Mathematical Sciences: Differential Equations of Mechanics and their Numerical Solution

研究人员将研究描述弹性材料行为的偏微分方程组。重点介绍弹性板壳弯曲的Reissner-Mindlin理论和相关模型。这些研究将包括对板方程解的定性分析，对其解的数值格式的设计和分析，以及对Reissner、Mindlin方程和三维弹性方程解之间的关系的分析和计算研究。具体目标包括分析自由和简支边界附近的Reissner-Mindlin模型的边界层，实现该模型的一种关于板厚度均匀的最优阶有限元方法，以及设计和分析高阶一致最优阶有限元方法。此外，研究人员将继续研究数值求解时间相关偏微分方程组的边界元方法，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此类时间相关方程的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.