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: * serial convergence evidence:
26: for M in 3 4 5 6 7; do ./ex9 -snes_grid_sequence $M -pc_type mg; done
27: * parallel full-cycle multigrid from enlarged coarse mesh:
28: mpiexec -n 4 ./ex9 -da_grid_x 12 -da_grid_y 12 -snes_converged_reason -snes_grid_sequence 4 -pc_type mg
29: */
31: #include <petsc.h>
33: /* z = psi(x,y) is the hemispherical obstacle, but made C^1 with "skirt" at r=r0 */
34: PetscReal psi(PetscReal x, PetscReal y)
35: {
36: const PetscReal r = x * x + y * y, r0 = 0.9, psi0 = PetscSqrtReal(1.0 - r0 * r0), dpsi0 = -r0 / psi0;
37: if (r <= r0) {
38: return PetscSqrtReal(1.0 - r);
39: } else {
40: return psi0 + dpsi0 * (r - r0);
41: }
42: }
44: /* This exact solution solves a 1D radial free-boundary problem for the
45: Laplace equation, on the interval 0 < r < 2, with above obstacle psi(x,y).
46: The Laplace equation applies where u(r) > psi(r),
47: u''(r) + r^-1 u'(r) = 0
48: with boundary conditions including free b.c.s at an unknown location r = a:
49: u(a) = psi(a), u'(a) = psi'(a), u(2) = 0
50: The solution is u(r) = - A log(r) + B on r > a. The boundary conditions
51: can then be reduced to a root-finding problem for a:
52: a^2 (log(2) - log(a)) = 1 - a^2
53: The solution is a = 0.697965148223374 (giving residual 1.5e-15). Then
54: A = a^2*(1-a^2)^(-0.5) and B = A*log(2) are as given below in the code. */
55: PetscReal u_exact(PetscReal x, PetscReal y)
56: {
57: const PetscReal afree = 0.697965148223374, A = 0.680259411891719, B = 0.471519893402112;
58: PetscReal r;
59: r = PetscSqrtReal(x * x + y * y);
60: return (r <= afree) ? psi(x, y) /* active set; on the obstacle */
61: : -A * PetscLogReal(r) + B; /* solves laplace eqn */
62: }
64: extern PetscErrorCode FormExactSolution(DMDALocalInfo *, Vec);
65: extern PetscErrorCode FormBounds(SNES, Vec, Vec);
66: extern PetscErrorCode FormFunctionLocal(DMDALocalInfo *, PetscReal **, PetscReal **, void *);
67: extern PetscErrorCode FormJacobianLocal(DMDALocalInfo *, PetscReal **, Mat, Mat, void *);
69: int main(int argc, char **argv)
70: {
71: SNES snes;
72: DM da, da_after;
73: Vec u, u_exact;
74: DMDALocalInfo info;
75: PetscReal error1, errorinf;
77: PetscFunctionBeginUser;
78: PetscCall(PetscInitialize(&argc, &argv, NULL, help));
80: 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 */
81: PETSC_DECIDE, PETSC_DECIDE, 1, 1, /* dof=1 and s = 1 (stencil extends out one cell) */
82: NULL, NULL, &da));
83: PetscCall(DMSetFromOptions(da));
84: PetscCall(DMSetUp(da));
85: PetscCall(DMDASetUniformCoordinates(da, -2.0, 2.0, -2.0, 2.0, 0.0, 1.0));
87: PetscCall(DMCreateGlobalVector(da, &u));
88: PetscCall(VecSet(u, 0.0));
90: PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
91: PetscCall(SNESSetDM(snes, da));
92: PetscCall(SNESSetType(snes, SNESVINEWTONRSLS));
93: PetscCall(SNESVISetComputeVariableBounds(snes, &FormBounds));
94: PetscCall(DMDASNESSetFunctionLocal(da, INSERT_VALUES, (DMDASNESFunctionFn *)FormFunctionLocal, NULL));
95: PetscCall(DMDASNESSetJacobianLocal(da, (DMDASNESJacobianFn *)FormJacobianLocal, NULL));
96: PetscCall(SNESSetFromOptions(snes));
98: /* solve nonlinear system */
99: PetscCall(SNESSolve(snes, NULL, u));
100: PetscCall(VecDestroy(&u));
101: PetscCall(DMDestroy(&da));
102: /* DMDA after solve may be different, e.g. with -snes_grid_sequence */
103: PetscCall(SNESGetDM(snes, &da_after));
104: PetscCall(SNESGetSolution(snes, &u)); /* do not destroy u */
105: PetscCall(DMDAGetLocalInfo(da_after, &info));
106: PetscCall(VecDuplicate(u, &u_exact));
107: PetscCall(FormExactSolution(&info, u_exact));
108: PetscCall(VecAXPY(u, -1.0, u_exact)); /* u <-- u - u_exact */
109: PetscCall(VecNorm(u, NORM_1, &error1));
110: error1 /= (PetscReal)info.mx * (PetscReal)info.my; /* average error */
111: PetscCall(VecNorm(u, NORM_INFINITY, &errorinf));
112: 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));
113: PetscCall(VecDestroy(&u_exact));
114: PetscCall(SNESDestroy(&snes));
115: PetscCall(DMDestroy(&da));
116: PetscCall(PetscFinalize());
117: return 0;
118: }
120: PetscErrorCode FormExactSolution(DMDALocalInfo *info, Vec u)
121: {
122: PetscInt i, j;
123: PetscReal **au, dx, dy, x, y;
125: PetscFunctionBeginUser;
126: dx = 4.0 / (PetscReal)(info->mx - 1);
127: dy = 4.0 / (PetscReal)(info->my - 1);
128: PetscCall(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: PetscCall(DMDAVecRestoreArray(info->da, u, &au));
137: PetscFunctionReturn(PETSC_SUCCESS);
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: PetscFunctionBeginUser;
148: PetscCall(SNESGetDM(snes, &da));
149: PetscCall(DMDAGetLocalInfo(da, &info));
150: dx = 4.0 / (PetscReal)(info.mx - 1);
151: dy = 4.0 / (PetscReal)(info.my - 1);
152: PetscCall(DMDAVecGetArray(da, Xl, &aXl));
153: for (j = info.ys; j < info.ys + info.ym; j++) {
154: y = -2.0 + j * dy;
155: for (i = info.xs; i < info.xs + info.xm; i++) {
156: x = -2.0 + i * dx;
157: aXl[j][i] = psi(x, y);
158: }
159: }
160: PetscCall(DMDAVecRestoreArray(da, Xl, &aXl));
161: PetscCall(VecSet(Xu, PETSC_INFINITY));
162: PetscFunctionReturn(PETSC_SUCCESS);
163: }
165: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **au, PetscScalar **af, void *user)
166: {
167: PetscInt i, j;
168: PetscReal dx, dy, x, y, ue, un, us, uw;
170: PetscFunctionBeginUser;
171: dx = 4.0 / (PetscReal)(info->mx - 1);
172: dy = 4.0 / (PetscReal)(info->my - 1);
173: for (j = info->ys; j < info->ys + info->ym; j++) {
174: y = -2.0 + j * dy;
175: for (i = info->xs; i < info->xs + info->xm; i++) {
176: x = -2.0 + i * dx;
177: if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
178: af[j][i] = 4.0 * (au[j][i] - u_exact(x, y));
179: } else {
180: uw = (i - 1 == 0) ? u_exact(x - dx, y) : au[j][i - 1];
181: ue = (i + 1 == info->mx - 1) ? u_exact(x + dx, y) : au[j][i + 1];
182: us = (j - 1 == 0) ? u_exact(x, y - dy) : au[j - 1][i];
183: un = (j + 1 == info->my - 1) ? u_exact(x, y + dy) : au[j + 1][i];
184: af[j][i] = -(dy / dx) * (uw - 2.0 * au[j][i] + ue) - (dx / dy) * (us - 2.0 * au[j][i] + un);
185: }
186: }
187: }
188: PetscCall(PetscLogFlops(12.0 * info->ym * info->xm));
189: PetscFunctionReturn(PETSC_SUCCESS);
190: }
192: PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **au, Mat A, Mat jac, void *user)
193: {
194: PetscInt i, j, n;
195: MatStencil col[5], row;
196: PetscReal v[5], dx, dy, oxx, oyy;
198: PetscFunctionBeginUser;
199: dx = 4.0 / (PetscReal)(info->mx - 1);
200: dy = 4.0 / (PetscReal)(info->my - 1);
201: oxx = dy / dx;
202: oyy = dx / dy;
203: for (j = info->ys; j < info->ys + info->ym; j++) {
204: for (i = info->xs; i < info->xs + info->xm; i++) {
205: row.j = j;
206: row.i = i;
207: if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) { /* boundary */
208: v[0] = 4.0;
209: PetscCall(MatSetValuesStencil(jac, 1, &row, 1, &row, v, INSERT_VALUES));
210: } else { /* interior grid points */
211: v[0] = 2.0 * (oxx + oyy);
212: col[0].j = j;
213: col[0].i = i;
214: n = 1;
215: if (i - 1 > 0) {
216: v[n] = -oxx;
217: col[n].j = j;
218: col[n++].i = i - 1;
219: }
220: if (i + 1 < info->mx - 1) {
221: v[n] = -oxx;
222: col[n].j = j;
223: col[n++].i = i + 1;
224: }
225: if (j - 1 > 0) {
226: v[n] = -oyy;
227: col[n].j = j - 1;
228: col[n++].i = i;
229: }
230: if (j + 1 < info->my - 1) {
231: v[n] = -oyy;
232: col[n].j = j + 1;
233: col[n++].i = i;
234: }
235: PetscCall(MatSetValuesStencil(jac, 1, &row, n, col, v, INSERT_VALUES));
236: }
237: }
238: }
240: /* Assemble matrix, using the 2-step process: */
241: PetscCall(MatAssemblyBegin(jac, MAT_FINAL_ASSEMBLY));
242: PetscCall(MatAssemblyEnd(jac, MAT_FINAL_ASSEMBLY));
243: if (A != jac) {
244: PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
245: PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
246: }
247: PetscCall(PetscLogFlops(2.0 * info->ym * info->xm));
248: PetscFunctionReturn(PETSC_SUCCESS);
249: }
251: /*TEST
253: build:
254: requires: !complex
256: test:
257: suffix: 1
258: requires: !single
259: nsize: 1
260: args: -da_refine 1 -snes_monitor_short -snes_type vinewtonrsls
262: test:
263: suffix: 2
264: requires: !single
265: nsize: 2
266: args: -da_refine 1 -snes_monitor_short -snes_type vinewtonssls
268: test:
269: suffix: 3
270: requires: !single
271: nsize: 2
272: args: -snes_grid_sequence 2 -snes_vi_monitor -snes_type vinewtonrsls
274: test:
275: suffix: mg
276: requires: !single
277: nsize: 4
278: args: -snes_grid_sequence 3 -snes_converged_reason -pc_type mg
280: test:
281: suffix: 4
282: nsize: 1
283: args: -mat_is_symmetric
285: test:
286: suffix: 5
287: nsize: 1
288: args: -ksp_converged_reason -snes_fd_color
290: test:
291: suffix: 6
292: requires: !single
293: nsize: 2
294: args: -snes_grid_sequence 2 -pc_type mg -snes_monitor_short -ksp_converged_reason
296: test:
297: suffix: 7
298: nsize: 2
299: 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
300: TODO: fix nasty memory leak in SNESCOMPOSITE
302: test:
303: suffix: 8
304: nsize: 2
305: args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additive -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
306: TODO: fix nasty memory leak in SNESCOMPOSITE
308: test:
309: suffix: 9
310: nsize: 2
311: args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additiveoptimal -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
312: TODO: fix nasty memory leak in SNESCOMPOSITE
314: TEST*/