求解双层弹性膜单侧接触问题的交替方向乘子法

An Alternating Direction M ultiplier Method for the Unilateral Contact Between Two Membranes

针对一类双层弹性膜单侧接触问题,提出了基于自适应罚参数的交替方向乘子法。利用辅助变量构造增广Lagrange函数,通过引入鞍点问题处理不等式约束。用交替方向乘子法对该问题进行数值求解,每次迭代包括2个线性椭圆问题,显式求解辅助变量和Lagrange乘子。基于迭代函数的自适应法则自动调整罚参数,展示了算法收敛性和自适应罚参数近似方法。最后用有限元离散化数值结果验证了算法的有效性。

It proposes an alternating direction method of multiplier for approximation solution of the unilateral contact problem of two membranes,with a self-adaptive penalty parameter.By using augmented Lagrange functional with an auxiliary unknown,a saddlepoint problem is introduced to deal with the inequality constrained.Then the alternating direction method of multiplier is applied to the corresponding problem,and each iteration consists of two linear elliptic problems while the auxiliary unknown and the Lagrange multiplier are computed explicitly.A self-adaptive rule based iterative functions is used to automatically adjust the penalty parameter.The convergence of the method is shown and the self-adaptive penalty parameter approximation is given.Finally,the numerical experiments with the finite element discretization are given to illustrate the efficiency of the proposed method.