接触力学中的非凸非光滑变分不等式问题研究

This project aims to study deeply several classes of variational inequalities in contact Mechanics, which involve nonconvexity and nonsmoothness. The main tasks of this project include (i) to build up several new mathematical models of variational inequalities with nonconvexity and nonsmoothness for frictional contact problems with different materials by using different constitutive laws, frictional conditions and contact conditions; (ii) to study properties of solutions, such as existence, uniqueness, regularity, and stability etc., for the variational inequalities with nonconvexity and nonsmoothness in the framework of variational inequality and nonsmooth analysis theory; (iii) to design approximation algorithms to solve the variational inequalities with nonconvexity and nonsmoothness by using the methods of error bound and discrete scheme, and to prove convergence results for the algorithms designed; (iv) to study well-possedness for the variational inequalities with nonconvexity and nonsmoothness, such as the metric characterizations for well-posedness and the equivalence between well-posedness and existence and uniqueness of their solution. The study of the present project could not only enrich and develop theory, methods, skills and algorithms for variational inequality, nonconvex optimization and nonsmooth analysis, but also be of great importance for both theory and application to many practical problems in Mechanics and Engineering.

本项目针对产生于接触力学中几类重要的非凸非光滑变分不等式问题进行系统和深入的研究。利用不同的本构定律和摩擦接触条件，建立涉及不同物理材料摩擦接触问题的几类非凸非光滑变分不等式新模型；在变分不等式和非光滑分析理论框架下，研究这些非凸非光滑变分不等式问题解的存在性、唯一性、正则性及稳定性等解的刻画性质；运用误差界和离散技巧等方法，设计逼近这些非凸非光滑变分不等式问题解的算法并证明算法的收敛性；研究这些非凸非光滑变分不等式问题的适定性，刻画适定性的度量性质，探讨适定性与解的存在唯一性之间的等价关系。本项目的研究不仅可以丰富和发展变分不等式、非凸优化和非光滑分析等相关学科的理论、方法、技巧和算法，而且可以用于解决力学和工程科学中的一些实际问题，具有重要的理论意义和实际应用价值。

项目针对接触力学中几类非凸非光滑变分不等式问题进行系统和深入的研究，主要研究成果包括：提出了接触力学中接触问题的非凸非光滑变分不等式新模型；获得了相关问题的可解性条件及解性质的刻画；设计了相关问题解的近似算法并研究了算法的收敛性分析；推广了相关问题适定性的研究；应用获得的研究成果于接触力学中的接触问题。本项目的研究不仅丰富和发展了变分不等式、非凸优化和非光滑分析等相关学科的理论、方法、技巧和算法，而且可以用于解决力学和工程科学中的一些实际问题，具有重要的理论意义和实际应用价值。