Actual source code: ex9.c

```  1: static const char help[] = "Solves obstacle problem in 2D as a variational inequality\n\
2: or nonlinear complementarity problem.  This is a form of the Laplace equation in\n\
3: which the solution u is constrained to be above a given function psi.  In the\n\
4: problem here an exact solution is known.\n";

6: /*  On a square S = {-2<x<2,-2<y<2}, the PDE
7:     u_{xx} + u_{yy} = 0
8: is solved on the set where membrane is above obstacle (u(x,y) >= psi(x,y)).
9: Here psi is the upper hemisphere of the unit ball.  On the boundary of S
10: we have Dirichlet boundary conditions from the exact solution.  Uses centered
11: FD scheme.  This example contributed by Ed Bueler.

13: Example usage:
14:   * get help:
15:     ./ex9 -help
16:   * monitor run:
17:     ./ex9 -da_refine 2 -snes_vi_monitor
18:   * use other SNESVI type (default is SNESVINEWTONRSLS):
19:     ./ex9 -da_refine 2 -snes_vi_monitor -snes_type vinewtonssls
20:   * use FD evaluation of Jacobian by coloring, instead of analytical:
21:     ./ex9 -da_refine 2 -snes_fd_color
22:   * X windows visualizations:
23:     ./ex9 -snes_monitor_solution draw -draw_pause 1 -da_refine 4
24:     ./ex9 -snes_vi_monitor_residual -draw_pause 1 -da_refine 4
25:   * full-cycle multigrid:
26:     ./ex9 -snes_converged_reason -snes_grid_sequence 4 -pc_type mg
27:   * serial convergence evidence:
28:     for M in 3 4 5 6 7; do ./ex9 -snes_grid_sequence \$M -pc_type mg; done
29:   * FIXME sporadic parallel bug:
30:     mpiexec -n 4 ./ex9 -snes_converged_reason -snes_grid_sequence 4 -pc_type mg
31: */

33: #include <petsc.h>

35: /* z = psi(x,y) is the hemispherical obstacle, but made C^1 with "skirt" at r=r0 */
36: PetscReal psi(PetscReal x, PetscReal y)
37: {
38:   const PetscReal r = x * x + y * y, r0 = 0.9, psi0 = PetscSqrtReal(1.0 - r0 * r0), dpsi0 = -r0 / psi0;
39:   if (r <= r0) {
40:     return PetscSqrtReal(1.0 - r);
41:   } else {
42:     return psi0 + dpsi0 * (r - r0);
43:   }
44: }

46: /*  This exact solution solves a 1D radial free-boundary problem for the
47: Laplace equation, on the interval 0 < r < 2, with above obstacle psi(x,y).
48: The Laplace equation applies where u(r) > psi(r),
49:     u''(r) + r^-1 u'(r) = 0
50: with boundary conditions including free b.c.s at an unknown location r = a:
51:     u(a) = psi(a),  u'(a) = psi'(a),  u(2) = 0
52: The solution is  u(r) = - A log(r) + B   on  r > a.  The boundary conditions
53: can then be reduced to a root-finding problem for a:
54:     a^2 (log(2) - log(a)) = 1 - a^2
55: The solution is a = 0.697965148223374 (giving residual 1.5e-15).  Then
56: A = a^2*(1-a^2)^(-0.5) and B = A*log(2) are as given below in the code.  */
57: PetscReal u_exact(PetscReal x, PetscReal y)
58: {
59:   const PetscReal afree = 0.697965148223374, A = 0.680259411891719, B = 0.471519893402112;
60:   PetscReal       r;
61:   r = PetscSqrtReal(x * x + y * y);
62:   return (r <= afree) ? psi(x, y)                 /* active set; on the obstacle */
63:                       : -A * PetscLogReal(r) + B; /* solves laplace eqn */
64: }

66: extern PetscErrorCode FormExactSolution(DMDALocalInfo *, Vec);
67: extern PetscErrorCode FormBounds(SNES, Vec, Vec);
68: extern PetscErrorCode FormFunctionLocal(DMDALocalInfo *, PetscReal **, PetscReal **, void *);
69: extern PetscErrorCode FormJacobianLocal(DMDALocalInfo *, PetscReal **, Mat, Mat, void *);

71: int main(int argc, char **argv)
72: {
73:   SNES          snes;
74:   DM            da, da_after;
75:   Vec           u, u_exact;
76:   DMDALocalInfo info;
77:   PetscReal     error1, errorinf;

79:   PetscFunctionBeginUser;
80:   PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));

82:   PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 5, 5, /* 5x5 coarse grid; override with -da_grid_x,_y */
83:                          PETSC_DECIDE, PETSC_DECIDE, 1, 1,                                              /* dof=1 and s = 1 (stencil extends out one cell) */
84:                          NULL, NULL, &da));
85:   PetscCall(DMSetFromOptions(da));
86:   PetscCall(DMSetUp(da));
87:   PetscCall(DMDASetUniformCoordinates(da, -2.0, 2.0, -2.0, 2.0, 0.0, 1.0));

89:   PetscCall(DMCreateGlobalVector(da, &u));
90:   PetscCall(VecSet(u, 0.0));

92:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
93:   PetscCall(SNESSetDM(snes, da));
94:   PetscCall(SNESSetType(snes, SNESVINEWTONRSLS));
95:   PetscCall(SNESVISetComputeVariableBounds(snes, &FormBounds));
96:   PetscCall(DMDASNESSetFunctionLocal(da, INSERT_VALUES, (DMDASNESFunctionFn *)FormFunctionLocal, NULL));
97:   PetscCall(DMDASNESSetJacobianLocal(da, (DMDASNESJacobianFn *)FormJacobianLocal, NULL));
98:   PetscCall(SNESSetFromOptions(snes));

100:   /* solve nonlinear system */
101:   PetscCall(SNESSolve(snes, NULL, u));
102:   PetscCall(VecDestroy(&u));
103:   PetscCall(DMDestroy(&da));
104:   /* DMDA after solve may be different, e.g. with -snes_grid_sequence */
105:   PetscCall(SNESGetDM(snes, &da_after));
106:   PetscCall(SNESGetSolution(snes, &u)); /* do not destroy u */
107:   PetscCall(DMDAGetLocalInfo(da_after, &info));
108:   PetscCall(VecDuplicate(u, &u_exact));
109:   PetscCall(FormExactSolution(&info, u_exact));
110:   PetscCall(VecAXPY(u, -1.0, u_exact)); /* u <-- u - u_exact */
111:   PetscCall(VecNorm(u, NORM_1, &error1));
112:   error1 /= (PetscReal)info.mx * (PetscReal)info.my; /* average error */
113:   PetscCall(VecNorm(u, NORM_INFINITY, &errorinf));
114:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "errors on %" PetscInt_FMT " x %" PetscInt_FMT " grid:  av |u-uexact|  = %.3e,  |u-uexact|_inf = %.3e\n", info.mx, info.my, (double)error1, (double)errorinf));
115:   PetscCall(VecDestroy(&u_exact));
116:   PetscCall(SNESDestroy(&snes));
117:   PetscCall(DMDestroy(&da));
118:   PetscCall(PetscFinalize());
119:   return 0;
120: }

122: PetscErrorCode FormExactSolution(DMDALocalInfo *info, Vec u)
123: {
124:   PetscInt    i, j;
125:   PetscReal **au, dx, dy, x, y;

127:   PetscFunctionBeginUser;
128:   dx = 4.0 / (PetscReal)(info->mx - 1);
129:   dy = 4.0 / (PetscReal)(info->my - 1);
130:   PetscCall(DMDAVecGetArray(info->da, u, &au));
131:   for (j = info->ys; j < info->ys + info->ym; j++) {
132:     y = -2.0 + j * dy;
133:     for (i = info->xs; i < info->xs + info->xm; i++) {
134:       x        = -2.0 + i * dx;
135:       au[j][i] = u_exact(x, y);
136:     }
137:   }
138:   PetscCall(DMDAVecRestoreArray(info->da, u, &au));
139:   PetscFunctionReturn(PETSC_SUCCESS);
140: }

142: PetscErrorCode FormBounds(SNES snes, Vec Xl, Vec Xu)
143: {
144:   DM            da;
145:   DMDALocalInfo info;
146:   PetscInt      i, j;
147:   PetscReal   **aXl, dx, dy, x, y;

149:   PetscFunctionBeginUser;
150:   PetscCall(SNESGetDM(snes, &da));
151:   PetscCall(DMDAGetLocalInfo(da, &info));
152:   dx = 4.0 / (PetscReal)(info.mx - 1);
153:   dy = 4.0 / (PetscReal)(info.my - 1);
154:   PetscCall(DMDAVecGetArray(da, Xl, &aXl));
155:   for (j = info.ys; j < info.ys + info.ym; j++) {
156:     y = -2.0 + j * dy;
157:     for (i = info.xs; i < info.xs + info.xm; i++) {
158:       x         = -2.0 + i * dx;
159:       aXl[j][i] = psi(x, y);
160:     }
161:   }
162:   PetscCall(DMDAVecRestoreArray(da, Xl, &aXl));
163:   PetscCall(VecSet(Xu, PETSC_INFINITY));
164:   PetscFunctionReturn(PETSC_SUCCESS);
165: }

167: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **au, PetscScalar **af, void *user)
168: {
169:   PetscInt  i, j;
170:   PetscReal dx, dy, x, y, ue, un, us, uw;

172:   PetscFunctionBeginUser;
173:   dx = 4.0 / (PetscReal)(info->mx - 1);
174:   dy = 4.0 / (PetscReal)(info->my - 1);
175:   for (j = info->ys; j < info->ys + info->ym; j++) {
176:     y = -2.0 + j * dy;
177:     for (i = info->xs; i < info->xs + info->xm; i++) {
178:       x = -2.0 + i * dx;
179:       if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
180:         af[j][i] = 4.0 * (au[j][i] - u_exact(x, y));
181:       } else {
182:         uw       = (i - 1 == 0) ? u_exact(x - dx, y) : au[j][i - 1];
183:         ue       = (i + 1 == info->mx - 1) ? u_exact(x + dx, y) : au[j][i + 1];
184:         us       = (j - 1 == 0) ? u_exact(x, y - dy) : au[j - 1][i];
185:         un       = (j + 1 == info->my - 1) ? u_exact(x, y + dy) : au[j + 1][i];
186:         af[j][i] = -(dy / dx) * (uw - 2.0 * au[j][i] + ue) - (dx / dy) * (us - 2.0 * au[j][i] + un);
187:       }
188:     }
189:   }
190:   PetscCall(PetscLogFlops(12.0 * info->ym * info->xm));
191:   PetscFunctionReturn(PETSC_SUCCESS);
192: }

194: PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **au, Mat A, Mat jac, void *user)
195: {
196:   PetscInt   i, j, n;
197:   MatStencil col[5], row;
198:   PetscReal  v[5], dx, dy, oxx, oyy;

200:   PetscFunctionBeginUser;
201:   dx  = 4.0 / (PetscReal)(info->mx - 1);
202:   dy  = 4.0 / (PetscReal)(info->my - 1);
203:   oxx = dy / dx;
204:   oyy = dx / dy;
205:   for (j = info->ys; j < info->ys + info->ym; j++) {
206:     for (i = info->xs; i < info->xs + info->xm; i++) {
207:       row.j = j;
208:       row.i = i;
209:       if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) { /* boundary */
210:         v[0] = 4.0;
211:         PetscCall(MatSetValuesStencil(jac, 1, &row, 1, &row, v, INSERT_VALUES));
212:       } else { /* interior grid points */
213:         v[0]     = 2.0 * (oxx + oyy);
214:         col[0].j = j;
215:         col[0].i = i;
216:         n        = 1;
217:         if (i - 1 > 0) {
218:           v[n]       = -oxx;
219:           col[n].j   = j;
220:           col[n++].i = i - 1;
221:         }
222:         if (i + 1 < info->mx - 1) {
223:           v[n]       = -oxx;
224:           col[n].j   = j;
225:           col[n++].i = i + 1;
226:         }
227:         if (j - 1 > 0) {
228:           v[n]       = -oyy;
229:           col[n].j   = j - 1;
230:           col[n++].i = i;
231:         }
232:         if (j + 1 < info->my - 1) {
233:           v[n]       = -oyy;
234:           col[n].j   = j + 1;
235:           col[n++].i = i;
236:         }
237:         PetscCall(MatSetValuesStencil(jac, 1, &row, n, col, v, INSERT_VALUES));
238:       }
239:     }
240:   }

242:   /* Assemble matrix, using the 2-step process: */
243:   PetscCall(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY));
244:   PetscCall(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY));
245:   if (A != jac) {
246:     PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
247:     PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
248:   }
249:   PetscCall(PetscLogFlops(2.0 * info->ym * info->xm));
250:   PetscFunctionReturn(PETSC_SUCCESS);
251: }

253: /*TEST

255:    build:
256:       requires: !complex

258:    test:
259:       suffix: 1
260:       requires: !single
261:       nsize: 1
262:       args: -da_refine 1 -snes_monitor_short -snes_type vinewtonrsls

264:    test:
265:       suffix: 2
266:       requires: !single
267:       nsize: 2
268:       args: -da_refine 1 -snes_monitor_short -snes_type vinewtonssls

270:    test:
271:       suffix: 3
272:       requires: !single
273:       nsize: 2
274:       args: -snes_grid_sequence 2 -snes_vi_monitor -snes_type vinewtonrsls

276:    test:
277:       suffix: mg
278:       requires: !single
279:       nsize: 4
280:       args: -snes_grid_sequence 3 -snes_converged_reason -pc_type mg

282:    test:
283:       suffix: 4
284:       nsize: 1
285:       args: -mat_is_symmetric

287:    test:
288:       suffix: 5
289:       nsize: 1
290:       args: -ksp_converged_reason -snes_fd_color

292:    test:
293:       suffix: 6
294:       requires: !single
295:       nsize: 2
296:       args: -snes_grid_sequence 2 -pc_type mg -snes_monitor_short -ksp_converged_reason

298:    test:
299:       suffix: 7
300:       nsize: 2
301:       args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type multiplicative -snes_composite_sneses vinewtonrsls,vinewtonssls -sub_0_snes_vi_monitor -sub_1_snes_vi_monitor
302:       TODO: fix nasty memory leak in SNESCOMPOSITE

304:    test:
305:       suffix: 8
306:       nsize: 2
307:       args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additive -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
308:       TODO: fix nasty memory leak in SNESCOMPOSITE

310:    test:
311:       suffix: 9
312:       nsize: 2
313:       args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additiveoptimal -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
314:       TODO: fix nasty memory leak in SNESCOMPOSITE

316: TEST*/
```