Research On Fictitious Domain Methods And Applications

The fictitious domain methods are efficient numerical methods for partial differential equations (PDES). The basic idea of these methods is to embed the original domain and problem into a large simply shaped domain where regular mesh and a more efficient solution method such as a fast direct solver for PDE can be used. Now popular fictitious domain methods mainly includ: boundary supported Lagrange multipliers method and immersed boundary method. The fictitious domain methods including their theories, algorithms and applications are studied in this thesis. The main results include:Theoretically, we analyze the existence of the approximate solutions of two-dimensional heat equation with immersed boundary method. We obtain the existence theorem of the solutions via finite element method, the Banach Fixed Point Theorem and a theorem in nonlinear ordinary differential equations in abstract spaces. We further analyze the existence of the approximate solutions of the two-dimensional Navier-Stokes equation with immersed boundary method. Because the source term in the Navier-Stokes equations involves a Dirac function and describes the elasity reaction of the immersed boundary, the problem is highly nonlinear and presents several difficulties related with the lack of regularity of the solutions of the Navier-Stokes equations due to such source term. No rigorous analysis of the problem was available so far. We obtain the existence theorem of the solutions via the Banach Fixed Point Theorem and the properties of the solutions to the Navier-Stokes equations of viscous incompressible fluids with space-periodic boundary conditions.The results perfect mathematical theory of the immersed boundary methods.In the aspects of numerical simulzations of the fictitious domain methods, we do some research as below:First, we outline the immersed boundary method in detail and do some numerical experiments; we use the technique of the bourndary approach of the immersed bourndary method to improve the boundary supported Lagrange multipliers fictitious domain method. The new method efficiently avoids the difficulty of computing integlral on the irregular mesh. It has a simpler structure and is easily programmable. Numerical simulation of two-dimensional incompressible inviscid uniform flows over a circular cylinder validates the methodology and the numerical procedure.Secondly, we give the method that introduces fictitious boundary condition to controll fictitious flows inside the body for the solution of symmetric and nozzle flows. We analyze the singularities near the corner of bodies like the trailing edge of an airfoil occurring in using fictitious domain method with Lagrange multipliers. Fictitious boundary conditions for controlling fictitious flows inside the body are presented in order to deal with the singularities. Such approach allows the use of fairly structured mesh such that the fast solvers can be used on it. Numerical results of fictitious domain method for solutions of incompressible inviscid flows around a symmetric airfoil on a decomposed domain are successfully obtained. A new application to flows in a nozzle is also presented. Numerical experiments show that the use of fictitious boundary conditions on a decomposed domain has good effect on controlling the magnitude of Lagrange multipliers.Then, we incorporate the fictitious method into POD framework for inverse design problems. This reduced order modeling approach solves basic functions by fictitious method. The"samples"are solved on the same regular mesh of the fictitious domain after disturbing the shape of the body. The whole mesh need not be regenerated for different configurations. Then the reduced model can facilitate inverse design problems. The method raises the speed and reduces the error of the algorithm. It is easily programmable. Numerical experiment of this reduced order modeling approach for uniform flows around bodies is presented. It is shown that the reduced model is efficient for inverse design problems.Finally, we discuss the numerical simulation of the scattering problems. We apply controllability techniques combining the developed fictitious domain methods to calculate solutions for wave propagation ploblem. Compared to the original one, it has a simpler structure and less computing cost. We deduce the computing formulations and give the detailed algorithm.