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Computational methods for a class of problems in acoustic, elastic and water waves.

机译：针对声波，弹性波和水波中的一类问题的计算方法。

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摘要

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.
机译：本文涉及一类声波，弹性波和非线性水波边值问题的数值解，它由两个独立的部分组成。第一部分研究了变分法的应用，包括有限元法。 ，边界元法及其耦合，以解决声学和弹性力学中三个特定的二维边界值问题。更精确地说，我们首先研究有限元方法在声学中解决外部诺伊曼问题的应用。通过引入人工边界将原始问题简化为有界域中的非局部边界值问题。我们分别采用直接边界积分方程法和傅里叶级数展开法在人工边界上定义相应的Dirichlet到Neumann映射。仔细研究了导致非局部边界值问题的弱公式。此后，我们采用边界元方法分别寻求声学和流体-结构相互作用中两种传输问题的解决方案。原始的传递问题被简化为耦合边界积分方程组。我们对他们的拙劣表述很感兴趣。在适当的Sobolev空间中仔细研究了弱解的唯一性和存在性。针对每个特定问题，进行了一系列数值测试，以说明求解过程的准确性和效率。在这些测试中，除标准边界元法外，还采用快速多极子法对涉及的边界积分方程进行数值处理。第二部分，我们提供了一种精确有效的数值模型，用于全面模拟在无限或有限深度上的非线性三维地面水波。数值方法基于将问题简化为仅涉及表面量的低维哈密顿系统。这是通过引入Dirichlet到Neumann映射来完成的，该映射以表面高度的均方次幂中的Taylor级数展开表示。通过模拟二维稳步前进波的长期演化以及深，浅层三维（短波）非线性波的发展，说明了模型的有效性和方法的有效性。水。

著录项

  • 作者

    Xu, Liwei.;

  • 作者单位

    University of Delaware.;

  • 授予单位 University of Delaware.;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 2009
  • 页码 202 p.
  • 总页数 202
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 数学;
  • 关键词

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