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解三维摩擦接触问题的一个二阶锥线性互补法

A second-order cone linear complementarity approach for three-dimensional frictional contact problems

  • 摘要: 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.

     

    Abstract: Frictional contact problems frequently arise in variousengineering applications, but their solutions, especially the solutions ofthree dimensional (3D) frictional contact problems, are challenging sincethe conditions for contact and friction are highly nonlinear and non-smooth.The 3D frictional contact problem is nonlinear and non-differentiable atleast in three aspects: 1) The unilateral contact law, combining a geometriccondition of impenetrability, a static condition of no-tension and an energycondition of complementarity, is represented by a multi-valuedforce-displacement relation. 2) The friction law, governed by a relationbetween reaction force and local relative velocity, is also multi-valued. 3)The Coulomb friction law in 3D space is expressed as a nonlinear inequalitythat is non-differentiable in the ordinary sense. In this paper, we proposea new linear second-order cone complementarity formulation for the numericalfinite element analysis of 3D frictional contact problem by using theparametric variational principle. Specifically, we develop a regularizationtechnique to resolve the multi-valued difficulty involved in the unilateralcontact law, and utilize a second-order cone complementarity condition tohandle the regularized Coulomb friction law in contact analysis. Wereformulate the governing equations of the 3D frictional contact problem asa linear second-order cone complementarity problem (SOCCP) via theparametric variational principle and the finite element method. Comparedwith the linear complementarity formulation of 3D frictional contactproblems, the proposal SOCCP formulation avoids the polyhedral approximationto the Coulomb friction cone so that the problem to be solved has muchsmaller size and the solution has better accuracy. A semismooth Newtonmethod is used to solve the obtained linear SOCCP. Numerical examples arecomputed and the results confirm the effectiveness and robustness of theSOCCP formulation developed.

     

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