several numerical experiments show that our theory is useful for practical purpose, and the proposed algorithm is feasible.     

Oseen Coupling Method for the Exterior Flow Part I: Oseen Coupling Approximation

Abstract In this paper, we consider the bidimensional exterior unsteady Navier-Stokes equations with nonhomogeneous boundary conditions and present an Oseen coupling problem which approximates the Navier-Stokes problem, obtained by coupling the Navier-Stokes equations in the inner region and the Oseen equations in the outer region. Moreover, we prove the existence, uniqueness and the approximate accuracy of the weak solution of the Oseen coupling equations. Project supported by NSF of China & State Major Key Project of Basic Research     

Shape optimization for Stokes flow

This paper is concerned with a shape sensitivity analysis of a viscous incompressible fluid driven by Stokes equations. The structures of continuous shape gradients with respect to the shape of the variable domain for some given cost functionals are established by introducing the Piola transformation and then deriving the state derivative and its associated adjoint state. Finally we give the finite element approximation of the problem and a gradient type algorithm is effectively used for our problem.     

外部流动的Oseen耦合方法,I:Oseen耦合逼近

这篇文章考虑了具有非齐次边界条件的二维非定常外部Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程．通过将内部区域的Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ方程和外部区域的Ｏｓｅｅｎ方程相耦合,得到了Ｎａｖｉｅｒ－Ｓｔｏｋｅｓ问题的逼近问题： Ｏｓｅｅｎ耦合问题,此外,我们证明了 Ｏｓｅｅｎ耦合方程弱解的存在性,唯一性和收敛精度．     

Oseen coupling method for the exterior flow. Part II: Well‐posedness analysis

In this paper, we recall the Oseen coupling method for solving the exterior unsteady Navier–Stokes equations with the non‐homogeneous boundary conditions. Moreover, we derive the coupling variational formulation of the Oseen coupling problem by using of the integral representations of the solution of the Oseen equations at an infinity domain. Finally, we provide some properties of the integral operators over the artificial boundary and the well‐posedness of the coupling variational formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

First‐order system least squares for the Oseen equations

Following earlier work for Stokes equations, a least squares functional is developed for two‐ and three‐dimensional Oseen equations. By introducing a velocity flux variable and associated curl and trace equations, ellipticity is established in an appropriate product norm. The form of Oseen equations examined here is obtained by linearizing the incompressible Navier–Stokes equations. An algorithm is presented for approximately solving steady‐state, incompressible Navier–Stokes equations with a nested iteration‐Newton‐FOSLS‐AMG iterative scheme, which involves solving a sequence of Oseen equations. Some numerical results for Kovasznay flow are provided. Copyright © 2006 John Wiley & Sons, Ltd.     

The local discontinuous Galerkin method for the Oseen equations

We introduce and analyze the local discontinuous Galerkin method for the Oseen equations of incompressible fluid flow. For a class of shape-regular meshes with hanging nodes, we derive optimal a priori estimates for the errors in the velocity and the pressure in - and negative-order norms. Numerical experiments are presented which verify these theoretical results and show that the method performs well for a wide range of Reynolds numbers.

     

Drag minimization for Navier–Stokes Flow

This paper investigates the drag minimization in a two‐dimensional flow which is governed by a nonhomogeneous Navier–Stokes equations. Two approaches are utilized to derive shape gradient of the cost functional. The first one is to use the shape derivative of the fluid state and its associated adjoint state; the second one is to utilize the differentiability of a minimax formulation involving a Lagrange functional with a function space parametrization technique. Finally, a gradient type algorithm is effectively formulated and implemented for the mentioned drag minimization problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

A weighted Lq‐approach to Oseen flow around a rotating body

We study time-periodic Oseen flows past a rotating body in ℝ³ proving weighted a priori estimates in L^q-spaces using Muckenhoupt weights. After a time-dependent change of coordinates the problem is reduced to a stationary Oseen equation with the additional terms (ω ∧ x) ⋅ ∇ u and −ω ∧ u in the equation of momentum where ω denotes the angular velocity. Due to the asymmetry of Oseen flow and to describe its wake we use anisotropic Muckenhoupt weights, a weighted theory of Littlewood–Paley decomposition and of maximal operators as well as one-sided univariate weights, one-sided maximal operators and a new version of Jones' factorization theorem. Copyright © 2007 John Wiley & Sons, Ltd.     

Shape optimization in two-dimensional viscous compressible fluids

We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids in two space dimensions. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier-Stokes system for a general viscous barotropic fluid with the pressure satisfying p（o） = aQlog^d（o） for large Q. Here d 〉 1 and a 〉 0.     

Shape optimization of materially non-linear bodies in contact

Optimal shape design problem for a deformable body in contact with a rigid foundation is studied. The body is made from material obeying a nonlinear Hooke's law. We study the existence of an optimal shape as well as its approximation with the finite element method. Practical realization with nonlinear programming is discussed. A numerical example is included.     

Lq‐estimates for the stationary Oseen equations on the exterior of a rotating obstacle

We study the Dirichlet problem for the stationary Oseen equations around a rotating body in an exterior domain. Our main results are the existence and uniqueness of weak and very weak solutions satisfying appropriate L^q‐estimates. The uniqueness of very weak solutions is shown by the method of cut‐off functions with an anisotropic decay. Then our existence result for very weak solutions is deduced by a duality argument from the existence and estimates of strong solutions. From this and interior regularity of very weak solutions, we finally establish the complete D^1,r‐result for weak solutions of the Oseen equations around a rotating body in an exterior domain, where 4/3<r <4. Here, D^1,r is the homogeneous Sobolev space.     

Compactness and dichotomy in nonlocal shape optimization

We prove a general result about the behaviour of minimizing sequences for nonlocal shape functionals satisfying suitable structural assumptions. Typical examples include functions of the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions. Exploiting a nonlocal version of Lions' concentration-compactness principle, we prove that either an optimal shape exists or there exists a minimizing sequence consisting of two “pieces” whose mutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur in the local case.     

The method of fundamental solutions for the Oseen steady‐state viscous flow past obstacles of known or unknown shapes

In this paper, the steady‐state Oseen viscous flow equations past a known or unknown obstacle are solved numerically using the method of fundamental solutions (MFS), which is free of meshes, singularities, and numerical integrations. The direct problem is linear and well‐posed, whereas the inverse problem is nonlinear and ill‐posed. For the direct problem, the MFS computations of the fluid flow characteristics (velocity, pressure, drag, and lift coefficients) are in very good agreement with the previously published results obtained using other methods for the Oseen flow past circular and elliptic cylinders, as well as past two circular cylinders. In the inverse obstacle problem the boundary data and the internal measurement of the fluid velocity are minimized using the MATLAB^© optimization toolbox lsqnonlin routine. Regularization was found necessary in the case the measured data are contaminated with noise. Numerical results show accurate and stable reconstructions of various star‐shaped obstacles of circular, bean, or peanut cross‐section.     

Shape reconstruction for the time‐dependent Navier‐Stokes flow

This article deals with the shape reconstruction of a bounded domain with a viscous incompressible fluid driven by the time‐dependent Navier‐Stokes equations. For the approximate solution of the ill‐posed and nonlinear problem we propose a regularized Newton method. A theoretical foundation for the Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of the domain derivative. Numerical examples indicate the feasibility of our method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Numerical Gradients for Shape Optimization Based on Embedding Domain Techniques

An embedding domain technique is proposed to characterize the gradient of shape optimization problems. A discussion of the numerical realization of the arising saddle point problems is given and numerical feasibility of the gradient information is discussed.     

Contact shape optimization based on the reciprocal variational formulation

The paper deals with a class of optimal shape design problems for elastic bodies unilaterally supported by a rigid foundation. Cost and constraint functionals defining the problem depend on contact stresses, i.e. their control is of primal interest. To this end, the so-called reciprocal variational formulation of contact problems making it possible to approximate directly the contact stresses is used. The existence and approximation results are established. The sensitivity analysis is carried out.