FEM2D_HEAT_RECTANGLE\
Finite Element Solution in 2D\
Time Dependent Heat Equation {#fem2d_heat_rectangle-finite-element-solution-in-2d-time-dependent-heat-equation align=”center”}
==============================
FEM2D_HEAT_RECTANGLE is a C++ program which solves the
time-dependent 2D heat equation using the finite element method in
space, and a method of lines in time with the backward Euler
approximation for the time derivative.
The computational region is a rectangle, with homogenous Dirichlet
boundary conditions applied along the boundary. The state variable
U(X,Y,T) is then constrained by:
Ut - ( Uxx + Uyy ) = F(x,y,t) in the box;
U(x,y,t) = G(x,y,t) for (x,y) on the boundary;
U(x,y,t) = H(x,y,t) for t = t_init.
The computational region is first covered with an NX by NY rectangular
array of points, creating (NX-1)*(NY-1) subrectangles. Each
subrectangle is divided into two triangles, creating a total of
2*(NX-1)*(NY-1) geometric “elements”. Because quadratic basis
functions are to be used, each triangle will be associated not only with
the three corner nodes that defined it, but with three extra midside
nodes. If we include these additional nodes, there are now a total of
(2*NX-1)*(2*NY-1) nodes in the region.
We now assume that, at any fixed time b, the unknown function U(x,y,t)
can be represented as a linear combination of the basis functions
associated with each node. The value of U at the boundary nodes is
obvious, so we concentrate on the NUNK interior nodes where U(x,y,t) is
unknown. For each node I, we determine a basis function PHI(I)(x,y), and
evaluate the following finite element integral:
Integral ( Ux(x,y,t) * dPHIdx(I)(x,y) + dUdy(x,y,t) * dPHIdy(I)(x,y) ) =
Integral ( F(x,y,t) * PHI(I)(x,y)
The time derivative is handled by the backward Euler approximation.
The program allows the user to supply two routines:
- rhs ( x, y, time ) returns the right hand side F(x,y,time) of
the heat equation.
- exact_u ( node_num, node_xy, time, u_exact ) returns the
exact solution U_EXACT evaluated at each of the NODE_NUM
points whose coordinates are stored in
NODE_XY(1:2,1:NODE_NUM), at time TIME.
There are a few variables that are easy to manipulate. In particular,
the user can change the variables NX and NY in the main program, to
change the number of nodes and elements. The variables (XL,YB) and
(XR,YT) define the location of the lower left and upper right corners of
the rectangular region, and these can also be changed in a single place
in the main program.
The program writes out a file containing an Encapsulated PostScript
image of the nodes and elements, with numbers. Unfortunately, for values
of NX and NY over 10, the plot is too cluttered to read. For lower
values, however, it is a valuable map of what is going on in the
geometry.
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.
Licensing: {#licensing align=”center”}
The computer code and data files described and made available on this
web page are distributed under the GNU LGPL
license.
Languages: {#languages align=”center”}
FEM2D_HEAT_RECTANGLE is available in a C++
version
and a FORTRAN90
version and
a MATLAB
version.
FEM2D_HEAT, a C++ program
which solves the time dependent heat equation on an arbitrary
triangulated region in 2D.
Author: {#author align=”center”}
Janet Peterson.
Reference: {#reference align=”center”}
- Hans Rudolf Schwarz,\
Finite Element Methods,\
Academic Press, 1988,\
ISBN: 0126330107,\
LC: TA347.F5.S3313..
- Gilbert Strang, George Fix,\
An Analysis of the Finite Element Method,\
Cambridge, 1973,\
ISBN: 096140888X,\
LC: TA335.S77.
- Olgierd Zienkiewicz,\
The Finite Element Method,\
Sixth Edition,\
Butterworth-Heinemann, 2005,\
ISBN: 0750663200,\
LC: TA640.2.Z54
Source Code: {#source-code align=”center”}
Examples and Tests: {#examples-and-tests align=”center”}
Data files created by the program:
- rectangle_output.txt, the printed output
from a run;
- fem2d_heat_rectangle_nodes.png, a PNG
image of the nodes, for NX = NY = 7 (the picture is hard to read for
larger values of NX and NY);
- rectangle_nodes.txt, a text file containing
a list, for each node, of its X and Y coordinates;
- rectangle_elements.png, a PNG image of
the elements, for NX = NY = 7 (the picture is hard to read for
larger values of NX and NY);
- rectangle_elements.txt, a text file
containing a list, for each element, of the six nodes that compose
it;
- rectangle_time.txt, a text file containing
the solution times;
- rectangle_u0000.txt, the solution U at time
step 0;
- rectangle_u0001.txt, the solution U at time
step 1;
- rectangle_u0002.txt, the solution U at time
step 2;
- rectangle_u0003.txt, the solution U at time
step 3;
- rectangle_u0004.txt, the solution U at time
step 4;
- rectangle_u0005.txt, the solution U at time
step 5;
- rectangle_u0006.txt, the solution U at time
step 6;
- rectangle_u0007.txt, the solution U at time
step 7;
- rectangle_u0008.txt, the solution U at time
step 8;
- rectangle_u0009.txt, the solution U at time
step 9;
- rectangle_u0010.txt, the solution U at time
step 10;
The MATLAB program CONTOUR_SEQUENCE4 can make contour plots from
the sequence of solutions:
- rectangle_u0000.png, the solution U at time
step 0;
- rectangle_u0001.png, the solution U at time
step 1;
- rectangle_u0002.png, the solution U at time
step 2;
- rectangle_u0003.png, the solution U at time
step 3;
- rectangle_u0004.png, the solution U at time
step 4;
- rectangle_u0005.png, the solution U at time
step 5;
- rectangle_u0006.png, the solution U at time
step 6;
- rectangle_u0007.png, the solution U at time
step 7;
- rectangle_u0008.png, the solution U at time
step 8;
- rectangle_u0009.png, the solution U at time
step 9;
- rectangle_u0010.png, the solution U at time
step 10;
List of Routines: {#list-of-routines align=”center”}
- MAIN is the main routine of the finite element program
FEM2D_HEAT_RECTANGLE.
- ADJUST_BACKWARD_EULER adjusts the system for the backward
Euler term.
- ADJUST_BOUNDARY modifies the linear system for boundary
conditions.
- AREA_SET sets the area of each element.
- ASSEMBLE assembles the matrix and right-hand side using
piecewise quadratics.
- BANDWIDTH determines the bandwidth of the coefficient matrix.
- COMPARE compares the exact and computed solution at the nodes.
- DGB_FA performs a LINPACK-style PLU factorization of an DGB
matrix.
- DGB_PRINT_SOME prints some of a DGB matrix.
- DGB_SL solves a system factored by DGB_FA.
- ELEMENT_WRITE writes the elements to a file.
- ERRORS calculates the error in the L2 and H1-seminorm.
- EXACT_U calculates the exact solution and its first
derivatives.
- FILE_NAME_INC increments a partially numeric file name.
- GRID_T6 produces a grid of pairs of 6 node triangles.
- I4_MAX returns the maximum of two ints.
- I4_MIN returns the smaller of two ints.
- I4VEC_PRINT_SOME prints “some” of an I4VEC.
- NODE_BOUNDARY_SET assigns an unknown value index at each node.
- NODES_PLOT plots a pointset.
- NODES_WRITE writes the nodes to a file.
- QBF evaluates the quadratic basis functions.
- QUAD_A sets the quadrature rule for assembly.
- QUAD_E sets a quadrature rule for the error calculation.
- R8_HUGE returns a “huge” R8.
- R8_MAX returns the maximum of two R8’s.
- R8_MIN returns the minimum of two R8’s.
- R8_NINT returns the nearest integer to an R8.
- R8VEC_PRINT_SOME prints “some” of an R8VEC.
- RHS gives the right-hand side of the differential equation.
- S_LEN_TRIM returns the length of a string to the last
nonblank.
- SOLUTION_WRITE writes the solution to a file.
- TIMESTAMP prints the current YMDHMS date as a time stamp.
- TRIANGULATION_ORDER6_PLOT plots a 6-node triangulation of a
pointset.
- XY_SET sets the XY coordinates of the nodes.
You can go up one level to the C++ source codes.
Last revised on 06 January 2011.