1 static char help[] = "Solves biharmonic equation in 1d.\n";
2
3 /*
4 Solves the equation biharmonic equation in split form
5
6 w = -kappa \Delta u
7 u_t = \Delta w
8 -1 <= u <= 1
9 Periodic boundary conditions
10
11 Evolve the biharmonic heat equation with bounds: (same as biharmonic)
12 ---------------
13 ./biharmonic3 -ts_monitor -snes_monitor -ts_monitor_draw_solution -pc_type lu -draw_pause .1 -snes_converged_reason -ts_type beuler -da_refine 5 -draw_fields 1 -ts_time_step 9.53674e-9
14
15 w = -kappa \Delta u + u^3 - u
16 u_t = \Delta w
17 -1 <= u <= 1
18 Periodic boundary conditions
19
20 Evolve the Cahn-Hillard equations:
21 ---------------
22 ./biharmonic3 -ts_monitor -snes_monitor -ts_monitor_draw_solution -pc_type lu -draw_pause .1 -snes_converged_reason -ts_type beuler -da_refine 6 -draw_fields 1 -kappa .00001 -ts_time_step 5.96046e-06 -cahn-hillard
23
24 */
25 #include <petscdm.h>
26 #include <petscdmda.h>
27 #include <petscts.h>
28 #include <petscdraw.h>
29
30 /*
31 User-defined routines
32 */
33 extern PetscErrorCode FormFunction(TS, PetscReal, Vec, Vec, Vec, void *), FormInitialSolution(DM, Vec, PetscReal);
34 typedef struct {
35 PetscBool cahnhillard;
36 PetscReal kappa;
37 PetscInt energy;
38 PetscReal tol;
39 PetscReal theta;
40 PetscReal theta_c;
41 } UserCtx;
42
main(int argc,char ** argv)43 int main(int argc, char **argv)
44 {
45 TS ts; /* nonlinear solver */
46 Vec x, r; /* solution, residual vectors */
47 Mat J; /* Jacobian matrix */
48 PetscInt steps, Mx;
49 DM da;
50 MatFDColoring matfdcoloring;
51 ISColoring iscoloring;
52 PetscReal dt;
53 PetscReal vbounds[] = {-100000, 100000, -1.1, 1.1};
54 SNES snes;
55 UserCtx ctx;
56
57 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
58 Initialize program
59 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
60 PetscFunctionBeginUser;
61 PetscCall(PetscInitialize(&argc, &argv, NULL, help));
62 ctx.kappa = 1.0;
63 PetscCall(PetscOptionsGetReal(NULL, NULL, "-kappa", &ctx.kappa, NULL));
64 ctx.cahnhillard = PETSC_FALSE;
65 PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL));
66 PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds));
67 PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600));
68 ctx.energy = 1;
69 PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL));
70 ctx.tol = 1.0e-8;
71 PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL));
72 ctx.theta = .001;
73 ctx.theta_c = 1.0;
74 PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL));
75 PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL));
76
77 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
78 Create distributed array (DMDA) to manage parallel grid and vectors
79 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
80 PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da));
81 PetscCall(DMSetFromOptions(da));
82 PetscCall(DMSetUp(da));
83 PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx"));
84 PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u"));
85 PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
86 dt = 1.0 / (10. * ctx.kappa * Mx * Mx * Mx * Mx);
87
88 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
89 Extract global vectors from DMDA; then duplicate for remaining
90 vectors that are the same types
91 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
92 PetscCall(DMCreateGlobalVector(da, &x));
93 PetscCall(VecDuplicate(x, &r));
94
95 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
96 Create timestepping solver context
97 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
98 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
99 PetscCall(TSSetDM(ts, da));
100 PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
101 PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx));
102 PetscCall(TSSetMaxTime(ts, .02));
103 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
104
105 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106 Create matrix data structure; set Jacobian evaluation routine
107
108 < Set Jacobian matrix data structure and default Jacobian evaluation
109 routine. User can override with:
110 -snes_mf : matrix-free Newton-Krylov method with no preconditioning
111 (unless user explicitly sets preconditioner)
112 -snes_mf_operator : form matrix used to construct the preconditioner as set by the user,
113 but use matrix-free approx for Jacobian-vector
114 products within Newton-Krylov method
115
116 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
117 PetscCall(TSGetSNES(ts, &snes));
118 PetscCall(SNESSetType(snes, SNESVINEWTONRSLS));
119 PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring));
120 PetscCall(DMSetMatType(da, MATAIJ));
121 PetscCall(DMCreateMatrix(da, &J));
122 PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring));
123 PetscCall(MatFDColoringSetFunction(matfdcoloring, (MatFDColoringFn *)SNESTSFormFunction, ts));
124 PetscCall(MatFDColoringSetFromOptions(matfdcoloring));
125 PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring));
126 PetscCall(ISColoringDestroy(&iscoloring));
127 PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring));
128
129 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
130 Customize nonlinear solver
131 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
132 PetscCall(TSSetType(ts, TSBEULER));
133
134 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
135 Set initial conditions
136 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
137 PetscCall(FormInitialSolution(da, x, ctx.kappa));
138 PetscCall(TSSetTimeStep(ts, dt));
139 PetscCall(TSSetSolution(ts, x));
140
141 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
142 Set runtime options
143 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
144 PetscCall(TSSetFromOptions(ts));
145
146 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
147 Solve nonlinear system
148 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
149 PetscCall(TSSolve(ts, x));
150 PetscCall(TSGetStepNumber(ts, &steps));
151
152 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
153 Free work space. All PETSc objects should be destroyed when they
154 are no longer needed.
155 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
156 PetscCall(MatDestroy(&J));
157 PetscCall(MatFDColoringDestroy(&matfdcoloring));
158 PetscCall(VecDestroy(&x));
159 PetscCall(VecDestroy(&r));
160 PetscCall(TSDestroy(&ts));
161 PetscCall(DMDestroy(&da));
162
163 PetscCall(PetscFinalize());
164 return 0;
165 }
166
167 typedef struct {
168 PetscScalar w, u;
169 } Field;
170 /* ------------------------------------------------------------------- */
171 /*
172 FormFunction - Evaluates nonlinear function, F(x).
173
174 Input Parameters:
175 . ts - the TS context
176 . X - input vector
177 . ptr - optional user-defined context, as set by SNESSetFunction()
178
179 Output Parameter:
180 . F - function vector
181 */
FormFunction(TS ts,PetscReal ftime,Vec X,Vec Xdot,Vec F,void * ptr)182 PetscErrorCode FormFunction(TS ts, PetscReal ftime, Vec X, Vec Xdot, Vec F, void *ptr)
183 {
184 DM da;
185 PetscInt i, Mx, xs, xm;
186 PetscReal hx, sx;
187 PetscScalar r, l;
188 Field *x, *xdot, *f;
189 Vec localX, localXdot;
190 UserCtx *ctx = (UserCtx *)ptr;
191
192 PetscFunctionBegin;
193 PetscCall(TSGetDM(ts, &da));
194 PetscCall(DMGetLocalVector(da, &localX));
195 PetscCall(DMGetLocalVector(da, &localXdot));
196 PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));
197
198 hx = 1.0 / (PetscReal)Mx;
199 sx = 1.0 / (hx * hx);
200
201 /*
202 Scatter ghost points to local vector,using the 2-step process
203 DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
204 By placing code between these two statements, computations can be
205 done while messages are in transition.
206 */
207 PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
208 PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
209 PetscCall(DMGlobalToLocalBegin(da, Xdot, INSERT_VALUES, localXdot));
210 PetscCall(DMGlobalToLocalEnd(da, Xdot, INSERT_VALUES, localXdot));
211
212 /*
213 Get pointers to vector data
214 */
215 PetscCall(DMDAVecGetArrayRead(da, localX, &x));
216 PetscCall(DMDAVecGetArrayRead(da, localXdot, &xdot));
217 PetscCall(DMDAVecGetArray(da, F, &f));
218
219 /*
220 Get local grid boundaries
221 */
222 PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
223
224 /*
225 Compute function over the locally owned part of the grid
226 */
227 for (i = xs; i < xs + xm; i++) {
228 f[i].w = x[i].w + ctx->kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;
229 if (ctx->cahnhillard) {
230 switch (ctx->energy) {
231 case 1: /* double well */
232 f[i].w += -x[i].u * x[i].u * x[i].u + x[i].u;
233 break;
234 case 2: /* double obstacle */
235 f[i].w += x[i].u;
236 break;
237 case 3: /* logarithmic */
238 if (x[i].u < -1.0 + 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogScalar(ctx->tol) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u;
239 else if (x[i].u > 1.0 - 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogScalar(ctx->tol)) + ctx->theta_c * x[i].u;
240 else f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u;
241 break;
242 case 4:
243 break;
244 }
245 }
246 f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx;
247 if (ctx->energy == 4) {
248 f[i].u = xdot[i].u;
249 /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */
250 r = (1.0 - x[i + 1].u * x[i + 1].u) * (x[i + 2].w - x[i].w) * .5 / hx;
251 l = (1.0 - x[i - 1].u * x[i - 1].u) * (x[i].w - x[i - 2].w) * .5 / hx;
252 f[i].u -= (r - l) * .5 / hx;
253 f[i].u += 2.0 * ctx->theta_c * x[i].u * (x[i + 1].u - x[i - 1].u) * (x[i + 1].u - x[i - 1].u) * .25 * sx - (ctx->theta - ctx->theta_c * (1 - x[i].u * x[i].u)) * (x[i + 1].u + x[i - 1].u - 2.0 * x[i].u) * sx;
254 }
255 }
256
257 /*
258 Restore vectors
259 */
260 PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot));
261 PetscCall(DMDAVecRestoreArrayRead(da, localX, &x));
262 PetscCall(DMDAVecRestoreArray(da, F, &f));
263 PetscCall(DMRestoreLocalVector(da, &localX));
264 PetscCall(DMRestoreLocalVector(da, &localXdot));
265 PetscFunctionReturn(PETSC_SUCCESS);
266 }
267
268 /* ------------------------------------------------------------------- */
FormInitialSolution(DM da,Vec X,PetscReal kappa)269 PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa)
270 {
271 PetscInt i, xs, xm, Mx, xgs, xgm;
272 Field *x;
273 PetscReal hx, xx, r, sx;
274 Vec Xg;
275
276 PetscFunctionBegin;
277 PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));
278
279 hx = 1.0 / (PetscReal)Mx;
280 sx = 1.0 / (hx * hx);
281
282 /*
283 Get pointers to vector data
284 */
285 PetscCall(DMCreateLocalVector(da, &Xg));
286 PetscCall(DMDAVecGetArray(da, Xg, &x));
287
288 /*
289 Get local grid boundaries
290 */
291 PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
292 PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL));
293
294 /*
295 Compute u function over the locally owned part of the grid including ghost points
296 */
297 for (i = xgs; i < xgs + xgm; i++) {
298 xx = i * hx;
299 r = PetscSqrtReal((xx - .5) * (xx - .5));
300 if (r < .125) x[i].u = 1.0;
301 else x[i].u = -.50;
302 /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
303 x[i].w = 0;
304 }
305 for (i = xs; i < xs + xm; i++) x[i].w = -kappa * (x[i - 1].u + x[i + 1].u - 2.0 * x[i].u) * sx;
306
307 /*
308 Restore vectors
309 */
310 PetscCall(DMDAVecRestoreArray(da, Xg, &x));
311
312 /* Grab only the global part of the vector */
313 PetscCall(VecSet(X, 0));
314 PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X));
315 PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X));
316 PetscCall(VecDestroy(&Xg));
317 PetscFunctionReturn(PETSC_SUCCESS);
318 }
319
320 /*TEST
321
322 build:
323 requires: !complex !single
324
325 test:
326 args: -ts_monitor -snes_monitor -pc_type lu -snes_converged_reason -ts_type beuler -da_refine 5 -ts_time_step 9.53674e-9 -ts_max_steps 50
327 requires: x
328
329 TEST*/
330