含裂缝线弹性问题的压缩预条件共轭梯度算法

Deflated Preconditioned Conjugate Gradient Solvers for Linear Elastic Crack Problems

基于几何型扩展有限元离散,研究含静态裂缝线弹性问题的高效压缩预条件算法。不仅构造适用于含裂缝线弹性问题的压缩子空间矩阵,而且给出压缩点的选点原则。为进一步提高计算效率,将该压缩技巧以乘性的方式与“裂尖型”区域分解预条件子相结合,提出一类高效的自适应压缩预条件共轭梯度算法,该算法能同时消去迭代求解中的高频误差和低频误差,数值实验验证了算法的有效性。

This paper focuses on some efficient deflated preconditioners for static elastic crack problems modelled by the geometrical extended finite element method.We not only construct the deflation subspace matrix which is suitable for linear elastic crack problems,but also give the principle for selecting the deflated mesh nodes.To further accelerate the convergence,we combine the deflation technique with the"crack tip"domain decomposition preconditioners through multiplicative way,and propose efficient adapted deflated preconditioned conjugate gradient solvers which can eliminate the high-frequency and low-frequency errors simultaneously in the iterations.Numerical experiments demonstrate the effectiveness of our algorithm.