Numerical Methods

Finite Element Approximation of Nonlocal Dynamic Fracture Models

In this work we estimate the convergence rate for time stepping schemes applied to nonlocal dynamic fracture modeling. Here we use the nonlocal formulation given by the bond based peridynamic equation of motion. We begin by establishing the existence …

Finite Element Convergence for State-Based Peridynamic Fracture Models

We establish the a priori convergence rate for finite element approximations of a class of nonlocal nonlinear fracture models. We consider state-based peridynamic models where the force at a material point is due to both the strain between two points …

Numerical Convergence of Finite Difference Approximations for State-Based Peridynamic Fracture Models

In this work, we study the finite difference approximation for a class of nonlocal fracture models. The nonlocal model is initially elastic but beyond a critical strain the material softens with increasing strain. This model is formulated as a …

Finite Differences and Finite Elements in Nonlocal Fracture Modeling: A Priori Convergence Rates

In this chapter we present a rigorous convergence analysis of finite difference and finite element approximation of nonlinear nonlocal models. In the previous chapter, we considered a differentiable version of the original bond-based model introduced …

Well-Posed Nonlinear Nonlocal Fracture Models Associated with Double-Well Potentials

In this chapter, we consider a generic class of bond-based nonlocal nonlinear potentials and formulate the evolution over suitable function spaces. The peridynamic potential considered in this work is a differentiable version of the original …

Numerical Convergence of Nonlinear Nonlocal Continuum Models to Local Elastodynamics

We quantify the numerical error and modeling error associated with replac- ing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamic model away from the fracture set. The non- local model treated …

Numerical Analysis of Nonlocal Fracture Models in Holder Space

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 …