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Connections Between the Meshfree Peridynamics Discretization and Graph Laplacian for Transient Diffusion Problems

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Abstract

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.

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Funding

This work was supported by the National Science Foundation under CDS&E CMMI award No. 1953346.

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  1. Department of Mechanical and Materials Engineering, University of Nebraska-Lincoln, Lincoln, NE, 68588-0526, USA

    Longzhen Wang & Florin Bobaru

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  1. Longzhen Wang
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Correspondence to Florin Bobaru.

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Wang, L., Bobaru, F. Connections Between the Meshfree Peridynamics Discretization and Graph Laplacian for Transient Diffusion Problems. J Peridyn Nonlocal Model 3, 307–326 (2021). https://doi.org/10.1007/s42102-021-00053-2

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