Actual source code: ex14.c

petsc-dev 2014-02-02
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  2: static char help[] = "Bratu nonlinear PDE in 3d.\n\
  3: We solve the  Bratu (SFI - solid fuel ignition) problem in a 3D rectangular\n\
  4: domain, using distributed arrays (DMDAs) to partition the parallel grid.\n\
  5: The command line options include:\n\
  6:   -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
  7:      problem SFI:  <parameter> = Bratu parameter (0 <= par <= 6.81)\n\n";

  9: /*T
 10:    Concepts: SNES^parallel Bratu example
 11:    Concepts: DMDA^using distributed arrays;
 12:    Processors: n
 13: T*/

 15: /* ------------------------------------------------------------------------

 17:     Solid Fuel Ignition (SFI) problem.  This problem is modeled by
 18:     the partial differential equation

 20:             -Laplacian u - lambda*exp(u) = 0,  0 < x,y < 1,

 22:     with boundary conditions

 24:              u = 0  for  x = 0, x = 1, y = 0, y = 1, z = 0, z = 1

 26:     A finite difference approximation with the usual 7-point stencil
 27:     is used to discretize the boundary value problem to obtain a nonlinear
 28:     system of equations.


 31:   ------------------------------------------------------------------------- */

 33: /*
 34:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
 35:    Include "petscsnes.h" so that we can use SNES solvers.  Note that this
 36:    file automatically includes:
 37:      petscsys.h       - base PETSc routines   petscvec.h - vectors
 38:      petscmat.h - matrices
 39:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
 40:      petscviewer.h - viewers               petscpc.h  - preconditioners
 41:      petscksp.h   - linear solvers
 42: */
 43: #include <petscdmda.h>
 44: #include <petscsnes.h>


 47: /*
 48:    User-defined application context - contains data needed by the
 49:    application-provided call-back routines, FormJacobian() and
 50:    FormFunction().
 51: */
 52: typedef struct {
 53:   PetscReal param;             /* test problem parameter */
 54:   DM        da;                /* distributed array data structure */
 55: } AppCtx;

 57: /*
 58:    User-defined routines
 59: */
 60: extern PetscErrorCode FormFunction(SNES,Vec,Vec,void*),FormInitialGuess(AppCtx*,Vec);
 61: extern PetscErrorCode FormJacobian(SNES,Vec,Mat*,Mat*,MatStructure*,void*);

 65: int main(int argc,char **argv)
 66: {
 67:   SNES           snes;                         /* nonlinear solver */
 68:   Vec            x,r;                          /* solution, residual vectors */
 69:   Mat            J;                            /* Jacobian matrix */
 70:   AppCtx         user;                         /* user-defined work context */
 71:   PetscInt       its;                          /* iterations for convergence */
 72:   MatFDColoring  matfdcoloring;
 73:   PetscBool      matrix_free = PETSC_FALSE,coloring = PETSC_FALSE;
 75:   PetscReal      bratu_lambda_max = 6.81,bratu_lambda_min = 0.,fnorm;

 77:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 78:      Initialize program
 79:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

 81:   PetscInitialize(&argc,&argv,(char*)0,help);

 83:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 84:      Initialize problem parameters
 85:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 86:   user.param = 6.0;
 87:   PetscOptionsGetReal(NULL,"-par",&user.param,NULL);
 88:   if (user.param >= bratu_lambda_max || user.param <= bratu_lambda_min) SETERRQ(PETSC_COMM_SELF,1,"Lambda is out of range");

 90:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 91:      Create nonlinear solver context
 92:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 93:   SNESCreate(PETSC_COMM_WORLD,&snes);

 95:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
 96:      Create distributed array (DMDA) to manage parallel grid and vectors
 97:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
 98:   DMDACreate3d(PETSC_COMM_WORLD,DMDA_BOUNDARY_NONE,DMDA_BOUNDARY_NONE,DMDA_BOUNDARY_NONE,DMDA_STENCIL_STAR,-4,-4,-4,PETSC_DECIDE,PETSC_DECIDE,
 99:                       PETSC_DECIDE,1,1,NULL,NULL,NULL,&user.da);

101:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
102:      Extract global vectors from DMDA; then duplicate for remaining
103:      vectors that are the same types
104:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
105:   DMCreateGlobalVector(user.da,&x);
106:   VecDuplicate(x,&r);

108:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
109:      Set function evaluation routine and vector
110:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
111:   SNESSetFunction(snes,r,FormFunction,(void*)&user);

113:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
114:      Create matrix data structure; set Jacobian evaluation routine

116:      Set Jacobian matrix data structure and default Jacobian evaluation
117:      routine. User can override with:
118:      -snes_mf : matrix-free Newton-Krylov method with no preconditioning
119:                 (unless user explicitly sets preconditioner)
120:      -snes_mf_operator : form preconditioning matrix as set by the user,
121:                          but use matrix-free approx for Jacobian-vector
122:                          products within Newton-Krylov method
123:      -fdcoloring : using finite differences with coloring to compute the Jacobian

125:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
126:   PetscOptionsGetBool(NULL,"-snes_mf",&matrix_free,NULL);
127:   PetscOptionsGetBool(NULL,"-fdcoloring",&coloring,NULL);
128:   if (!matrix_free) {
129:     DMSetMatType(user.da,MATAIJ);
130:     DMCreateMatrix(user.da,&J);
131:     if (coloring) {
132:       ISColoring iscoloring;
133:       DMCreateColoring(user.da,IS_COLORING_GLOBAL,&iscoloring);
134:       MatFDColoringCreate(J,iscoloring,&matfdcoloring);
135:       MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))FormFunction,&user);
136:       MatFDColoringSetFromOptions(matfdcoloring);
137:       MatFDColoringSetUp(J,iscoloring,matfdcoloring);
138:       SNESSetJacobian(snes,J,J,SNESComputeJacobianDefaultColor,matfdcoloring);
139:       ISColoringDestroy(&iscoloring);
140:     } else {
141:       SNESSetJacobian(snes,J,J,FormJacobian,&user);
142:     }
143:   }

145:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
146:      Customize nonlinear solver; set runtime options
147:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
148:   SNESSetDM(snes,user.da);
149:   SNESSetFromOptions(snes);

151:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
152:      Evaluate initial guess
153:      Note: The user should initialize the vector, x, with the initial guess
154:      for the nonlinear solver prior to calling SNESSolve().  In particular,
155:      to employ an initial guess of zero, the user should explicitly set
156:      this vector to zero by calling VecSet().
157:   */
158:   FormInitialGuess(&user,x);

160:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
161:      Solve nonlinear system
162:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
163:   SNESSolve(snes,NULL,x);
164:   SNESGetIterationNumber(snes,&its);

166:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
167:      Explicitly check norm of the residual of the solution
168:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
169:   FormFunction(snes,x,r,(void*)&user);
170:   VecNorm(r,NORM_2,&fnorm);
171:   PetscPrintf(PETSC_COMM_WORLD,"Number of SNES iterations = %D fnorm %g\n",its,(double)fnorm);

173:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
174:      Free work space.  All PETSc objects should be destroyed when they
175:      are no longer needed.
176:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

178:   if (!matrix_free) {
179:     MatDestroy(&J);
180:   }
181:   VecDestroy(&x);
182:   VecDestroy(&r);
183:   SNESDestroy(&snes);
184:   DMDestroy(&user.da);
185:   if (coloring) {MatFDColoringDestroy(&matfdcoloring);}
186:   PetscFinalize();
187:   return(0);
188: }
189: /* ------------------------------------------------------------------- */
192: /*
193:    FormInitialGuess - Forms initial approximation.

195:    Input Parameters:
196:    user - user-defined application context
197:    X - vector

199:    Output Parameter:
200:    X - vector
201:  */
202: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
203: {
204:   PetscInt       i,j,k,Mx,My,Mz,xs,ys,zs,xm,ym,zm;
206:   PetscReal      lambda,temp1,hx,hy,hz,tempk,tempj;
207:   PetscScalar    ***x;

210:   DMDAGetInfo(user->da,PETSC_IGNORE,&Mx,&My,&Mz,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

212:   lambda = user->param;
213:   hx     = 1.0/(PetscReal)(Mx-1);
214:   hy     = 1.0/(PetscReal)(My-1);
215:   hz     = 1.0/(PetscReal)(Mz-1);
216:   temp1  = lambda/(lambda + 1.0);

218:   /*
219:      Get a pointer to vector data.
220:        - For default PETSc vectors, VecGetArray() returns a pointer to
221:          the data array.  Otherwise, the routine is implementation dependent.
222:        - You MUST call VecRestoreArray() when you no longer need access to
223:          the array.
224:   */
225:   DMDAVecGetArray(user->da,X,&x);

227:   /*
228:      Get local grid boundaries (for 3-dimensional DMDA):
229:        xs, ys, zs   - starting grid indices (no ghost points)
230:        xm, ym, zm   - widths of local grid (no ghost points)

232:   */
233:   DMDAGetCorners(user->da,&xs,&ys,&zs,&xm,&ym,&zm);

235:   /*
236:      Compute initial guess over the locally owned part of the grid
237:   */
238:   for (k=zs; k<zs+zm; k++) {
239:     tempk = (PetscReal)(PetscMin(k,Mz-k-1))*hz;
240:     for (j=ys; j<ys+ym; j++) {
241:       tempj = PetscMin((PetscReal)(PetscMin(j,My-j-1))*hy,tempk);
242:       for (i=xs; i<xs+xm; i++) {
243:         if (i == 0 || j == 0 || k == 0 || i == Mx-1 || j == My-1 || k == Mz-1) {
244:           /* boundary conditions are all zero Dirichlet */
245:           x[k][j][i] = 0.0;
246:         } else {
247:           x[k][j][i] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,Mx-i-1))*hx,tempj));
248:         }
249:       }
250:     }
251:   }

253:   /*
254:      Restore vector
255:   */
256:   DMDAVecRestoreArray(user->da,X,&x);
257:   return(0);
258: }
259: /* ------------------------------------------------------------------- */
262: /*
263:    FormFunction - Evaluates nonlinear function, F(x).

265:    Input Parameters:
266: .  snes - the SNES context
267: .  X - input vector
268: .  ptr - optional user-defined context, as set by SNESSetFunction()

270:    Output Parameter:
271: .  F - function vector
272:  */
273: PetscErrorCode FormFunction(SNES snes,Vec X,Vec F,void *ptr)
274: {
275:   AppCtx         *user = (AppCtx*)ptr;
277:   PetscInt       i,j,k,Mx,My,Mz,xs,ys,zs,xm,ym,zm;
278:   PetscReal      two = 2.0,lambda,hx,hy,hz,hxhzdhy,hyhzdhx,hxhydhz,sc;
279:   PetscScalar    u_north,u_south,u_east,u_west,u_up,u_down,u,u_xx,u_yy,u_zz,***x,***f;
280:   Vec            localX;
281:   DM             da;

284:   SNESGetDM(snes,&da);
285:   DMGetLocalVector(da,&localX);
286:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,&Mz,PETSC_IGNORE,PETSC_IGNORE,
287:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

289:   lambda  = user->param;
290:   hx      = 1.0/(PetscReal)(Mx-1);
291:   hy      = 1.0/(PetscReal)(My-1);
292:   hz      = 1.0/(PetscReal)(Mz-1);
293:   sc      = hx*hy*hz*lambda;
294:   hxhzdhy = hx*hz/hy;
295:   hyhzdhx = hy*hz/hx;
296:   hxhydhz = hx*hy/hz;

298:   /*
299:      Scatter ghost points to local vector,using the 2-step process
300:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
301:      By placing code between these two statements, computations can be
302:      done while messages are in transition.
303:   */
304:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
305:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);

307:   /*
308:      Get pointers to vector data
309:   */
310:   DMDAVecGetArray(da,localX,&x);
311:   DMDAVecGetArray(da,F,&f);

313:   /*
314:      Get local grid boundaries
315:   */
316:   DMDAGetCorners(da,&xs,&ys,&zs,&xm,&ym,&zm);

318:   /*
319:      Compute function over the locally owned part of the grid
320:   */
321:   for (k=zs; k<zs+zm; k++) {
322:     for (j=ys; j<ys+ym; j++) {
323:       for (i=xs; i<xs+xm; i++) {
324:         if (i == 0 || j == 0 || k == 0 || i == Mx-1 || j == My-1 || k == Mz-1) {
325:           f[k][j][i] = x[k][j][i];
326:         } else {
327:           u          = x[k][j][i];
328:           u_east     = x[k][j][i+1];
329:           u_west     = x[k][j][i-1];
330:           u_north    = x[k][j+1][i];
331:           u_south    = x[k][j-1][i];
332:           u_up       = x[k+1][j][i];
333:           u_down     = x[k-1][j][i];
334:           u_xx       = (-u_east + two*u - u_west)*hyhzdhx;
335:           u_yy       = (-u_north + two*u - u_south)*hxhzdhy;
336:           u_zz       = (-u_up + two*u - u_down)*hxhydhz;
337:           f[k][j][i] = u_xx + u_yy + u_zz - sc*PetscExpScalar(u);
338:         }
339:       }
340:     }
341:   }

343:   /*
344:      Restore vectors
345:   */
346:   DMDAVecRestoreArray(da,localX,&x);
347:   DMDAVecRestoreArray(da,F,&f);
348:   DMRestoreLocalVector(da,&localX);
349:   PetscLogFlops(11.0*ym*xm);
350:   return(0);
351: }
352: /* ------------------------------------------------------------------- */
355: /*
356:    FormJacobian - Evaluates Jacobian matrix.

358:    Input Parameters:
359: .  snes - the SNES context
360: .  x - input vector
361: .  ptr - optional user-defined context, as set by SNESSetJacobian()

363:    Output Parameters:
364: .  A - Jacobian matrix
365: .  B - optionally different preconditioning matrix
366: .  flag - flag indicating matrix structure

368: */
369: PetscErrorCode FormJacobian(SNES snes,Vec X,Mat *J,Mat *B,MatStructure *flag,void *ptr)
370: {
371:   AppCtx         *user = (AppCtx*)ptr;  /* user-defined application context */
372:   Mat            jac   = *B;              /* Jacobian matrix */
373:   Vec            localX;
375:   PetscInt       i,j,k,Mx,My,Mz;
376:   MatStencil     col[7],row;
377:   PetscInt       xs,ys,zs,xm,ym,zm;
378:   PetscScalar    lambda,v[7],hx,hy,hz,hxhzdhy,hyhzdhx,hxhydhz,sc,***x;
379:   DM             da;

382:   SNESGetDM(snes,&da);
383:   DMGetLocalVector(da,&localX);
384:   DMDAGetInfo(da,PETSC_IGNORE,&Mx,&My,&Mz,PETSC_IGNORE,PETSC_IGNORE,
385:                      PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);

387:   lambda  = user->param;
388:   hx      = 1.0/(PetscReal)(Mx-1);
389:   hy      = 1.0/(PetscReal)(My-1);
390:   hz      = 1.0/(PetscReal)(Mz-1);
391:   sc      = hx*hy*hz*lambda;
392:   hxhzdhy = hx*hz/hy;
393:   hyhzdhx = hy*hz/hx;
394:   hxhydhz = hx*hy/hz;

396:   /*
397:      Scatter ghost points to local vector, using the 2-step process
398:         DMGlobalToLocalBegin(), DMGlobalToLocalEnd().
399:      By placing code between these two statements, computations can be
400:      done while messages are in transition.
401:   */
402:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
403:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);

405:   /*
406:      Get pointer to vector data
407:   */
408:   DMDAVecGetArray(da,localX,&x);

410:   /*
411:      Get local grid boundaries
412:   */
413:   DMDAGetCorners(da,&xs,&ys,&zs,&xm,&ym,&zm);

415:   /*
416:      Compute entries for the locally owned part of the Jacobian.
417:       - Currently, all PETSc parallel matrix formats are partitioned by
418:         contiguous chunks of rows across the processors.
419:       - Each processor needs to insert only elements that it owns
420:         locally (but any non-local elements will be sent to the
421:         appropriate processor during matrix assembly).
422:       - Here, we set all entries for a particular row at once.
423:       - We can set matrix entries either using either
424:         MatSetValuesLocal() or MatSetValues(), as discussed above.
425:   */
426:   for (k=zs; k<zs+zm; k++) {
427:     for (j=ys; j<ys+ym; j++) {
428:       for (i=xs; i<xs+xm; i++) {
429:         row.k = k; row.j = j; row.i = i;
430:         /* boundary points */
431:         if (i == 0 || j == 0 || k == 0|| i == Mx-1 || j == My-1 || k == Mz-1) {
432:           v[0] = 1.0;
433:           MatSetValuesStencil(jac,1,&row,1,&row,v,INSERT_VALUES);
434:         } else {
435:           /* interior grid points */
436:           v[0] = -hxhydhz; col[0].k=k-1;col[0].j=j;  col[0].i = i;
437:           v[1] = -hxhzdhy; col[1].k=k;  col[1].j=j-1;col[1].i = i;
438:           v[2] = -hyhzdhx; col[2].k=k;  col[2].j=j;  col[2].i = i-1;
439:           v[3] = 2.0*(hyhzdhx+hxhzdhy+hxhydhz)-sc*PetscExpScalar(x[k][j][i]);col[3].k=row.k;col[3].j=row.j;col[3].i = row.i;
440:           v[4] = -hyhzdhx; col[4].k=k;  col[4].j=j;  col[4].i = i+1;
441:           v[5] = -hxhzdhy; col[5].k=k;  col[5].j=j+1;col[5].i = i;
442:           v[6] = -hxhydhz; col[6].k=k+1;col[6].j=j;  col[6].i = i;
443:           MatSetValuesStencil(jac,1,&row,7,col,v,INSERT_VALUES);
444:         }
445:       }
446:     }
447:   }
448:   DMDAVecRestoreArray(da,localX,&x);
449:   DMRestoreLocalVector(da,&localX);

451:   /*
452:      Assemble matrix, using the 2-step process:
453:        MatAssemblyBegin(), MatAssemblyEnd().
454:   */
455:   MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
456:   MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);

458:   /*
459:      Normally since the matrix has already been assembled above; this
460:      would do nothing. But in the matrix free mode -snes_mf_operator
461:      this tells the "matrix-free" matrix that a new linear system solve
462:      is about to be done.
463:   */

465:   MatAssemblyBegin(*J,MAT_FINAL_ASSEMBLY);
466:   MatAssemblyEnd(*J,MAT_FINAL_ASSEMBLY);

468:   /*
469:      Set flag to indicate that the Jacobian matrix retains an identical
470:      nonzero structure throughout all nonlinear iterations (although the
471:      values of the entries change). Thus, we can save some work in setting
472:      up the preconditioner (e.g., no need to redo symbolic factorization for
473:      ILU/ICC preconditioners).
474:       - If the nonzero structure of the matrix is different during
475:         successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
476:         must be used instead.  If you are unsure whether the matrix
477:         structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
478:       - Caution:  If you specify SAME_NONZERO_PATTERN, PETSc
479:         believes your assertion and does not check the structure
480:         of the matrix.  If you erroneously claim that the structure
481:         is the same when it actually is not, the new preconditioner
482:         will not function correctly.  Thus, use this optimization
483:         feature with caution!
484:   */
485:   *flag = SAME_NONZERO_PATTERN;


488:   /*
489:      Tell the matrix we will never add a new nonzero location to the
490:      matrix. If we do, it will generate an error.
491:   */
492:   MatSetOption(jac,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
493:   return(0);
494: }