Convergence of Nonlocal Integral Operator in Peridynamics

Gwanghyun Jo; Youn Doh Ha

doi:10.7734/COSEIK.2021.34.3.151

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2021 Vol.34, Issue 3 Preview Page Next Page

Research Paper

Convergence of Nonlocal Integral Operator in Peridynamics 비국부 적분 연산기로 표현되는 페리다이나믹 방정식의 수렴성

30 June 2021. pp. 151-157

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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

본 연구에서는 비국부 적분 연산기로 표현되는 페리다이나믹 방정식의 수렴성을 검토한다. 정적/준정적 손상 해석 문제를 효율적으로 해석하기 위해 페리다이나믹 방정식의 implicit 정식화가 필요하다. 이 과정에서 페리다이나믹 비국부 적분 방정식으로부터 대수방정식 형태가 나타나게 되어 시스템 행렬 계산을 위해 많은 시간이 소요되기 때문에, 효율적인 계산을 위해 수렴성이 중요한 요소가 된다. 특히 radial influence 함수를 적분 kernel로 사용하는 경우 fractional Laplacian 적분 방정식이 유도된다. 비국부 적분 연산기의 교윳값 성질에 의해 대수방정식의 condition number가 radial influence 함수의 차수 및 비국부 영역의 크기에 영향을 받는 것이 수학적으로 확인되었다. 본 연구에서는 이를 토대로 균열이 있는 페리다이나믹 정적 해석 문제를 Newton-Raphson 방법으로 해석할 때 적분 커널의 차수, 비국부 영역의 크기 등이 대수방정식의 condition number와 preconditioned conjugate gradient (PCG) 방법으로 계산 시 수렴성 및 계산 시간에 미치는 영향을 수치적으로 분석한다.

키워드

페리다이나믹 해석법

Fractional Laplacian kernel

비국부 적분 연산기

condition number

implicit formulation

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Information

Publisher :Computational Structural Engineering Institute of Korea
Publisher(Ko) :한국전산구조공학회
Journal Title :Journal of the Computational Structural Engineering Institute of Korea
Journal Title(Ko) :한국전산구조공학회 논문집
Volume : 34
No :3
Pages :151-157
Received Date : 2021-05-06
Revised Date : 2021-05-25
Accepted Date : 2021-05-26
DOI :https://doi.org/10.7734/COSEIK.2021.34.3.151

Journal of the Computational Structural Engineering Institute of Korea ISSN:1229-3059(Print) 2287-2302(Online) 한국전산구조공학회 논문집

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