Abstract
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.
Highlights
The contact problems exist everywhere in life
We will use three numerical examples to examine the efficiency of the contact analysis while combining the cell-based smoothed finite element method (CS-finite element method (FEM)) and the bi-potential formulation
Since the largest penetration produced by CS-FEM-1D is less than 3 10−20 mm, it can be neglected in the calculation
Summary
The contact problems exist everywhere in life. For example, the contact between the tire and the road, the contact between the shaft and the bearing, the contact between the knife and the flesh in virtual surgery, etc. The widely used contact algorithms in engineering applications are the penalty function method [5]-[8], the Lagrange multiplier method [9]-[12] and the linear complementarity techniques [13]-[16]. The finite element method (FEM) is one of the most effective numerical methods for solving contact problems. The smoothed finite element method (S-FEM) proposed by Liu et al is a weakened weak form method based on the G-space theory [29]. The S-FEM has been applied to solve contact problems [16][26][39][40]. The CS-FEM with quadrilateral elements is used to solve the two-dimensional contact problems within the bi-potential framework. Three numerical examples are presented to examine the numerical accuracy of the proposed method
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