Browse > Article
http://dx.doi.org/10.7734/COSEIK.2021.34.3.151

Convergence of Nonlocal Integral Operator in Peridynamics  

Jo, Gwanghyun (Department of Mathematics, Kunsan National University)
Ha, Youn Doh (Department of Naval Architecture and Ocean Engineering, Kunsan National University)
Publication Information
Journal of the Computational Structural Engineering Institute of Korea / v.34, no.3, 2021 , pp. 151-157 More about this Journal
Abstract
This paper is devoted to a convergence study of the nonlocal integral operator in peridynamics. The implicit formulation can be an efficient approach to obtain the static/quasi-static solution of crack propagation problems. Implicit methods require constly large-matrix operations. Therefore, convergence is important for improving computational efficiency. When the radial influence function is utilized in the nonlocal integral equation, the fractional Laplacian integral equation is obtained. It has been mathematically proved that the condition number of the system matrix is affected by the order of the radial influence function and nonlocal horizon size. We formulate the static crack problem with peridynamics and utilize Newton-Raphson methods with a preconditioned conjugate gradient scheme to solve this nonlinear stationary system. The convergence behavior and the computational time for solving the implicit algebraic system have been studied with respect to the order of the radial influence function and nonlocal horizon size.
Keywords
peridynamics; fractional laplacian kernel; nonlocal integral operator; condition number; implicit formulation;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Silling, S.A. (2000) Reformulation of Elasticity Theory for Discontinuities and Long-Range Forces, J. Mech. & Phys. Solids, 48(1), pp.175~209.   DOI
2 Seleson, P., Parks, M.L. (2011) On the Role of the Influence Function in the Peridynamic Theory, Int. J. Multiscale. Comput. Eng., 9(6), pp.689~706.   DOI
3 Jo, G., Ha, Y.D. (2021) Effective Multigrid Algorithms for Algebraic System Arising from Static Peridynamic Systems, Numer. Algorithms, pp.1~20.
4 Aksoylu, B., Parks, M.L. (2011) Variational Theory and Domain Decomposition for Nonlocal Problems, Appl. Math. Comput., 217(14), pp.6498~6515.   DOI
5 Aksoylu, B., Unlu, Z. (2014) Conditioning Analysis of Nonlocal Integral Operators in Fractional Sobolev Spaces, SIAM. J. Numer. Anal., 52(2), pp.653~677.   DOI
6 Ha, Y.D., Bobaru, F. (2011) Characteristics of Dynamic Brittle Fracture Captured with Peridynamics, Eng. Fract. Mech., 78(6), pp.1156~1168.   DOI
7 Ha, Y.D., Cho, S. (2011) Dynamic Brittle Fracture Captured with Peridynamics: Crack Branching Angle & Crack Propagation Speed, J. Comput. Strcut. Eng. Inst. Korea, 24(6), pp.637~643.
8 Hu, W ., Ha, Y.D., Bobaru, F. (2012) Peridynamic Model for Dynamic Fracture in Unidirectional Fiber-Reinforced Composites, Comput. Methods Appl. Mech. Eng., 217-220, pp.247~261.   DOI
9 Ni, T., Zaccariotto, M., Zhu, Q-Z., Galvanetto, U. (2019) Static Solution of Crack Propagation Problems in Peridynamics, Comput. Methods Appl. Mech. Eng., 346, 126~151.   DOI
10 Ha, Y.D., Lee, J., Hong, J.W. (2015) Fracturing Patterns of Rock-Like Materials in Compression Captured with Peridynamics, Eng. Fract. Mech., 144, pp.176~193.   DOI
11 Silling, S., Askari, E. (2005) A Meshfree Method Based on the Peridynamic Model of Solid Mechanics, Comput. Struct., 83(17-18), pp.1526~1535.   DOI
12 Ha, Y.D. (2020) An Extended Ghost Interlayer Model in Peridynamic Theory for High-Velocity Impact Fracture of Laminated Glass Structures, Comput. Math. Appl., 80(5), pp.744~761.   DOI
13 Hashim, N.R., Coombs, W.M., Augarde, C.E., Hattori, G. (2020) An Impliciti Non-Ordinary State-Based Peridynamics with Stabilized Correspondence Material Model for Finite Deformation Analysis, Comput. Methods Appl. Mech. Eng., 371(1), p.113304.   DOI