Connections Between the Meshfree Peridynamics Discretization and Graph Laplacian for Transient Diffusion Problems
Connections Between the Meshfree Peridynamics Discretization and Graph Laplacian for Transient Diffusion Problems
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DOI：
10.1007/s42102-021-00053-2
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Longzhen Wang;F. Bobaru
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文献类型：
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作者：
Longzhen Wang;F. Bobaru
最近，基于图拉普拉斯核的扩散过程公式已被用于用基于谱的半解析方法求解具有绝缘边界条件的线性瞬态传热问题。这种方法被称为“谱图”（SG）方法。在本文中，我们证明了相应的周动力（PD）模型的无网格离散化导致了一个图结构，允许我们使用SG方法中采用的半解析方法引入“图拉普拉斯PD”（GL-PD）方法来解决瞬态热扩散问题。新方法GL-PD可以看作是SG和PD之间的“混合”，具有无网格离散化。我们讨论了GL-PD和SG方法之间的异同。主要的区别与校准和离散化程序有关。我们使用一个一维热扩散的例子来强调SG方法中基于光谱的半解析方法与通常用于计算瞬态扩散问题解的直接时间积分方法相比的一些局限性。然后，我们提出了半解析方法的扩展，以解决瞬态扩散问题与Dirichlet边界条件，使用最近的缩放和平方算法，计算矩阵指数的非对称矩阵。
Formulations for diffusion processes based on graph Laplacian kernels have been recently used to solve linear transient heat transfer problems with insulated boundary conditions by way of a spectral-based semi-analytical approach. This has been called the “spectral graph” (SG) approach. In this paper, we show that the meshfree discretization for corresponding peridynamic (PD) models leads to a graph structure that allows us to introduce the “graph Laplacian PD” (GL-PD) method using the semi-analytical approach employed in the SG approach to solve the transient heat diffusion problems. The new method, GL-PD, can be seen as a “hybrid” between SG and PD with meshfree discretization. We discuss the similarities and differences between the GL-PD and the SG approaches. Main differences are related to calibration and discretization procedures. We use a 1D heat diffusion example to highlight some limitations the spectral-based semi-analytical method in the SG approach has compared with the direct time-integration normally used in computing solutions to transient diffusion problems. We then propose an extension of the semi-analytical approach to solve transient diffusion problems with Dirichlet boundary conditions, using a recent scaling and squaring algorithm that calculates the matrix exponential of non-symmetric matrices.