Actual source code: ex8.c

petsc-3.6.1 2015-08-06
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  1: #include <petscsnes.h>
  2: #include <petscdm.h>
  3: #include <petscdmda.h>

  5: static char help[] = "Parallel version of the minimum surface area problem using DMs.\n\
  6: See ex10.c for the serial version. It solves a system of nonlinear equations in mixed\n\
  7: complementarity form using semismooth newton algorithm.This example is based on a\n\
  8: problem from the MINPACK-2 test suite.  Given a rectangular 2-D domain and\n\
  9: boundary values along the edges of the domain, the objective is to find the\n\
 10: surface with the minimal area that satisfies the boundary conditions.\n\
 11: This application solves this problem using complimentarity -- We are actually\n\
 12: solving the system  (grad f)_i >= 0, if x_i == l_i \n\
 13:                     (grad f)_i = 0, if l_i < x_i < u_i \n\
 14:                     (grad f)_i <= 0, if x_i == u_i  \n\
 15: where f is the function to be minimized. \n\
 16: \n\
 17: The command line options are:\n\
 18:   -da_grid_x <nx>, where <nx> = number of grid points in the 1st coordinate direction\n\
 19:   -da_grid_y <ny>, where <ny> = number of grid points in the 2nd coordinate direction\n\
 20:   -start <st>, where <st> =0 for zero vector, and an average of the boundary conditions otherwise\n\
 21:   -lb <value>, lower bound on the variables\n\
 22:   -ub <value>, upper bound on the variables\n\n";

 24: /*
 25:    User-defined application context - contains data needed by the
 26:    application-provided call-back routines, FormJacobian() and
 27:    FormFunction().
 28: */

 30: typedef struct {
 31:   DM          da;
 32:   PetscScalar *bottom, *top, *left, *right;
 33:   PetscInt    mx,my;
 34: } AppCtx;


 37: /* -------- User-defined Routines --------- */

 39: extern PetscErrorCode MSA_BoundaryConditions(AppCtx*);
 40: extern PetscErrorCode MSA_InitialPoint(AppCtx*, Vec);
 41: extern PetscErrorCode FormGradient(SNES, Vec, Vec, void*);
 42: extern PetscErrorCode FormJacobian(SNES, Vec, Mat, Mat, void*);

 46: int main(int argc, char **argv)
 47: {
 49:   Vec            x,r;               /* solution and residual vectors */
 50:   Vec            xl,xu;             /* Bounds on the variables */
 51:   PetscBool      flg_l,flg_u;       /* flags to check if the bounds are set */
 52:   SNES           snes;              /* nonlinear solver context */
 53:   Mat            J;                 /* Jacobian matrix */
 54:   PetscInt       N;                 /* Number of elements in vector */
 55:   PetscScalar    lb = .05;
 56:   PetscScalar    ub = PETSC_INFINITY;
 57:   AppCtx         user;              /* user-defined work context */
 58:   PetscBool      flg;

 60:   /* Initialize PETSc */
 61:   PetscInitialize(&argc, &argv, (char*)0, help);

 63: #if defined(PETSC_USE_COMPLEX)
 64:   SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"This example does not work for scalar type complex\n");
 65: #endif

 67:   /* Check if lower and upper bounds are set */
 68:   PetscOptionsGetScalar(NULL, "-lb", &lb, &flg_l);
 69:   PetscOptionsGetScalar(NULL, "-ub", &ub, &flg_u);

 71:   /* Create distributed array to manage the 2d grid */
 72:   DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,-4,-4,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&user.da);
 73:   DMDAGetIerr(user.da,PETSC_IGNORE,&user.mx,&user.my,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
 74:   /* Extract global vectors from DMDA; */
 75:   DMCreateGlobalVector(user.da,&x);
 76:   VecDuplicate(x, &r);

 78:   N    = user.mx*user.my;
 79:   DMSetMatType(user.da,MATAIJ);
 80:   DMCreateMatrix(user.da,&J);

 82:   /* Create nonlinear solver context */
 83:   SNESCreate(PETSC_COMM_WORLD,&snes);

 85:   /*  Set function evaluation and Jacobian evaluation  routines */
 86:   SNESSetFunction(snes,r,FormGradient,&user);
 87:   SNESSetJacobian(snes,J,J,FormJacobian,&user);

 89:   /* Set the boundary conditions */
 90:   MSA_BoundaryConditions(&user);

 92:   /* Set initial solution guess */
 93:   MSA_InitialPoint(&user, x);


 96:   /* Set Bounds on variables */
 97:   VecDuplicate(x, &xl);
 98:   VecDuplicate(x, &xu);
 99:   VecSet(xl, lb);
100:   VecSet(xu, ub);

102:   SNESVISetVariableBounds(snes,xl,xu);

104:   SNESSetFromOptions(snes);

106:   /* Solve the application */
107:   SNESSolve(snes,NULL,x);

109:   PetscOptionsHasName(NULL,"-view_sol",&flg);
110:   if (flg) { VecView(x,PETSC_VIEWER_STDOUT_WORLD); }

112:   /* Free memory */
113:   VecDestroy(&x);
114:   VecDestroy(&xl);
115:   VecDestroy(&xu);
116:   VecDestroy(&r);
117:   MatDestroy(&J);
118:   SNESDestroy(&snes);

120:   /* Free user-created data structures */
121:   DMDestroy(&user.da);
122:   PetscFree(user.bottom);
123:   PetscFree(user.top);
124:   PetscFree(user.left);
125:   PetscFree(user.right);

127:   PetscFinalize();

129:   return 0;
130: }

132: /* -------------------------------------------------------------------- */

136: /*  FormGradient - Evaluates gradient of f.

138:     Input Parameters:
139: .   snes  - the SNES context
140: .   X     - input vector
141: .   ptr   - optional user-defined context, as set by SNESSetFunction()

143:     Output Parameters:
144: .   G - vector containing the newly evaluated gradient
145: */
146: PetscErrorCode FormGradient(SNES snes, Vec X, Vec G, void *ptr)
147: {
148:   AppCtx      *user = (AppCtx*) ptr;
149:   int         ierr;
150:   PetscInt    i,j;
151:   PetscInt    mx=user->mx, my=user->my;
152:   PetscScalar hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
153:   PetscScalar f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
154:   PetscScalar df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
155:   PetscScalar **g, **x;
156:   PetscInt    xs,xm,ys,ym;
157:   Vec         localX;

160:   /* Initialize vector to zero */
161:   VecSet(G,0.0);

163:   /* Get local vector */
164:   DMGetLocalVector(user->da,&localX);
165:   /* Get ghost points */
166:   DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
167:   DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
168:   /* Get pointer to local vector data */
169:   DMDAVecGetArrayRead(user->da,localX, &x);
170:   DMDAVecGetArray(user->da,G, &g);

172:   DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
173:   /* Compute function over the locally owned part of the mesh */
174:   for (j=ys; j < ys+ym; j++) {
175:     for (i=xs; i< xs+xm; i++) {

177:       xc = x[j][i];
178:       xlt=xrb=xl=xr=xb=xt=xc;

180:       if (i==0) { /* left side */
181:         xl  = user->left[j+1];
182:         xlt = user->left[j+2];
183:       } else xl = x[j][i-1];

185:       if (j==0) { /* bottom side */
186:         xb  = user->bottom[i+1];
187:         xrb = user->bottom[i+2];
188:       } else xb = x[j-1][i];

190:       if (i+1 == mx) { /* right side */
191:         xr  = user->right[j+1];
192:         xrb = user->right[j];
193:       } else xr = x[j][i+1];

195:       if (j+1==0+my) { /* top side */
196:         xt  = user->top[i+1];
197:         xlt = user->top[i];
198:       } else xt = x[j+1][i];

200:       if (i>0 && j+1<my) xlt = x[j+1][i-1]; /* left top side */
201:       if (j>0 && i+1<mx) xrb = x[j-1][i+1]; /* right bottom */

203:       d1 = (xc-xl);
204:       d2 = (xc-xr);
205:       d3 = (xc-xt);
206:       d4 = (xc-xb);
207:       d5 = (xr-xrb);
208:       d6 = (xrb-xb);
209:       d7 = (xlt-xl);
210:       d8 = (xt-xlt);

212:       df1dxc = d1*hydhx;
213:       df2dxc = (d1*hydhx + d4*hxdhy);
214:       df3dxc = d3*hxdhy;
215:       df4dxc = (d2*hydhx + d3*hxdhy);
216:       df5dxc = d2*hydhx;
217:       df6dxc = d4*hxdhy;

219:       d1 /= hx;
220:       d2 /= hx;
221:       d3 /= hy;
222:       d4 /= hy;
223:       d5 /= hy;
224:       d6 /= hx;
225:       d7 /= hy;
226:       d8 /= hx;

228:       f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
229:       f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
230:       f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
231:       f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
232:       f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
233:       f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);

235:       df1dxc /= f1;
236:       df2dxc /= f2;
237:       df3dxc /= f3;
238:       df4dxc /= f4;
239:       df5dxc /= f5;
240:       df6dxc /= f6;

242:       g[j][i] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0;

244:     }
245:   }

247:   /* Restore vectors */
248:   DMDAVecRestoreArrayRead(user->da,localX, &x);
249:   DMDAVecRestoreArray(user->da,G, &g);
250:   DMRestoreLocalVector(user->da,&localX);
251:   PetscLogFlops(67*mx*my);
252:   return(0);
253: }

255: /* ------------------------------------------------------------------- */
258: /*
259:    FormJacobian - Evaluates Jacobian matrix.

261:    Input Parameters:
262: .  snes - SNES context
263: .  X    - input vector
264: .  ptr  - optional user-defined context, as set by SNESSetJacobian()

266:    Output Parameters:
267: .  tH    - Jacobian matrix

269: */
270: PetscErrorCode FormJacobian(SNES snes, Vec X, Mat H, Mat tHPre, void *ptr)
271: {
272:   AppCtx         *user = (AppCtx*) ptr;
274:   PetscInt       i,j,k;
275:   PetscInt       mx=user->mx, my=user->my;
276:   MatStencil     row,col[7];
277:   PetscScalar    hx=1.0/(mx+1), hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
278:   PetscScalar    f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
279:   PetscScalar    hl,hr,ht,hb,hc,htl,hbr;
280:   PetscScalar    **x, v[7];
281:   PetscBool      assembled;
282:   PetscInt       xs,xm,ys,ym;
283:   Vec            localX;

286:   /* Set various matrix options */
287:   MatAssembled(H,&assembled);
288:   if (assembled) {MatZeroEntries(H);}

290:   /* Get local vector */
291:   DMGetLocalVector(user->da,&localX);
292:   /* Get ghost points */
293:   DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
294:   DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);

296:   /* Get pointers to vector data */
297:   DMDAVecGetArrayRead(user->da,localX, &x);

299:   DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
300:   /* Compute Jacobian over the locally owned part of the mesh */
301:   for (j=ys; j< ys+ym; j++) {
302:     for (i=xs; i< xs+xm; i++) {
303:       xc = x[j][i];
304:       xlt=xrb=xl=xr=xb=xt=xc;

306:       /* Left */
307:       if (i==0) {
308:         xl  = user->left[j+1];
309:         xlt = user->left[j+2];
310:       } else xl = x[j][i-1];

312:       /* Bottom */
313:       if (j==0) {
314:         xb  = user->bottom[i+1];
315:         xrb = user->bottom[i+2];
316:       } else xb = x[j-1][i];

318:       /* Right */
319:       if (i+1 == mx) {
320:         xr  = user->right[j+1];
321:         xrb = user->right[j];
322:       } else xr = x[j][i+1];

324:       /* Top */
325:       if (j+1==my) {
326:         xt  = user->top[i+1];
327:         xlt = user->top[i];
328:       } else xt = x[j+1][i];

330:       /* Top left */
331:       if (i>0 && j+1<my) xlt = x[j+1][i-1];

333:       /* Bottom right */
334:       if (j>0 && i+1<mx) xrb = x[j-1][i+1];

336:       d1 = (xc-xl)/hx;
337:       d2 = (xc-xr)/hx;
338:       d3 = (xc-xt)/hy;
339:       d4 = (xc-xb)/hy;
340:       d5 = (xrb-xr)/hy;
341:       d6 = (xrb-xb)/hx;
342:       d7 = (xlt-xl)/hy;
343:       d8 = (xlt-xt)/hx;

345:       f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
346:       f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
347:       f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
348:       f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
349:       f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
350:       f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);


353:       hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
354:            (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
355:       hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
356:            (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
357:       ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
358:            (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
359:       hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
360:            (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);

362:       hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
363:       htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);

365:       hc = hydhx*(1.0+d7*d7)/(f1*f1*f1) + hxdhy*(1.0+d8*d8)/(f3*f3*f3) +
366:            hydhx*(1.0+d5*d5)/(f5*f5*f5) + hxdhy*(1.0+d6*d6)/(f6*f6*f6) +
367:            (hxdhy*(1.0+d1*d1)+hydhx*(1.0+d4*d4)-2*d1*d4)/(f2*f2*f2) +
368:            (hxdhy*(1.0+d2*d2)+hydhx*(1.0+d3*d3)-2*d2*d3)/(f4*f4*f4);

370:       hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0;  hc/=2.0;

372:       k     =0;
373:       row.i = i;row.j= j;
374:       /* Bottom */
375:       if (j>0) {
376:         v[k]     =hb;
377:         col[k].i = i; col[k].j=j-1; k++;
378:       }

380:       /* Bottom right */
381:       if (j>0 && i < mx -1) {
382:         v[k]     =hbr;
383:         col[k].i = i+1; col[k].j = j-1; k++;
384:       }

386:       /* left */
387:       if (i>0) {
388:         v[k]     = hl;
389:         col[k].i = i-1; col[k].j = j; k++;
390:       }

392:       /* Centre */
393:       v[k]= hc; col[k].i= row.i; col[k].j = row.j; k++;

395:       /* Right */
396:       if (i < mx-1) {
397:         v[k]    = hr;
398:         col[k].i= i+1; col[k].j = j;k++;
399:       }

401:       /* Top left */
402:       if (i>0 && j < my-1) {
403:         v[k]     = htl;
404:         col[k].i = i-1;col[k].j = j+1; k++;
405:       }

407:       /* Top */
408:       if (j < my-1) {
409:         v[k]     = ht;
410:         col[k].i = i; col[k].j = j+1; k++;
411:       }

413:       MatSetValuesStencil(H,1,&row,k,col,v,INSERT_VALUES);
414:     }
415:   }

417:   /* Assemble the matrix */
418:   MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
419:   DMDAVecRestoreArrayRead(user->da,localX,&x);
420:   MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
421:   DMRestoreLocalVector(user->da,&localX);

423:   PetscLogFlops(199*mx*my);
424:   return(0);
425: }

427: /* ------------------------------------------------------------------- */
430: /*
431:    MSA_BoundaryConditions -  Calculates the boundary conditions for
432:    the region.

434:    Input Parameter:
435: .  user - user-defined application context

437:    Output Parameter:
438: .  user - user-defined application context
439: */
440: PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
441: {
443:   PetscInt       i,j,k,limit=0,maxits=5;
444:   PetscInt       mx   =user->mx,my=user->my;
445:   PetscInt       bsize=0, lsize=0, tsize=0, rsize=0;
446:   PetscScalar    one  =1.0, two=2.0, three=3.0, tol=1e-10;
447:   PetscScalar    fnorm,det,hx,hy,xt=0,yt=0;
448:   PetscScalar    u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
449:   PetscScalar    b=-0.5, t=0.5, l=-0.5, r=0.5;
450:   PetscScalar    *boundary;

453:   bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;

455:   PetscMalloc1(bsize, &user->bottom);
456:   PetscMalloc1(tsize, &user->top);
457:   PetscMalloc1(lsize, &user->left);
458:   PetscMalloc1(rsize, &user->right);

460:   hx= (r-l)/(mx+1); hy=(t-b)/(my+1);

462:   for (j=0; j<4; j++) {
463:     if (j==0) {
464:       yt       = b;
465:       xt       = l;
466:       limit    = bsize;
467:       boundary = user->bottom;
468:     } else if (j==1) {
469:       yt       = t;
470:       xt       = l;
471:       limit    = tsize;
472:       boundary = user->top;
473:     } else if (j==2) {
474:       yt       = b;
475:       xt       = l;
476:       limit    = lsize;
477:       boundary = user->left;
478:     } else { /* if  (j==3) */
479:       yt       = b;
480:       xt       = r;
481:       limit    = rsize;
482:       boundary = user->right;
483:     }

485:     for (i=0; i<limit; i++) {
486:       u1=xt;
487:       u2=-yt;
488:       for (k=0; k<maxits; k++) {
489:         nf1   = u1 + u1*u2*u2 - u1*u1*u1/three-xt;
490:         nf2   = -u2 - u1*u1*u2 + u2*u2*u2/three-yt;
491:         fnorm = PetscSqrtReal(nf1*nf1+nf2*nf2);
492:         if (fnorm <= tol) break;
493:         njac11 = one+u2*u2-u1*u1;
494:         njac12 = two*u1*u2;
495:         njac21 = -two*u1*u2;
496:         njac22 = -one - u1*u1 + u2*u2;
497:         det    = njac11*njac22-njac21*njac12;
498:         u1     = u1-(njac22*nf1-njac12*nf2)/det;
499:         u2     = u2-(njac11*nf2-njac21*nf1)/det;
500:       }

502:       boundary[i]=u1*u1-u2*u2;
503:       if (j==0 || j==1) xt=xt+hx;
504:       else yt=yt+hy; /* if (j==2 || j==3) */
505:     }
506:   }
507:   return(0);
508: }

510: /* ------------------------------------------------------------------- */
513: /*
514:    MSA_InitialPoint - Calculates the initial guess in one of three ways.

516:    Input Parameters:
517: .  user - user-defined application context
518: .  X - vector for initial guess

520:    Output Parameters:
521: .  X - newly computed initial guess
522: */
523: PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
524: {
526:   PetscInt       start=-1,i,j;
527:   PetscScalar    zero =0.0;
528:   PetscBool      flg;

531:   PetscOptionsGetInt(NULL,"-start",&start,&flg);

533:   if (flg && start==0) { /* The zero vector is reasonable */

535:     VecSet(X, zero);
536:     /* PLogIerr(user,"Min. Surface Area Problem: Start with 0 vector \n"); */


539:   } else { /* Take an average of the boundary conditions */
540:     PetscInt    mx=user->mx,my=user->my;
541:     PetscScalar **x;
542:     PetscInt    xs,xm,ys,ym;

544:     /* Get pointers to vector data */
545:     DMDAVecGetArray(user->da,X,&x);
546:     DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);

548:     /* Perform local computations */
549:     for (j=ys; j<ys+ym; j++) {
550:       for (i=xs; i< xs+xm; i++) {
551:         x[j][i] = (((j+1)*user->bottom[i+1]+(my-j+1)*user->top[i+1])/(my+2)+
552:                    ((i+1)*user->left[j+1]+(mx-i+1)*user->right[j+1])/(mx+2))/2.0;
553:       }
554:     }

556:     /* Restore vectors */
557:     DMDAVecRestoreArray(user->da,X,&x);

559:   }
560:   return(0);
561: }