This paper presents a cell-based smoothed finite element method (CS-FEM) for solving two-dimensional contact problems with the bi-potential formulation. The contact force and the relative displacements on the contact surface are coupled with each other. The Uzawa algorithm, which is a local iterative technique, is used to solve the contact force. The classic Coulomb friction rule and a unilateral contact relationship are considered. There is no need to select any user-defined parameter in the whole process. Three contact states are investigated accurately, which can be stated as sticking, separating and sliding, respectively. The CS-FEM is performed with six different kinds of smoothing domains which are constructed by dividing the background element into different regions. Only boundary integrations instead of domain integral are required in the calculation, and no coordinate mapping is needed. Three numerical examples are presented to verify the effectiveness of the method. The effect of the friction coefficient for the contact is also investigated. All the obtained numerical solutions agree well with the reference values. The results produced by the CS-FEM are more accurate than those of the traditional FEM. Moreover, the CS-FEM can provide both an upper bound and a lower bound of the strain energy solutions while using different smoothing domains.

本文提出了一种基于单元的平滑有限元方法 (CS-FEM)，用于使用双势公式解决二维接触问题。接触力和接触面上的相对位移相互耦合。Uzawa 算法是一种局部迭代技术，用于求解接触力。考虑了经典的库仑摩擦规则和单边接触关系。整个过程无需选择任何用户自定义参数。准确研究了三种接触状态，分别为粘着、分离和滑动。CS-FEM 使用六种不同类型的平滑域执行，这些域通过将背景元素划分为不同的区域来构建。计算中只需要边界积分而不是域积分，不需要坐标映射。给出了三个数值例子来验证该方法的有效性。还研究了接触摩擦系数的影响。所有得到的数值解都与参考值吻合得很好。CS-FEM 产生的结果比传统 FEM 的结果更准确。此外，CS-FEM 可以在使用不同的平滑域时同时提供应变能解的上限和下限。所有得到的数值解都与参考值吻合得很好。CS-FEM 产生的结果比传统 FEM 的结果更准确。此外，CS-FEM 可以在使用不同的平滑域时同时提供应变能解的上限和下限。所有得到的数值解都与参考值吻合得很好。CS-FEM 产生的结果比传统 FEM 的结果更准确。此外，CS-FEM 可以在使用不同的平滑域时同时提供应变能解的上限和下限。