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 ./biharmonic2 -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: (this fails after a few timesteps 12/17/2017)
21 ---------------
22 ./biharmonic2 -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
66 PetscCall(PetscOptionsGetBool(NULL, NULL, "-cahn-hillard", &ctx.cahnhillard, NULL));
67 PetscCall(PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 2, vbounds));
68 PetscCall(PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD), 600, 600));
69 ctx.energy = 1;
70 /*PetscCall(PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL));*/
71 PetscCall(PetscOptionsGetInt(NULL, NULL, "-energy", &ctx.energy, NULL));
72 ctx.tol = 1.0e-8;
73 PetscCall(PetscOptionsGetReal(NULL, NULL, "-tol", &ctx.tol, NULL));
74 ctx.theta = .001;
75 ctx.theta_c = 1.0;
76 PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta", &ctx.theta, NULL));
77 PetscCall(PetscOptionsGetReal(NULL, NULL, "-theta_c", &ctx.theta_c, NULL));
78
79 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
80 Create distributed array (DMDA) to manage parallel grid and vectors
81 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
82 PetscCall(DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10, 2, 2, NULL, &da));
83 PetscCall(DMSetFromOptions(da));
84 PetscCall(DMSetUp(da));
85 PetscCall(DMDASetFieldName(da, 0, "Biharmonic heat equation: w = -kappa*u_xx"));
86 PetscCall(DMDASetFieldName(da, 1, "Biharmonic heat equation: u"));
87 PetscCall(DMDAGetInfo(da, 0, &Mx, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0));
88 dt = 1.0 / (10. * ctx.kappa * Mx * Mx * Mx * Mx);
89
90 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
91 Extract global vectors from DMDA; then duplicate for remaining
92 vectors that are the same types
93 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
94 PetscCall(DMCreateGlobalVector(da, &x));
95 PetscCall(VecDuplicate(x, &r));
96
97 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
98 Create timestepping solver context
99 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
100 PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
101 PetscCall(TSSetDM(ts, da));
102 PetscCall(TSSetProblemType(ts, TS_NONLINEAR));
103 PetscCall(TSSetIFunction(ts, NULL, FormFunction, &ctx));
104 PetscCall(TSSetMaxTime(ts, .02));
105 PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_INTERPOLATE));
106
107 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
108 Create matrix data structure; set Jacobian evaluation routine
109
110 < Set Jacobian matrix data structure and default Jacobian evaluation
111 routine. User can override with:
112 -snes_mf : matrix-free Newton-Krylov method with no preconditioning
113 (unless user explicitly sets preconditioner)
114 -snes_mf_operator : form matrix used to construct the preconditioner as set by the user,
115 but use matrix-free approx for Jacobian-vector
116 products within Newton-Krylov method
117
118 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
119 PetscCall(TSGetSNES(ts, &snes));
120 PetscCall(DMCreateColoring(da, IS_COLORING_GLOBAL, &iscoloring));
121 PetscCall(DMSetMatType(da, MATAIJ));
122 PetscCall(DMCreateMatrix(da, &J));
123 PetscCall(MatFDColoringCreate(J, iscoloring, &matfdcoloring));
124 PetscCall(MatFDColoringSetFunction(matfdcoloring, (MatFDColoringFn *)SNESTSFormFunction, ts));
125 PetscCall(MatFDColoringSetFromOptions(matfdcoloring));
126 PetscCall(MatFDColoringSetUp(J, iscoloring, matfdcoloring));
127 PetscCall(ISColoringDestroy(&iscoloring));
128 PetscCall(SNESSetJacobian(snes, J, J, SNESComputeJacobianDefaultColor, matfdcoloring));
129
130 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
131 Customize nonlinear solver
132 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
133 PetscCall(TSSetType(ts, TSBEULER));
134
135 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136 Set initial conditions
137 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
138 PetscCall(FormInitialSolution(da, x, ctx.kappa));
139 PetscCall(TSSetTimeStep(ts, dt));
140 PetscCall(TSSetSolution(ts, x));
141
142 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
143 Set runtime options
144 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
145 PetscCall(TSSetFromOptions(ts));
146
147 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
148 Solve nonlinear system
149 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
150 PetscCall(TSSolve(ts, x));
151 PetscCall(TSGetStepNumber(ts, &steps));
152
153 /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
154 Free work space. All PETSc objects should be destroyed when they
155 are no longer needed.
156 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
157 PetscCall(MatDestroy(&J));
158 PetscCall(MatFDColoringDestroy(&matfdcoloring));
159 PetscCall(VecDestroy(&x));
160 PetscCall(VecDestroy(&r));
161 PetscCall(TSDestroy(&ts));
162 PetscCall(DMDestroy(&da));
163
164 PetscCall(PetscFinalize());
165 return 0;
166 }
167
168 typedef struct {
169 PetscScalar w, u;
170 } Field;
171 /* ------------------------------------------------------------------- */
172 /*
173 FormFunction - Evaluates nonlinear function, F(x).
174
175 Input Parameters:
176 . ts - the TS context
177 . X - input vector
178 . ptr - optional user-defined context, as set by SNESSetFunction()
179
180 Output Parameter:
181 . F - function vector
182 */
FormFunction(TS ts,PetscReal ftime,Vec X,Vec Xdot,Vec F,void * ptr)183 PetscErrorCode FormFunction(TS ts, PetscReal ftime, Vec X, Vec Xdot, Vec F, void *ptr)
184 {
185 DM da;
186 PetscInt i, Mx, xs, xm;
187 PetscReal hx, sx;
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 (PetscRealPart(x[i].u) < -1.0 + 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogReal(ctx->tol) + PetscLogScalar((1.0 - x[i].u) / 2.0)) + ctx->theta_c * x[i].u;
239 else if (PetscRealPart(x[i].u) > 1.0 - 2.0 * ctx->tol) f[i].w += .5 * ctx->theta * (-PetscLogScalar((1.0 + x[i].u) / 2.0) + PetscLogReal(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 }
243 }
244 f[i].u = xdot[i].u - (x[i - 1].w + x[i + 1].w - 2.0 * x[i].w) * sx;
245 }
246
247 /*
248 Restore vectors
249 */
250 PetscCall(DMDAVecRestoreArrayRead(da, localXdot, &xdot));
251 PetscCall(DMDAVecRestoreArrayRead(da, localX, &x));
252 PetscCall(DMDAVecRestoreArray(da, F, &f));
253 PetscCall(DMRestoreLocalVector(da, &localX));
254 PetscCall(DMRestoreLocalVector(da, &localXdot));
255 PetscFunctionReturn(PETSC_SUCCESS);
256 }
257
258 /* ------------------------------------------------------------------- */
FormInitialSolution(DM da,Vec X,PetscReal kappa)259 PetscErrorCode FormInitialSolution(DM da, Vec X, PetscReal kappa)
260 {
261 PetscInt i, xs, xm, Mx, xgs, xgm;
262 Field *x;
263 PetscReal hx, xx, r, sx;
264 Vec Xg;
265
266 PetscFunctionBegin;
267 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));
268
269 hx = 1.0 / (PetscReal)Mx;
270 sx = 1.0 / (hx * hx);
271
272 /*
273 Get pointers to vector data
274 */
275 PetscCall(DMCreateLocalVector(da, &Xg));
276 PetscCall(DMDAVecGetArray(da, Xg, &x));
277
278 /*
279 Get local grid boundaries
280 */
281 PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &xm, NULL, NULL));
282 PetscCall(DMDAGetGhostCorners(da, &xgs, NULL, NULL, &xgm, NULL, NULL));
283
284 /*
285 Compute u function over the locally owned part of the grid including ghost points
286 */
287 for (i = xgs; i < xgs + xgm; i++) {
288 xx = i * hx;
289 r = PetscSqrtReal((xx - .5) * (xx - .5));
290 if (r < .125) x[i].u = 1.0;
291 else x[i].u = -.50;
292 /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
293 x[i].w = 0;
294 }
295 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;
296
297 /*
298 Restore vectors
299 */
300 PetscCall(DMDAVecRestoreArray(da, Xg, &x));
301
302 /* Grab only the global part of the vector */
303 PetscCall(VecSet(X, 0));
304 PetscCall(DMLocalToGlobalBegin(da, Xg, ADD_VALUES, X));
305 PetscCall(DMLocalToGlobalEnd(da, Xg, ADD_VALUES, X));
306 PetscCall(VecDestroy(&Xg));
307 PetscFunctionReturn(PETSC_SUCCESS);
308 }
309
310 /*TEST
311
312 build:
313 requires: !complex !single
314
315 test:
316 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
317 requires: x
318
319 TEST*/
320