Journal of Scientific Computing

, Volume 74, Issue 2, pp 728–742 | Cite as

A Fast Finite Difference Method for a Continuous Static Linear Bond-Based Peridynamics Model of Mechanics

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Abstract

The peridynamic nonlocal continuum model for solid mechanics is an integro-differential equation that does not involve spatial derivatives of the displacement field. Several numerical methods such as finite element method and collocation method have been developed and analyzed in many articles. However, there is no theory to give a finite difference method because the model does not involve spatial derivatives of the displacement field. Here, we consider a finite difference scheme to solve a continuous static bond-based peridynamics model of mechanics based on its equivalent partial integro-differential equations. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization respectively. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from \(O(N^{3})\) required by traditional methods to O(Nlog\(^{2}N)\) and the memory requirement from \(O(N^{2})\) to O(N) without using any lossy compression, where N is the number of unknowns. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.

Keywords

Peridynamic Finite difference method Topelitz Fast Fourier transform 

Mathematics Subject Classification

65M06 65M12 65M15 26A33 

Notes

Acknowledgements

We would like to acknowledge the assistance of volunteers in putting together this example manuscript and supplement.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina

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