Mathematical Sciences: Differential Equations of Mechanics and their Numerical Solution

研究者将研究描述弹性材料行为的偏微分方程组。特别强调的是赖斯纳。弹性板壳弯曲的Mindlin理论及相关模型。研究将包括板方程解的定性分析，其解的数值格式的设计和分析，以及Reissner解之间关系的分析和计算研究。Mindlin方程和三维弹性方程。具体目标包括对赖斯纳的边界层进行分析。明林模型接近自由和简单。支持边，对该模型实施了最近发展的相对于板厚均匀的最优阶有限元方法，并对更高程度的均匀最优阶有限元方法进行了设计和分析。此外，研究者将继续研究时变偏微分方程数值解的边界元方法，重点是将最新的理论进展应用于实际计算，并将研究边界元方法在时变偏微分方程精确控制问题中的应用。

The investigator will study systems of partial differential equations which describe the behavior of elastic materials. Particular emphasis will be on the Reissner.Mindlin theory and related models for the bending of elastic plates and shells. The investigations will include a qualitative analysis of the solutions of the plate equations, design and analysis of numerical schemes for their solution, and both analytical and computational studies of the relation between solutions of the Reissner.Mindlin equations and the equations of three dimensional elasticity. Specific goals include an analysis of the boundary layer for the Reissner.Mindlin model near free and simply.supported edges, the implementation of a recently developed optimal order finite element method for this model which is uniform with respect to the plate thickness, and the design and analysis of uniformly optimal order finite element methods of higher degree. In addition the investigator will continue work on boundary element methods for the numerical solution of time.dependent partial differential equations, with an emphasis on applying recent theoretical advances to practical computation, and will study applications of the boundary element approach to exact control problems for such time.dependent equations.