Volume 27, Issue 1
A Fast State-Based Peridynamic Numerical Model

Ning Du, Xu GuoHong Wang

Commun. Comput. Phys., 27 (2020), pp. 274-291.

Published online: 2019-10

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  • Abstract

The peridynamic (PD) theory is a reformulation of the classical theory of continuum solid mechanics and is particularly suitable for the representation of discontinuities in displacement fields and the description of cracks and their evolution in materials, which the classical partial differential equation (PDE) models tend to fail to apply. However, the PD models yield numerical methods with dense stiffness matrices which requires O(N2) memory and O(N3) computational complexity where N is the number of spatial unknowns. Consequently, the PD models are deemed to be computationally very expensive especially for problems in multiple space dimensions. State-based PD models, which were developed lately, can be treated as a great improvement of the previous bond-based PD models. The state-based PD models have more complicated structures than the bond-based PD models. In this paper we develop a fast collocation method for a state-based linear PD model by exploring the structure of the stiffness matrix of the numerical method. The method has an O(N) memory requirement and computational complexity of O(NlogN) per Krylov subspace iteration. Numerical methods are presented to show the utility of the method.

  • Keywords

State-based peridynamic model, fast algorithm, collocation method, stiffness matrix.

  • AMS Subject Headings

65F10, 65M70, 65T50, 74B15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

duning@sdu.edu.cn (Ning Du)

guoxu5861@gmail.com (Xu Guo)

hwang@math.sc.edu (Hong Wang)

  • BibTex
  • RIS
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@Article{CiCP-27-274, author = {Du , Ning and Guo , Xu and Wang , Hong}, title = {A Fast State-Based Peridynamic Numerical Model}, journal = {Communications in Computational Physics}, year = {2019}, volume = {27}, number = {1}, pages = {274--291}, abstract = {

The peridynamic (PD) theory is a reformulation of the classical theory of continuum solid mechanics and is particularly suitable for the representation of discontinuities in displacement fields and the description of cracks and their evolution in materials, which the classical partial differential equation (PDE) models tend to fail to apply. However, the PD models yield numerical methods with dense stiffness matrices which requires O(N2) memory and O(N3) computational complexity where N is the number of spatial unknowns. Consequently, the PD models are deemed to be computationally very expensive especially for problems in multiple space dimensions. State-based PD models, which were developed lately, can be treated as a great improvement of the previous bond-based PD models. The state-based PD models have more complicated structures than the bond-based PD models. In this paper we develop a fast collocation method for a state-based linear PD model by exploring the structure of the stiffness matrix of the numerical method. The method has an O(N) memory requirement and computational complexity of O(NlogN) per Krylov subspace iteration. Numerical methods are presented to show the utility of the method.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2018-0288}, url = {http://global-sci.org/intro/article_detail/cicp/13322.html} }
TY - JOUR T1 - A Fast State-Based Peridynamic Numerical Model AU - Du , Ning AU - Guo , Xu AU - Wang , Hong JO - Communications in Computational Physics VL - 1 SP - 274 EP - 291 PY - 2019 DA - 2019/10 SN - 27 DO - http://doi.org/10.4208/cicp.OA-2018-0288 UR - https://global-sci.org/intro/article_detail/cicp/13322.html KW - State-based peridynamic model, fast algorithm, collocation method, stiffness matrix. AB -

The peridynamic (PD) theory is a reformulation of the classical theory of continuum solid mechanics and is particularly suitable for the representation of discontinuities in displacement fields and the description of cracks and their evolution in materials, which the classical partial differential equation (PDE) models tend to fail to apply. However, the PD models yield numerical methods with dense stiffness matrices which requires O(N2) memory and O(N3) computational complexity where N is the number of spatial unknowns. Consequently, the PD models are deemed to be computationally very expensive especially for problems in multiple space dimensions. State-based PD models, which were developed lately, can be treated as a great improvement of the previous bond-based PD models. The state-based PD models have more complicated structures than the bond-based PD models. In this paper we develop a fast collocation method for a state-based linear PD model by exploring the structure of the stiffness matrix of the numerical method. The method has an O(N) memory requirement and computational complexity of O(NlogN) per Krylov subspace iteration. Numerical methods are presented to show the utility of the method.

Ning Du, Xu Guo & Hong Wang. (2019). A Fast State-Based Peridynamic Numerical Model. Communications in Computational Physics. 27 (1). 274-291. doi:10.4208/cicp.OA-2018-0288
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