Mathematical Sciences: Duality Theory in Finite Deformation Nonsmooth Mechanics and Numerical Approaches

高9400565 本文以Gao-斯特朗在1988-89年发展的互补理论为基础，对非光滑力学中的对偶理论及其相应的有限差分数值方法进行了较全面的研究。 变形系统 主题将包括： (1)非光滑塑性流动与结构极限分析;（2）板壳的非线性屈曲分析;（3）智能材料与结构的控制问题与优化设计;（4）非光滑优化的互补有限元法。 有限变形非光滑力学在智能结构分析、民用和航空航天工程中有着越来越重要的应用。求解非线性、非光滑方程组的直接方法一直是数学家和工程师的一大难题。 双 逼近与互补有限元 然而，该方法具有很大的优点。我们可以预期，本文的研究结果将为非光滑力学、变分方法、有限变形理论、非线性优化理论和数值方法提供新的思路。

Gao 9400565 Based on the complementarity theory developed by Gao- Strang in 1988-89, the objective of this proposed research is to offer a comprehensive study on the duality theory in nonsmooth mechanics and the related numerical methods for finite deformation systems. Topics will include: (1) Nonsmooth plastic flow and structural limit analysis; (2) Nonlinear buckling analysis for plates and shells; (3) Control problems and optimal design of smart materials and structures; (4) Complementary finite element method for nonsmooth optimization. Finite deformation nonsmooth mechanics has been found more and more important applications in smart structural analysis, civil and aerospace engineering. Direct method for solving nonlinear and nonsmooth systems has always presented serious difficulties for mathematicians and engineers. The dual approach andcomplementary finite element method, however, possess a great of advantages. We can expect that the results of this proposed research will provide new insight to the nonsmooth mechanics, variational methods, finite deformation theory, nonlinear optimization theory and numerical methods.