fem2d_heat

fem2d_heat, a C++ code which applies the finite element method (FEM) to solve a form of the time-dependent heat equation over an arbitrary triangulated region.

The computational region is initially unknown by the program. The user specifies it by preparing a file containing the coordinates of the nodes, and a file containing the indices of nodes that make up triangles that form a triangulation of the region.

Normally, the user does not type in this information by hand, but has a program fill in the nodes, and perhaps another program that constructs the triangulation. However, in the simplest case, the user might construct a very crude triangulation by hand, and have TRIANGULATION_REFINE refine it to something more reasonable.

For the following ridiculously small example:

       10-11-12
        |\   |\
        | \  | \
        6  7 8  9
        |   \|   \
        1-2--3--4-5
      

         0.0  0.0
         1.0  0.0
         2.0  0.0
         3.0  0.0
         4.0  0.0
         0.0  1.0
         1.0  1.0
         2.0  1.0
         3.0  1.0
         0.0  2.0
         1.0  2.0
         2.0  2.0
      

         1  3 10  2  7  6
         3  5 12  4  9  8
        12 10  3 11  7  8
      

The program is set up to handle the time dependent heat equation with a right hand side function, and nonhomogeneous Dirichlet boundary conditions. The state variable U(T,X,Y) is then constrained by:

        Ut - ( Uxx + Uyy ) + K(x,y,t) * U = F(x,y,t)  in the region
                                        U = G(x,y,t)  on the boundary
                                        U = H(x,y,t)  at initial time TINIT.
      

To specify the right hand side function F(x,y,t), the linear coefficient K(x,y,t), the boundary condition function G(x,y,t), and the initial condition H(x,y,t), the user has to supply a file, perhaps called myprog.C, containing several functions:

• double rhs ( int node_num, double node_xy[], double time ) evaluates the right hand side forcing term F(x,y,t).
• double k_coef ( int node_num, double node_xy[], double time ) evaluates K(x,y,t);
• double *dirichlet_condition ( int node_num, double node_xy[], double time ) evaluates G(x,y,t) for all nodes on the boundary;
• double *initial_condition ( int node_num, double node_xy[], double time ) evaluates H(x,y,t) for all nodes at the initial time.

The program is also able to write out a file containing the solution value at every node. This file may be used to create contour plots of the solution.

Usage:

g++ fem2d_heat.cpp myprog.cpp
mv a.out fem2d_heat
fem2d_heat prefix

• "prefix"_nodes.txt, contains the node coordinates.
• "prefix"_elements.txt, contains the indices of nodes that form elements.

Licensing:

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

Languages:

fem2d_heat is available in a C++ version and a FORTRAN90 version and a MATLAB version.

Related Programs:

FD2D_HEAT_STEADY, a C++ code which uses the finite difference method (FDM) to solve the steady (time independent) heat equation in 2D.

FEM1D_HEAT_STEADY, a C++ code which uses the finite element method to solve the steady (time independent) heat equation in 1D.

fem2d_heat_test

STOCHASTIC_HEAT2D, a C++ code which implements a finite difference method (FDM) for the steady (time independent) 2D heat equation, with a stochastic heat diffusivity coefficient, using gnuplot to illustrate the results.

Reference:

1. Hans Rudolf Schwarz,
   Finite Element Methods,
   Academic Press, 1988,
   ISBN: 0126330107,
   LC: TA347.F5.S3313.
2. Gilbert Strang, George Fix,
   An Analysis of the Finite Element Method,
   Cambridge, 1973,
   ISBN: 096140888X,
   LC: TA335.S77.
3. Olgierd Zienkiewicz,
   The Finite Element Method,
   Sixth Edition,
   Butterworth-Heinemann, 2005,
   ISBN: 0750663200,
   LC: TA640.2.Z54

Source Code:

• fem2d_heat.cpp, the source code;
• fem2d_heat.sh, compiles the source code;