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首页> 外文期刊>Journal of Computational Physics >A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model
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A fast Galerkin method with efficient matrix assembly and storage for a peridynamic model

机译：快速Galerkin方法，具有有效的矩阵组装和存储功能，可用于动力学模型

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Peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities, which cannot be described properly by classical theory of solid mechanics. However, the operators in the peridynamic models are nonlocal, so the resulting numerical methods generate dense or full stiffness matrices. Gaussian types of direct solvers were traditionally used to solve these problems, which requires O(N ~3) of operations and O(N ~2) of memory where N is the number of spatial nodes. This imposes significant computational and memory challenge for a peridynamic model, especially for problems in multiple space dimensions. A simplified model, which assumes that the horizon of the material δ=O(N -1), was proposed to reduce the computational cost and memory requirement to O(N). However, the drawback is that the corresponding error estimate becomes one-order suboptimal. Furthermore, the assumption of δ=O(N ~(-1)) does not seem to be physically reasonable since the horizon δ represents a physical property of the material that should not depend on computational mesh size. We develop a fast Galerkin method for the(non-simplified) peridynamic model by exploiting the structure of the stiffness matrix. The new method reduces the computational work from O(N ~3) required by the traditional methods to O(Nlog ~2N) and the memory requirement from O(N ~2) to O(N) without using any lossy compression. The significant computational and memory reduction of the fast method is better reflected in numerical experiments. When solving a one-dimensional peridynamic model with 2 ~(14)=16, 384 unknowns, the traditional method consumed CPU time of 6days and 11h while the fast method used only 3.3s. In addition, on the same computer(with 128GB memory), the traditional method with a Gaussian elimination or conjugate gradient method ran out of memory when solving the problem with 2 ~(16)=131, 072 unknowns. In contrast, the fast method was able to solve the problem with 2 ~(28)=268, 435, 456 unknowns using 3days and 11h of CPU time. This shows the benefit of the significantly reduced memory requirement of the fast method.
机译：绕动力学理论提供了关于包含不连续性或其他奇异性的连续物体变形的适当描述，而经典的固体力学理论无法正确描述。但是，围动力模型中的算子是非局部的，因此所得的数值方法会生成密集或完全刚度的矩阵。传统上使用高斯类型的直接求解器来解决这些问题，这需要操作的O（N〜3）和内存的O（N〜2），其中N是空间节点的数量。这给围动态模型，特别是对于多个空间维度的问题，带来了显着的计算和存储挑战。提出了一个简化模型，该模型假定材料的视界为δ= O（N -1），以将计算成本和内存需求降低到O（N）。然而，缺点是相应的误差估计变为一阶次优。此外，δ= O（N〜（-1））的假设在物理上似乎并不合理，因为水平δ代表了不依赖于计算网格大小的材料的物理特性。通过利用刚度矩阵的结构，我们为（非简化的）绕动力学模型开发了一种快速的Galerkin方法。新方法将计算方法从传统方法所需的O（N〜3）减少到O（Nlog〜2N），并将内存需求从O（N〜2）减少到O（N），而无需使用任何有损压缩。数值实验更好地反映了快速方法的显着计算和内存减少。在求解2〜（14）= 16、384个未知数的一维绕动力学模型时，传统方法消耗的CPU时间分别为6天和11h，而快速方法仅消耗3.3s。另外，在同一台计算机上（具有128GB内存），当解决2〜（16）= 131，072个未知数的问题时，传统的采用高斯消除或共轭梯度法的方法用完了内存。相比之下，快速方法能够使用3天和11小时的CPU时间解决2〜（28）= 268、435、456个未知数的问题。这表明快速方法大大减少了内存需求。

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