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*/