Actual source code: ex14.c
2: /* Program usage: mpirun -np <procs> ex14 [-help] [all PETSc options] */
4: static char help[] = "Bratu nonlinear PDE in 3d.\n\
5: We solve the Bratu (SFI - solid fuel ignition) problem in a 3D rectangular\n\
6: domain, using distributed arrays (DAs) to partition the parallel grid.\n\
7: The command line options include:\n\
8: -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
9: problem SFI: <parameter> = Bratu parameter (0 <= par <= 6.81)\n\n";
11: /*T
12: Concepts: SNES^parallel Bratu example
13: Concepts: DA^using distributed arrays;
14: Processors: n
15: T*/
17: /* ------------------------------------------------------------------------
19: Solid Fuel Ignition (SFI) problem. This problem is modeled by
20: the partial differential equation
21:
22: -Laplacian u - lambda*exp(u) = 0, 0 < x,y < 1,
23:
24: with boundary conditions
25:
26: u = 0 for x = 0, x = 1, y = 0, y = 1, z = 0, z = 1
27:
28: A finite difference approximation with the usual 7-point stencil
29: is used to discretize the boundary value problem to obtain a nonlinear
30: system of equations.
33: ------------------------------------------------------------------------- */
35: /*
36: Include "petscda.h" so that we can use distributed arrays (DAs).
37: Include "petscsnes.h" so that we can use SNES solvers. Note that this
38: file automatically includes:
39: petsc.h - base PETSc routines petscvec.h - vectors
40: petscsys.h - system routines petscmat.h - matrices
41: petscis.h - index sets petscksp.h - Krylov subspace methods
42: petscviewer.h - viewers petscpc.h - preconditioners
43: petscksp.h - linear solvers
44: */
45: #include petscda.h
46: #include petscsnes.h
49: /*
50: User-defined application context - contains data needed by the
51: application-provided call-back routines, FormJacobian() and
52: FormFunction().
53: */
54: typedef struct {
55: PetscReal param; /* test problem parameter */
56: DA da; /* distributed array data structure */
57: } AppCtx;
59: /*
60: User-defined routines
61: */
67: int main(int argc,char **argv)
68: {
69: SNES snes; /* nonlinear solver */
70: Vec x,r; /* solution, residual vectors */
71: Mat J; /* Jacobian matrix */
72: AppCtx user; /* user-defined work context */
73: PetscInt its; /* iterations for convergence */
74: PetscTruth matrix_free,coloring;
75: PetscErrorCode ierr;
76: PetscReal bratu_lambda_max = 6.81,bratu_lambda_min = 0.,fnorm;
77: MatFDColoring matfdcoloring;
79: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
80: Initialize program
81: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
83: PetscInitialize(&argc,&argv,(char *)0,help);
85: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
86: Initialize problem parameters
87: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
88: user.param = 6.0;
89: PetscOptionsGetReal(PETSC_NULL,"-par",&user.param,PETSC_NULL);
90: if (user.param >= bratu_lambda_max || user.param <= bratu_lambda_min) {
91: SETERRQ(1,"Lambda is out of range");
92: }
94: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
95: Create nonlinear solver context
96: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
97: SNESCreate(PETSC_COMM_WORLD,&snes);
99: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
100: Create distributed array (DA) to manage parallel grid and vectors
101: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
102: DACreate3d(PETSC_COMM_WORLD,DA_NONPERIODIC,DA_STENCIL_STAR,4,4,4,PETSC_DECIDE,PETSC_DECIDE,
103: PETSC_DECIDE,1,1,PETSC_NULL,PETSC_NULL,PETSC_NULL,&user.da);
105: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
106: Extract global vectors from DA; then duplicate for remaining
107: vectors that are the same types
108: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
109: DACreateGlobalVector(user.da,&x);
110: VecDuplicate(x,&r);
112: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
113: Set function evaluation routine and vector
114: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
115: SNESSetFunction(snes,r,FormFunction,(void*)&user);
117: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
118: Create matrix data structure; set Jacobian evaluation routine
120: Set Jacobian matrix data structure and default Jacobian evaluation
121: routine. User can override with:
122: -snes_mf : matrix-free Newton-Krylov method with no preconditioning
123: (unless user explicitly sets preconditioner)
124: -snes_mf_operator : form preconditioning matrix as set by the user,
125: but use matrix-free approx for Jacobian-vector
126: products within Newton-Krylov method
127: -fdcoloring : using finite differences with coloring to compute the Jacobian
129: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
130: PetscOptionsHasName(PETSC_NULL,"-snes_mf",&matrix_free);
131: PetscOptionsHasName(PETSC_NULL,"-fdcoloring",&coloring);
132: if (!matrix_free) {
133: if (coloring) {
134: ISColoring iscoloring;
136: DAGetColoring(user.da,IS_COLORING_GLOBAL,&iscoloring);
137: DAGetMatrix(user.da,MATAIJ,&J);
138: MatFDColoringCreate(J,iscoloring,&matfdcoloring);
139: ISColoringDestroy(iscoloring);
140: MatFDColoringSetFunction(matfdcoloring,(PetscErrorCode (*)(void))FormFunction,&user);
141: MatFDColoringSetFromOptions(matfdcoloring);
142: SNESSetJacobian(snes,J,J,SNESDefaultComputeJacobianColor,matfdcoloring);
143: } else {
144: DAGetMatrix(user.da,MATAIJ,&J);
145: SNESSetJacobian(snes,J,J,FormJacobian,&user);
146: }
147: }
149: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
150: Customize nonlinear solver; set runtime options
151: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
152: SNESSetFromOptions(snes);
154: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
155: Evaluate initial guess
156: Note: The user should initialize the vector, x, with the initial guess
157: for the nonlinear solver prior to calling SNESSolve(). In particular,
158: to employ an initial guess of zero, the user should explicitly set
159: this vector to zero by calling VecSet().
160: */
161: FormInitialGuess(&user,x);
163: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
164: Solve nonlinear system
165: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
166: SNESSolve(snes,PETSC_NULL,x);
167: SNESGetIterationNumber(snes,&its);
169: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
170: Explicitly check norm of the residual of the solution
171: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
172: FormFunction(snes,x,r,(void*)&user);
173: VecNorm(r,NORM_2,&fnorm);
174: PetscPrintf(PETSC_COMM_WORLD,"Number of Newton iterations = %D fnorm %G\n",its,fnorm);
176: /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
177: Free work space. All PETSc objects should be destroyed when they
178: are no longer needed.
179: - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
181: if (!matrix_free) {
182: MatDestroy(J);
183: }
184: if (coloring) {
185: MatFDColoringDestroy(matfdcoloring);
186: }
187: VecDestroy(x);
188: VecDestroy(r);
189: SNESDestroy(snes);
190: DADestroy(user.da);
191: PetscFinalize();
193: return(0);
194: }
195: /* ------------------------------------------------------------------- */
198: /*
199: FormInitialGuess - Forms initial approximation.
201: Input Parameters:
202: user - user-defined application context
203: X - vector
205: Output Parameter:
206: X - vector
207: */
208: PetscErrorCode FormInitialGuess(AppCtx *user,Vec X)
209: {
210: PetscInt i,j,k,Mx,My,Mz,xs,ys,zs,xm,ym,zm;
212: PetscReal lambda,temp1,hx,hy,hz,tempk,tempj;
213: PetscScalar ***x;
216: DAGetInfo(user->da,PETSC_IGNORE,&Mx,&My,&Mz,PETSC_IGNORE,PETSC_IGNORE,
217: PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
219: lambda = user->param;
220: hx = 1.0/(PetscReal)(Mx-1);
221: hy = 1.0/(PetscReal)(My-1);
222: hz = 1.0/(PetscReal)(Mz-1);
223: temp1 = lambda/(lambda + 1.0);
225: /*
226: Get a pointer to vector data.
227: - For default PETSc vectors, VecGetArray() returns a pointer to
228: the data array. Otherwise, the routine is implementation dependent.
229: - You MUST call VecRestoreArray() when you no longer need access to
230: the array.
231: */
232: DAVecGetArray(user->da,X,&x);
234: /*
235: Get local grid boundaries (for 3-dimensional DA):
236: xs, ys, zs - starting grid indices (no ghost points)
237: xm, ym, zm - widths of local grid (no ghost points)
239: */
240: DAGetCorners(user->da,&xs,&ys,&zs,&xm,&ym,&zm);
242: /*
243: Compute initial guess over the locally owned part of the grid
244: */
245: for (k=zs; k<zs+zm; k++) {
246: tempk = (PetscReal)(PetscMin(k,Mz-k-1))*hz;
247: for (j=ys; j<ys+ym; j++) {
248: tempj = PetscMin((PetscReal)(PetscMin(j,My-j-1))*hy,tempk);
249: for (i=xs; i<xs+xm; i++) {
250: if (i == 0 || j == 0 || k == 0 || i == Mx-1 || j == My-1 || k == Mz-1) {
251: /* boundary conditions are all zero Dirichlet */
252: x[k][j][i] = 0.0;
253: } else {
254: x[k][j][i] = temp1*sqrt(PetscMin((PetscReal)(PetscMin(i,Mx-i-1))*hx,tempj));
255: }
256: }
257: }
258: }
260: /*
261: Restore vector
262: */
263: DAVecRestoreArray(user->da,X,&x);
264: return(0);
265: }
266: /* ------------------------------------------------------------------- */
269: /*
270: FormFunction - Evaluates nonlinear function, F(x).
272: Input Parameters:
273: . snes - the SNES context
274: . X - input vector
275: . ptr - optional user-defined context, as set by SNESSetFunction()
277: Output Parameter:
278: . F - function vector
279: */
280: PetscErrorCode FormFunction(SNES snes,Vec X,Vec F,void *ptr)
281: {
282: AppCtx *user = (AppCtx*)ptr;
284: PetscInt i,j,k,Mx,My,Mz,xs,ys,zs,xm,ym,zm;
285: PetscReal two = 2.0,lambda,hx,hy,hz,hxhzdhy,hyhzdhx,hxhydhz,sc;
286: PetscScalar u_north,u_south,u_east,u_west,u_up,u_down,u,u_xx,u_yy,u_zz,***x,***f;
287: Vec localX;
290: DAGetLocalVector(user->da,&localX);
291: DAGetInfo(user->da,PETSC_IGNORE,&Mx,&My,&Mz,PETSC_IGNORE,PETSC_IGNORE,
292: PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
294: lambda = user->param;
295: hx = 1.0/(PetscReal)(Mx-1);
296: hy = 1.0/(PetscReal)(My-1);
297: hz = 1.0/(PetscReal)(Mz-1);
298: sc = hx*hy*hz*lambda;
299: hxhzdhy = hx*hz/hy;
300: hyhzdhx = hy*hz/hx;
301: hxhydhz = hx*hy/hz;
303: /*
304: Scatter ghost points to local vector,using the 2-step process
305: DAGlobalToLocalBegin(),DAGlobalToLocalEnd().
306: By placing code between these two statements, computations can be
307: done while messages are in transition.
308: */
309: DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
310: DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
312: /*
313: Get pointers to vector data
314: */
315: DAVecGetArray(user->da,localX,&x);
316: DAVecGetArray(user->da,F,&f);
318: /*
319: Get local grid boundaries
320: */
321: DAGetCorners(user->da,&xs,&ys,&zs,&xm,&ym,&zm);
323: /*
324: Compute function over the locally owned part of the grid
325: */
326: for (k=zs; k<zs+zm; k++) {
327: for (j=ys; j<ys+ym; j++) {
328: for (i=xs; i<xs+xm; i++) {
329: if (i == 0 || j == 0 || k == 0 || i == Mx-1 || j == My-1 || k == Mz-1) {
330: f[k][j][i] = x[k][j][i];
331: } else {
332: u = x[k][j][i];
333: u_east = x[k][j][i+1];
334: u_west = x[k][j][i-1];
335: u_north = x[k][j+1][i];
336: u_south = x[k][j-1][i];
337: u_up = x[k+1][j][i];
338: u_down = x[k-1][j][i];
339: u_xx = (-u_east + two*u - u_west)*hyhzdhx;
340: u_yy = (-u_north + two*u - u_south)*hxhzdhy;
341: u_zz = (-u_up + two*u - u_down)*hxhydhz;
342: f[k][j][i] = u_xx + u_yy + u_zz - sc*PetscExpScalar(u);
343: }
344: }
345: }
346: }
348: /*
349: Restore vectors
350: */
351: DAVecRestoreArray(user->da,localX,&x);
352: DAVecRestoreArray(user->da,F,&f);
353: DARestoreLocalVector(user->da,&localX);
354: PetscLogFlops(11*ym*xm);
355: return(0);
356: }
357: /* ------------------------------------------------------------------- */
360: /*
361: FormJacobian - Evaluates Jacobian matrix.
363: Input Parameters:
364: . snes - the SNES context
365: . x - input vector
366: . ptr - optional user-defined context, as set by SNESSetJacobian()
368: Output Parameters:
369: . A - Jacobian matrix
370: . B - optionally different preconditioning matrix
371: . flag - flag indicating matrix structure
373: */
374: PetscErrorCode FormJacobian(SNES snes,Vec X,Mat *J,Mat *B,MatStructure *flag,void *ptr)
375: {
376: AppCtx *user = (AppCtx*)ptr; /* user-defined application context */
377: Mat jac = *B; /* Jacobian matrix */
378: Vec localX;
380: PetscInt i,j,k,Mx,My,Mz;
381: MatStencil col[7],row;
382: PetscInt xs,ys,zs,xm,ym,zm;
383: PetscScalar lambda,v[7],hx,hy,hz,hxhzdhy,hyhzdhx,hxhydhz,sc,***x;
387: DAGetLocalVector(user->da,&localX);
388: DAGetInfo(user->da,PETSC_IGNORE,&Mx,&My,&Mz,PETSC_IGNORE,PETSC_IGNORE,
389: PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE,PETSC_IGNORE);
391: lambda = user->param;
392: hx = 1.0/(PetscReal)(Mx-1);
393: hy = 1.0/(PetscReal)(My-1);
394: hz = 1.0/(PetscReal)(Mz-1);
395: sc = hx*hy*hz*lambda;
396: hxhzdhy = hx*hz/hy;
397: hyhzdhx = hy*hz/hx;
398: hxhydhz = hx*hy/hz;
400: /*
401: Scatter ghost points to local vector, using the 2-step process
402: DAGlobalToLocalBegin(), DAGlobalToLocalEnd().
403: By placing code between these two statements, computations can be
404: done while messages are in transition.
405: */
406: DAGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
407: DAGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
409: /*
410: Get pointer to vector data
411: */
412: DAVecGetArray(user->da,localX,&x);
414: /*
415: Get local grid boundaries
416: */
417: DAGetCorners(user->da,&xs,&ys,&zs,&xm,&ym,&zm);
419: /*
420: Compute entries for the locally owned part of the Jacobian.
421: - Currently, all PETSc parallel matrix formats are partitioned by
422: contiguous chunks of rows across the processors.
423: - Each processor needs to insert only elements that it owns
424: locally (but any non-local elements will be sent to the
425: appropriate processor during matrix assembly).
426: - Here, we set all entries for a particular row at once.
427: - We can set matrix entries either using either
428: MatSetValuesLocal() or MatSetValues(), as discussed above.
429: */
430: for (k=zs; k<zs+zm; k++) {
431: for (j=ys; j<ys+ym; j++) {
432: for (i=xs; i<xs+xm; i++) {
433: row.k = k; row.j = j; row.i = i;
434: /* boundary points */
435: if (i == 0 || j == 0 || k == 0|| i == Mx-1 || j == My-1 || k == Mz-1) {
436: v[0] = 1.0;
437: MatSetValuesStencil(jac,1,&row,1,&row,v,INSERT_VALUES);
438: } else {
439: /* interior grid points */
440: v[0] = -hxhydhz; col[0].k=k-1;col[0].j=j; col[0].i = i;
441: v[1] = -hxhzdhy; col[1].k=k; col[1].j=j-1;col[1].i = i;
442: v[2] = -hyhzdhx; col[2].k=k; col[2].j=j; col[2].i = i-1;
443: 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;
444: v[4] = -hyhzdhx; col[4].k=k; col[4].j=j; col[4].i = i+1;
445: v[5] = -hxhzdhy; col[5].k=k; col[5].j=j+1;col[5].i = i;
446: v[6] = -hxhydhz; col[6].k=k+1;col[6].j=j; col[6].i = i;
447: MatSetValuesStencil(jac,1,&row,7,col,v,INSERT_VALUES);
448: }
449: }
450: }
451: }
452: DAVecRestoreArray(user->da,localX,&x);
453: DARestoreLocalVector(user->da,&localX);
455: /*
456: Assemble matrix, using the 2-step process:
457: MatAssemblyBegin(), MatAssemblyEnd().
458: */
459: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
460: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
462: /*
463: Normally since the matrix has already been assembled above; this
464: would do nothing. But in the matrix free mode -snes_mf_operator
465: this tells the "matrix-free" matrix that a new linear system solve
466: is about to be done.
467: */
469: MatAssemblyBegin(*J,MAT_FINAL_ASSEMBLY);
470: MatAssemblyEnd(*J,MAT_FINAL_ASSEMBLY);
472: /*
473: Set flag to indicate that the Jacobian matrix retains an identical
474: nonzero structure throughout all nonlinear iterations (although the
475: values of the entries change). Thus, we can save some work in setting
476: up the preconditioner (e.g., no need to redo symbolic factorization for
477: ILU/ICC preconditioners).
478: - If the nonzero structure of the matrix is different during
479: successive linear solves, then the flag DIFFERENT_NONZERO_PATTERN
480: must be used instead. If you are unsure whether the matrix
481: structure has changed or not, use the flag DIFFERENT_NONZERO_PATTERN.
482: - Caution: If you specify SAME_NONZERO_PATTERN, PETSc
483: believes your assertion and does not check the structure
484: of the matrix. If you erroneously claim that the structure
485: is the same when it actually is not, the new preconditioner
486: will not function correctly. Thus, use this optimization
487: feature with caution!
488: */
489: *flag = SAME_NONZERO_PATTERN;
492: /*
493: Tell the matrix we will never add a new nonzero location to the
494: matrix. If we do, it will generate an error.
495: */
496: MatSetOption(jac,MAT_NEW_NONZERO_LOCATION_ERR);
497: return(0);
498: }