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 * 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 */ 32 33 #include <petsc.h> 34 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 } 45 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 } 67 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*); 72 73 int main(int argc,char **argv) 74 { 75 SNES snes; 76 DM da, da_after; 77 Vec u, u_exact; 78 DMDALocalInfo info; 79 PetscReal error1,errorinf; 80 81 PetscFunctionBeginUser; 82 PetscCall(PetscInitialize(&argc,&argv,(char*)0,help)); 83 84 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 */ 85 PETSC_DECIDE,PETSC_DECIDE, 1,1, /* dof=1 and s = 1 (stencil extends out one cell) */ 86 NULL,NULL,&da)); 87 PetscCall(DMSetFromOptions(da)); 88 PetscCall(DMSetUp(da)); 89 PetscCall(DMDASetUniformCoordinates(da,-2.0,2.0,-2.0,2.0,0.0,1.0)); 90 91 PetscCall(DMCreateGlobalVector(da,&u)); 92 PetscCall(VecSet(u,0.0)); 93 94 PetscCall(SNESCreate(PETSC_COMM_WORLD,&snes)); 95 PetscCall(SNESSetDM(snes,da)); 96 PetscCall(SNESSetType(snes,SNESVINEWTONRSLS)); 97 PetscCall(SNESVISetComputeVariableBounds(snes,&FormBounds)); 98 PetscCall(DMDASNESSetFunctionLocal(da,INSERT_VALUES,(DMDASNESFunction)FormFunctionLocal,NULL)); 99 PetscCall(DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)FormJacobianLocal,NULL)); 100 PetscCall(SNESSetFromOptions(snes)); 101 102 /* solve nonlinear system */ 103 PetscCall(SNESSolve(snes,NULL,u)); 104 PetscCall(VecDestroy(&u)); 105 PetscCall(DMDestroy(&da)); 106 /* DMDA after solve may be different, e.g. with -snes_grid_sequence */ 107 PetscCall(SNESGetDM(snes,&da_after)); 108 PetscCall(SNESGetSolution(snes,&u)); /* do not destroy u */ 109 PetscCall(DMDAGetLocalInfo(da_after,&info)); 110 PetscCall(VecDuplicate(u,&u_exact)); 111 PetscCall(FormExactSolution(&info,u_exact)); 112 PetscCall(VecAXPY(u,-1.0,u_exact)); /* u <-- u - u_exact */ 113 PetscCall(VecNorm(u,NORM_1,&error1)); 114 error1 /= (PetscReal)info.mx * (PetscReal)info.my; /* average error */ 115 PetscCall(VecNorm(u,NORM_INFINITY,&errorinf)); 116 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)); 117 PetscCall(VecDestroy(&u_exact)); 118 PetscCall(SNESDestroy(&snes)); 119 PetscCall(DMDestroy(&da)); 120 PetscCall(PetscFinalize()); 121 return 0; 122 } 123 124 PetscErrorCode FormExactSolution(DMDALocalInfo *info, Vec u) 125 { 126 PetscInt i,j; 127 PetscReal **au, dx, dy, x, y; 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 return 0; 140 } 141 142 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; 148 149 PetscCall(SNESGetDM(snes,&da)); 150 PetscCall(DMDAGetLocalInfo(da,&info)); 151 dx = 4.0 / (PetscReal)(info.mx-1); 152 dy = 4.0 / (PetscReal)(info.my-1); 153 PetscCall(DMDAVecGetArray(da, Xl, &aXl)); 154 for (j=info.ys; j<info.ys+info.ym; j++) { 155 y = -2.0 + j * dy; 156 for (i=info.xs; i<info.xs+info.xm; i++) { 157 x = -2.0 + i * dx; 158 aXl[j][i] = psi(x,y); 159 } 160 } 161 PetscCall(DMDAVecRestoreArray(da, Xl, &aXl)); 162 PetscCall(VecSet(Xu,PETSC_INFINITY)); 163 return 0; 164 } 165 166 PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **au, PetscScalar **af, void *user) 167 { 168 PetscInt i,j; 169 PetscReal dx,dy,x,y,ue,un,us,uw; 170 171 PetscFunctionBeginUser; 172 dx = 4.0 / (PetscReal)(info->mx-1); 173 dy = 4.0 / (PetscReal)(info->my-1); 174 for (j=info->ys; j<info->ys+info->ym; j++) { 175 y = -2.0 + j * dy; 176 for (i=info->xs; i<info->xs+info->xm; i++) { 177 x = -2.0 + i * dx; 178 if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) { 179 af[j][i] = 4.0 * (au[j][i] - u_exact(x,y)); 180 } else { 181 uw = (i-1 == 0) ? u_exact(x-dx,y) : au[j][i-1]; 182 ue = (i+1 == info->mx-1) ? u_exact(x+dx,y) : au[j][i+1]; 183 us = (j-1 == 0) ? u_exact(x,y-dy) : au[j-1][i]; 184 un = (j+1 == info->my-1) ? u_exact(x,y+dy) : au[j+1][i]; 185 af[j][i] = - (dy/dx) * (uw - 2.0 * au[j][i] + ue) - (dx/dy) * (us - 2.0 * au[j][i] + un); 186 } 187 } 188 } 189 PetscCall(PetscLogFlops(12.0*info->ym*info->xm)); 190 PetscFunctionReturn(0); 191 } 192 193 PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **au, Mat A, Mat jac, void *user) 194 { 195 PetscInt i,j,n; 196 MatStencil col[5],row; 197 PetscReal v[5],dx,dy,oxx,oyy; 198 199 PetscFunctionBeginUser; 200 dx = 4.0 / (PetscReal)(info->mx-1); 201 dy = 4.0 / (PetscReal)(info->my-1); 202 oxx = dy / dx; 203 oyy = dx / dy; 204 for (j=info->ys; j<info->ys+info->ym; j++) { 205 for (i=info->xs; i<info->xs+info->xm; i++) { 206 row.j = j; 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); col[0].j = j; col[0].i = i; 212 n = 1; 213 if (i-1 > 0) { 214 v[n] = -oxx; col[n].j = j; col[n++].i = i-1; 215 } 216 if (i+1 < info->mx-1) { 217 v[n] = -oxx; col[n].j = j; col[n++].i = i+1; 218 } 219 if (j-1 > 0) { 220 v[n] = -oyy; col[n].j = j-1; col[n++].i = i; 221 } 222 if (j+1 < info->my-1) { 223 v[n] = -oyy; col[n].j = j+1; col[n++].i = i; 224 } 225 PetscCall(MatSetValuesStencil(jac,1,&row,n,col,v,INSERT_VALUES)); 226 } 227 } 228 } 229 230 /* Assemble matrix, using the 2-step process: */ 231 PetscCall(MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY)); 232 PetscCall(MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY)); 233 if (A != jac) { 234 PetscCall(MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY)); 235 PetscCall(MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY)); 236 } 237 PetscCall(PetscLogFlops(2.0*info->ym*info->xm)); 238 PetscFunctionReturn(0); 239 } 240 241 /*TEST 242 243 build: 244 requires: !complex 245 246 test: 247 suffix: 1 248 requires: !single 249 nsize: 1 250 args: -da_refine 1 -snes_monitor_short -snes_type vinewtonrsls 251 252 test: 253 suffix: 2 254 requires: !single 255 nsize: 2 256 args: -da_refine 1 -snes_monitor_short -snes_type vinewtonssls 257 258 test: 259 suffix: 3 260 requires: !single 261 nsize: 2 262 args: -snes_grid_sequence 2 -snes_vi_monitor -snes_type vinewtonrsls 263 264 test: 265 suffix: mg 266 requires: !single 267 nsize: 4 268 args: -snes_grid_sequence 3 -snes_converged_reason -pc_type mg 269 270 test: 271 suffix: 4 272 nsize: 1 273 args: -mat_is_symmetric 274 275 test: 276 suffix: 5 277 nsize: 1 278 args: -ksp_converged_reason -snes_fd_color 279 280 test: 281 suffix: 6 282 requires: !single 283 nsize: 2 284 args: -snes_grid_sequence 2 -pc_type mg -snes_monitor_short -ksp_converged_reason 285 286 test: 287 suffix: 7 288 nsize: 2 289 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 290 TODO: fix nasty memory leak in SNESCOMPOSITE 291 292 test: 293 suffix: 8 294 nsize: 2 295 args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additive -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor 296 TODO: fix nasty memory leak in SNESCOMPOSITE 297 298 test: 299 suffix: 9 300 nsize: 2 301 args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additiveoptimal -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor 302 TODO: fix nasty memory leak in SNESCOMPOSITE 303 304 TEST*/ 305