Actual source code: ex9.c

petsc-master 2019-10-15
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  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);if (ierr) return ierr;

 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 ierr;
124: }

126: PetscErrorCode FormExactSolution(DMDALocalInfo *info, Vec u)
127: {
129:   PetscInt       i,j;
130:   PetscReal      **au, dx, dy, x, y;
131:   dx = 4.0 / (PetscReal)(info->mx-1);
132:   dy = 4.0 / (PetscReal)(info->my-1);
133:   DMDAVecGetArray(info->da, u, &au);
134:   for (j=info->ys; j<info->ys+info->ym; j++) {
135:     y = -2.0 + j * dy;
136:     for (i=info->xs; i<info->xs+info->xm; i++) {
137:       x = -2.0 + i * dx;
138:       au[j][i] = u_exact(x,y);
139:     }
140:   }
141:   DMDAVecRestoreArray(info->da, u, &au);
142:   return 0;
143: }

145: PetscErrorCode FormBounds(SNES snes, Vec Xl, Vec Xu)
146: {
148:   DM             da;
149:   DMDALocalInfo  info;
150:   PetscInt       i, j;
151:   PetscReal      **aXl, dx, dy, x, y;

153:   SNESGetDM(snes,&da);
154:   DMDAGetLocalInfo(da,&info);
155:   dx = 4.0 / (PetscReal)(info.mx-1);
156:   dy = 4.0 / (PetscReal)(info.my-1);
157:   DMDAVecGetArray(da, Xl, &aXl);
158:   for (j=info.ys; j<info.ys+info.ym; j++) {
159:     y = -2.0 + j * dy;
160:     for (i=info.xs; i<info.xs+info.xm; i++) {
161:       x = -2.0 + i * dx;
162:       aXl[j][i] = psi(x,y);
163:     }
164:   }
165:   DMDAVecRestoreArray(da, Xl, &aXl);
166:   VecSet(Xu,PETSC_INFINITY);
167:   return 0;
168: }

170: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **au, PetscScalar **af, void *user)
171: {
173:   PetscInt       i,j;
174:   PetscReal      dx,dy,x,y,ue,un,us,uw;

177:   dx = 4.0 / (PetscReal)(info->mx-1);
178:   dy = 4.0 / (PetscReal)(info->my-1);
179:   for (j=info->ys; j<info->ys+info->ym; j++) {
180:     y = -2.0 + j * dy;
181:     for (i=info->xs; i<info->xs+info->xm; i++) {
182:       x = -2.0 + i * dx;
183:       if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) {
184:         af[j][i] = 4.0 * (au[j][i] - u_exact(x,y));
185:       } else {
186:         uw = (i-1 == 0)          ? u_exact(x-dx,y) : au[j][i-1];
187:         ue = (i+1 == info->mx-1) ? u_exact(x+dx,y) : au[j][i+1];
188:         us = (j-1 == 0)          ? u_exact(x,y-dy) : au[j-1][i];
189:         un = (j+1 == info->my-1) ? u_exact(x,y+dy) : au[j+1][i];
190:         af[j][i] = - (dy/dx) * (uw - 2.0 * au[j][i] + ue) - (dx/dy) * (us - 2.0 * au[j][i] + un);
191:       }
192:     }
193:   }
194:   PetscLogFlops(12.0*info->ym*info->xm);
195:   return(0);
196: }

198: PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **au, Mat A, Mat jac, void *user)
199: {
201:   PetscInt       i,j,n;
202:   MatStencil     col[5],row;
203:   PetscReal      v[5],dx,dy,oxx,oyy;

206:   dx  = 4.0 / (PetscReal)(info->mx-1);
207:   dy  = 4.0 / (PetscReal)(info->my-1);
208:   oxx = dy / dx;
209:   oyy = dx / dy;
210:   for (j=info->ys; j<info->ys+info->ym; j++) {
211:     for (i=info->xs; i<info->xs+info->xm; i++) {
212:       row.j = j; row.i = i;
213:       if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) { /* boundary */
214:         v[0] = 4.0;
215:         MatSetValuesStencil(jac,1,&row,1,&row,v,INSERT_VALUES);
216:       } else { /* interior grid points */
217:         v[0] = 2.0 * (oxx + oyy);  col[0].j = j;  col[0].i = i;
218:         n = 1;
219:         if (i-1 > 0) {
220:           v[n] = -oxx;  col[n].j = j;  col[n++].i = i-1;
221:         }
222:         if (i+1 < info->mx-1) {
223:           v[n] = -oxx;  col[n].j = j;  col[n++].i = i+1;
224:         }
225:         if (j-1 > 0) {
226:           v[n] = -oyy;  col[n].j = j-1;  col[n++].i = i;
227:         }
228:         if (j+1 < info->my-1) {
229:           v[n] = -oyy;  col[n].j = j+1;  col[n++].i = i;
230:         }
231:         MatSetValuesStencil(jac,1,&row,n,col,v,INSERT_VALUES);
232:       }
233:     }
234:   }

236:   /* Assemble matrix, using the 2-step process: */
237:   MatAssemblyBegin(jac,MAT_FINAL_ASSEMBLY);
238:   MatAssemblyEnd(jac,MAT_FINAL_ASSEMBLY);
239:   if (A != jac) {
240:     MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
241:     MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
242:   }
243:   PetscLogFlops(2.0*info->ym*info->xm);
244:   return(0);
245: }

247: /*TEST

249:    build:
250:       requires: !complex

252:    test:
253:       suffix: 1
254:       requires: !single
255:       nsize: 1
256:       args: -da_refine 1 -snes_monitor_short -snes_type vinewtonrsls

258:    test:
259:       suffix: 2
260:       requires: !single
261:       nsize: 2
262:       args: -da_refine 1 -snes_monitor_short -snes_type vinewtonssls

264:    test:
265:       suffix: 3
266:       requires: !single
267:       nsize: 2
268:       args: -snes_grid_sequence 2 -snes_vi_monitor -snes_type vinewtonrsls

270:    test:
271:       suffix: mg
272:       requires: !single
273:       nsize: 4
274:       args: -snes_grid_sequence 3 -snes_converged_reason -pc_type mg

276:    test:
277:       suffix: 4
278:       nsize: 1
279:       args: -mat_is_symmetric

281:    test:
282:       suffix: 5
283:       nsize: 1
284:       args: -ksp_converged_reason -snes_fd_color

286:    test:
287:       suffix: 6
288:       requires: !single
289:       nsize: 2
290:       args: -snes_grid_sequence 2 -pc_type mg -snes_monitor_short -ksp_converged_reason

292:    test:
293:       suffix: 7
294:       nsize: 2
295:       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
296:       TODO: fix nasty memory leak in SNESCOMPOSITE

298:    test:
299:       suffix: 8
300:       nsize: 2
301:       args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additive -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
302:       TODO: fix nasty memory leak in SNESCOMPOSITE

304:    test:
305:       suffix: 9
306:       nsize: 2
307:       args: -da_refine 1 -snes_monitor_short -snes_type composite -snes_composite_type additiveoptimal -snes_composite_sneses vinewtonrsls -sub_0_snes_vi_monitor
308:       TODO: fix nasty memory leak in SNESCOMPOSITE

310: TEST*/