forked from johannesgerer/jburkardt-cpp
-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathfem2d_heat_rectangle.html
459 lines (423 loc) · 13.8 KB
/
fem2d_heat_rectangle.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
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
<html>
<head>
<title>
FEM2D_HEAT_RECTANGLE - Finite Element Solution of the 2D Heat Equation
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
FEM2D_HEAT_RECTANGLE <br>
Finite Element Solution in 2D <br>
Time Dependent Heat Equation
</h1>
<hr>
<p>
<b>FEM2D_HEAT_RECTANGLE</b>
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.
</p>
<p>
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:
<pre>
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.
</pre>
</p>
<p>
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.
</p>
<p>
We now assume that, at any fixed time <t>b</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:
<pre>
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)
</pre>
</p>
<p>
The time derivative is handled by the backward Euler approximation.
</p>
<p>
The program allows the user to supply two routines:
<ul>
<li>
<b>rhs ( x, y, time )</b> returns the right hand side
F(x,y,time) of the heat equation.
</li>
<li>
<b>exact_u ( node_num, node_xy, time, u_exact )</b>
returns the exact solution <b>U_EXACT</b> evaluated at each of
the <b>NODE_NUM</b> points whose coordinates are stored in
<b>NODE_XY(1:2,1:NODE_NUM)</b>, at time <b>TIME</b>.
</li>
</ul>
</p>
<p>
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.
</p>
<p>
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.
</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">
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>
<b>FEM2D_HEAT_RECTANGLE</b> is available in
<a href = "../../cpp_src/fem2d_heat_rectangle/fem2d_heat_rectangle.html">a C++ version</a> and
<a href = "../../f_src/fem2d_heat_rectangle/fem2d_heat_rectangle.html">a FORTRAN90 version</a> and
<a href = "../../m_src/fem2d_heat_rectangle/fem2d_heat_rectangle.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/fem2d_heat/fem2d_heat.html">
FEM2D_HEAT</a>,
a C++ program which
solves the
time dependent heat equation on an arbitrary triangulated
region in 2D.
</p>
<h3 align = "center">
Author:
</h3>
<p>
Janet Peterson.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313..
</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,<br>
LC: TA640.2.Z54
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "fem2d_heat_rectangle.cpp">fem2d_heat_rectangle.cpp</a>, the source code;
</li>
<li>
<a href = "fem2d_heat_rectangle.sh">fem2d_heat_rectangle.sh</a>,
commands to compile and run the program;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
Data files created by the program:
<ul>
<li>
<a href = "rectangle_output.txt">rectangle_output.txt</a>,
the printed output from a run;
</li>
<li>
<a href = "rectangle_nodes.png">fem2d_heat_rectangle_nodes.png</a>,
a PNG image of
the nodes, for NX = NY = 7 (the picture is
hard to read for larger values of NX and NY);
</li>
<li>
<a href = "rectangle_nodes.txt">rectangle_nodes.txt</a>,
a text file containing a list, for each node, of its X and Y
coordinates;
</li>
<li>
<a href = "rectangle_elements.png">rectangle_elements.png</a>,
a PNG image of
the elements, for NX = NY = 7 (the picture is
hard to read for larger values of NX and NY);
</li>
<li>
<a href = "rectangle_elements.txt">rectangle_elements.txt</a>,
a text file containing a list, for each element, of the six
nodes that compose it;
</li>
<li>
<a href = "rectangle_time.txt">rectangle_time.txt</a>,
a text file containing the solution times;
</li>
<li>
<a href = "rectangle_u0000.txt">rectangle_u0000.txt</a>,
the solution U at time step 0;
</li>
<li>
<a href = "rectangle_u0001.txt">rectangle_u0001.txt</a>,
the solution U at time step 1;
</li>
<li>
<a href = "rectangle_u0002.txt">rectangle_u0002.txt</a>,
the solution U at time step 2;
</li>
<li>
<a href = "rectangle_u0003.txt">rectangle_u0003.txt</a>,
the solution U at time step 3;
</li>
<li>
<a href = "rectangle_u0004.txt">rectangle_u0004.txt</a>,
the solution U at time step 4;
</li>
<li>
<a href = "rectangle_u0005.txt">rectangle_u0005.txt</a>,
the solution U at time step 5;
</li>
<li>
<a href = "rectangle_u0006.txt">rectangle_u0006.txt</a>,
the solution U at time step 6;
</li>
<li>
<a href = "rectangle_u0007.txt">rectangle_u0007.txt</a>,
the solution U at time step 7;
</li>
<li>
<a href = "rectangle_u0008.txt">rectangle_u0008.txt</a>,
the solution U at time step 8;
</li>
<li>
<a href = "rectangle_u0009.txt">rectangle_u0009.txt</a>,
the solution U at time step 9;
</li>
<li>
<a href = "rectangle_u0010.txt">rectangle_u0010.txt</a>,
the solution U at time step 10;
</li>
</ul>
</p>
<p>
The MATLAB program <b>CONTOUR_SEQUENCE4</b> can make contour
plots from the sequence of solutions:
<ul>
<li>
<a href = "rectangle_u0000.png">rectangle_u0000.png</a>,
the solution U at time step 0;
</li>
<li>
<a href = "rectangle_u0001.png">rectangle_u0001.png</a>,
the solution U at time step 1;
</li>
<li>
<a href = "rectangle_u0002.png">rectangle_u0002.png</a>,
the solution U at time step 2;
</li>
<li>
<a href = "rectangle_u0003.png">rectangle_u0003.png</a>,
the solution U at time step 3;
</li>
<li>
<a href = "rectangle_u0004.png">rectangle_u0004.png</a>,
the solution U at time step 4;
</li>
<li>
<a href = "rectangle_u0005.png">rectangle_u0005.png</a>,
the solution U at time step 5;
</li>
<li>
<a href = "rectangle_u0006.png">rectangle_u0006.png</a>,
the solution U at time step 6;
</li>
<li>
<a href = "rectangle_u0007.png">rectangle_u0007.png</a>,
the solution U at time step 7;
</li>
<li>
<a href = "rectangle_u0008.png">rectangle_u0008.png</a>,
the solution U at time step 8;
</li>
<li>
<a href = "rectangle_u0009.png">rectangle_u0009.png</a>,
the solution U at time step 9;
</li>
<li>
<a href = "rectangle_u0010.png">rectangle_u0010.png</a>,
the solution U at time step 10;
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>MAIN</b> is the main routine of the finite element program FEM2D_HEAT_RECTANGLE.
</li>
<li>
<b>ADJUST_BACKWARD_EULER</b> adjusts the system for the backward Euler term.
</li>
<li>
<b>ADJUST_BOUNDARY</b> modifies the linear system for boundary conditions.
</li>
<li>
<b>AREA_SET</b> sets the area of each element.
</li>
<li>
<b>ASSEMBLE</b> assembles the matrix and right-hand side using piecewise quadratics.
</li>
<li>
<b>BANDWIDTH</b> determines the bandwidth of the coefficient matrix.
</li>
<li>
<b>COMPARE</b> compares the exact and computed solution at the nodes.
</li>
<li>
<b>DGB_FA</b> performs a LINPACK-style PLU factorization of an DGB matrix.
</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>ELEMENT_WRITE</b> writes the elements to a file.
</li>
<li>
<b>ERRORS</b> calculates the error in the L2 and H1-seminorm.
</li>
<li>
<b>EXACT_U</b> calculates the exact solution and its first derivatives.
</li>
<li>
<b>FILE_NAME_INC</b> increments a partially numeric file name.
</li>
<li>
<b>GRID_T6</b> produces a grid of pairs of 6 node triangles.
</li>
<li>
<b>I4_MAX</b> returns the maximum of two ints.
</li>
<li>
<b>I4_MIN</b> returns the smaller of two ints.
</li>
<li>
<b>I4VEC_PRINT_SOME</b> prints "some" of an I4VEC.
</li>
<li>
<b>NODE_BOUNDARY_SET</b> assigns an unknown value index at each node.
</li>
<li>
<b>NODES_PLOT</b> plots a pointset.
</li>
<li>
<b>NODES_WRITE</b> writes the nodes to a file.
</li>
<li>
<b>QBF</b> evaluates the quadratic basis functions.
</li>
<li>
<b>QUAD_A</b> sets the quadrature rule for assembly.
</li>
<li>
<b>QUAD_E</b> sets a quadrature rule for the error calculation.
</li>
<li>
<b>R8_HUGE</b> returns a "huge" R8.
</li>
<li>
<b>R8_MAX</b> returns the maximum of two R8's.
</li>
<li>
<b>R8_MIN</b> returns the minimum of two R8's.
</li>
<li>
<b>R8_NINT</b> returns the nearest integer to an R8.
</li>
<li>
<b>R8VEC_PRINT_SOME</b> prints "some" of an R8VEC.
</li>
<li>
<b>RHS</b> gives the right-hand side of the differential equation.
</li>
<li>
<b>S_LEN_TRIM</b> returns the length of a string to the last nonblank.
</li>
<li>
<b>SOLUTION_WRITE</b> writes the solution to a file.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
<li>
<b>TRIANGULATION_ORDER6_PLOT</b> plots a 6-node triangulation of a pointset.
</li>
<li>
<b>XY_SET</b> sets the XY coordinates of the nodes.
</li>
</ul>
<p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 06 January 2011.
</i>
<!-- John Burkardt -->
</body>
</html>