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A Fast Discontinuous Galerkin Method for a Bond-Based Linear Peridynamic Model Discretized on a Locally Refined Composite Mesh

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Abstract

We develop a family of fast discontinuous Galerkin (DG) finite element methods for a bond-based linear peridynamic (PD) model in one space dimension. More precisely, we develop a preconditioned fast piecewise-constant DG scheme on a geometrically graded locally refined composite mesh which is suited for the scenario in which the jump discontinuity of the displacement field occurs at the one of the nodes in the original uniform partition. We also develop a preconditioned fast piecewise-linear DG scheme on a uniform mesh that has a second-order convergence rate when the jump discontinuity occurs at one of the computational nodes or has a convergence rate of one-half order otherwise. Motivated by these results, we develop a preconditioned fast hybrid DG scheme that is discretized on a locally uniformly refined composite mesh to numerically simulate the PD model where the jump discontinuity of the displacement field occurs inside a computational cell. We use a piecewise-constant DG scheme on a uniform mesh and a piecewise-linear DG scheme on a locally uniformly refined mesh of mesh size \(O(h^2)\), which has an overall convergence rate of O(h). Because of their nonlocal nature, numerical methods for PD models generate dense stiffness matrices which have \(O(N^2)\) memory requirement and \(O(N^3)\) computational complexity, where N is the number of computational nodes. In this paper, we explore the structure of the stiffness matrices to develop three preconditioned fast Krylov subspace iterative solvers for the DG method. Consequently, the methods have significantly reduced computational complexity and memory requirement. Numerical results show the utility of the numerical methods.

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Acknowledgements

The authors would like to express their sincere thanks to the editor and referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. This work was supported in part by the OSD/ARO MURI Grant W911NF-15-1-0562, by the National Natural Science Foundation of China under Grants 11471194, 11571115, and 91630207, by the National Science Foundation under Grant DMS-1620194, by the National Science and Technology Major Project of China under Grants 2011ZX05052 and 2011ZX05011-004, and by Shandong Provincial Natural Science Foundation, China under Grant ZR2011AM015, and by Taishan Scholars Program of Shandong Province of China.

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, China

    Huan Liu & Aijie Cheng

  2. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    Hong Wang

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  1. Huan Liu
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  2. Aijie Cheng
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  3. Hong Wang
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Correspondence to Aijie Cheng or Hong Wang.

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Liu, H., Cheng, A. & Wang, H. A Fast Discontinuous Galerkin Method for a Bond-Based Linear Peridynamic Model Discretized on a Locally Refined Composite Mesh. J Sci Comput 76, 913–942 (2018). https://doi.org/10.1007/s10915-018-0645-6

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