张红梅, 肖映雄. 三维弹性问题高次有限元离散线性系统的块对角逆预条件PCG法[J]. 工程力学, 2010, 27(7): 62-066.
引用本文: 张红梅, 肖映雄. 三维弹性问题高次有限元离散线性系统的块对角逆预条件PCG法[J]. 工程力学, 2010, 27(7): 62-066.
ZHANG Hong-mei, XIAO Ying-xiong. A BLOCK PRECONDITIONING METHOD FOR HIGHER-ORDER FINITE ELEMENT DISCRETIZATIONS OF LINEAR ELASTICITY EQUATIONS IN THREE DIMENSIONS[J]. Engineering Mechanics, 2010, 27(7): 62-066.
Citation: ZHANG Hong-mei, XIAO Ying-xiong. A BLOCK PRECONDITIONING METHOD FOR HIGHER-ORDER FINITE ELEMENT DISCRETIZATIONS OF LINEAR ELASTICITY EQUATIONS IN THREE DIMENSIONS[J]. Engineering Mechanics, 2010, 27(7): 62-066.

三维弹性问题高次有限元离散线性系统的块对角逆预条件PCG法

A BLOCK PRECONDITIONING METHOD FOR HIGHER-ORDER FINITE ELEMENT DISCRETIZATIONS OF LINEAR ELASTICITY EQUATIONS IN THREE DIMENSIONS

  • 摘要: 高次有限元由于对问题具有更好的逼近效果及某些特殊的优点,如能解决弹性问题的闭锁现象 (Poisson’s ratio locking),使得它们在实际计算中被广泛使用。但与线性元相比,它具有更高的计算复杂性。该文基于标量椭圆问题高次有限元离散化系统的代数多层网格(AMG)法,针对三维弹性问题高次有限元离散化线性系统的求解,设计了一种以块对角逆为预条件子的共轭梯度法(AMG-BPCG)。数值实验表明,该文设计的AMG-BPCG法较标准的ILU-型PCG法具有更好的计算效率和鲁棒性。

     

    Abstract: The higher-order finite elements have been often used in practical computations in that they are superior and necessary under certain conditions over the commonly used low-order ones, for example, they can overcome the Poisson’s ratio locking in linear elasticity. However, they have much higher computational complexity than the low-order elements. A block preconditioned conjugate gradient (BPCG) algorithm with a block diagonal preconditioner is proposed for solving the linear systems derived from higher-order finite element discretizations of linear elasticity equations in three dimensions. In the BPCG method, an algebraic multigrid (AMG) V-cycle, firstly introduced for high-order discretizations of the scalar elliptic problems, is employed for computing approximately the action of the inverse of each diagonal block. The results of various numerical experiments show that the resulting BPCG method based on AMG V-cycle (AMG-BPCG) is more robust and efficient than the usual ILU-type PCG methods.

     

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