Robert Lipton

Ivo Babuska, Robert Lipton

The paper addresses a numerical method for solving second order elliptic partial differential equations that describe fields inside heterogeneous media. The scope is general and treats the case of rough coefficients, i.e. coefficients with values in $L^\infty(Ω)$. This class of coefficients includes as examples media with micro-structure as well as media with multiple non-separated length scales. The approach taken here is based on the the generalized finite element method (GFEM) introduced in \cite{107}, and elaborated in \cite{102}, \cite{103} and \cite{104}. The GFEM is constructed by partitioning the computational domain $Ω$ into to a collection of preselected subsets $ω_{i},i=1,2,..m$ and constructing finite dimensional approximation spaces $Ψ_{i}$ over each subset using local information. The notion of the Kolmogorov $n$-width is used to identify the optimal local approximation spaces. These spaces deliver local approximations with errors that decay almost exponentially with the degrees of freedom $N_{i}$ in the energy norm over $ω_i$. The local spaces $% Ψ_{i}$ are used within the GFEM scheme to produce a finite dimensional subspace $S^N$ of $H^{1}(Ω)$ which is then employed in the Galerkin method. It is shown that the error in the Galerkin approximation decays in the energy norm almost exponentially (i.e., super-algebraicly) with respect to the degrees of freedom $N$. When length scales "`separate" and the microstructure is sufficiently fine with respect to the length scale of the domain $ω_i$ it is shown that homogenization theory can be used to construct local approximation spaces with exponentially decreasing error in the pre-asymtotic regime.

Prashant K. Jha, Robert Lipton

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 $γ\in (0,1]$. The rate of convergence of the finite difference approximation depends on the factor $C_s h^γ/ε^2$ where $ε$ gives the length scale of nonlocal interaction, $h$ is the discretization length and $C_s$ 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.

Prashant K. Jha, Robert Lipton

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 state-based perydynamic model using two potentials: one associated with hydrostatic strain and the other associated with tensile strain. We show that the dynamic evolution is well-posed in the space of Hölder continuous functions $C^{0,γ}$ with Hölder exponent $γ\in (0,1]$. Here the length scale of nonlocality is $ε$, the size of time step is $Δt$ and the mesh size is $h$. The finite difference approximations are seen to converge to the Hölder solution at the rate $C_t Δt + C_s h^γ/ε^2$ where the constants $C_t$ and $C_s$ are independent of the discretization. The semi-discrete approximations are found to be stable with time. We present numerical simulations for crack propagation that computationally verify the theoretically predicted convergence rate. We also present numerical simulations for crack propagation in precracked samples subject to a bending load.

Prashant K. Jha, Robert Lipton

We quantify the numerical error and modeling error associated with replacing a nonlinear nonlocal bond-based peridynamic model with a local elasticity model or a linearized peridynamics model away from the fracture set. The nonlocal model treated here is characterized by a double well potential and is a smooth version of the peridynamic model introduced in n Silling (J Mech Phys Solids 48(1), 2000). The solutions of nonlinear peridynamics are shown to converge to the solution of linear elastodynamics at a rate linear with respect to the length scale $ε$ of non local interaction. This rate also holds for the convergence of solutions of the linearized peridynamic model to the solution of the local elastodynamic model. For local linear Lagrange interpolation the consistency error for the numerical approximation is found to depend on the ratio between mesh size $h$ and $ε$. More generally for local Lagrange interpolation of order $p\geq 1$ the consistency error is of order $h^p/ε$. A new stability theory for the time discretization is provided and an explicit generalization of the CFL condition on the time step and its relation to mesh size $h$ is given. Numerical simulations are provided illustrating the consistency error associated with the convergence of nonlinear and linearized peridynamics to linear elastodynamics.

Prashant K. Jha, Robert Lipton

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 and the change in volume inside the domain of nonlocal interaction. The pairwise interactions between points are mediated by a bond potential of multi-well type while multi point interactions are associated with volume change mediated by a hydrostatic strain potential. The hydrostatic potential can either be a quadratic function, delivering a linear force-strain relation, or a multi-well type that can be associated with material degradation and cavitation. We first show the well-posedness of the peridynamic formulation and that peridynamic evolutions exist in the Sobolev space $H^2$. We show that the finite element approximations converge to the $H^2$ solutions uniformly as measured in the mean square norm. For linear continuous finite elements the convergence rate is shown to be $C_t Δt + C_s h^2/ε^2$, where $ε$ is the size of horizon, $h$ is the mesh size, and $Δt$ is the size of time step. The constants $C_t$ and $C_s$ are independent of $Δt$ and $h$ and may depend on $ε$ through the norm of the exact solution. We demonstrate the stability of the semi-discrete approximation. The stability of the fully discrete approximation is shown for the linearized peridynamic force. We present numerical simulations with dynamic crack propagation that support the theoretical convergence rate.

Prashant K. Jha, Robert Lipton

We consider nonlocal nonlinear potentials and estimate the rate of convergence of time stepping schemes to the peridynamic equation of motion. We begin by establishing the existence of $H^2$ solutions over any finite time interval. Here spatial approximation by finite element interpolations are considered. The energy stability of the associated semi-discrete time stepping scheme is established and the approximation of strong and weak formulations of the evolution using FE interpolations of $H^2$ solutions are investigated. The strong and weak form of approximations are shown to converge to the actual solution in the mean square norm at the rate $C_tΔt +C_s h^2/ε^2$ where $h$ is the mesh size, $ε$ is the size of nonlocal interaction and $Δt$ is the time step. The constants $C_t$ and $C_s$ are independent of $Δt$, and $h$. In the absence of nonlinearity a CFL like condition for the energy stability of the central difference time discretization scheme is developed.

Robert Lipton, Paul Sinz, Michael Stuebner

We introduce a systematic method for identifying the worst case load among all boundary loads of fixed energy. Here the worst case load is defined to be the one that delivers the largest fraction of input energy to a prescribed subdomain of interest. The worst case load is identified with the first eigenfunction of a suitably defined eigenvalue problem. The first eigenvalue for this problem is the maximum fraction of boundary energy that can be delivered to the subdomain. We compute worst case boundary loads and associated energy contained inside a prescribed subdomain through the numerical solution of the eigenvalue problem. We apply this computational method to bound the worst case load associated with an ensemble of random boundary loads given by a second order random process. Several examples are carried out on heterogeneous structures to illustrate the method.