Numerical Analysis of Nonlocal Fracture Models in Hölder Space

Abstract

In this work, we calculate the convergence rate of the finite difference approximation for a class of nonlocal fracture models. We consider two point force interactions characterized by a double well potential. We show the existence of a evolving displacement field in Hölder space with Hölder exponent γ ∈ (0, 1]. The rate of convergence of the finite difference approximation depends on the factor Csh γ / 2 where gives the length scale of nonlocal interaction, h is the discretization length and Cs is the maximum of Hölder norm of the solution and its second derivatives during the evolution. It is shown that the rate of convergence holds for both the forward Euler scheme as well as general single step implicit schemes. A stability result is established for the semi-discrete approximation. The Hölder continuous evolutions are seen to converge to a brittle fracture evolution in the limit of vanishing nonlocality.

Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics

“…However, it is found that the semidiscrete approximation of the nonlinear model is stable in the energy norm, see our other work. 25 In Section 5, we present numerical simulations that confirm the error estimates for both linearized and nonlinear peridynamics. The numerical experiments show that the discretization error can be reduced by choosing the ratio h∕ suitably small for every choice of as → 0, see Figure 4.…”

“…Earlier related work analyzed the model considered here but for less regular nondifferentiable Hölder continuous solutions. For that case, solutions can approach discontinuous deformations (fracture‐like solutions) as ε →0 and it is shown that the numerical approximation of the nonlinear model in dimension d =1,2,3 converges to the exact solution at the rate O (Δ t + h γ / ε 2 ), where γ ∈(0,1] is the Hölder exponent, h is the size of mesh, ε is the size of horizon, and Δ t is the size of time step.…”

“…One no longer has an explicit stability condition for the nonlinear model. However, it is found that the semidiscrete approximation of the nonlinear model is stable in the energy norm, see our other work …”

“…Earlier related work analyzes the nonlinear model and establishes the existence of nondifferentiable Hölder continuous solutions. It is shown there that the rate of convergence of the discrete model to the continuum nonlocal model is of the order h γ / ε , where 0< γ ≤1 is the Hölder exponent.…”

“…The limiting displacement field evolution has bounded Griffith fracture energy and away from the fracture set satisfies classic local elastodynamics. [23][24][25] This model motivates adaptive implementations of peridynamics for brittle fracture. In regions of the body where brittle fracture is anticipated, one would apply the nonlinear nonlocal model, but in regions where no fracture is to be anticipated, one would like to apply the linear elastic model.…”