Actual source code: biharmonic2.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: ./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_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: (this fails after a few timesteps 12/17/2017)
 22: ---------------
 23: ./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_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;

 60:   PetscOptionsGetBool(NULL,NULL,"-cahn-hillard",&ctx.cahnhillard,NULL);
 61:   PetscViewerDrawSetBounds(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),2,vbounds);
 62:   PetscViewerDrawResize(PETSC_VIEWER_DRAW_(PETSC_COMM_WORLD),600,600);
 63:   ctx.energy = 1;
 64:   /*PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);*/
 65:   PetscOptionsGetInt(NULL,NULL,"-energy",&ctx.energy,NULL);
 66:   ctx.tol     = 1.0e-8;
 67:   PetscOptionsGetReal(NULL,NULL,"-tol",&ctx.tol,NULL);
 68:   ctx.theta   = .001;
 69:   ctx.theta_c = 1.0;
 70:   PetscOptionsGetReal(NULL,NULL,"-theta",&ctx.theta,NULL);
 71:   PetscOptionsGetReal(NULL,NULL,"-theta_c",&ctx.theta_c,NULL);

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

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

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

101:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
102:      Create matrix data structure; set Jacobian evaluation routine

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

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

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

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

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

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

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

158:   PetscFinalize();
159:   return ierr;
160: }

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

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

172:    Output Parameter:
173: .  F - function vector
174:  */
175: PetscErrorCode FormFunction(TS ts,PetscReal ftime,Vec X,Vec Xdot,Vec F,void *ptr)
176: {
177:   DM             da;
179:   PetscInt       i,Mx,xs,xm;
180:   PetscReal      hx,sx;
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 (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;
231:         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;
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:       }
235:     }
236:     f[i].u = xdot[i].u - (x[i-1].w + x[i+1].w - 2.0*x[i].w)*sx;
237:   }

239:   /*
240:      Restore vectors
241:   */
242:   DMDAVecRestoreArrayRead(da,localXdot,&xdot);
243:   DMDAVecRestoreArrayRead(da,localX,&x);
244:   DMDAVecRestoreArray(da,F,&f);
245:   DMRestoreLocalVector(da,&localX);
246:   DMRestoreLocalVector(da,&localXdot);
247:   return(0);
248: }

250: /* ------------------------------------------------------------------- */
251: PetscErrorCode FormInitialSolution(DM da,Vec X,PetscReal kappa)
252: {
254:   PetscInt       i,xs,xm,Mx,xgs,xgm;
255:   Field          *x;
256:   PetscReal      hx,xx,r,sx;
257:   Vec            Xg;

260:   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);

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

265:   /*
266:      Get pointers to vector data
267:   */
268:   DMCreateLocalVector(da,&Xg);
269:   DMDAVecGetArray(da,Xg,&x);

271:   /*
272:      Get local grid boundaries
273:   */
274:   DMDAGetCorners(da,&xs,NULL,NULL,&xm,NULL,NULL);
275:   DMDAGetGhostCorners(da,&xgs,NULL,NULL,&xgm,NULL,NULL);

277:   /*
278:      Compute u function over the locally owned part of the grid including ghost points
279:   */
280:   for (i=xgs; i<xgs+xgm; i++) {
281:     xx = i*hx;
282:     r = PetscSqrtReal((xx-.5)*(xx-.5));
283:     if (r < .125) x[i].u = 1.0;
284:     else          x[i].u = -.50;
285:     /* fill in x[i].w so that valgrind doesn't detect use of uninitialized memory */
286:     x[i].w = 0;
287:   }
288:   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;

290:   /*
291:      Restore vectors
292:   */
293:   DMDAVecRestoreArray(da,Xg,&x);

295:   /* Grab only the global part of the vector */
296:   VecSet(X,0);
297:   DMLocalToGlobalBegin(da,Xg,ADD_VALUES,X);
298:   DMLocalToGlobalEnd(da,Xg,ADD_VALUES,X);
299:   VecDestroy(&Xg);
300:   return(0);
301: }

303: /*TEST

305:    build:
306:      requires: !complex !single

308:    test:
309:      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
310:      requires: x

312: TEST*/