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Two solution strategies to improve the computational performance of Sequentially Linear Analysis for quasi-brittle structures
International Journal for Numerical Methods in Engineering ( IF 2.9 ) Pub Date : 2020-05-30 , DOI: 10.1002/nme.6302
M. Pari 1 , W. Swart 2 , M.B. Gijzen 2 , M.A.N. Hendriks 1, 3 , J.G. Rots 1
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Sequentially Linear Analysis (SLA), an event-by-event procedure for finite element (FE) simulation of quasi-brittle materials, is based on sequentially identifying a critical integration point in the FE model, to reduce its strength and stiffness, and the corresponding critical load multiplier (λcrit ), to scale the linear analysis results. In this article, two strategies are proposed to efficiently reuse previous stiffness matrix factorisations and their corresponding solutions in subsequent linear analyses, since the global system of linear equations representing the FE model changes only locally. The first is based on a direct solution method in combination with the Woodbury matrix identity, to compute the inverse of a low-rank corrected stiffness matrix relatively cheaply. The second is a variation of the traditional incomplete LU preconditioned conjugate gradient method (CG), wherein the preconditioner is the complete factorisation of a previous analysis step’s stiffness matrix. For both the approaches, optimal points at which the factorisation is recomputed are determined such that the total analysis time is minimized. Comparison and validation against a traditional parallel direct sparse solver, with regard to a 2D and 3D benchmark study, illustrates the improved performance of the Woodbury based direct solver over its counterparts, especially for large 3D problems.

中文翻译:

提高准脆性结构序列线性分析计算性能的两种求解策略

顺序线性分析 (SLA) 是准脆性材料有限元 (FE) 模拟的逐事件程序,它基于顺序识别 FE 模型中的关键积分点,以降低其强度和刚度,以及相应的临界载荷乘数 (λcrit),用于缩放线性分析结果。在本文中,提出了两种策略,以在随后的线性分析中有效地重用先前的刚度矩阵分解及其相应的解,因为代表有限元模型的全局线性方程组仅在局部发生变化。第一种是基于直接求解方法并结合伍德伯里矩阵恒等式,以相对便宜的方式计算低秩校正刚度矩阵的逆。第二个是传统的不完全 LU 预处理共轭梯度法 (CG) 的变体,其中预处理器是前一个分析步骤刚度矩阵的完全分解。对于这两种方法,确定重新计算因式分解的最佳点,从而使总分析时间最小化。在 2D 和 3D 基准研究方面,与传统并行直接稀疏求解器的比较和验证表明,基于 Woodbury 的直接求解器的性能优于同类直接求解器,尤其是对于大型 3D 问题。确定重新计算因式分解的最佳点,从而使总分析时间最小化。在 2D 和 3D 基准研究方面,与传统并行直接稀疏求解器的比较和验证表明,基于 Woodbury 的直接求解器的性能优于同类直接求解器,尤其是对于大型 3D 问题。确定重新计算因式分解的最佳点,从而使总分析时间最小化。在 2D 和 3D 基准研究方面,与传统并行直接稀疏求解器的比较和验证表明,基于 Woodbury 的直接求解器的性能优于同类直接求解器,尤其是对于大型 3D 问题。
更新日期:2020-05-30
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