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Refined Isogeometric Analysis for a preconditioned conjugate gradient solver

机译：预处理共轭梯度求解器的精细等几何分析

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Starting from a highly continuous Isogeometric Analysis (IGA) discretization, refined Isogeometric Analysis (rIGA) introduces C-0 hyperplanes that act as separators for the direct LU factorization solver. As a result, the total computational cost required to solve the corresponding system of equations using a direct LU factorization solver dramatically reduces (up to a factor of 55) (Garcia et al., 2017). At the same time, rIGA enriches the IGA spaces, thus improving the best approximation error. In this work, we extend the complexity analysis of rIGA to the case of iterative solvers. We build an iterative solver as follows: we first construct the Schur complements using a direct solver over small subdomains (macro-elements). We then assemble those Schur complements into a global skeleton system. Subsequently, we solve this system iteratively using Conjugate Gradients (CG) with an incomplete LU (ILU) preconditioner. For a 2D Poisson model problem with a structured mesh and a uniform polynomial degree of approximation, rIGA achieves moderate savings with respect to IGA in terms of the number of Floating Point Operations (FLOPs) and computational time (in seconds) required to solve the resulting system of linear equations. For instance, for a mesh with four million elements and polynomial degree p = 3, the iterative solver is approximately 2.6 times faster (in time) when applied to the rIGA system than to the IGA one. These savings occur because the skeleton rIGA system contains fewer non-zero entries than the IGA one. The opposite situation occurs for 3D problems, and as a result, 3D rIGA discretizations provide no gains with respect to their IGA counterparts when considering iterative solvers. (C) 2018 Elsevier B.V. All rights reserved.
机译：从高度连续的等几何分析（IGA）离散化开始，改进的等几何分析（rIGA）引入了C-0超平面，该平面用作直接LU分解求解器的分隔符。结果，使用直接LU分解求解器求解相应方程组所需的总计算成本显着降低了（高达55倍）（Garcia et al。，2017）。同时，rIGA丰富了IGA空间，从而改善了最佳逼近误差。在这项工作中，我们将rIGA的复杂性分析扩展到迭代求解器的情况。我们按以下方式构建迭代求解器：首先，在小的子域（宏元素）上使用直接求解器构造Schur补码。然后，我们将这些Schur补语装配到一个全局骨架系统中。随后，我们使用具有不完整LU（ILU）预调节器的共轭梯度（CG）迭代求解该系统。对于具有结构化网格和统一多项式逼近度的二维Poisson模型问题，相对于IGA，rIGA在浮点运算（FLOP）数量和计算结果所需的计算时间（以秒为单位）方面实现了适度的节省线性方程组。例如，对于具有四百万个元素且多项式度p = 3的网格，当应用于rIGA系统时，迭代求解器（在时间上）的速度比IGA系统快约2.6倍。产生这些节省的原因是，框架rIGA系统包含的非零条目比IGA少。对于3D问题，情况恰恰相反，因此，在考虑迭代求解器时，相对于其IGA同类，3D rIGA离散化不会带来任何收益。 （C）2018 Elsevier B.V.保留所有权利。

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