一种基于自由度凝聚的求解接触计算中鞍点系统的算法
A DOFs condensation based algorithm for solving saddle point systems in contact computation
AI总结：
针对接触计算中的大规模离散鞍点系统，提出一种基于mortar矩阵三对角特性的自由度凝聚技术，消元后得到对称正定系统并用PCG高效求解。
摘要：
在接触力学计算中，接触面上的约束条件通常通过Lagrange乘子法施加，从而产生一个鞍点系统。通常采用mortar有限元方法对网格化接触面上的变分形式进行离散，导致大规模的离散鞍点系统。由于离散系统的不定性，求解鞍点代数系统是一项挑战。针对二维绑定接触问题，开发了一种高效的自由度凝聚技术。该方法的核心是利用mortar矩阵的三对角特性进行自由度消元。自由度消元后得到的线性系统规模更小，且矩阵为对称正定矩阵。通过使用预处理共轭梯度法（PCG），可以高效求解该线性系统。数值结果证明了该方法的有效性。
英文摘要：
In contact mechanics computation, the constraint conditions on the contact surfaces are typically enforced by the Lagrange multiplier method, resulting in a saddle point system. The mortar finite element method is usually employed to discretize the variational form on the meshed contact surfaces, leading to a large-scale discretized saddle point system. Due to the indefiniteness of the discretized system, it is a challenge to solve the saddle point algebraic system. For two-dimensional tied contact problem, an efficient DOFs condensation technique is developed. The essential of the proposed method is to carry out the DOFs elimination by using the tridiagonal characteristic of the mortar matrix. The scale of the linear system obtained after DOFs elimination is smaller, and the matrix is symmetric positive definite. By using the preconditioned conjugate gradient (PCG) method, the linear system can be solved efficiently. Numerical results show the effectiveness of the method.
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