forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
Copy pathfem2d_poisson.html
406 lines (370 loc) · 11.4 KB
/
fem2d_poisson.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
<html>
<head>
<title>
FEM2D_POISSON - Finite Element Solution on Arbitrary 2D Region
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_POISSON <br>
Finite Element Solution of Poisson's Equation <br>
on a Triangulated Region
</h1>
<hr>
<p>
<b>FEM2D_POISSON</b>
is a FORTRAN90 program which
applies the finite element method to solve
a form of Poisson's equation over an arbitrary triangulated region.
</p>
<p>
The computational region is 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.
</p>
<p>
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 <a href = "../../f_src/triangulation_refine/triangulation_refine.html">
TRIANGULATION_REFINE</a> refine it to something more reasonable.
</p>
<p>
For the following ridiculously small example:
<pre>
4----5
|\ |\
| \ | \
| \ | \
| \| \
1----2----3
</pre>
the node file would be:
<pre>
0.0 0.0
1.0 0.0
2.0 0.0
0.0 1.0
1.0 1.0
</pre>
and the triangle file would be
<pre>
1 2 4
5 4 2
2 3 5
</pre>
</p>
<p>
The program is set up to handle the linear Poisson
equation with a right hand side function, and nonhomogeneous
Dirichlet boundary conditions. The state variable
U(X,Y) is then constrained by:
<pre>
- Del H(x,y) Del U(x,y) + K(x,y) * U(x,y) = F(x,y) inside the region;
U(x,y) = G(x,y) on the boundary.
</pre>
</p>
<p>
An improved version of the program would
handle a more interesting nonlinear PDE, and include optional Neumann
boundary conditions.
</p>
<p>
To specify the boundary condition function G(x,y),
the coefficients H(x,y) and K(x,y) and the right hand side function F(x,y),
the user has to modify a file containing three subroutines,
<ul>
<li>
<b>SUBROUTINE DIRICHLET_CONDITION ( NODE_NUM, NODE_XY, NODE_BC )</b>
evaluates G(X,Y).
</li>
<li>
<b>SUBROUTINE H_COEF ( NODE_NUM, NODE_XY, NODE_H )</b>
evaluates H(x,y).
</li>
<li>
<b>SUBROUTINE K_COEF ( NODE_NUM, NODE_XY, NODE_K )</b>
evaluates K(x,y).
</li>
<li>
<b>SUBROUTINE RHS ( NODE_NUM, NODE_XY, NODE_F )</b>
evaluates F(x,y).
</li>
</ul>
</p>
<p>
To run the program, the user compiles the user routines,
links them with <b>FEM2D_POISSON</b>, and runs the executable.
</p>
<p>
The program 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.
</p>
<p>
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.
</p>
<h3 align = "center">
Usage:
</h3>
<p>
The user must create an executable by compiling the user routines
and linking them with the main program, perhaps by commands like:
<pre>
gfortran -c fem2d_poisson.f90
gfortran -c user.f90
gfortran fem2d_poisson.o user.o
mv a.out fem2d_poisson
</pre>
</p>
<p>
Assuming the executable program is called "fem2d_poisson", then
the program is executed by
<blockquote>
<b>fem2d_poisson</b> <i>prefix</i>
</blockquote>
where prefix is the common filename prefix, so that
<ul>
<li>
<i>prefix</i><b>_nodes.txt</b>, is a file containing the node coordinates;
</li>
<li>
<i>prefix</i><b>_elements.txt</b>, is a file listing the 3 nodes that
make up each element;
</li>
</ul>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
FEM2D_POISSON is available in
<a href = "../../cpp_src/fem2d_poisson/fem2d_poisson.html">a C++ version</a> and
<a href = "../../f_src/fem2d_poisson/fem2d_poisson.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_poisson/fem2d_poisson.html">a MATLAB version.</a>
</p>
<h3 align = "center">
Related Programs:
</h3>
<p>
<a href = "../../f_src/fem2d_poisson_ell/fem2d_poisson_ell.html">
FEM2D_POISSON_ELL</a>,
a FORTRAN90 library which
defines the geometry of an L-shaped region, as well as boundary
conditions for a given Poisson problem, and is called by FEM2D_POISSON
as part of a solution procedure.
</p>
<p>
<a href = "../../f_src/fem2d_poisson_lake/fem2d_poisson_lake.html">
FEM2D_POISSON_LAKE</a>,
a FORTRAN90 library which
defines the geometry of a lake-shaped region, as well as boundary
conditions for a given Poisson problem, and is called by FEM2D_POISSON
as part of a solution procedure.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Hans Rudolf Schwarz,<br>
Methode der Finiten Elemente,<br>
Teubner Studienbuecher, 1980,<br>
ISBN: 3-519-02349-0.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_poisson.f90">fem2d_poisson.f90</a>,
the source code;
</li>
<li>
<a href = "fem2d_poisson.sh">fem2d_poisson.sh</a>,
commands to compile the partial program;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main program of FEM2D_POISSON.
</li>
<li>
<b>ASSEMBLE_POISSON</b> assembles the system for the Poisson equation.
</li>
<li>
<b>BANDWIDTH</b> determines the bandwidth of the coefficient matrix.
</li>
<li>
<b>BASIS_ONE_T3</b> evaluates a linear basis function.
</li>
<li>
<b>CH_CAP</b> capitalizes a single character.
</li>
<li>
<b>CH_EQI</b> is a case insensitive comparison of two characters for equality.
</li>
<li>
<b>CH_TO_DIGIT</b> returns the integer value of a base 10 digit.
</li>
<li>
<b>DGB_FA</b> performs a LINPACK-style PLU factorization of an DGB matrix.
</li>
<li>
<b>DGB_MXV</b> multiplies a DGB matrix times a vector.
</li>
<li>
<b>DGB_PRINT_SOME</b> prints some of a DGB matrix.
</li>
<li>
<b>DGB_SL</b> solves a system factored by DGB_FA.
</li>
<li>
<b>DIRICHLET_APPLY</b> accounts for Dirichlet boundary conditions.
</li>
<li>
<b>FILE_COLUMN_COUNT</b> counts the number of columns in the first line of a file.
</li>
<li>
<b>FILE_ROW_COUNT</b> counts the number of row records in a file.
</li>
<li>
<b>GET_UNIT</b> returns a free FORTRAN unit number.
</li>
<li>
<b>I4_MODP</b> returns the nonnegative remainder of integer division.
</li>
<li>
<b>I4_WRAP</b> forces an integer to lie between given limits by wrapping.
</li>
<li>
<b>I4COL_COMPARE</b> compares columns I and J of an I4COL.
</li>
<li>
<b>I4COL_SORT_A</b> ascending sorts an I4COL.
</li>
<li>
<b>I4COL_SWAP</b> swaps columns I and J of an I4COL.
</li>
<li>
<b>I4MAT_TRANSPOSE_PRINT_SOME</b> prints some of the transpose of an I4mat.
</li>
<li>
<b>I4MAT_DATA_READ</b> reads data from an I4MAT file.
</li>
<li>
<b>I4MAT_HEADER_READ</b> reads the header from an I4MAT.
</li>
<li>
<b>LVEC_PRINT</b> prints a logical vector.
</li>
<li>
<b>POINTS_PLOT</b> plots a pointset.
</li>
<li>
<b>QUAD_RULE</b> sets the quadrature rule for assembly.
</li>
<li>
<b>R8MAT_DATA_READ</b> reads data from an R8MAT file.
</li>
<li>
<b>R8MAT_HEADER_READ</b> reads the header from an R8MAT file.
</li>
<li>
<b>R8MAT_TRANSPOSE_PRINT_SOME</b> prints some of an R8MAT, transposed.
</li>
<li>
<b>R8MAT_WRITE</b> writes an R8MAT file.
</li>
<li>
<b>R8VEC_PRINT_SOME</b> prints "some" of an R8VEC.
</li>
<li>
<b>REFERENCE_TO_PHYSICAL_T3</b> maps reference points to physical points.
</li>
<li>
<b>RESIDUAL_POISSON</b> evaluates the residual for the Poisson equation.
</li>
<li>
<b>S_TO_I4</b> reads an I4 from a string.
</li>
<li>
<b>S_TO_I4VEC</b> reads an I4VEC from a string.
</li>
<li>
<b>S_TO_R8</b> reads an R8 from a string.
</li>
<li>
<b>S_TO_R8VEC</b> reads an R8VEC from a string.
</li>
<li>
<b>S_WORD_COUNT</b> counts the number of "words" in a string.
</li>
<li>
<b>SOLUTION_EVALUATE</b> evaluates the solution at a point in an element.
</li>
<li>
<b>SORT_HEAP_EXTERNAL</b> externally sorts a list of items into ascending order.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGLE_AREA_2D</b> computes the area of a triangle in 2D.
</li>
<li>
<b>TRIANGULATION_ORDER3_BOUNDARY_NODE</b> indicates which nodes are on the boundary.
</li>
<li>
<b>TRIANGULATION_ORDER3_PLOT</b> plots a 3-node triangulation of a pointset.
</li>
</ul>
<p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 01 January 2011.
</i>
<!-- John Burkardt -->
</body>
</html>