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A high-order multi-time-step scheme for bond-based peridynamics

Chenguang Liu, Jie Sun, Hao Tian, WaiSun Don, Lili Ju

TL;DR

This work develops a high-order multi-time-step (MTS) scheme for bond-based peridynamics to efficiently handle crack discontinuities. By coupling meshfree spatial discretization with explicit Runge-Kutta time integrators and using coarse and fine time steps in different subdomains, the method concentrates temporal resolution where needed while reducing overall cost. The authors prove that, under smoothness, the MTS schemes achieve global convergence orders of 3 or 4 and validate these results with 2D and 3D numerical experiments, showing accuracy comparable to undecomposed PD but with substantial speedups. The approach enables scalable, crack-aware PD simulations suitable for large-scale problems in fracture mechanics.

Abstract

A high-order multi-time-step (MTS) scheme for the bond-based peridynamic (PD) model, an extension of classical continuous mechanics widely used for analyzing discontinuous problems like cracks, is proposed. The MTS scheme discretizes the spatial domain with a meshfree method and advances in time with a high-order Runge-Kutta method. To effectively handle discontinuities (cracks) that appear in a local subdomain in the solution, the scheme employs the Taylor expansion and Lagrange interpolation polynomials with a finer time step size, that is, coarse and fine time step sizes for smooth and discontinuous subdomains, respectively, to achieve accurate and efficient simulations. By eliminating unnecessary fine-scale resolution imposed on the entire domain, the MTS scheme outperforms the standard PD scheme by significantly reducing computational costs, particularly for problems with discontinuous solutions, as demonstrated by comprehensive theoretical analysis and numerical experiments.

A high-order multi-time-step scheme for bond-based peridynamics

TL;DR

This work develops a high-order multi-time-step (MTS) scheme for bond-based peridynamics to efficiently handle crack discontinuities. By coupling meshfree spatial discretization with explicit Runge-Kutta time integrators and using coarse and fine time steps in different subdomains, the method concentrates temporal resolution where needed while reducing overall cost. The authors prove that, under smoothness, the MTS schemes achieve global convergence orders of 3 or 4 and validate these results with 2D and 3D numerical experiments, showing accuracy comparable to undecomposed PD but with substantial speedups. The approach enables scalable, crack-aware PD simulations suitable for large-scale problems in fracture mechanics.

Abstract

A high-order multi-time-step (MTS) scheme for the bond-based peridynamic (PD) model, an extension of classical continuous mechanics widely used for analyzing discontinuous problems like cracks, is proposed. The MTS scheme discretizes the spatial domain with a meshfree method and advances in time with a high-order Runge-Kutta method. To effectively handle discontinuities (cracks) that appear in a local subdomain in the solution, the scheme employs the Taylor expansion and Lagrange interpolation polynomials with a finer time step size, that is, coarse and fine time step sizes for smooth and discontinuous subdomains, respectively, to achieve accurate and efficient simulations. By eliminating unnecessary fine-scale resolution imposed on the entire domain, the MTS scheme outperforms the standard PD scheme by significantly reducing computational costs, particularly for problems with discontinuous solutions, as demonstrated by comprehensive theoretical analysis and numerical experiments.
Paper Structure (11 sections, 1 theorem, 13 equations, 7 figures, 6 tables)

This paper contains 11 sections, 1 theorem, 13 equations, 7 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Fundamentals of bond-based peridynamics and its discretization
  3. Temporal discretization schemes based on the Runge-Kutta methods
  4. Multi-time-step schemes
  5. The third and fourth order multi-time-step scheme
  6. Convergence analysis
  7. Numerical results
  8. Plate under transverse loading
  9. Block under transverse loading
  10. Plate with a pre-existing crack
  11. Conclusions

Key Result

Theorem 1

If the operator $\mathcal{L}$ is $C^5$, the global error of the r-order MTS scheme under the uniform time step size $\Delta t$ is $O(\Delta t^{r})$, where $\mathbf{U}(\mathbf{x},t)$ is the exact solution of the semi-discrete system in pd:dis1. This holds for $r=3,4$.

Figures (7)

  • Figure 1: Different time step size for the subdomains $\hat{\Omega}_{F}$ (fine time step size $\Delta t_k$) and $\hat{\Omega}_{C}$ (coarse time step size $\Delta t$).
  • Figure 2: Four subdomains of the physical domain $\Omega$. The subdomain $\Omega_{F}$, BL subdomain $\Omega_{FI}$, BL subdomain $\Omega_{CI}$, and subdomain$\Omega_{C}$ are colored in yellow, blue, red, and green, respectively. Here $\hat{\Omega}_C=\Omega_{C}\cup\Omega_{CI}$ and $\hat{\Omega}_F=\Omega_{F}\cup\Omega_{FI}$.
  • Figure 3: Geometry of a 2D plate under transverse loading and its discretization.
  • Figure 4: Geometry of a 3D plate under transverse loading and its discretization.
  • Figure 5: Geometry of a 2D plate with a pre-existing crack .
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof