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"; 5 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. 12 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 */ 30 31 #include <petsc.h> 32 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 } 43 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 } 63 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 *); 68 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; 76 77 PetscFunctionBeginUser; 78 PetscCall(PetscInitialize(&argc, &argv, NULL, help)); 79 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)); 86 87 PetscCall(DMCreateGlobalVector(da, &u)); 88 PetscCall(VecSet(u, 0.0)); 89 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)); 97 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 } 119 120 PetscErrorCode FormExactSolution(DMDALocalInfo *info, Vec u) 121 { 122 PetscInt i, j; 123 PetscReal **au, dx, dy, x, y; 124 125 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 } 139 140 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; 146 147 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 } 164 165 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; 169 170 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 } 191 192 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; 197 198 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 } 239 240 /* 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 } 250 251 /*TEST 252 253 build: 254 requires: !complex 255 256 test: 257 suffix: 1 258 requires: !single 259 nsize: 1 260 args: -da_refine 1 -snes_monitor_short -snes_type vinewtonrsls 261 262 test: 263 suffix: 2 264 requires: !single 265 nsize: 2 266 args: -da_refine 1 -snes_monitor_short -snes_type vinewtonssls 267 268 test: 269 suffix: 3 270 requires: !single 271 nsize: 2 272 args: -snes_grid_sequence 2 -snes_vi_monitor -snes_type vinewtonrsls 273 274 test: 275 suffix: mg 276 requires: !single 277 nsize: 4 278 args: -snes_grid_sequence 3 -snes_converged_reason -pc_type mg 279 280 test: 281 suffix: 4 282 nsize: 1 283 args: -mat_is_symmetric 284 285 test: 286 suffix: 5 287 nsize: 1 288 args: -ksp_converged_reason -snes_fd_color 289 290 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 295 296 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 301 302 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 307 308 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 313 314 TEST*/ 315