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Nodal finite element approximation of peridynamics

Prashant K. Jha, Patrick Diehl, Robert Lipton

Abstract

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.

Nodal finite element approximation of peridynamics

Abstract

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 norm at the rate . Here, , , and are time step size, mesh size, and the size of the horizon or nonlocal length scale, respectively. Constants and are independent of and 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.
Paper Structure (27 sections, 4 theorems, 101 equations, 22 figures, 2 tables, 1 algorithm)

This paper contains 27 sections, 4 theorems, 101 equations, 22 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Outline of the article.
  3. Bond-based peridynamics
  4. Peridynamics equation of motion using the RNP model
  5. Finite element approximation
  6. Projection onto $V_h$
  7. Clément interpolation
  8. Nodal finite element approximation
  9. Comparison of NFEA with the standard FEA
  10. A-priori convergence of nodal FEA for nonlinear peridynamics models
  11. Key estimates
  12. Proof
  13. A-priori convergence
  14. An alternate nodal formulation based on the Clément interpolation
  15. Asymptotic compatibility of Clément NFEA
  16. ...and 12 more sections

Key Result

Lemma 4.2

Consistency of the peridynamics force For $\boldsymbol{u}^k$ in $H^2(D)\cap C^2(D)$, the following estimates hold Here, $c_i$, $i=1,2,3,4$, are constants depending only on the triangulation $\mathcal{T}_h$, see eq:interpolationerror, eq:interpolationerrorpointwise, eq:l2L2relation. Moreover, $L_i$, $i=1,2,3$, are constants that only depend on the influence function $J$ and the peridynamics force

Figures (22)

  • Figure 1: Profile of pairwise potential $\mathcal{W}^\epsilon(S)$ defined in \ref{['eq:rnppotential']}. Here $C = \lim_{S \to \infty}\mathcal{W}^\epsilon(S)$.
  • Figure 2: (a) Prototype microelastic brittle (PMB) material. Here, $f_{pmb}(S)$ is a scalar such that the bond force is $\boldsymbol{f}_{pmb}(\boldsymbol{y}, \boldsymbol{x}, t;\boldsymbol{u}) = f_{pmb}(S)\frac{\boldsymbol{y} - \boldsymbol{x}}{|\boldsymbol{y} - \boldsymbol{x}|}$; see \ref{['eq:pmbbondforcesmall']}. (b) Regularized nonlinear peridynamics (RNP) material.
  • Figure 3: Typical mesh node $\boldsymbol{x}_i$ and one of the neighboring nodes $\boldsymbol{x}_j$ in an example 2-d finite element mesh. All the red nodes contribute to the force at $\boldsymbol{x}_i$. The set $E_j = \{e^j_k\}_{k=1}^5$ of elements with the node $\boldsymbol{x}_j$ as the vertex is shown in grey.
  • Figure 4: (a) Convergence test: Setup. The horizon is fixed to $\epsilon = 0.05$ m. (b) A representative view of the mesh consisting of linear triangle elements. (c) Rate of convergence at discrete times using \ref{['eq:convergencerate']}.
  • Figure 5: Convergence test: Comparing the magnitude of displacement at the final simulation time $t_F$ for all four meshes. Unless otherwise stated, all the plots, including this figure, are in the deformed configuration and based on the finite element representation.
  • ...and 17 more figures

Theorems & Definitions (9)

  • Remark 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Remark 5.2
  • Theorem 6.1
  • Remark 6.2
  • Definition 7.1: Damage
  • Remark 7.2