Highly Accurate And Efficient Shape Gradient Algorithms For Shape Optimization In Fluids

Shape design and control in fluid flows are widely used in engineering,such as design of aircraft wings,arterial bypass in hemodynamics,etc.With the rapid develop-ment of computational fluid dynamics and the improvement of computer performance in the past decade,shape design in fluid flows has currently become an important re-search field of computational fluid dynamics.This thesis studies algorithms with high accuracy and efficiency for shape optimization problems arising from steady,viscous,and incompressible Stokes flows,Navier-Stokes flows,and Stokes eigenvalues.We propose distributed and boundary corrected Eulerian derivatives to improve numerical accuracy of the corresponding discrete shape gradients and numerical performance of optimization algorithms.In Chapter 1,we introduce research background of shape optimization and related mathematical definitions.The contents of the thesis can be organized as follows.In Chapter 2,we present convergence analysis of mixed finite element approxi-mations to the shape gradient flow in Stokes flows.In recent years,more and more evidence has shown that distributed Eulerian derivatives have significant advantages in accuracy over boundary Eulerian derivatives.We find that for two types of shape op-timization problems:shape reconstruction and energy dissipation,distributed Eulerian derivatives have higher accuracy,require less domain smoothness,and thus applicable to more general domains.We present error estimates of the shape gradient in general spaces and show a priori error estimates of the distributed and boundary types of shape gradient flows.Both mathematical theory and numerical experiments show that the dis-tributed shape gradient flow has better accuracy than boundary type of gradient flow.In Chapter 3,we consider shape optimization of nonlinear Navier-Stokes flows.The Stokes operator is introduced to analyze the errors of the adjoint equation in energy dissipation problems of Navier-Stokes flows.By presenting error estimates of mixed finite element approximations to distributed and boundary types of shape gradients for studying performance of the shape gradient algorithms.Theoretical analysis and numer-ical experimental results show that the distributed shape gradient algorithm has better convergence and robustness for shape optimization.In Chapter 4,we study the Stokes eigenvalue optimization problem.We derive distributed and boundary types of Eulerian derivatives by shape sensitivity analysis.We prove that finite element approximations to distributed shape gradients have better accuracy and present numerical verification.In Chapter 5,we combine the two-grid algorithm with the distributed Eulerian derivative to design optimization algorithms and improve computational efficiency.For shape optimization algorithms,the cost of solving the nonlinear partial differential equa-tion each time in the iterative process is high.Two-grid method is an effective and fast algorithm for dealing with nonlinearities.The error estimate is asymptotically optimal in the energy norm.We use two-grid algorithms in shape optimization of Navier-Stokes flows and Stokes eigenvalues.We analyze error estimates of the distributed shape gra-dient by a two-grid algorithm for the energy dissipation in shape optimization of Navier-Stokes flows.Numerical examples in 2D and 3D are given for the two-grid based op-timization algorithms used in Stokes eigenvalue optimization and shape optimization of Navier-Stokes flow.The efficiency of two-grid method for solving Navier-Stokes equations is further accelerated by introducing the Uzawa iteration method.In order to improve the quality of the meshes during the deformation process,conformal mapping is introduced in the gradient flow to ensure that the shapes of triangular elements remain unchanged during the deformation process.In Chapter 6,to correct the low accuracy and the discontinuity of the traditional dis-cretized boundary-type shape gradient,we propose two continuity-preserving methods using the boundary correction and the gradient recovery technique for the shape opti-mization problems under the Dirichlet boundary condition and the Neumann boundary condition,respectively.The shape gradient by the boundary correction method im-proves the numerical accuracy,and theoretically possesses almost the same accuracy as the distributed shape gradient.We apply the boundary corrected shape gradient to shape optimization of the Laplace eigenvalue problem,the Stokes eigenvalue problem,the Stokes/Navier-Stokes flow,and the interface identification problem.Through nu-merical experiments,we find that the shape optimization algorithm with the proposed scalar-decomposed H~1gradient flow combined with the boundary correction method has highly computational efficiency.In Chapter 7,we propose a level set method to solve the Stokes eigenvalue op-timization problem.Firstly,a relaxed approach is adopted to approximate the Stokes eigenvalue problem,and then transform the original shape optimization problem into a topology optimization model.The distributed shape gradient is then applied in the level set method based on the transport equation.Both single-grid and two-grid algo-rithms are proposed for the relaxed model.During the optimization process,an accu-racy reliable and asymptotically optimal two-grid mixed finite element discrete scheme is introduced to improve the efficiency of the Stokes eigenvalue solver and thus save computational costs of the entire optimization process.Two and three-dimensional nu-merical results show fastness and effectiveness of the algorithms.Finally,we make a summary of the thesis and outlook future research directions.