Robert Lipton
14 papers 243 citations h-index 4 2010–2024
Nodal finite element approximation of peridynamics
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-based discrete element method (PeriDEM) model of granular systems involving breakage of arbitrarily shaped particles
Usage, manipulation, transport, delivery, and mixing of granular or particulate media, comprised of spherical or polyhedral particles, is commonly encountered in industrial sectors of construction...
A Multiscale Fracture Model using Peridynamic Enrichment of Finite Elements within an Adaptive Partition of Unity: Experimental Validation
Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Within the PUM global-local enrichment scheme [1,...
A Fracture Multiscale Model for Peridynamic enrichment within the Partition of Unity Method
Partition of unity methods (PUM) are of domain decomposition type and provide the opportunity for multiscale and multiphysics numerical modeling. Different physical models can exist within a PUM...
Quasistatic fracture evolution
Nonlocal quasistatic fracture evolution for interacting cracks is developed and supporting numerical examples are presented. The approach is implicit and is based on local stationarity and fixed...
Multiscale-Spectral GFEM and Optimal Oversampling
In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [I. Babu\v{s}ka and R. Lipton, Multiscale Modeling and Simulation 9 (2011), pp. 373--406]....
An asynchronous and task-based implementation of Peridynamics utilizing HPX -- the C++ standard library for parallelism and concurrency
On modern supercomputers, asynchronous many task systems are emerging to address the new architecture of computational nodes. Through this shift of increasing cores per node, a new programming...
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...
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...
Finite element approximation of nonlocal fracture models
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 $H^2$...
Numerical convergence of nonlinear nonlocal continuum models to local elastodynamics
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...
Numerical analysis of nonlocal fracture models in H\"older 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...
Uncertain Loading and Quantifying Maximum Energy Concentration within Composite Structures
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...
Optimal Local Approximation Spaces for Generalized Finite Element Methods with Application to Multiscale Problems
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...