Parallel Domain Decomposition Algorithms For Shape Optimization Problems

Shape optimization, or optimal shape design, aims to optimize an objective function by changing the shape of the computational domain. It arises in a variety of engineering applications, e.g., aerodynamic shape design, artery bypass design, microfluidic biochip design and so on. Most of these optimization problems have constraints imposed by partial differential equations or other physical and geo-metrical conditions. They are considerably more difficult and expensive to solve than the corresponding simulation problems, and often require large scale parallel computers for their memory capacity and processing speed. The objective of this thesis is to study the parallel algorithm for shape optimization problems. A new parallel one-shot Lagrange-Newton-Krylov-Schwarz (LNKSz) algorithm for shape optimization discretized by finite element methods on unstructured moving mesh-es was introduced in this thesis. Most existing algorithms for shape optimization problems solve iteratively the three components of the optimality system:the state equations for the constraints, the adjoint equations for the Lagrange multipliers, and the design equations for the shape parameters. Such approaches are relative-ly easy to implement, but generally not easy to converge as they are basically nonlinear Gauss-Seidel algorithms with three large blocks. In this thesis, a fully coupled, or the so-called one-shot, approach which solves the three components simultaneously was introduced. First, a moving mesh finite element method for the shape optimization problems was introduced, in which the mesh equations are implicitly coupled with the optimization problems. Second, a LNKSz framework based on an overlapping domain decomposition method for solving the fully cou-pled problem was introduced. Such an approach doesn’t involve any sequential steps that are necessary for the Gauss-Seidel type reduced space methods. The main challenges in full space approaches are that the corresponding nonlinear sys-tem is much harder to solve because it is two to three times larger and its indefinite Jacobian problems are also much more ill-conditioned. Effective preconditioning becomes the most important component of the method. We need to design pre-conditioners which can substantially reduce the condition number of the large fully coupled system and, at the same time, provide the scalability for parallel comput-ing. The one-level and two-level restricted Schwarz preconditioners are introduced for shape optimization problems in this thesis. Due to the pollution effects of the coarse to fine interpolation, direct extensions of the one-level method to two-level do not work. Then a pollution removing coarse to fine interpolation scheme was introduced in this thesis. As an application, we consider the shape optimization of a cannula problem and an artery bypass problem in2D. Numerically, we show that LNKSz deals with these challenges quite well. Numerical experiments show that our algorithms perform well a supercomputer with a thousand of processors.As the complexity of the shape optimization problems considered in Chapter1to Chapter4, there is no convergence and convergence rate estimate for the algorithms yet. In the last Chapter, we try to do some elementary researches on the convergence and convergence rate estimate. We first simplify the problems and just consider a simplified nonlinear optimization problem. A new subspace correction method is constructed for it and the convergence as well as a convergence rate estimate for the algorithm are obtained under some reasonable assumptions.