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,
60: A = 0.680259411891719,
61: B = 0.471519893402112;
62: PetscReal r;
63: r = PetscSqrtReal(x * x + y * y);
64: return (r <= afree) ? psi(x,y) /* active set; on the obstacle */
65: : - A * PetscLogReal(r) + B; /* solves laplace eqn */
66: }
68: extern PetscErrorCode FormExactSolution(DMDALocalInfo*,Vec);
69: extern PetscErrorCode FormBounds(SNES,Vec,Vec);
70: extern PetscErrorCode FormFunctionLocal(DMDALocalInfo*,PetscReal**,PetscReal**,void*);
71: extern PetscErrorCode FormJacobianLocal(DMDALocalInfo*,PetscReal**,Mat,Mat,void*);
73: int main(int argc,char **argv)
74: {
75: PetscErrorCode ierr;
76: SNES snes;
77: DM da, da_after;
78: Vec u, u_exact;
79: DMDALocalInfo info;
80: PetscReal error1,errorinf;
82: PetscInitialize(&argc,&argv,(char*)0,help);
84: DMDACreate2d(PETSC_COMM_WORLD,DM_BOUNDARY_NONE,DM_BOUNDARY_NONE,
85: DMDA_STENCIL_STAR,5,5, /* 5x5 coarse grid; override with -da_grid_x,_y */
86: PETSC_DECIDE,PETSC_DECIDE,
87: 1,1, /* dof=1 and s = 1 (stencil extends out one cell) */
88: NULL,NULL,&da);
89: DMSetFromOptions(da);
90: DMSetUp(da);
91: DMDASetUniformCoordinates(da,-2.0,2.0,-2.0,2.0,0.0,1.0);
93: DMCreateGlobalVector(da,&u);
94: VecSet(u,0.0);
96: SNESCreate(PETSC_COMM_WORLD,&snes);
97: SNESSetDM(snes,da);
98: SNESSetType(snes,SNESVINEWTONRSLS);
99: SNESVISetComputeVariableBounds(snes,&FormBounds);
100: DMDASNESSetFunctionLocal(da,INSERT_VALUES,(DMDASNESFunction)FormFunctionLocal,NULL);
101: DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)FormJacobianLocal,NULL);
102: SNESSetFromOptions(snes);
104: /* solve nonlinear system */
105: SNESSolve(snes,NULL,u);
106: VecDestroy(&u);
107: DMDestroy(&da);
108: /* DMDA after solve may be different, e.g. with -snes_grid_sequence */
109: SNESGetDM(snes,&da_after);
110: SNESGetSolution(snes,&u); /* do not destroy u */
111: DMDAGetLocalInfo(da_after,&info);
112: VecDuplicate(u,&u_exact);
113: FormExactSolution(&info,u_exact);
114: VecAXPY(u,-1.0,u_exact); /* u <-- u - u_exact */
115: VecNorm(u,NORM_1,&error1);
116: error1 /= (PetscReal)info.mx * (PetscReal)info.my; /* average error */
117: VecNorm(u,NORM_INFINITY,&errorinf);
118: PetscPrintf(PETSC_COMM_WORLD,"errors on %D x %D grid: av |u-uexact| = %.3e, |u-uexact|_inf = %.3e\n",info.mx,info.my,(double)error1,(double)errorinf);
119: VecDestroy(&u_exact);
120: SNESDestroy(&snes);
121: DMDestroy(&da);
122: PetscFinalize();
123: return 0;
124: }
126: PetscErrorCode FormExactSolution(DMDALocalInfo *info, Vec u)
127: {
128: PetscInt i,j;
129: PetscReal **au, dx, dy, x, y;
130: dx = 4.0 / (PetscReal)(info->mx-1);
131: dy = 4.0 / (PetscReal)(info->my-1);
132: DMDAVecGetArray(info->da, u, &au);
133: for (j=info->ys; j<info->ys+info->ym; j++) {
134: y = -2.0 + j * dy;
135: for (i=info->xs; i<info->xs+info->xm; i++) {
136: x = -2.0 + i * dx;
137: au[j][i] = u_exact(x,y);
138: }
139: }
140: DMDAVecRestoreArray(info->da, u, &au);
141: return 0;
142: }
144: PetscErrorCode FormBounds(SNES snes, Vec Xl, Vec Xu)
145: {
146: DM da;
147: DMDALocalInfo info;
148: PetscInt i, j;
149: PetscReal **aXl, dx, dy, x, y;
151: SNESGetDM(snes,&da);
152: DMDAGetLocalInfo(da,&info);
153: dx = 4.0 / (PetscReal)(info.mx-1);
154: dy = 4.0 / (PetscReal)(info.my-1);
155: DMDAVecGetArray(da, Xl, &aXl);
156: for (j=info.ys; j<info.ys+info.ym; j++) {
157: y = -2.0 + j * dy;
158: for (i=info.xs; i<info.xs+info.xm; i++) {
159: x = -2.0 + i * dx;
160: aXl[j][i] = psi(x,y);
161: }
162: }
163: DMDAVecRestoreArray(da, Xl, &aXl);
164: VecSet(Xu,PETSC_INFINITY);
165: return 0;
166: }
168: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **au, PetscScalar **af, void *user)
169: {
170: PetscInt i,j;
171: PetscReal dx,dy,x,y,ue,un,us,uw;
174: dx = 4.0 / (PetscReal)(info->mx-1);
175: dy = 4.0 / (PetscReal)(info->my-1);
176: for (j=info->ys; j<info->ys+info->ym; j++) {
177: y = -2.0 + j * dy;
178: for (i=info->xs; i<info->xs+info->xm; i++) {
179: x = -2.0 + i * dx;
180: if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) {
181: af[j][i] = 4.0 * (au[j][i] - u_exact(x,y));
182: } else {
183: uw = (i-1 == 0) ? u_exact(x-dx,y) : au[j][i-1];
184: ue = (i+1 == info->mx-1) ? u_exact(x+dx,y) : au[j][i+1];
185: us = (j-1 == 0) ? u_exact(x,y-dy) : au[j-1][i];
186: un = (j+1 == info->my-1) ? u_exact(x,y+dy) : au[j+1][i];
187: af[j][i] = - (dy/dx) * (uw - 2.0 * au[j][i] + ue) - (dx/dy) * (us - 2.0 * au[j][i] + un);
188: }
189: }
190: }
191: PetscLogFlops(12.0*info->ym*info->xm);
192: return 0;
193: }
195: PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **au, Mat A, Mat jac, void *user)
196: {
197: PetscInt i,j,n;
198: MatStencil col[5],row;
199: PetscReal v[5],dx,dy,oxx,oyy;
202: dx = 4.0 / (PetscReal)(info->mx-1);
203: dy = 4.0 / (PetscReal)(info->my-1);
204: oxx = dy / dx;
205: oyy = dx / dy;
206: for (j=info->ys; j<info->ys+info->ym; j++) {
207: for (i=info->xs; i<info->xs+info->xm; i++) {
208: row.j = j; row.i = i;
209: if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) { /* boundary */
210: v[0] = 4.0;
211: MatSetValuesStencil(jac,1,&row,1,&row,v,INSERT_VALUES);
212: } else { /* interior grid points */
213: v[0] = 2.0 * (oxx + oyy); col[0].j = j; col[0].i = i;
214: n = 1;
215: if (i-1 > 0) {
216: v[n] = -oxx; col[n].j = j; col[n++].i = i-1;
217: }
218: if (i+1 < info->mx-1) {
219: v[n] = -oxx; col[n].j = j; col[n++].i = i+1;
220: }
221: if (j-1 > 0) {
222: v[n] = -oyy; col[n].j = j-1; col[n++].i = i;
223: }
224: if (j+1 < info->my-1) {
225: v[n] = -oyy; col[n].j = j+1; col[n++].i = i;
226: }
227: MatSetValuesStencil(jac,1,&row,n,col,v,INSERT_VALUES);
228: }
229: }
230: }
232: /* Assemble matrix, using the 2-step process: */
233: MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
234: MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
235: if (A != jac) {
236: MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
237: MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
238: }
239: PetscLogFlops(2.0*info->ym*info->xm);
240: return 0;
241: }
243: /*TEST
245: build:
246: requires: !complex
248: test:
249: suffix: 1
250: requires: !single
251: nsize: 1
252: args: -da_refine 1 -snes_monitor_short -snes_type vinewtonrsls
254: test:
255: suffix: 2
256: requires: !single
257: nsize: 2
258: args: -da_refine 1 -snes_monitor_short -snes_type vinewtonssls
260: test:
261: suffix: 3
262: requires: !single
263: nsize: 2
264: args: -snes_grid_sequence 2 -snes_vi_monitor -snes_type vinewtonrsls
266: test:
267: suffix: mg
268: requires: !single
269: nsize: 4
270: args: -snes_grid_sequence 3 -snes_converged_reason -pc_type mg
272: test:
273: suffix: 4
274: nsize: 1
275: args: -mat_is_symmetric
277: test:
278: suffix: 5
279: nsize: 1
280: args: -ksp_converged_reason -snes_fd_color
282: test:
283: suffix: 6
284: requires: !single
285: nsize: 2
286: args: -snes_grid_sequence 2 -pc_type mg -snes_monitor_short -ksp_converged_reason
288: test:
289: suffix: 7
290: nsize: 2
291: 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
292: TODO: fix nasty memory leak in SNESCOMPOSITE
294: test:
295: suffix: 8
296: nsize: 2
297: args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additive -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
298: TODO: fix nasty memory leak in SNESCOMPOSITE
300: test:
301: suffix: 9
302: nsize: 2
303: args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additiveoptimal -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
304: TODO: fix nasty memory leak in SNESCOMPOSITE
306: TEST*/