Actual source code: ex8.c
petsc-dev 2014-02-02
1: #include <petscsnes.h>
2: #include <petscdmda.h>
4: static char help[] = "Parallel version of the minimum surface area problem using DMs.\n\
5: See ex10.c for the serial version. It solves a system of nonlinear equations in mixed\n\
6: complementarity form using semismooth newton algorithm.This example is based on a\n\
7: problem from the MINPACK-2 test suite. Given a rectangular 2-D domain and\n\
8: boundary values along the edges of the domain, the objective is to find the\n\
9: surface with the minimal area that satisfies the boundary conditions.\n\
10: This application solves this problem using complimentarity -- We are actually\n\
11: solving the system (grad f)_i >= 0, if x_i == l_i \n\
12: (grad f)_i = 0, if l_i < x_i < u_i \n\
13: (grad f)_i <= 0, if x_i == u_i \n\
14: where f is the function to be minimized. \n\
15: \n\
16: The command line options are:\n\
17: -da_grid_x <nx>, where <nx> = number of grid points in the 1st coordinate direction\n\
18: -da_grid_y <ny>, where <ny> = number of grid points in the 2nd coordinate direction\n\
19: -start <st>, where <st> =0 for zero vector, and an average of the boundary conditions otherwise\n\
20: -lb <value>, lower bound on the variables\n\
21: -ub <value>, upper bound on the variables\n\n";
23: /*
24: User-defined application context - contains data needed by the
25: application-provided call-back routines, FormJacobian() and
26: FormFunction().
27: */
29: typedef struct {
30: DM da;
31: PetscScalar *bottom, *top, *left, *right;
32: PetscInt mx,my;
33: } AppCtx;
36: /* -------- User-defined Routines --------- */
38: extern PetscErrorCode MSA_BoundaryConditions(AppCtx*);
39: extern PetscErrorCode MSA_InitialPoint(AppCtx*, Vec);
40: extern PetscErrorCode FormGradient(SNES, Vec, Vec, void*);
41: extern PetscErrorCode FormJacobian(SNES, Vec, Mat*, Mat*, MatStructure*,void*);
45: int main(int argc, char **argv)
46: {
48: Vec x,r; /* solution and residual vectors */
49: Vec xl,xu; /* Bounds on the variables */
50: PetscBool flg_l,flg_u; /* flags to check if the bounds are set */
51: SNES snes; /* nonlinear solver context */
52: Mat J; /* Jacobian matrix */
53: PetscInt N; /* Number of elements in vector */
54: PetscScalar lb = .05;
55: PetscScalar ub = PETSC_INFINITY;
56: AppCtx user; /* user-defined work context */
57: PetscBool flg;
59: /* Initialize PETSc */
60: PetscInitialize(&argc, &argv, (char*)0, help);
62: #if defined(PETSC_USE_COMPLEX)
63: SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_SUP,"This example does not work for scalar type complex\n");
64: #endif
66: /* Check if lower and upper bounds are set */
67: PetscOptionsGetScalar(NULL, "-lb", &lb, &flg_l);
68: PetscOptionsGetScalar(NULL, "-ub", &ub, &flg_u);
70: /* Create distributed array to manage the 2d grid */
71: DMDACreate2d(PETSC_COMM_WORLD, DMDA_BOUNDARY_NONE, DMDA_BOUNDARY_NONE,DMDA_STENCIL_BOX,-4,-4,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&user.da);
72: 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);
73: /* Extract global vectors from DMDA; */
74: DMCreateGlobalVector(user.da,&x);
75: VecDuplicate(x, &r);
77: N = user.mx*user.my;
78: DMSetMatType(user.da,MATAIJ);
79: DMCreateMatrix(user.da,&J);
81: /* Create nonlinear solver context */
82: SNESCreate(PETSC_COMM_WORLD,&snes);
84: /* Set function evaluation and Jacobian evaluation routines */
85: SNESSetFunction(snes,r,FormGradient,&user);
86: SNESSetJacobian(snes,J,J,FormJacobian,&user);
88: /* Set the boundary conditions */
89: MSA_BoundaryConditions(&user);
91: /* Set initial solution guess */
92: MSA_InitialPoint(&user, x);
95: /* Set Bounds on variables */
96: VecDuplicate(x, &xl);
97: VecDuplicate(x, &xu);
98: VecSet(xl, lb);
99: VecSet(xu, ub);
101: SNESVISetVariableBounds(snes,xl,xu);
103: SNESSetFromOptions(snes);
105: /* Solve the application */
106: SNESSolve(snes,NULL,x);
108: PetscOptionsHasName(NULL,"-view_sol",&flg);
109: if (flg) { VecView(x,PETSC_VIEWER_STDOUT_WORLD); }
111: /* Free memory */
112: VecDestroy(&x);
113: VecDestroy(&xl);
114: VecDestroy(&xu);
115: VecDestroy(&r);
116: MatDestroy(&J);
117: SNESDestroy(&snes);
119: /* Free user-created data structures */
120: DMDestroy(&user.da);
121: PetscFree(user.bottom);
122: PetscFree(user.top);
123: PetscFree(user.left);
124: PetscFree(user.right);
126: PetscFinalize();
128: return 0;
129: }
131: /* -------------------------------------------------------------------- */
135: /* FormGradient - Evaluates gradient of f.
137: Input Parameters:
138: . snes - the SNES context
139: . X - input vector
140: . ptr - optional user-defined context, as set by SNESSetFunction()
142: Output Parameters:
143: . G - vector containing the newly evaluated gradient
144: */
145: PetscErrorCode FormGradient(SNES snes, Vec X, Vec G, void *ptr)
146: {
147: AppCtx *user = (AppCtx*) ptr;
148: int ierr;
149: PetscInt i,j;
150: PetscInt mx=user->mx, my=user->my;
151: PetscScalar hx=1.0/(mx+1),hy=1.0/(my+1), hydhx=hy/hx, hxdhy=hx/hy;
152: PetscScalar f1,f2,f3,f4,f5,f6,d1,d2,d3,d4,d5,d6,d7,d8,xc,xl,xr,xt,xb,xlt,xrb;
153: PetscScalar df1dxc,df2dxc,df3dxc,df4dxc,df5dxc,df6dxc;
154: PetscScalar **g, **x;
155: PetscInt xs,xm,ys,ym;
156: Vec localX;
159: /* Initialize vector to zero */
160: VecSet(G,0.0);
162: /* Get local vector */
163: DMGetLocalVector(user->da,&localX);
164: /* Get ghost points */
165: DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
166: DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
167: /* Get pointer to local vector data */
168: DMDAVecGetArray(user->da,localX, &x);
169: DMDAVecGetArray(user->da,G, &g);
171: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
172: /* Compute function over the locally owned part of the mesh */
173: for (j=ys; j < ys+ym; j++) {
174: for (i=xs; i< xs+xm; i++) {
176: xc = x[j][i];
177: xlt=xrb=xl=xr=xb=xt=xc;
179: if (i==0) { /* left side */
180: xl = user->left[j+1];
181: xlt = user->left[j+2];
182: } else xl = x[j][i-1];
184: if (j==0) { /* bottom side */
185: xb = user->bottom[i+1];
186: xrb = user->bottom[i+2];
187: } else xb = x[j-1][i];
189: if (i+1 == mx) { /* right side */
190: xr = user->right[j+1];
191: xrb = user->right[j];
192: } else xr = x[j][i+1];
194: if (j+1==0+my) { /* top side */
195: xt = user->top[i+1];
196: xlt = user->top[i];
197: } else xt = x[j+1][i];
199: if (i>0 && j+1<my) xlt = x[j+1][i-1]; /* left top side */
200: if (j>0 && i+1<mx) xrb = x[j-1][i+1]; /* right bottom */
202: d1 = (xc-xl);
203: d2 = (xc-xr);
204: d3 = (xc-xt);
205: d4 = (xc-xb);
206: d5 = (xr-xrb);
207: d6 = (xrb-xb);
208: d7 = (xlt-xl);
209: d8 = (xt-xlt);
211: df1dxc = d1*hydhx;
212: df2dxc = (d1*hydhx + d4*hxdhy);
213: df3dxc = d3*hxdhy;
214: df4dxc = (d2*hydhx + d3*hxdhy);
215: df5dxc = d2*hydhx;
216: df6dxc = d4*hxdhy;
218: d1 /= hx;
219: d2 /= hx;
220: d3 /= hy;
221: d4 /= hy;
222: d5 /= hy;
223: d6 /= hx;
224: d7 /= hy;
225: d8 /= hx;
227: f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
228: f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
229: f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
230: f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
231: f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
232: f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);
234: df1dxc /= f1;
235: df2dxc /= f2;
236: df3dxc /= f3;
237: df4dxc /= f4;
238: df5dxc /= f5;
239: df6dxc /= f6;
241: g[j][i] = (df1dxc+df2dxc+df3dxc+df4dxc+df5dxc+df6dxc)/2.0;
243: }
244: }
246: /* Restore vectors */
247: DMDAVecRestoreArray(user->da,localX, &x);
248: DMDAVecRestoreArray(user->da,G, &g);
249: DMRestoreLocalVector(user->da,&localX);
250: PetscLogFlops(67*mx*my);
251: return(0);
252: }
254: /* ------------------------------------------------------------------- */
257: /*
258: FormJacobian - Evaluates Jacobian matrix.
260: Input Parameters:
261: . snes - SNES context
262: . X - input vector
263: . ptr - optional user-defined context, as set by SNESSetJacobian()
265: Output Parameters:
266: . tH - Jacobian matrix
268: */
269: PetscErrorCode FormJacobian(SNES snes, Vec X, Mat *tH, Mat *tHPre, MatStructure *flag, void *ptr)
270: {
271: AppCtx *user = (AppCtx*) ptr;
272: Mat H = *tH;
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);}
289: *flag=SAME_NONZERO_PATTERN;
291: /* Get local vector */
292: DMGetLocalVector(user->da,&localX);
293: /* Get ghost points */
294: DMGlobalToLocalBegin(user->da,X,INSERT_VALUES,localX);
295: DMGlobalToLocalEnd(user->da,X,INSERT_VALUES,localX);
297: /* Get pointers to vector data */
298: DMDAVecGetArray(user->da,localX, &x);
300: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
301: /* Compute Jacobian over the locally owned part of the mesh */
302: for (j=ys; j< ys+ym; j++) {
303: for (i=xs; i< xs+xm; i++) {
304: xc = x[j][i];
305: xlt=xrb=xl=xr=xb=xt=xc;
307: /* Left */
308: if (i==0) {
309: xl = user->left[j+1];
310: xlt = user->left[j+2];
311: } else xl = x[j][i-1];
313: /* Bottom */
314: if (j==0) {
315: xb = user->bottom[i+1];
316: xrb = user->bottom[i+2];
317: } else xb = x[j-1][i];
319: /* Right */
320: if (i+1 == mx) {
321: xr = user->right[j+1];
322: xrb = user->right[j];
323: } else xr = x[j][i+1];
325: /* Top */
326: if (j+1==my) {
327: xt = user->top[i+1];
328: xlt = user->top[i];
329: } else xt = x[j+1][i];
331: /* Top left */
332: if (i>0 && j+1<my) xlt = x[j+1][i-1];
334: /* Bottom right */
335: if (j>0 && i+1<mx) xrb = x[j-1][i+1];
337: d1 = (xc-xl)/hx;
338: d2 = (xc-xr)/hx;
339: d3 = (xc-xt)/hy;
340: d4 = (xc-xb)/hy;
341: d5 = (xrb-xr)/hy;
342: d6 = (xrb-xb)/hx;
343: d7 = (xlt-xl)/hy;
344: d8 = (xlt-xt)/hx;
346: f1 = PetscSqrtReal(1.0 + d1*d1 + d7*d7);
347: f2 = PetscSqrtReal(1.0 + d1*d1 + d4*d4);
348: f3 = PetscSqrtReal(1.0 + d3*d3 + d8*d8);
349: f4 = PetscSqrtReal(1.0 + d3*d3 + d2*d2);
350: f5 = PetscSqrtReal(1.0 + d2*d2 + d5*d5);
351: f6 = PetscSqrtReal(1.0 + d4*d4 + d6*d6);
354: hl = (-hydhx*(1.0+d7*d7)+d1*d7)/(f1*f1*f1)+
355: (-hydhx*(1.0+d4*d4)+d1*d4)/(f2*f2*f2);
356: hr = (-hydhx*(1.0+d5*d5)+d2*d5)/(f5*f5*f5)+
357: (-hydhx*(1.0+d3*d3)+d2*d3)/(f4*f4*f4);
358: ht = (-hxdhy*(1.0+d8*d8)+d3*d8)/(f3*f3*f3)+
359: (-hxdhy*(1.0+d2*d2)+d2*d3)/(f4*f4*f4);
360: hb = (-hxdhy*(1.0+d6*d6)+d4*d6)/(f6*f6*f6)+
361: (-hxdhy*(1.0+d1*d1)+d1*d4)/(f2*f2*f2);
363: hbr = -d2*d5/(f5*f5*f5) - d4*d6/(f6*f6*f6);
364: htl = -d1*d7/(f1*f1*f1) - d3*d8/(f3*f3*f3);
366: hc = hydhx*(1.0+d7*d7)/(f1*f1*f1) + hxdhy*(1.0+d8*d8)/(f3*f3*f3) +
367: hydhx*(1.0+d5*d5)/(f5*f5*f5) + hxdhy*(1.0+d6*d6)/(f6*f6*f6) +
368: (hxdhy*(1.0+d1*d1)+hydhx*(1.0+d4*d4)-2*d1*d4)/(f2*f2*f2) +
369: (hxdhy*(1.0+d2*d2)+hydhx*(1.0+d3*d3)-2*d2*d3)/(f4*f4*f4);
371: hl/=2.0; hr/=2.0; ht/=2.0; hb/=2.0; hbr/=2.0; htl/=2.0; hc/=2.0;
373: k =0;
374: row.i = i;row.j= j;
375: /* Bottom */
376: if (j>0) {
377: v[k] =hb;
378: col[k].i = i; col[k].j=j-1; k++;
379: }
381: /* Bottom right */
382: if (j>0 && i < mx -1) {
383: v[k] =hbr;
384: col[k].i = i+1; col[k].j = j-1; k++;
385: }
387: /* left */
388: if (i>0) {
389: v[k] = hl;
390: col[k].i = i-1; col[k].j = j; k++;
391: }
393: /* Centre */
394: v[k]= hc; col[k].i= row.i; col[k].j = row.j; k++;
396: /* Right */
397: if (i < mx-1) {
398: v[k] = hr;
399: col[k].i= i+1; col[k].j = j;k++;
400: }
402: /* Top left */
403: if (i>0 && j < my-1) {
404: v[k] = htl;
405: col[k].i = i-1;col[k].j = j+1; k++;
406: }
408: /* Top */
409: if (j < my-1) {
410: v[k] = ht;
411: col[k].i = i; col[k].j = j+1; k++;
412: }
414: MatSetValuesStencil(H,1,&row,k,col,v,INSERT_VALUES);
415: }
416: }
418: /* Assemble the matrix */
419: MatAssemblyBegin(H,MAT_FINAL_ASSEMBLY);
420: DMDAVecRestoreArray(user->da,localX,&x);
421: MatAssemblyEnd(H,MAT_FINAL_ASSEMBLY);
422: DMRestoreLocalVector(user->da,&localX);
424: PetscLogFlops(199*mx*my);
425: return(0);
426: }
428: /* ------------------------------------------------------------------- */
431: /*
432: MSA_BoundaryConditions - Calculates the boundary conditions for
433: the region.
435: Input Parameter:
436: . user - user-defined application context
438: Output Parameter:
439: . user - user-defined application context
440: */
441: PetscErrorCode MSA_BoundaryConditions(AppCtx * user)
442: {
444: PetscInt i,j,k,limit=0,maxits=5;
445: PetscInt mx =user->mx,my=user->my;
446: PetscInt bsize=0, lsize=0, tsize=0, rsize=0;
447: PetscScalar one =1.0, two=2.0, three=3.0, tol=1e-10;
448: PetscScalar fnorm,det,hx,hy,xt=0,yt=0;
449: PetscScalar u1,u2,nf1,nf2,njac11,njac12,njac21,njac22;
450: PetscScalar b=-0.5, t=0.5, l=-0.5, r=0.5;
451: PetscScalar *boundary;
454: bsize=mx+2; lsize=my+2; rsize=my+2; tsize=mx+2;
456: PetscMalloc1(bsize, &user->bottom);
457: PetscMalloc1(tsize, &user->top);
458: PetscMalloc1(lsize, &user->left);
459: PetscMalloc1(rsize, &user->right);
461: hx= (r-l)/(mx+1); hy=(t-b)/(my+1);
463: for (j=0; j<4; j++) {
464: if (j==0) {
465: yt = b;
466: xt = l;
467: limit = bsize;
468: boundary = user->bottom;
469: } else if (j==1) {
470: yt = t;
471: xt = l;
472: limit = tsize;
473: boundary = user->top;
474: } else if (j==2) {
475: yt = b;
476: xt = l;
477: limit = lsize;
478: boundary = user->left;
479: } else { /* if (j==3) */
480: yt = b;
481: xt = r;
482: limit = rsize;
483: boundary = user->right;
484: }
486: for (i=0; i<limit; i++) {
487: u1=xt;
488: u2=-yt;
489: for (k=0; k<maxits; k++) {
490: nf1 = u1 + u1*u2*u2 - u1*u1*u1/three-xt;
491: nf2 = -u2 - u1*u1*u2 + u2*u2*u2/three-yt;
492: fnorm = PetscSqrtReal(nf1*nf1+nf2*nf2);
493: if (fnorm <= tol) break;
494: njac11 = one+u2*u2-u1*u1;
495: njac12 = two*u1*u2;
496: njac21 = -two*u1*u2;
497: njac22 = -one - u1*u1 + u2*u2;
498: det = njac11*njac22-njac21*njac12;
499: u1 = u1-(njac22*nf1-njac12*nf2)/det;
500: u2 = u2-(njac11*nf2-njac21*nf1)/det;
501: }
503: boundary[i]=u1*u1-u2*u2;
504: if (j==0 || j==1) xt=xt+hx;
505: else yt=yt+hy; /* if (j==2 || j==3) */
506: }
507: }
508: return(0);
509: }
511: /* ------------------------------------------------------------------- */
514: /*
515: MSA_InitialPoint - Calculates the initial guess in one of three ways.
517: Input Parameters:
518: . user - user-defined application context
519: . X - vector for initial guess
521: Output Parameters:
522: . X - newly computed initial guess
523: */
524: PetscErrorCode MSA_InitialPoint(AppCtx * user, Vec X)
525: {
527: PetscInt start=-1,i,j;
528: PetscScalar zero =0.0;
529: PetscBool flg;
532: PetscOptionsGetInt(NULL,"-start",&start,&flg);
534: if (flg && start==0) { /* The zero vector is reasonable */
536: VecSet(X, zero);
537: /* PLogIerr(user,"Min. Surface Area Problem: Start with 0 vector \n"); */
540: } else { /* Take an average of the boundary conditions */
541: PetscInt mx=user->mx,my=user->my;
542: PetscScalar **x;
543: PetscInt xs,xm,ys,ym;
545: /* Get pointers to vector data */
546: DMDAVecGetArray(user->da,X,&x);
547: DMDAGetCorners(user->da,&xs,&ys,NULL,&xm,&ym,NULL);
549: /* Perform local computations */
550: for (j=ys; j<ys+ym; j++) {
551: for (i=xs; i< xs+xm; i++) {
552: x[j][i] = (((j+1)*user->bottom[i+1]+(my-j+1)*user->top[i+1])/(my+2)+
553: ((i+1)*user->left[j+1]+(mx-i+1)*user->right[j+1])/(mx+2))/2.0;
554: }
555: }
557: /* Restore vectors */
558: DMDAVecRestoreArray(user->da,X,&x);
560: }
561: return(0);
562: }