Date: 2022-03-31T04:48:33+00:00

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*/