Meshfree quadrature rule

This repository contains the code for the paper

• An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics

There are five test problems: static nonlocal diffusion(converge to nonlocal limit), static nonlocal diffusion(converge to local limit), dynamic nonlocal diffusion, static bond-based peridynamics, and Kalthoff-Winkler experiment. The first four problems focus on heterogeneous materials. For KW experiment we're interested in homogeneous case. All the problems are in two-dimensional physical space.

Requirements

• gcc 7.5.0 or newer
• Install BLAS and LAPACK packages, you may need to change Makefile to direct to BLAS and LAPACK libraries.

Files

• nonlocaldiff_static.cpp is the code for the static nonlocal diffusion problem with Dirichlet boundary condition.
• nonlocaldiff.cpp is the code for the dynamic nonlocal diffusion problem with Dirichlet boundary condition and backward Euler scheme.
• PMB_2Dweight.cpp is the code for the static bond-based peridynamics with Dirichlet boundary condition.
• KW_2Dweight_dynamic.cpp is the code for the Kalthoff-Winkler experiment simulation.

Quick start

static nonlocal diffusion problems (converging to local limit)

compile the run code with the following command:

1. make Nldiff
2. ./nldiff.ex <number of particles><ratio of delta/h><poly_order><case>

Example:

./nldiff.ex 20 3.5 3 1

dynamic nonlocal diffusion problems

compile the run code with the following command:

1. make Nldiffd
2. ./nldiffd.ex <number of particles><ratio of delta/h><poly_order><dt><timestep>

Example:

./nldiffd.ex 20 3.5 3 0.01 10

static bond-based peridynamics

compile the run code with the following command:

1. make PD
2. ./PMB_2d.ex <number of particles><ratio of delta/h><poly_order><random perturbation coefficient>

Example:

./PMB_2d.ex 20 3.5 3 0.0

Kalthoff-Winkler experiment

compile the run code with the following command:

1. make KW
2. ./KW.ex <number of particles><ratio of delta/h><poly_order><dt><timestep>

Example:

./KW.ex 32 3.0 3 2e-4 1000

Citations

If you find this code or method useful for your project, please cite

• An asymptotically compatible meshfree quadrature rule for nonlocal problems with applications to peridynamics
• An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems
• An asymptotically compatible formulation for local-to-nonlocal coupling problems without overlapping regions