Actual source code: biharmonic3.c


  2: static char help[] = "Solves biharmonic equation in 1d.\n";

  4: /*
  5:   Solves the equation biharmonic equation in split form

  7:     w = -kappa \Delta u
  8:     u_t =  \Delta w
  9:     -1  <= u <= 1
 10:     Periodic boundary conditions

 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

 16:     w = -kappa \Delta u  + u^3  - u
 17:     u_t =  \Delta w
 18:     -1  <= u <= 1
 19:     Periodic boundary conditions

 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

 25: */
 26: #include <petscdm.h>
 27: #include <petscdmda.h>
 28: #include <petscts.h>
 29: #include <petscdraw.h>

 31: /*
 32:    User-defined routines
 33: */
 34: extern PetscErrorCode FormFunction(TS,PetscReal,Vec,Vec,Vec,void*),FormInitialSolution(DM,Vec,PetscReal);
 35: typedef struct {PetscBool cahnhillard;PetscReal kappa;PetscInt energy;PetscReal tol;PetscReal theta;PetscReal theta_c;} UserCtx;

 37: int main(int argc,char **argv)
 38: {
 39:   TS             ts;                           /* nonlinear solver */
 40:   Vec            x,r;                          /* solution, residual vectors */
 41:   Mat            J;                            /* Jacobian matrix */
 42:   PetscInt       steps,Mx;
 44:   DM             da;
 45:   MatFDColoring  matfdcoloring;
 46:   ISColoring     iscoloring;
 47:   PetscReal      dt;
 48:   PetscReal      vbounds[] = {-100000,100000,-1.1,1.1};
 49:   SNES           snes;
 50:   UserCtx        ctx;

 52:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 53:      Initialize program
 54:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 55:   PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;
 56:   ctx.kappa       = 1.0;
 57:   PetscOptionsGetReal(NULL,NULL,"-kappa",&ctx.kappa,NULL);
 58:   ctx.cahnhillard = PETSC_FALSE;
 59:   PetscOptionsGetBool(NULL,NULL,"-cahn-hillard",&ctx.cahnhillard,NULL);
 60:   PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),2,vbounds);
 61:   PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),600,600);
 62:   ctx.energy      = 1;
 63:   PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);
 64:   ctx.tol     = 1.0e-8;
 65:   PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL);
 66:   ctx.theta   = .001;
 67:   ctx.theta_c = 1.0;
 68:   PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL);
 69:   PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL);

 71:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 72:      Create distributed array (DMDA) to manage parallel grid and vectors
 73:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 74:   DMDACreate1d(PETSC_COMM_WORLD, DM_BOUNDARY_PERIODIC, 10,2,2,NULL,&da);
 75:   DMSetFromOptions(da);
 76:   DMSetUp(da);
 77:   DMDASetFieldName(da,0,"Biharmonic heat equation: w = -kappa*u_xx");
 78:   DMDASetFieldName(da,1,"Biharmonic heat equation: u");
 79:   DMDAGetInfo(da,0,&Mx,0,0,0,0,0,0,0,0,0,0,0);
 80:   dt   = 1.0/(10.*ctx.kappa*Mx*Mx*Mx*Mx);

 82:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 83:      Extract global vectors from DMDA; then duplicate for remaining
 84:      vectors that are the same types
 85:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 86:   DMCreateGlobalVector(da,&x);
 87:   VecDuplicate(x,&r);

 89:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 90:      Create timestepping solver context
 91:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 92:   TSCreate(PETSC_COMM_WORLD,&ts);
 93:   TSSetDM(ts,da);
 94:   TSSetProblemType(ts,TS_NONLINEAR);
 95:   TSSetIFunction(ts,NULL,FormFunction,&ctx);
 96:   TSSetMaxTime(ts,.02);
 97:   TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER);

 99:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
100:      Create matrix data structure; set Jacobian evaluation routine

102: <     Set Jacobian matrix data structure and default Jacobian evaluation
103:      routine. User can override with:
104:      -snes_mf : matrix-free Newton-Krylov method with no preconditioning
105:                 (unless user explicitly sets preconditioner)
106:      -snes_mf_operator : form preconditioning matrix as set by the user,
107:                          but use matrix-free approx for Jacobian-vector
108:                          products within Newton-Krylov method

110:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111:   TSGetSNES(ts,&snes);
112:   SNESSetType(snes,SNESVINEWTONRSLS);
113:   DMCreateColoring(da,IS_COLORING_GLOBAL,&iscoloring);
114:   DMSetMatType(da,MATAIJ);
115:   DMCreateMatrix(da,&J);
116:   MatFDColoringCreate(J,iscoloring,&matfdcoloring);
117:   MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))SNESTSFormFunction,ts);
118:   MatFDColoringSetFromOptions(matfdcoloring);
119:   MatFDColoringSetUp(J,iscoloring,matfdcoloring);
120:   ISColoringDestroy(&iscoloring);
121:   SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,matfdcoloring);

123:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
124:      Customize nonlinear solver
125:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
126:   TSSetType(ts,TSBEULER);

128:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
129:      Set initial conditions
130:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
131:   FormInitialSolution(da,x,ctx.kappa);
132:   TSSetTimeStep(ts,dt);
133:   TSSetSolution(ts,x);

135:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
136:      Set runtime options
137:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
138:   TSSetFromOptions(ts);

140:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
141:      Solve nonlinear system
142:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
143:   TSSolve(ts,x);
144:   TSGetStepNumber(ts,&steps);

146:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
147:      Free work space.  All PETSc objects should be destroyed when they
148:      are no longer needed.
149:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
150:   MatDestroy(&J);
151:   MatFDColoringDestroy(&matfdcoloring);
152:   VecDestroy(&x);
153:   VecDestroy(&r);
154:   TSDestroy(&ts);
155:   DMDestroy(&da);

157:   PetscFinalize();
158:   return ierr;
159: }

161: typedef struct {PetscScalar w,u;} Field;
162: /* ------------------------------------------------------------------- */
163: /*
164:    FormFunction - Evaluates nonlinear function, F(x).

166:    Input Parameters:
167: .  ts - the TS context
168: .  X - input vector
169: .  ptr - optional user-defined context, as set by SNESSetFunction()

171:    Output Parameter:
172: .  F - function vector
173:  */
174: PetscErrorCode FormFunction(TS ts,PetscReal ftime,Vec X,Vec Xdot,Vec F,void *ptr)
175: {
176:   DM             da;
178:   PetscInt       i,Mx,xs,xm;
179:   PetscReal      hx,sx;
180:   PetscScalar    r,l;
181:   Field          *x,*xdot,*f;
182:   Vec            localX,localXdot;
183:   UserCtx        *ctx = (UserCtx*)ptr;

186:   TSGetDM(ts,&da);
187:   DMGetLocalVector(da,&localX);
188:   DMGetLocalVector(da,&localXdot);
189:   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);

191:   hx = 1.0/(PetscReal)Mx; sx = 1.0/(hx*hx);

193:   /*
194:      Scatter ghost points to local vector,using the 2-step process
195:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
196:      By placing code between these two statements, computations can be
197:      done while messages are in transition.
198:   */
199:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
200:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
201:   DMGlobalToLocalBegin(da,Xdot,INSERT_VALUES,localXdot);
202:   DMGlobalToLocalEnd(da,Xdot,INSERT_VALUES,localXdot);

204:   /*
205:      Get pointers to vector data
206:   */
207:   DMDAVecGetArrayRead(da,localX,&x);
208:   DMDAVecGetArrayRead(da,localXdot,&xdot);
209:   DMDAVecGetArray(da,F,&f);

211:   /*
212:      Get local grid boundaries
213:   */
214:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);

216:   /*
217:      Compute function over the locally owned part of the grid
218:   */
219:   for (i=xs; i<xs+xm; i++) {
220:     f[i].w =  x[i].w + ctx->kappa*(x[i-1].u + x[i+1].u - 2.0*x[i].u)*sx;
221:     if (ctx->cahnhillard) {
222:       switch (ctx->energy) {
223:       case 1: /* double well */
224:         f[i].w += -x[i].u*x[i].u*x[i].u + x[i].u;
225:         break;
226:       case 2: /* double obstacle */
227:         f[i].w += x[i].u;
228:         break;
229:       case 3: /* logarithmic */
230:         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;
231:         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;
232:         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;
233:         break;
234:       case 4:
235:         break;
236:       }
237:     }
238:     f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx;
239:     if (ctx->energy==4) {
240:       f[i].u = xdot[i].u;
241:       /* approximation of \grad (M(u) \grad w), where M(u) = (1-u^2) */
242:       r       = (1.0 - x[i+1].u*x[i+1].u)*(x[i+2].w-x[i].w)*.5/hx;
243:       l       = (1.0 - x[i-1].u*x[i-1].u)*(x[i].w-x[i-2].w)*.5/hx;
244:       f[i].u -= (r - l)*.5/hx;
245:       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;
246:     }
247:   }

249:   /*
250:      Restore vectors
251:   */
252:   DMDAVecRestoreArrayRead(da,localXdot,&xdot);
253:   DMDAVecRestoreArrayRead(da,localX,&x);
254:   DMDAVecRestoreArray(da,F,&f);
255:   DMRestoreLocalVector(da,&localX);
256:   DMRestoreLocalVector(da,&localXdot);
257:   return(0);
258: }

260: /* ------------------------------------------------------------------- */
261: PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa)
262: {
264:   PetscInt       i,xs,xm,Mx,xgs,xgm;
265:   Field          *x;
266:   PetscReal      hx,xx,r,sx;
267:   Vec            Xg;

270:   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);

272:   hx = 1.0/(PetscReal)Mx;
273:   sx = 1.0/(hx*hx);

275:   /*
276:      Get pointers to vector data
277:   */
278:   DMCreateLocalVector(da,&Xg);
279:   DMDAVecGetArray(da,Xg,&x);

281:   /*
282:      Get local grid boundaries
283:   */
284:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
285:   DMDAGetGhostCorners(da,&xgs,NULL,NULL,&xgm,NULL,NULL);

287:   /*
288:      Compute u function over the locally owned part of the grid including ghost points
289:   */
290:   for (i=xgs; i<xgs+xgm; i++) {
291:     xx = i*hx;
292:     r = PetscSqrtReal((xx-.5)*(xx-.5));
293:     if (r < .125) x[i].u = 1.0;
294:     else          x[i].u = -.50;
295:     /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
296:     x[i].w = 0;
297:   }
298:   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;

300:   /*
301:      Restore vectors
302:   */
303:   DMDAVecRestoreArray(da,Xg,&x);

305:   /* Grab only the global part of the vector */
306:   VecSet(X,0);
307:   DMLocalToGlobalBegin(da,Xg,ADD_VALUES,X);
308:   DMLocalToGlobalEnd(da,Xg,ADD_VALUES,X);
309:   VecDestroy(&Xg);
310:   return(0);
311: }

313: /*TEST

315:    build:
316:      requires: !complex !single

318:    test:
319:      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
320:      requires: x

322: TEST*/