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;
80: 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: DMSetFromOptions(da);
86: DMSetUp(da);
87: DMDASetUniformCoordinates(da, -2.0, 2.0, -2.0, 2.0, 0.0, 1.0);
89: DMCreateGlobalVector(da, &u);
90: VecSet(u, 0.0);
92: SNESCreate(PETSC_COMM_WORLD, &snes);
93: SNESSetDM(snes, da);
94: SNESSetType(snes, SNESVINEWTONRSLS);
95: SNESVISetComputeVariableBounds(snes, &FormBounds);
96: DMDASNESSetFunctionLocal(da, INSERT_VALUES, (DMDASNESFunction)FormFunctionLocal, NULL);
97: DMDASNESSetJacobianLocal(da, (DMDASNESJacobian)FormJacobianLocal, NULL);
98: SNESSetFromOptions(snes);
100: /* solve nonlinear system */
101: SNESSolve(snes, NULL, u);
102: VecDestroy(&u);
103: DMDestroy(&da);
104: /* DMDA after solve may be different, e.g. with -snes_grid_sequence */
105: SNESGetDM(snes, &da_after);
106: SNESGetSolution(snes, &u); /* do not destroy u */
107: DMDAGetLocalInfo(da_after, &info);
108: VecDuplicate(u, &u_exact);
109: FormExactSolution(&info, u_exact);
110: VecAXPY(u, -1.0, u_exact); /* u <-- u - u_exact */
111: VecNorm(u, NORM_1, &error1);
112: error1 /= (PetscReal)info.mx * (PetscReal)info.my; /* average error */
113: VecNorm(u, NORM_INFINITY, &errorinf);
114: 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: VecDestroy(&u_exact);
116: SNESDestroy(&snes);
117: DMDestroy(&da);
118: PetscFinalize();
119: return 0;
120: }
122: PetscErrorCode FormExactSolution(DMDALocalInfo *info, Vec u)
123: {
124: PetscInt i, j;
125: PetscReal **au, dx, dy, x, y;
126: dx = 4.0 / (PetscReal)(info->mx - 1);
127: dy = 4.0 / (PetscReal)(info->my - 1);
128: DMDAVecGetArray(info->da, u, &au);
129: for (j = info->ys; j < info->ys + info->ym; j++) {
130: y = -2.0 + j * dy;
131: for (i = info->xs; i < info->xs + info->xm; i++) {
132: x = -2.0 + i * dx;
133: au[j][i] = u_exact(x, y);
134: }
135: }
136: DMDAVecRestoreArray(info->da, u, &au);
137: return 0;
138: }
140: PetscErrorCode FormBounds(SNES snes, Vec Xl, Vec Xu)
141: {
142: DM da;
143: DMDALocalInfo info;
144: PetscInt i, j;
145: PetscReal **aXl, dx, dy, x, y;
147: SNESGetDM(snes, &da);
148: DMDAGetLocalInfo(da, &info);
149: dx = 4.0 / (PetscReal)(info.mx - 1);
150: dy = 4.0 / (PetscReal)(info.my - 1);
151: DMDAVecGetArray(da, Xl, &aXl);
152: for (j = info.ys; j < info.ys + info.ym; j++) {
153: y = -2.0 + j * dy;
154: for (i = info.xs; i < info.xs + info.xm; i++) {
155: x = -2.0 + i * dx;
156: aXl[j][i] = psi(x, y);
157: }
158: }
159: DMDAVecRestoreArray(da, Xl, &aXl);
160: VecSet(Xu, PETSC_INFINITY);
161: return 0;
162: }
164: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **au, PetscScalar **af, void *user)
165: {
166: PetscInt i, j;
167: PetscReal dx, dy, x, y, ue, un, us, uw;
170: dx = 4.0 / (PetscReal)(info->mx - 1);
171: dy = 4.0 / (PetscReal)(info->my - 1);
172: for (j = info->ys; j < info->ys + info->ym; j++) {
173: y = -2.0 + j * dy;
174: for (i = info->xs; i < info->xs + info->xm; i++) {
175: x = -2.0 + i * dx;
176: if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
177: af[j][i] = 4.0 * (au[j][i] - u_exact(x, y));
178: } else {
179: uw = (i - 1 == 0) ? u_exact(x - dx, y) : au[j][i - 1];
180: ue = (i + 1 == info->mx - 1) ? u_exact(x + dx, y) : au[j][i + 1];
181: us = (j - 1 == 0) ? u_exact(x, y - dy) : au[j - 1][i];
182: un = (j + 1 == info->my - 1) ? u_exact(x, y + dy) : au[j + 1][i];
183: af[j][i] = -(dy / dx) * (uw - 2.0 * au[j][i] + ue) - (dx / dy) * (us - 2.0 * au[j][i] + un);
184: }
185: }
186: }
187: PetscLogFlops(12.0 * info->ym * info->xm);
188: return 0;
189: }
191: PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **au, Mat A, Mat jac, void *user)
192: {
193: PetscInt i, j, n;
194: MatStencil col[5], row;
195: PetscReal v[5], dx, dy, oxx, oyy;
198: dx = 4.0 / (PetscReal)(info->mx - 1);
199: dy = 4.0 / (PetscReal)(info->my - 1);
200: oxx = dy / dx;
201: oyy = dx / dy;
202: for (j = info->ys; j < info->ys + info->ym; j++) {
203: for (i = info->xs; i < info->xs + info->xm; i++) {
204: row.j = j;
205: row.i = i;
206: if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) { /* boundary */
207: v[0] = 4.0;
208: MatSetValuesStencil(jac, 1, &row, 1, &row, v, INSERT_VALUES);
209: } else { /* interior grid points */
210: v[0] = 2.0 * (oxx + oyy);
211: col[0].j = j;
212: col[0].i = i;
213: n = 1;
214: if (i - 1 > 0) {
215: v[n] = -oxx;
216: col[n].j = j;
217: col[n++].i = i - 1;
218: }
219: if (i + 1 < info->mx - 1) {
220: v[n] = -oxx;
221: col[n].j = j;
222: col[n++].i = i + 1;
223: }
224: if (j - 1 > 0) {
225: v[n] = -oyy;
226: col[n].j = j - 1;
227: col[n++].i = i;
228: }
229: if (j + 1 < info->my - 1) {
230: v[n] = -oyy;
231: col[n].j = j + 1;
232: col[n++].i = i;
233: }
234: MatSetValuesStencil(jac, 1, &row, n, col, v, INSERT_VALUES);
235: }
236: }
237: }
239: /* Assemble matrix, using the 2-step process: */
240: MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY);
241: MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY);
242: if (A != jac) {
243: MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY);
244: MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY);
245: }
246: PetscLogFlops(2.0 * info->ym * info->xm);
247: return 0;
248: }
250: /*TEST
252: build:
253: requires: !complex
255: test:
256: suffix: 1
257: requires: !single
258: nsize: 1
259: args: -da_refine 1 -snes_monitor_short -snes_type vinewtonrsls
261: test:
262: suffix: 2
263: requires: !single
264: nsize: 2
265: args: -da_refine 1 -snes_monitor_short -snes_type vinewtonssls
267: test:
268: suffix: 3
269: requires: !single
270: nsize: 2
271: args: -snes_grid_sequence 2 -snes_vi_monitor -snes_type vinewtonrsls
273: test:
274: suffix: mg
275: requires: !single
276: nsize: 4
277: args: -snes_grid_sequence 3 -snes_converged_reason -pc_type mg
279: test:
280: suffix: 4
281: nsize: 1
282: args: -mat_is_symmetric
284: test:
285: suffix: 5
286: nsize: 1
287: args: -ksp_converged_reason -snes_fd_color
289: test:
290: suffix: 6
291: requires: !single
292: nsize: 2
293: args: -snes_grid_sequence 2 -pc_type mg -snes_monitor_short -ksp_converged_reason
295: test:
296: suffix: 7
297: nsize: 2
298: 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
299: TODO: fix nasty memory leak in SNESCOMPOSITE
301: test:
302: suffix: 8
303: nsize: 2
304: args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additive -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
305: TODO: fix nasty memory leak in SNESCOMPOSITE
307: test:
308: suffix: 9
309: nsize: 2
310: args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additiveoptimal -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
311: TODO: fix nasty memory leak in SNESCOMPOSITE
313: TEST*/