PERIDYNAMICS_1D_STEADY
Steady 1D Poisson Equation
Nonlocal Peridynamics Model

PERIDYNAMICS_1D_STEADY a MATLAB library which solves a 1D steady version of the Poisson equation, using the non-local peridynamics model, by Marta D'Elia.

The problem data is specified by a user-supplied file which evaluates:

• the exact solution u(x);
• the derivative of the exact solution u'(x).
• the right hand side of the Poisson equation, -u"(x).
• the lifting, a function which adjusts the problem for nonzero Dirichlet boundary conditions;

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Related Data and Programs:

PERI1D, a C program which sets up and solves a 1D time-dependent peridynamics problem, by Miroslav Stoyanov;

Author:

Marta D'Elia

Reference:

1. Qiang Du, Max Gunzburger, Rich Lehoucq, Kun Zhou,
   Analysis and approximation of nonlocal diffusion problems with volume constraints,
   SIAM Review,
   Volume 54, Number 4, pages 667-696, 2012.
2. Max Gunzburger, Rich Lehoucq,
   A nonlocal vector calculus with application to nonlocal boundary value problems,
   Multiscale Modeling and Simulation,
   Volume 8, Number 5, 2010, pages 1581-1598.

Source Code:

• nonlocal_diffusion.m, solves the steady 1D Poisson equation for given problem data.

Examples and Tests:

• pd1d_steady_test.m, calls both convergence tests for all problems. You probably don't want to call this function, but it shows you how to call either test below for any of the five sample problems.
• pd1d_steady_test01.m, for a given problem (1, 2, 3, 4 or 5), this test computes convergence results as EPSILON and H go to zero in fixed ratio. You specify the problem as an input quantity that is a MATLAB function handle, such as "@problem1".
• pd1d_steady_test02.m, for a given problem (1, 2, 3, 4 or 5), this test computes convergence results for fixed EPSILON, while H goes to zero. You specify the problem as an input quantity that is a MATLAB function handle, such as "@problem5".

PROBLEM 1 has the solution U(X) = X^2, on the domain [0,1].

• problem1.m, specifies the data for problem 1.
• problem1_output.txt, the output file.
• problem1_plot1.png, convergence results as EPSILON and H go to zero in fixed ratio.
• problem1_plot2.png, convergence results for fixed EPSILON, while H goes to zero.
• problem1_plot3.png, comparison of computed and exact solutions.

PROBLEM 2 has the solution U(X) = X^2*(1-X^2), on the domain [0,1].

• problem2.m, specifies the data for problem 2.
• problem2_output.txt, the output file.
• problem2_plot1.png, convergence results as EPSILON and H go to zero in fixed ratio.
• problem2_plot2.png, convergence results for fixed EPSILON, while H goes to zero.
• problem2_plot3.png, comparison of computed and exact solutions.

PROBLEM 3 has the solution U(X) = X^2, on the domain [0,1].

• problem3.m, specifies the data for problem 3.
• problem3_output.txt, the output file.
• problem3_plot1.png, convergence results as EPSILON and H go to zero in fixed ratio.
• problem3_plot2.png, convergence results for fixed EPSILON, while H goes to zero.
• problem3_plot3.png, comparison of computed and exact solutions.

PROBLEM 4 has the solution U(X)=X-1/4 left of 1/2, and U(X)=X-1/2 to the right of 1/2, on the domain [0,1]. This problem was devised to study the behavior of singularities.

• problem4.m, specifies the data for problem 4.
• problem4_output.txt, the output file.
• problem4_plot1.png, convergence results as EPSILON and H go to zero in fixed ratio.
• problem4_plot2.png, convergence results for fixed EPSILON, while H goes to zero.
• problem4_plot3.png, comparison of computed and exact solutions.

PROBLEM 5 has the solution U(X) = 1+X, on the domain [0,1]. This problem was devised for a simple accuracy check.

• problem5.m, specifies the data for problem 5.
• problem5_output.txt, the output file.
• problem5_plot1.png, convergence results as EPSILON and H go to zero in fixed ratio.
• problem5_plot2.png, convergence results for fixed EPSILON, while H goes to zero.
• problem5_plot3.png, comparison of computed and exact solutions.

You can go up one level to the MATLAB source codes.