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GAMG method for higher-order finite element discretizations of modeling weak discontinuities problems[J].计算力学学报,2017,(1):35~42

GAMG method for higher-order finite element discretizations of modeling weak discontinuities problems
GAMG method for higher-order finite element discretizations of modeling weak discontinuities problems

DOI：10.7511/jslx201701004

 作者 单位 肖映雄 湘潭大学 土木工程与力学学院, 湘潭 411105 王彪 湘潭大学 土木工程与力学学院, 湘潭 411105 李真有 湘潭大学 土木工程与力学学院, 湘潭 411105

弱不连续问题（如含夹杂问题）是固体力学计算中的一类重要问题。高阶有限元方法由于其具有更好的逼近效果，是确保数值解在界面保持较高精度的计算方法之一。但与线性元相比，高阶单元需要更多的计算机存储单元，具有更高的计算复杂性。本文利用两水平算法的思想，将高阶有限元离散系统化归于线性元离散系统的求解，为弱不连续问题高阶有限元离散系统设计了一种新的基于几何与分析信息的代数多重网格（GAMG）法，并应用于圆形求解域含单夹杂问题的高阶有限元离散系统的求解。数值试验结果表明，相比于常用GAMG法，新方法的迭代次数基本不依赖于问题规模、单元阶次以及杨氏模量的间断性，CPU计算时间得到明显改善，具有更好的计算效率和鲁棒性，可大大提高弱不连续问题有限元分析的整体效率。

Weak discontinuities problems (such as inclusion problems) are important problems in solid mechanics calculation.Higher-order finite element method is a method which can ensure the accuracy of the numerical solutions near the interfaces.However,they have much higher computational complexity than the linear elements.In this paper,we present a new algebraic multigrid method (GAMG) for higher-order finite element discretizations of the weak discontinuous problems based on some geometric and analytical information by using two-level method.The resulting GAMG method is then applied to the solution of the single inclusion problem in a circular domain.Numerical results have been shown that the iteration counts of the new GAMG method do not substantially depend on the size of the problem,the number of elements and the discontinuity of Young's modulus with compared to those commonly used GAMG methods,and the CPU time is also improved obviously.Thus,the overall efficiency of the finite element analysis is greatly improved for modeling weak discontinuities problems.