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