# fem2d_heat

fem2d_heat, a FORTRAN90 code which applies the finite element method to solve a form of the time-dependent heat equation over an arbitrary triangulated region.

The computational region is initially unknown by the code. 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 code fill in the nodes, and perhaps another code 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
```
the node file would be:
```         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
```
and the triangle file would be
```         1  3 10  2  7  6
3  5 12  4  9  8
12 10  3 11  7  8
```

The code 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 containing four subroutines,

• SUBROUTINE RHS ( N, X, Y, TIME, U ) evaluates the right hand side forcing term F(x,y,t);
• SUBROUTINE K_COEF ( N, X, Y, TIME, U ) evaluates Kx,y,t);
• SUBROUTINE DIRICHLET_CONDITION ( N, X, Y, TIME, U ) evaluates G(x,y,t), and is only called at nodes on the boundary;
• SUBROUTINE INITIAL_CONDITION ( N, X, Y, TIME, U ) evaluates H(x,y,t), and is only called for TIME = TINIT.

To run the code, the user compiles the user routines, links them with FEM2D_HEAT, and runs the executable.

The code writes out a file containing an Encapsulated PostScript image of the nodes and elements, with numbers. If there are too many nodes, the plot may be too cluttered to read. For lower values, however, it is a valuable map of what is going on in the geometry.

The code 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.

### Languages:

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

### Related codes:

FD2D_HEAT_STEADY, a FORTRAN90 code which uses the finite difference method (FDM) to solve the steady (time independent) heat equation in 2D.

FEM1D_HEAT_STEADY, a FORTRAN90 code which uses the finite element method to solve the steady (time independent) heat equation in 1D.

FEM2D_HEAT_RECTANGLE, a FORTRAN90 code which solves the 2D time dependent heat equation on the unit square, using a uniform grid of triangular elements.

FEM2D_HEAT_SQUARE, a FORTRAN90 code which defines the geometry of a square region, as well as boundary and initial conditions for a given heat problem, and is called by FEM2D_HEAT as part of a solution procedure.

STOCHASTIC_HEAT2D, a FORTRAN90 code which implements a finite difference method (FDM) for the steady (time independent) 2D heat equation, with a stochastic heat diffusivity coefficient.

### Reference:

1. Hans Rudolf Schwarz,
Finite Element Methods,
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:

Last revised on 08 July 2020.