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Effective multigrid algorithms for algebraic system arising from static peridynamic systems

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Abstract

Peridynamics is a nonlocal continuum theory that uses integral equations with no assumption of differentiability of displacement fields. One of the advantages of implicit type schemes for peridynamic systems is that they guarantee the equilibrium of the numerical solution, regardless of existing discontinuities on displacement fields. In this work, we propose multigrid algorithms for implicit discretizations of static peridynamic systems. The considerations are emphasized on the selection of horizon parameters at coarse levels. Among some versions, the multigrid algorithms which fix the horizon size at the coarse level are shown to be most efficient. In such case, the computational complexity of a smoothing (say Jacobi or Gauss-Seidel) at coarse level decreases by 1/16 as the level decreases. To enhance the efficiency of multigrid algorithms, the interpolation operator is modified near the crack on the underlying domain so that the energy-like norm does not blow up near the crack. In the numerical experiments, we report the performance of preconditioned conjugated gradient (PCG) preconditioned by our multigrid algorithms (MG-PCG). The computational complexity of MG-PCG is \(\mathcal {O}(N)\) where N implies the number of the unknowns.

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Funding

The first author, G. Jo, received financial support from the National Research Foundation of Korea(NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01005396). The corresponding author, Y.D. Ha, received financial support from the National Research Foundation of Korea (NRF) funded by the Korea government(MSIT) (No. 2018R1D1A1B07049124) and the Human Resources Development Program (No. 20194010201800) of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) funded by the Ministry of Trade, Industry, and Energy.

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Authors and Affiliations

  1. Department of Mathematics, Kunsan National University, Gunsan-si, Jeollabuk-do, Republic of Korea

    Gwanghyun Jo

  2. Department of Naval Architecture and Ocean Engineering, Kunsan National University, Gunsan-si, Jeollabuk-do, Republic of Korea

    Youn Doh Ha

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  1. Gwanghyun Jo
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  2. Youn Doh Ha
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Correspondence to Youn Doh Ha.

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Jo, G., Ha, Y.D. Effective multigrid algorithms for algebraic system arising from static peridynamic systems. Numer Algor 89, 885–904 (2022). https://doi.org/10.1007/s11075-021-01138-1

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