Nodal finite element approximation of peridynamics
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912130799763456 |
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| author | Jha, Prashant K. Diehl, Patrick Lipton, Robert |
| author_facet | Jha, Prashant K. Diehl, Patrick Lipton, Robert |
| contents | This work considers the nodal finite element approximation of peridynamics, in which the nodal displacements satisfy the peridynamics equation at each mesh node. For the nonlinear bond-based peridynamics model, it is shown that, under the suitable assumptions on an exact solution, the discretized solution associated with the central-in-time and nodal finite element discretization converges to the exact solution in $L^2$ norm at the rate $C_1 Δt + C_2 h^2/ε^2$. Here, $Δt$, $h$, and $ε$ are time step size, mesh size, and the size of the horizon or nonlocal length scale, respectively. Constants $C_1$ and $C_2$ are independent of $h$ and $Δt$ and depend on norms of the solution and nonlocal length scale. Several numerical examples involving pre-crack, void, and notch are considered, and the efficacy of the proposed nodal finite element discretization is analyzed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_05501 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nodal finite element approximation of peridynamics Jha, Prashant K. Diehl, Patrick Lipton, Robert Numerical Analysis 65M12, 65M60, 65M15, 74A70 This work considers the nodal finite element approximation of peridynamics, in which the nodal displacements satisfy the peridynamics equation at each mesh node. For the nonlinear bond-based peridynamics model, it is shown that, under the suitable assumptions on an exact solution, the discretized solution associated with the central-in-time and nodal finite element discretization converges to the exact solution in $L^2$ norm at the rate $C_1 Δt + C_2 h^2/ε^2$. Here, $Δt$, $h$, and $ε$ are time step size, mesh size, and the size of the horizon or nonlocal length scale, respectively. Constants $C_1$ and $C_2$ are independent of $h$ and $Δt$ and depend on norms of the solution and nonlocal length scale. Several numerical examples involving pre-crack, void, and notch are considered, and the efficacy of the proposed nodal finite element discretization is analyzed. |
| title | Nodal finite element approximation of peridynamics |
| topic | Numerical Analysis 65M12, 65M60, 65M15, 74A70 |
| url | https://arxiv.org/abs/2403.05501 |