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Abstract
This dissertation is concerned with numerical solutions of a class of boundary value problems in acoustic, elastic and nonlinear water waves, and consists of two independent parts.
In the first part, we deal with the application of variational methods, including the finite element method, the boundary element method as well as their coupling, to solutions of three specific two-dimensional boundary value problems in acoustics and elastodynamics. To be more precise, we first study the application of finite element methods to the solution of exterior Neumann problems in acoustics. The original problem is reduced to a nonlocal boundary value problem in a bounded domain by introducing an artificial boundary. We employ, respectively, a direct boundary integral equation method and a Foureier series expansion method to define corresponding Dirichlet-to-Neumann mappings on the artificial boundary. Weak formulations for the resulting nonlocal boundary value problems are carefully studied. Thereafter, we employ the boundary element methods to seek solutions of two type of transmission problems in acoustics and fluid-structure interaction, respectively. The original transmission problems are reduced to a system of coupled boundary integral equations. We are interested in their weak formulations. Uniqueness and existence for the weak solutions are carefully investigated in appropriate Sobolev spaces.
For each specific problem, a sequence of numerical tests are implemented to illustrate the accuracy and efficiency of the solution procedures. During these tests, in addition to the standard boundary element method, fast multipole methods are also employed for the numerical treatment of boundary integral equations to be involved.
In the second part, we present an accurate and efficient numerical model for the simulation of fully nonlinear, three-dimensional surface water waves on infinite or finite depth. The numerical method is based on the reduction of the problem to a lower-dimensional Hamiltonian system involving surface quantities alone. This is accomplished by introducing an Dirichlet-to-Neumann mapping which is represented in terms of its Taylor series expansion in homogeneous powers of the surface elevation. The validity of the model and the efficiency of the method are illustrated by simulating the long-time evolution of two-dimensional steadily progressing waves, as well as the development of three-dimensional (short-crested) nonlinear waves, both in deep and shallow water.
