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 引用本文: 万军 唐国金 李道奎. 弹塑性摩擦接触问题形状设计灵敏度分析[J]. 力学学报, 2009, 41(4): 503-517.
Wan Jun, Guo-jin Tang, Dao-kui Li. Shape design sensitivity analysis of elastoplastic frictional contact problems with finite deformation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(4): 503-517.
 Citation: Wan Jun, Guo-jin Tang, Dao-kui Li. Shape design sensitivity analysis of elastoplastic frictional contact problems with finite deformation[J]. Chinese Journal of Theoretical and Applied Mechanics, 2009, 41(4): 503-517.

## Shape design sensitivity analysis of elastoplastic frictional contact problems with finite deformation

• 摘要: 提出了一种计算二维有限变形弹塑性摩擦接触问题形状设计灵敏度的算法. 采用主动集策略和mortar方法处理接触边线上的约束条件. 在mortar接触边线的切线和法线方向上采用相同的名义罚函数，提出基于名义罚函数的移动摩擦锥算法来正则化接触约束条件，发展了一种新的二维多体有限变形摩擦接触算法. 在此基础上, 通过将离散形式的摩擦接触问题控制方程对形状设计变量微分，得到了该路径相关问题的直接微分法解析设计灵敏度计算格式, 其节点位移灵敏度方程在每个增量步不用迭代、直接求解. 与国际上现有的二维多体有限变形摩擦接触问题的解析设计灵敏度算法相比，本算法不需分解为法向和切向推导，表达式较简洁，便于编程实现. 数值算例验证了算法的精度和有效性.

Abstract: A new shape design sensitivity analysis algorithm oftwo-dimensional multi-body elastoplastic frictional contact problems withfinite deformation was presented in this paper. In the direct analysis ofcontact problems, the variational inequality of contact constraints wereanalyzed with the active set strategies, and the contact interface wasdiscretized by the mortar method. The same nominal penalty parameters wereadopted in the normal and tangential directions of mortar surface'ssegments, and the normal and tangential contact conditions were regularizedby the moving friction cone algorithm based on the nominal penaltyformulation. A new two-dimensional multi-body finite deformation frictionalcontact algorithm was proposed, and the algorithm could inherit the advantagesof the moving friction algorithm and mortar method. In the shape designsensitivity analysis of contact problems, the incremental (path-dependent)sensitivity problem was derived by the direct differentiation of thediscretized equations governing the direct problem. The shape designsensitivity equation was linear and could be solved at each increment stepwithout iterations. In contrast to the classical shape design sensitivityalgorithm, normal and tangential directions was not to be divided in thepresent algorithm and its formulation was more concise to program. Numericalexamples were presented to illustrate the accuracy and efficiency of thisapproach.

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