解三维摩擦接触问题的一个二阶锥线性互补法

针对三维摩擦接触问题的求解，给出了一种基于参变量变分原理的二阶锥线性互补法. 首先， 基于三维Coulomb摩擦锥在数学表述上属于二阶锥的事实，利用二阶锥规划对偶理论，建立 了三维Coulomb摩擦接触条件的参变量二阶锥线性互补模型，它是二维Coulomb摩擦接触条 件参变量线性互补模型在三维情形下的自然推广；随后，利用参变量变分原理与有限元方法， 建立了求解三维摩擦接触问题的二阶锥线性互补法. 较之于将三维Coulomb摩擦锥进行显式 线性化的线性互补法，该方法无需对三维Coulomb摩擦锥进行线性化，因而在保证精度的 前提下所解问题的规模要小很多. 最后通过算例展示了该方法的特点.

A SECOND-ORDER CONE LINEAR COMPLEMENTARITY APPROACH FOR THREE-DIMENSIONAL FRICTIONAL CONTACT PROBLEMS

Frictional contact problems frequently arise in various engineering applications, but their solutions, especially the solutions of three dimensional (3D) frictional contact problems, are challenging since the conditions for contact and friction are highly nonlinear and non-smooth. The 3D frictional contact problem is nonlinear and non-differentiable at least in three aspects: 1) The unilateral contact law, combining a geometric condition of impenetrability, a static condition of no-tension and an energy condition of complementarity, is represented by a multi-valued force-displacement relation. 2) The friction law, governed by a relation between reaction force and local relative velocity, is also multi-valued. 3) The Coulomb friction law in 3D space is expressed as a nonlinear inequality that is non-differentiable in the ordinary sense. In this paper, we propose a new linear second-order cone complementarity formulation for the numerical finite element analysis of 3D frictional contact problem by using the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the unilateral contact law, and utilize a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. We reformulate the governing equations of the 3D frictional contact problem as a linear second-order cone complementarity problem (SOCCP) via the parametric variational principle and the finite element method. Compared with the linear complementarity formulation of 3D frictional contact problems, the proposal SOCCP formulation avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. A semismooth Newton method is used to solve the obtained linear SOCCP. Numerical examples are computed and the results confirm the effectiveness and robustness of the SOCCP formulation developed.