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 引用本文: 薛冰寒, 林皋, 胡志强, 庞林. 摩擦接触问题的比例边界等几何B可微方程组方法[J]. 力学学报, 2016, 48(3): 615-623.
Xue Binghan, Lin Gao, Hu Zhiqiang, Pang Lin. ANALYSIS OF FRICTIONAL CONTACT PROBLEMS BY SBIGA-BDE METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 615-623.
 Citation: Xue Binghan, Lin Gao, Hu Zhiqiang, Pang Lin. ANALYSIS OF FRICTIONAL CONTACT PROBLEMS BY SBIGA-BDE METHOD[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3): 615-623.

## ANALYSIS OF FRICTIONAL CONTACT PROBLEMS BY SBIGA-BDE METHOD

• 摘要: 摩擦接触问题是计算力学领域最具挑战性的问题之一,接触系统的泛函具有非线性、非光滑的特点,导致接触算法的收敛性与精确性难以保证.因此将比例边界等几何分析(scaled boundary isogeometric analysis,SBIGA)与B可微方程组(B dierential equation,BDE)相结合,提出了求解二维摩擦接触问题的比例边界等几何B可微方程组方法.在比例边界等几何坐标变换的基础上,通过虚功原理推导了关于边界控制点变量的接触平衡方程,表示成B可微方程组形式的接触条件可被严格满足,求解B可微方程组的算法的收敛性有理论保证.此比例边界等几何B可微方程组方法(SBIGA-BDE)只需在接触体边界进行等几何离散,使问题降低一维,能精确描述接触边界,并可通过节点插入算法进行真实接触区域的识别.此外,由于几何建模和数值分析使用相同的基函数,节约了划分网格的时间.以赫兹接触问题和悬臂梁摩擦接触问题为例,通过与解析解及数值计算软件ANSYS计算结果进行对比,验证了该方法求解二维摩擦接触问题的有效性及高精度等特点.

Abstract: Frictional contact analysis is one of the most challenging problems in computational mechanics. The functional system of the contact problem is not only nonlinear, but also non-smooth, so in general the convergence and accuracy of contact algorithms are di cult to be guaranteed. For 2D elastic frictional contact problem, the scaled boundary isogeometric analysis combined with B di erential equation method (SBIGA-BDE method) is developed. Based on the scaled boundary isogeometric transformation, the contact equilibrium equation is derived by using virtual principle. The contact conditions are formulated as B di erential equation and satisfied rigorously. The convergence of the algorithm to solve the B di erential equation is guaranteed by the theory of mathematical programming. In the proposed method, only the outer boundary including the contact boundary need to be discretized isogeometrically, which reduces the spatial dimension by one and the boundary are represented accurately. The real contact length can be detected by the knot insertion algorithm. In addition, as the interpolatory functions used in geometry modeling and numerical analyzing are the same, the time costs in mesh generation is saved. The numerical examples, including Hertz contact problem and cantilever beams frictional contact problem, are presented and compared with analytic solution and ANSYS results. It validates the e ectiveness and accuracy of the proposed method in solving 2D elastic frictional contact problem.

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