A fast convolution-based method for peridynamic transient diffusion in arbitrary domains
A fast convolution-based method for peridynamic transient diffusion in arbitrary domains
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DOI：
10.1016/j.cma.2020.113633
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发表时间：
2021-03
影响因子：
7.2
通讯作者：
S. Jafarzadeh;Longzhen Wang;Adam Larios;F. Bobaru
中科院分区：
文献类型：
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作者：
S. Jafarzadeh;Longzhen Wang;Adam Larios;F. Bobaru
介绍了一种基于卷积的快速方法(FCBM)，用于求解一维、二维和三维中的线性和一类非线性动力学(PD)暂态扩散问题。该方法利用PD扩散算子的卷积结构，利用快速傅立叶变换(FFT)有效地计算了PD扩散算子的卷积结构。一种新的“嵌入约束”(EC)策略允许在不规则区域上使用傅里叶变换并施加任意的非局部边界条件。新方法的计算复杂度为O(N Log2 N)，而传统的动力无网格或有限元方法的计算复杂度为O(N 2)。通过空间离散化，我们得到了一维和二维问题中精确非局部解的二次收敛速度。数值试验表明，对于非局部区域内节点数目较多的问题，使用FCBM-EC方法比无网格离散方法有显著的效率提高。通过在GPU或多个CPU上进行计算时使用MatLab固有的FFT函数，进一步提高了计算速度。一个三维算例，在数万个时间步长上有超过10亿个自由度，用新方法在单个CPU上只需几天时间就能求解。同样的问题需要一个多世纪才能用常用的无网格离散化来完成。
We introduce a fast convolution-based method (FCBM) for solving linear and a certain class of nonlinear peridynamic (PD) transient diffusion problems in 1D, 2D, and 3D. The method exploits the convolutional structure of the PD diffusion operator to compute it efficiently by using the fast Fourier transform (FFT). A new “embedded constraint”(EC) strategy allows using the Fourier transform on irregular domains and imposing arbitrary nonlocal boundary conditions. The complexity of the new method is O (N log 2 N), compared with O (N 2) for the conventional peridynamic meshfree or finite element solvers of the same problem. We find quadratic convergence rates in terms of spatial discretization to the exact nonlocal solutions in 1D and 2D problems. Numerical tests show substantial efficiency gains when using the FCBM-EC method compared to the meshfree discretization method for problems with a larger number of nodes inside the nonlocal region. Further speedup is achieved with Matlab’s intrinsic FFT functions for computations on GPUs or multiple CPUs. An example in 3D with over 1 billion degrees of freedom over tens of thousands of time-steps, is solved by the new method on a single CPU in a matter of days. The same problem would have required over a century to complete with the commonly used meshfree discretization.