Actual source code: ex49.c

petsc-3.6.0 2015-06-09
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  1: static char help[] =  "   Solves the compressible plane strain elasticity equations in 2d on the unit domain using Q1 finite elements. \n\
  2:    Material properties E (Youngs modulus) and nu (Poisson ratio) may vary as a function of space. \n\
  3:    The model utilises boundary conditions which produce compression in the x direction. \n\
  4: Options: \n"
  5: "\
  6:      -mx : number of elements in x-direction \n\
  7:      -my : number of elements in y-direction \n\
  8:      -c_str : indicates the structure of the coefficients to use. \n"
  9: "\
 10:           -c_str 0 => Setup for an isotropic material with constant coefficients. \n\
 11:                          Parameters: \n\
 12:                              -iso_E  : Youngs modulus \n\
 13:                              -iso_nu : Poisson ratio \n\
 14:           -c_str 1 => Setup for a step function in the material properties in x. \n\
 15:                          Parameters: \n\
 16:                               -step_E0  : Youngs modulus to the left of the step \n\
 17:                               -step_nu0 : Poisson ratio to the left of the step \n\
 18:                               -step_E1  : Youngs modulus to the right of the step \n\
 19:                               -step_n1  : Poisson ratio to the right of the step \n\
 20:                               -step_xc  : x coordinate of the step \n"
 21: "\
 22:           -c_str 2 => Setup for a checkerboard material with alternating properties. \n\
 23:                       Repeats the following pattern throughout the domain. For example with 4 materials specified, we would heve \n\
 24:                       -------------------------\n\
 25:                       |  D  |  A  |  B  |  C  |\n\
 26:                       ------|-----|-----|------\n\
 27:                       |  C  |  D  |  A  |  B  |\n\
 28:                       ------|-----|-----|------\n\
 29:                       |  B  |  C  |  D  |  A  |\n\
 30:                       ------|-----|-----|------\n\
 31:                       |  A  |  B  |  C  |  D  |\n\
 32:                       -------------------------\n\
 33:                       \n\
 34:                          Parameters: \n\
 35:                               -brick_E    : a comma separated list of Young's modulii \n\
 36:                               -brick_nu   : a comma separated list of Poisson ratios  \n\
 37:                               -brick_span : the number of elements in x and y each brick will span \n\
 38:           -c_str 3 => Setup for a sponge-like material with alternating properties. \n\
 39:                       Repeats the following pattern throughout the domain \n"
 40: "\
 41:                       -----------------------------\n\
 42:                       |       [background]        |\n\
 43:                       |          E0,nu0           |\n\
 44:                       |     -----------------     |\n\
 45:                       |     |  [inclusion]  |     |\n\
 46:                       |     |    E1,nu1     |     |\n\
 47:                       |     |               |     |\n\
 48:                       |     | <---- w ----> |     |\n\
 49:                       |     |               |     |\n\
 50:                       |     |               |     |\n\
 51:                       |     -----------------     |\n\
 52:                       |                           |\n\
 53:                       |                           |\n\
 54:                       -----------------------------\n\
 55:                       <--------  t + w + t ------->\n\
 56:                       \n\
 57:                          Parameters: \n\
 58:                               -sponge_E0  : Youngs modulus of the surrounding material \n\
 59:                               -sponge_E1  : Youngs modulus of the inclusion \n\
 60:                               -sponge_nu0 : Poisson ratio of the surrounding material \n\
 61:                               -sponge_nu1 : Poisson ratio of the inclusion \n\
 62:                               -sponge_t   : the number of elements defining the border around each inclusion \n\
 63:                               -sponge_w   : the number of elements in x and y each inclusion will span\n\
 64:      -use_gp_coords : Evaluate the Youngs modulus, Poisson ratio and the body force at the global coordinates of the quadrature points.\n\
 65:      By default, E, nu and the body force are evaulated at the element center and applied as a constant over the entire element.\n\
 66:      -use_nonsymbc : Option to use non-symmetric boundary condition imposition. This choice will use less memory.";

 68: /* Contributed by Dave May */

 70: #include <petscksp.h>
 71: #include <petscdm.h>
 72: #include <petscdmda.h>

 74: static PetscErrorCode DMDABCApplyCompression(DM,Mat,Vec);
 75: static PetscErrorCode DMDABCApplySymmetricCompression(DM elas_da,Mat A,Vec f,IS *dofs,Mat *AA,Vec *ff);


 78: #define NSD            2 /* number of spatial dimensions */
 79: #define NODES_PER_EL   4 /* nodes per element */
 80: #define U_DOFS         2 /* degrees of freedom per displacement node */
 81: #define GAUSS_POINTS   4

 83: /* cell based evaluation */
 84: typedef struct {
 85:   PetscScalar E,nu,fx,fy;
 86: } Coefficients;

 88: /* Gauss point based evaluation 8+4+4+4 = 20 */
 89: typedef struct {
 90:   PetscScalar gp_coords[2*GAUSS_POINTS];
 91:   PetscScalar E[GAUSS_POINTS];
 92:   PetscScalar nu[GAUSS_POINTS];
 93:   PetscScalar fx[GAUSS_POINTS];
 94:   PetscScalar fy[GAUSS_POINTS];
 95: } GaussPointCoefficients;

 97: typedef struct {
 98:   PetscScalar ux_dof;
 99:   PetscScalar uy_dof;
100: } ElasticityDOF;


103: /*

105:  D = E/((1+nu)(1-2nu)) * [ 1-nu   nu        0     ]
106:                          [  nu   1-nu       0     ]
107:                          [  0     0   0.5*(1-2nu) ]

109:  B = [ d_dx   0   ]
110:      [  0    d_dy ]
111:      [ d_dy  d_dx ]

113:  */

115: /* FEM routines */
116: /*
117:  Element: Local basis function ordering
118:  1-----2
119:  |     |
120:  |     |
121:  0-----3
122:  */
123: static void ConstructQ12D_Ni(PetscScalar _xi[],PetscScalar Ni[])
124: {
125:   PetscScalar xi  = _xi[0];
126:   PetscScalar eta = _xi[1];

128:   Ni[0] = 0.25*(1.0-xi)*(1.0-eta);
129:   Ni[1] = 0.25*(1.0-xi)*(1.0+eta);
130:   Ni[2] = 0.25*(1.0+xi)*(1.0+eta);
131:   Ni[3] = 0.25*(1.0+xi)*(1.0-eta);
132: }

134: static void ConstructQ12D_GNi(PetscScalar _xi[],PetscScalar GNi[][NODES_PER_EL])
135: {
136:   PetscScalar xi  = _xi[0];
137:   PetscScalar eta = _xi[1];

139:   GNi[0][0] = -0.25*(1.0-eta);
140:   GNi[0][1] = -0.25*(1.0+eta);
141:   GNi[0][2] =   0.25*(1.0+eta);
142:   GNi[0][3] =   0.25*(1.0-eta);

144:   GNi[1][0] = -0.25*(1.0-xi);
145:   GNi[1][1] =   0.25*(1.0-xi);
146:   GNi[1][2] =   0.25*(1.0+xi);
147:   GNi[1][3] = -0.25*(1.0+xi);
148: }

150: static void ConstructQ12D_GNx(PetscScalar GNi[][NODES_PER_EL],PetscScalar GNx[][NODES_PER_EL],PetscScalar coords[],PetscScalar *det_J)
151: {
152:   PetscScalar J00,J01,J10,J11,J;
153:   PetscScalar iJ00,iJ01,iJ10,iJ11;
154:   PetscInt    i;

156:   J00 = J01 = J10 = J11 = 0.0;
157:   for (i = 0; i < NODES_PER_EL; i++) {
158:     PetscScalar cx = coords[2*i+0];
159:     PetscScalar cy = coords[2*i+1];

161:     J00 = J00+GNi[0][i]*cx;      /* J_xx = dx/dxi */
162:     J01 = J01+GNi[0][i]*cy;      /* J_xy = dy/dxi */
163:     J10 = J10+GNi[1][i]*cx;      /* J_yx = dx/deta */
164:     J11 = J11+GNi[1][i]*cy;      /* J_yy = dy/deta */
165:   }
166:   J = (J00*J11)-(J01*J10);

168:   iJ00 =  J11/J;
169:   iJ01 = -J01/J;
170:   iJ10 = -J10/J;
171:   iJ11 =  J00/J;


174:   for (i = 0; i < NODES_PER_EL; i++) {
175:     GNx[0][i] = GNi[0][i]*iJ00+GNi[1][i]*iJ01;
176:     GNx[1][i] = GNi[0][i]*iJ10+GNi[1][i]*iJ11;
177:   }

179:   if (det_J != NULL) *det_J = J;
180: }

182: static void ConstructGaussQuadrature(PetscInt *ngp,PetscScalar gp_xi[][2],PetscScalar gp_weight[])
183: {
184:   *ngp         = 4;
185:   gp_xi[0][0]  = -0.57735026919;gp_xi[0][1] = -0.57735026919;
186:   gp_xi[1][0]  = -0.57735026919;gp_xi[1][1] =  0.57735026919;
187:   gp_xi[2][0]  =  0.57735026919;gp_xi[2][1] =  0.57735026919;
188:   gp_xi[3][0]  =  0.57735026919;gp_xi[3][1] = -0.57735026919;
189:   gp_weight[0] = 1.0;
190:   gp_weight[1] = 1.0;
191:   gp_weight[2] = 1.0;
192:   gp_weight[3] = 1.0;
193: }


196: /* procs to the left claim the ghost node as their element */
199: static PetscErrorCode DMDAGetLocalElementSize(DM da,PetscInt *mxl,PetscInt *myl,PetscInt *mzl)
200: {
202:   PetscInt       m,n,p,M,N,P;
203:   PetscInt       sx,sy,sz;

206:   DMDAGetInfo(da,0,&M,&N,&P,0,0,0,0,0,0,0,0,0);
207:   DMDAGetCorners(da,&sx,&sy,&sz,&m,&n,&p);

209:   if (mxl != NULL) {
210:     *mxl = m;
211:     if ((sx+m) == M) *mxl = m-1;    /* last proc */
212:   }
213:   if (myl != NULL) {
214:     *myl = n;
215:     if ((sy+n) == N) *myl = n-1;  /* last proc */
216:   }
217:   if (mzl != NULL) {
218:     *mzl = p;
219:     if ((sz+p) == P) *mzl = p-1;  /* last proc */
220:   }
221:   return(0);
222: }

226: static PetscErrorCode DMDAGetElementCorners(DM da,PetscInt *sx,PetscInt *sy,PetscInt *sz,PetscInt *mx,PetscInt *my,PetscInt *mz)
227: {
229:   PetscInt       si,sj,sk;

232:   DMDAGetGhostCorners(da,&si,&sj,&sk,0,0,0);

234:   if (sx) {
235:     *sx = si;
236:     if (si != 0) *sx = si+1;
237:   }
238:   if (sy) {
239:     *sy = sj;
240:     if (sj != 0) *sy = sj+1;
241:   }

243:   if (sk) {
244:     *sz = sk;
245:     if (sk != 0) *sz = sk+1;
246:   }

248:   DMDAGetLocalElementSize(da,mx,my,mz);
249:   return(0);
250: }

254: static PetscErrorCode DMDAGetElementOwnershipRanges2d(DM da,PetscInt **_lx,PetscInt **_ly)
255: {
257:   PetscMPIInt    rank;
258:   PetscInt       proc_I,proc_J;
259:   PetscInt       cpu_x,cpu_y;
260:   PetscInt       local_mx,local_my;
261:   Vec            vlx,vly;
262:   PetscInt       *LX,*LY,i;
263:   PetscScalar    *_a;
264:   Vec            V_SEQ;
265:   VecScatter     ctx;

268:   MPI_Comm_rank(PETSC_COMM_WORLD,&rank);

270:   DMDAGetInfo(da,0,0,0,0,&cpu_x,&cpu_y,0,0,0,0,0,0,0);

272:   proc_J = rank/cpu_x;
273:   proc_I = rank-cpu_x*proc_J;

275:   PetscMalloc1(cpu_x,&LX);
276:   PetscMalloc1(cpu_y,&LY);

278:   DMDAGetLocalElementSize(da,&local_mx,&local_my,NULL);
279:   VecCreate(PETSC_COMM_WORLD,&vlx);
280:   VecSetSizes(vlx,PETSC_DECIDE,cpu_x);
281:   VecSetFromOptions(vlx);

283:   VecCreate(PETSC_COMM_WORLD,&vly);
284:   VecSetSizes(vly,PETSC_DECIDE,cpu_y);
285:   VecSetFromOptions(vly);

287:   VecSetValue(vlx,proc_I,(PetscScalar)(local_mx+1.0e-9),INSERT_VALUES);
288:   VecSetValue(vly,proc_J,(PetscScalar)(local_my+1.0e-9),INSERT_VALUES);
289:   VecAssemblyBegin(vlx);VecAssemblyEnd(vlx);
290:   VecAssemblyBegin(vly);VecAssemblyEnd(vly);



294:   VecScatterCreateToAll(vlx,&ctx,&V_SEQ);
295:   VecScatterBegin(ctx,vlx,V_SEQ,INSERT_VALUES,SCATTER_FORWARD);
296:   VecScatterEnd(ctx,vlx,V_SEQ,INSERT_VALUES,SCATTER_FORWARD);
297:   VecGetArray(V_SEQ,&_a);
298:   for (i = 0; i < cpu_x; i++) LX[i] = (PetscInt)PetscRealPart(_a[i]);
299:   VecRestoreArray(V_SEQ,&_a);
300:   VecScatterDestroy(&ctx);
301:   VecDestroy(&V_SEQ);

303:   VecScatterCreateToAll(vly,&ctx,&V_SEQ);
304:   VecScatterBegin(ctx,vly,V_SEQ,INSERT_VALUES,SCATTER_FORWARD);
305:   VecScatterEnd(ctx,vly,V_SEQ,INSERT_VALUES,SCATTER_FORWARD);
306:   VecGetArray(V_SEQ,&_a);
307:   for (i = 0; i < cpu_y; i++) LY[i] = (PetscInt)PetscRealPart(_a[i]);
308:   VecRestoreArray(V_SEQ,&_a);
309:   VecScatterDestroy(&ctx);
310:   VecDestroy(&V_SEQ);

312:   *_lx = LX;
313:   *_ly = LY;

315:   VecDestroy(&vlx);
316:   VecDestroy(&vly);
317:   return(0);
318: }

322: static PetscErrorCode DMDACoordViewGnuplot2d(DM da,const char prefix[])
323: {
324:   DM             cda;
325:   Vec            coords;
326:   DMDACoor2d     **_coords;
327:   PetscInt       si,sj,nx,ny,i,j;
328:   FILE           *fp;
329:   char           fname[PETSC_MAX_PATH_LEN];
330:   PetscMPIInt    rank;

334:   MPI_Comm_rank(PETSC_COMM_WORLD,&rank);
335:   PetscSNPrintf(fname,sizeof(fname),"%s-p%1.4d.dat",prefix,rank);
336:   PetscFOpen(PETSC_COMM_SELF,fname,"w",&fp);
337:   if (!fp) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER,"Cannot open file");
338:   PetscFPrintf(PETSC_COMM_SELF,fp,"### Element geometry for processor %1.4d ### \n",rank);

340:   DMGetCoordinateDM(da,&cda);
341:   DMGetCoordinatesLocal(da,&coords);
342:   DMDAVecGetArray(cda,coords,&_coords);
343:   DMDAGetGhostCorners(cda,&si,&sj,0,&nx,&ny,0);
344:   for (j = sj; j < sj+ny-1; j++) {
345:     for (i = si; i < si+nx-1; i++) {
346:       PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e \n",PetscRealPart(_coords[j][i].x),PetscRealPart(_coords[j][i].y));
347:       PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e \n",PetscRealPart(_coords[j+1][i].x),PetscRealPart(_coords[j+1][i].y));
348:       PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e \n",PetscRealPart(_coords[j+1][i+1].x),PetscRealPart(_coords[j+1][i+1].y));
349:       PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e \n",PetscRealPart(_coords[j][i+1].x),PetscRealPart(_coords[j][i+1].y));
350:       PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e \n\n",PetscRealPart(_coords[j][i].x),PetscRealPart(_coords[j][i].y));
351:     }
352:   }
353:   DMDAVecRestoreArray(cda,coords,&_coords);

355:   PetscFClose(PETSC_COMM_SELF,fp);
356:   return(0);
357: }

361: static PetscErrorCode DMDAViewGnuplot2d(DM da,Vec fields,const char comment[],const char prefix[])
362: {
363:   DM             cda;
364:   Vec            coords,local_fields;
365:   DMDACoor2d     **_coords;
366:   FILE           *fp;
367:   char           fname[PETSC_MAX_PATH_LEN];
368:   const char     *field_name;
369:   PetscMPIInt    rank;
370:   PetscInt       si,sj,nx,ny,i,j;
371:   PetscInt       n_dofs,d;
372:   PetscScalar    *_fields;

376:   MPI_Comm_rank(PETSC_COMM_WORLD,&rank);
377:   PetscSNPrintf(fname,sizeof(fname),"%s-p%1.4d.dat",prefix,rank);
378:   PetscFOpen(PETSC_COMM_SELF,fname,"w",&fp);
379:   if (!fp) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER,"Cannot open file");

381:   PetscFPrintf(PETSC_COMM_SELF,fp,"### %s (processor %1.4d) ### \n",comment,rank);
382:   DMDAGetInfo(da,0,0,0,0,0,0,0,&n_dofs,0,0,0,0,0);
383:   PetscFPrintf(PETSC_COMM_SELF,fp,"### x y ");
384:   for (d = 0; d < n_dofs; d++) {
385:     DMDAGetFieldName(da,d,&field_name);
386:     PetscFPrintf(PETSC_COMM_SELF,fp,"%s ",field_name);
387:   }
388:   PetscFPrintf(PETSC_COMM_SELF,fp,"###\n");


391:   DMGetCoordinateDM(da,&cda);
392:   DMGetCoordinatesLocal(da,&coords);
393:   DMDAVecGetArray(cda,coords,&_coords);
394:   DMDAGetGhostCorners(cda,&si,&sj,0,&nx,&ny,0);

396:   DMCreateLocalVector(da,&local_fields);
397:   DMGlobalToLocalBegin(da,fields,INSERT_VALUES,local_fields);
398:   DMGlobalToLocalEnd(da,fields,INSERT_VALUES,local_fields);
399:   VecGetArray(local_fields,&_fields);

401:   for (j = sj; j < sj+ny; j++) {
402:     for (i = si; i < si+nx; i++) {
403:       PetscScalar coord_x,coord_y;
404:       PetscScalar field_d;

406:       coord_x = _coords[j][i].x;
407:       coord_y = _coords[j][i].y;

409:       PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e ",PetscRealPart(coord_x),PetscRealPart(coord_y));
410:       for (d = 0; d < n_dofs; d++) {
411:         field_d = _fields[n_dofs*((i-si)+(j-sj)*(nx))+d];
412:         PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e ",PetscRealPart(field_d));
413:       }
414:       PetscFPrintf(PETSC_COMM_SELF,fp,"\n");
415:     }
416:   }
417:   VecRestoreArray(local_fields,&_fields);
418:   VecDestroy(&local_fields);

420:   DMDAVecRestoreArray(cda,coords,&_coords);

422:   PetscFClose(PETSC_COMM_SELF,fp);
423:   return(0);
424: }

428: static PetscErrorCode DMDAViewCoefficientsGnuplot2d(DM da,Vec fields,const char comment[],const char prefix[])
429: {
430:   DM                     cda;
431:   Vec                    local_fields;
432:   FILE                   *fp;
433:   char                   fname[PETSC_MAX_PATH_LEN];
434:   const char             *field_name;
435:   PetscMPIInt            rank;
436:   PetscInt               si,sj,nx,ny,i,j,p;
437:   PetscInt               n_dofs,d;
438:   GaussPointCoefficients **_coefficients;
439:   PetscErrorCode         ierr;

442:   MPI_Comm_rank(PETSC_COMM_WORLD,&rank);
443:   PetscSNPrintf(fname,sizeof(fname),"%s-p%1.4d.dat",prefix,rank);
444:   PetscFOpen(PETSC_COMM_SELF,fname,"w",&fp);
445:   if (!fp) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER,"Cannot open file");

447:   PetscFPrintf(PETSC_COMM_SELF,fp,"### %s (processor %1.4d) ### \n",comment,rank);
448:   DMDAGetInfo(da,0,0,0,0,0,0,0,&n_dofs,0,0,0,0,0);
449:   PetscFPrintf(PETSC_COMM_SELF,fp,"### x y ");
450:   for (d = 0; d < n_dofs; d++) {
451:     DMDAGetFieldName(da,d,&field_name);
452:     PetscFPrintf(PETSC_COMM_SELF,fp,"%s ",field_name);
453:   }
454:   PetscFPrintf(PETSC_COMM_SELF,fp,"###\n");


457:   DMGetCoordinateDM(da,&cda);
458:   DMDAGetGhostCorners(cda,&si,&sj,0,&nx,&ny,0);

460:   DMCreateLocalVector(da,&local_fields);
461:   DMGlobalToLocalBegin(da,fields,INSERT_VALUES,local_fields);
462:   DMGlobalToLocalEnd(da,fields,INSERT_VALUES,local_fields);
463:   DMDAVecGetArray(da,local_fields,&_coefficients);

465:   for (j = sj; j < sj+ny; j++) {
466:     for (i = si; i < si+nx; i++) {
467:       PetscScalar coord_x,coord_y;

469:       for (p = 0; p < GAUSS_POINTS; p++) {
470:         coord_x = _coefficients[j][i].gp_coords[2*p];
471:         coord_y = _coefficients[j][i].gp_coords[2*p+1];

473:         PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e ",PetscRealPart(coord_x),PetscRealPart(coord_y));

475:         PetscFPrintf(PETSC_COMM_SELF,fp,"%1.6e %1.6e %1.6e %1.6e",
476:                             PetscRealPart(_coefficients[j][i].E[p]),PetscRealPart(_coefficients[j][i].nu[p]),
477:                             PetscRealPart(_coefficients[j][i].fx[p]),PetscRealPart(_coefficients[j][i].fy[p]));
478:         PetscFPrintf(PETSC_COMM_SELF,fp,"\n");
479:       }
480:     }
481:   }
482:   DMDAVecRestoreArray(da,local_fields,&_coefficients);
483:   VecDestroy(&local_fields);

485:   PetscFClose(PETSC_COMM_SELF,fp);
486:   return(0);
487: }

489: static void FormStressOperatorQ1(PetscScalar Ke[],PetscScalar coords[],PetscScalar E[],PetscScalar nu[])
490: {
491:   PetscInt    ngp;
492:   PetscScalar gp_xi[GAUSS_POINTS][2];
493:   PetscScalar gp_weight[GAUSS_POINTS];
494:   PetscInt    p,i,j,k,l;
495:   PetscScalar GNi_p[NSD][NODES_PER_EL],GNx_p[NSD][NODES_PER_EL];
496:   PetscScalar J_p;
497:   PetscScalar B[3][U_DOFS*NODES_PER_EL];
498:   PetscScalar prop_E,prop_nu,factor,constit_D[3][3];

500:   /* define quadrature rule */
501:   ConstructGaussQuadrature(&ngp,gp_xi,gp_weight);

503:   /* evaluate integral */
504:   for (p = 0; p < ngp; p++) {
505:     ConstructQ12D_GNi(gp_xi[p],GNi_p);
506:     ConstructQ12D_GNx(GNi_p,GNx_p,coords,&J_p);

508:     for (i = 0; i < NODES_PER_EL; i++) {
509:       PetscScalar d_dx_i = GNx_p[0][i];
510:       PetscScalar d_dy_i = GNx_p[1][i];

512:       B[0][2*i] = d_dx_i;  B[0][2*i+1] = 0.0;
513:       B[1][2*i] = 0.0;     B[1][2*i+1] = d_dy_i;
514:       B[2][2*i] = d_dy_i;  B[2][2*i+1] = d_dx_i;
515:     }

517:     /* form D for the quadrature point */
518:     prop_E          = E[p];
519:     prop_nu         = nu[p];
520:     factor          = prop_E / ((1.0+prop_nu)*(1.0-2.0*prop_nu));
521:     constit_D[0][0] = 1.0-prop_nu;  constit_D[0][1] = prop_nu;      constit_D[0][2] = 0.0;
522:     constit_D[1][0] = prop_nu;      constit_D[1][1] = 1.0-prop_nu;  constit_D[1][2] = 0.0;
523:     constit_D[2][0] = 0.0;          constit_D[2][1] = 0.0;          constit_D[2][2] = 0.5*(1.0-2.0*prop_nu);
524:     for (i = 0; i < 3; i++) {
525:       for (j = 0; j < 3; j++) {
526:         constit_D[i][j] = factor * constit_D[i][j] * gp_weight[p] * J_p;
527:       }
528:     }

530:     /* form Bt tildeD B */
531:     /*
532:      Ke_ij = Bt_ik . D_kl . B_lj
533:      = B_ki . D_kl . B_lj
534:      */
535:     for (i = 0; i < 8; i++) {
536:       for (j = 0; j < 8; j++) {
537:         for (k = 0; k < 3; k++) {
538:           for (l = 0; l < 3; l++) {
539:             Ke[8*i+j] = Ke[8*i+j] + B[k][i] * constit_D[k][l] * B[l][j];
540:           }
541:         }
542:       }
543:     }

545:   } /* end quadrature */
546: }

548: static void FormMomentumRhsQ1(PetscScalar Fe[],PetscScalar coords[],PetscScalar fx[],PetscScalar fy[])
549: {
550:   PetscInt    ngp;
551:   PetscScalar gp_xi[GAUSS_POINTS][2];
552:   PetscScalar gp_weight[GAUSS_POINTS];
553:   PetscInt    p,i;
554:   PetscScalar Ni_p[NODES_PER_EL];
555:   PetscScalar GNi_p[NSD][NODES_PER_EL],GNx_p[NSD][NODES_PER_EL];
556:   PetscScalar J_p,fac;

558:   /* define quadrature rule */
559:   ConstructGaussQuadrature(&ngp,gp_xi,gp_weight);

561:   /* evaluate integral */
562:   for (p = 0; p < ngp; p++) {
563:     ConstructQ12D_Ni(gp_xi[p],Ni_p);
564:     ConstructQ12D_GNi(gp_xi[p],GNi_p);
565:     ConstructQ12D_GNx(GNi_p,GNx_p,coords,&J_p);
566:     fac = gp_weight[p]*J_p;

568:     for (i = 0; i < NODES_PER_EL; i++) {
569:       Fe[NSD*i]   += fac*Ni_p[i]*fx[p];
570:       Fe[NSD*i+1] += fac*Ni_p[i]*fy[p];
571:     }
572:   }
573: }

575: /*
576:  i,j are the element indices
577:  The unknown is a vector quantity.
578:  The s[].c is used to indicate the degree of freedom.
579:  */
582: static PetscErrorCode DMDAGetElementEqnums_u(MatStencil s_u[],PetscInt i,PetscInt j)
583: {
585:   /* displacement */
586:   /* node 0 */
587:   s_u[0].i = i;s_u[0].j = j;s_u[0].c = 0;          /* Ux0 */
588:   s_u[1].i = i;s_u[1].j = j;s_u[1].c = 1;          /* Uy0 */

590:   /* node 1 */
591:   s_u[2].i = i;s_u[2].j = j+1;s_u[2].c = 0;        /* Ux1 */
592:   s_u[3].i = i;s_u[3].j = j+1;s_u[3].c = 1;        /* Uy1 */

594:   /* node 2 */
595:   s_u[4].i = i+1;s_u[4].j = j+1;s_u[4].c = 0;      /* Ux2 */
596:   s_u[5].i = i+1;s_u[5].j = j+1;s_u[5].c = 1;      /* Uy2 */

598:   /* node 3 */
599:   s_u[6].i = i+1;s_u[6].j = j;s_u[6].c = 0;        /* Ux3 */
600:   s_u[7].i = i+1;s_u[7].j = j;s_u[7].c = 1;        /* Uy3 */
601:   return(0);
602: }

606: static PetscErrorCode GetElementCoords(DMDACoor2d **_coords,PetscInt ei,PetscInt ej,PetscScalar el_coords[])
607: {
609:   /* get coords for the element */
610:   el_coords[NSD*0+0] = _coords[ej][ei].x;      el_coords[NSD*0+1] = _coords[ej][ei].y;
611:   el_coords[NSD*1+0] = _coords[ej+1][ei].x;    el_coords[NSD*1+1] = _coords[ej+1][ei].y;
612:   el_coords[NSD*2+0] = _coords[ej+1][ei+1].x;  el_coords[NSD*2+1] = _coords[ej+1][ei+1].y;
613:   el_coords[NSD*3+0] = _coords[ej][ei+1].x;    el_coords[NSD*3+1] = _coords[ej][ei+1].y;
614:   return(0);
615: }

619: static PetscErrorCode AssembleA_Elasticity(Mat A,DM elas_da,DM properties_da,Vec properties)
620: {
621:   DM                     cda;
622:   Vec                    coords;
623:   DMDACoor2d             **_coords;
624:   MatStencil             u_eqn[NODES_PER_EL*U_DOFS]; /* 2 degrees of freedom */
625:   PetscInt               sex,sey,mx,my;
626:   PetscInt               ei,ej;
627:   PetscScalar            Ae[NODES_PER_EL*U_DOFS*NODES_PER_EL*U_DOFS];
628:   PetscScalar            el_coords[NODES_PER_EL*NSD];
629:   Vec                    local_properties;
630:   GaussPointCoefficients **props;
631:   PetscScalar            *prop_E,*prop_nu;
632:   PetscErrorCode         ierr;

635:   /* setup for coords */
636:   DMGetCoordinateDM(elas_da,&cda);
637:   DMGetCoordinatesLocal(elas_da,&coords);
638:   DMDAVecGetArray(cda,coords,&_coords);

640:   /* setup for coefficients */
641:   DMCreateLocalVector(properties_da,&local_properties);
642:   DMGlobalToLocalBegin(properties_da,properties,INSERT_VALUES,local_properties);
643:   DMGlobalToLocalEnd(properties_da,properties,INSERT_VALUES,local_properties);
644:   DMDAVecGetArray(properties_da,local_properties,&props);

646:   DMDAGetElementCorners(elas_da,&sex,&sey,0,&mx,&my,0);
647:   for (ej = sey; ej < sey+my; ej++) {
648:     for (ei = sex; ei < sex+mx; ei++) {
649:       /* get coords for the element */
650:       GetElementCoords(_coords,ei,ej,el_coords);

652:       /* get coefficients for the element */
653:       prop_E  = props[ej][ei].E;
654:       prop_nu = props[ej][ei].nu;

656:       /* initialise element stiffness matrix */
657:       PetscMemzero(Ae,sizeof(PetscScalar)*NODES_PER_EL*U_DOFS*NODES_PER_EL*U_DOFS);

659:       /* form element stiffness matrix */
660:       FormStressOperatorQ1(Ae,el_coords,prop_E,prop_nu);

662:       /* insert element matrix into global matrix */
663:       DMDAGetElementEqnums_u(u_eqn,ei,ej);
664:       MatSetValuesStencil(A,NODES_PER_EL*U_DOFS,u_eqn,NODES_PER_EL*U_DOFS,u_eqn,Ae,ADD_VALUES);
665:     }
666:   }
667:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
668:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);

670:   DMDAVecRestoreArray(cda,coords,&_coords);

672:   DMDAVecRestoreArray(properties_da,local_properties,&props);
673:   VecDestroy(&local_properties);
674:   return(0);
675: }


680: static PetscErrorCode DMDASetValuesLocalStencil_ADD_VALUES(ElasticityDOF **fields_F,MatStencil u_eqn[],PetscScalar Fe_u[])
681: {
682:   PetscInt n;

685:   for (n = 0; n < 4; n++) {
686:     fields_F[u_eqn[2*n].j][u_eqn[2*n].i].ux_dof     = fields_F[u_eqn[2*n].j][u_eqn[2*n].i].ux_dof+Fe_u[2*n];
687:     fields_F[u_eqn[2*n+1].j][u_eqn[2*n+1].i].uy_dof = fields_F[u_eqn[2*n+1].j][u_eqn[2*n+1].i].uy_dof+Fe_u[2*n+1];
688:   }
689:   return(0);
690: }

694: static PetscErrorCode AssembleF_Elasticity(Vec F,DM elas_da,DM properties_da,Vec properties)
695: {
696:   DM                     cda;
697:   Vec                    coords;
698:   DMDACoor2d             **_coords;
699:   MatStencil             u_eqn[NODES_PER_EL*U_DOFS]; /* 2 degrees of freedom */
700:   PetscInt               sex,sey,mx,my;
701:   PetscInt               ei,ej;
702:   PetscScalar            Fe[NODES_PER_EL*U_DOFS];
703:   PetscScalar            el_coords[NODES_PER_EL*NSD];
704:   Vec                    local_properties;
705:   GaussPointCoefficients **props;
706:   PetscScalar            *prop_fx,*prop_fy;
707:   Vec                    local_F;
708:   ElasticityDOF          **ff;
709:   PetscErrorCode         ierr;

712:   /* setup for coords */
713:   DMGetCoordinateDM(elas_da,&cda);
714:   DMGetCoordinatesLocal(elas_da,&coords);
715:   DMDAVecGetArray(cda,coords,&_coords);

717:   /* setup for coefficients */
718:   DMGetLocalVector(properties_da,&local_properties);
719:   DMGlobalToLocalBegin(properties_da,properties,INSERT_VALUES,local_properties);
720:   DMGlobalToLocalEnd(properties_da,properties,INSERT_VALUES,local_properties);
721:   DMDAVecGetArray(properties_da,local_properties,&props);

723:   /* get acces to the vector */
724:   DMGetLocalVector(elas_da,&local_F);
725:   VecZeroEntries(local_F);
726:   DMDAVecGetArray(elas_da,local_F,&ff);


729:   DMDAGetElementCorners(elas_da,&sex,&sey,0,&mx,&my,0);
730:   for (ej = sey; ej < sey+my; ej++) {
731:     for (ei = sex; ei < sex+mx; ei++) {
732:       /* get coords for the element */
733:       GetElementCoords(_coords,ei,ej,el_coords);

735:       /* get coefficients for the element */
736:       prop_fx = props[ej][ei].fx;
737:       prop_fy = props[ej][ei].fy;

739:       /* initialise element stiffness matrix */
740:       PetscMemzero(Fe,sizeof(PetscScalar)*NODES_PER_EL*U_DOFS);

742:       /* form element stiffness matrix */
743:       FormMomentumRhsQ1(Fe,el_coords,prop_fx,prop_fy);

745:       /* insert element matrix into global matrix */
746:       DMDAGetElementEqnums_u(u_eqn,ei,ej);

748:       DMDASetValuesLocalStencil_ADD_VALUES(ff,u_eqn,Fe);
749:     }
750:   }

752:   DMDAVecRestoreArray(elas_da,local_F,&ff);
753:   DMLocalToGlobalBegin(elas_da,local_F,ADD_VALUES,F);
754:   DMLocalToGlobalEnd(elas_da,local_F,ADD_VALUES,F);
755:   DMRestoreLocalVector(elas_da,&local_F);

757:   DMDAVecRestoreArray(cda,coords,&_coords);

759:   DMDAVecRestoreArray(properties_da,local_properties,&props);
760:   DMRestoreLocalVector(properties_da,&local_properties);
761:   return(0);
762: }

766: static PetscErrorCode solve_elasticity_2d(PetscInt mx,PetscInt my)
767: {
768:   DM                     elas_da,da_prop;
769:   PetscInt               u_dof,dof,stencil_width;
770:   Mat                    A;
771:   PetscInt               mxl,myl;
772:   DM                     prop_cda,vel_cda;
773:   Vec                    prop_coords,vel_coords;
774:   PetscInt               si,sj,nx,ny,i,j,p;
775:   Vec                    f,X;
776:   PetscInt               prop_dof,prop_stencil_width;
777:   Vec                    properties,l_properties;
778:   MatNullSpace           matnull;
779:   PetscReal              dx,dy;
780:   PetscInt               M,N;
781:   DMDACoor2d             **_prop_coords,**_vel_coords;
782:   GaussPointCoefficients **element_props;
783:   KSP                    ksp_E;
784:   PetscInt               coefficient_structure = 0;
785:   PetscInt               cpu_x,cpu_y,*lx = NULL,*ly = NULL;
786:   PetscBool              use_gp_coords = PETSC_FALSE;
787:   PetscBool              use_nonsymbc  = PETSC_FALSE;
788:   PetscBool              no_view       = PETSC_FALSE;
789:   PetscBool              flg;
790:   PetscErrorCode         ierr;

793:   /* Generate the da for velocity and pressure */
794:   /*
795:    We use Q1 elements for the temperature.
796:    FEM has a 9-point stencil (BOX) or connectivity pattern
797:    Num nodes in each direction is mx+1, my+1
798:    */
799:   u_dof         = U_DOFS; /* Vx, Vy - velocities */
800:   dof           = u_dof;
801:   stencil_width = 1;
802:   DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,
803:                                mx+1,my+1,PETSC_DECIDE,PETSC_DECIDE,dof,stencil_width,NULL,NULL,&elas_da);
804:   DMDASetFieldName(elas_da,0,"Ux");
805:   DMDASetFieldName(elas_da,1,"Uy");

807:   /* unit box [0,1] x [0,1] */
808:   DMDASetUniformCoordinates(elas_da,0.0,1.0,0.0,1.0,0.0,1.0);


811:   /* Generate element properties, we will assume all material properties are constant over the element */
812:   /* local number of elements */
813:   DMDAGetLocalElementSize(elas_da,&mxl,&myl,NULL);

815:   /* !!! IN PARALLEL WE MUST MAKE SURE THE TWO DMDA's ALIGN !!! */
816:   DMDAGetInfo(elas_da,0,0,0,0,&cpu_x,&cpu_y,0,0,0,0,0,0,0);
817:   DMDAGetElementOwnershipRanges2d(elas_da,&lx,&ly);

819:   prop_dof           = (PetscInt)(sizeof(GaussPointCoefficients)/sizeof(PetscScalar)); /* gauss point setup */
820:   prop_stencil_width = 0;
821:   DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_BOX,
822:                                     mx,my,cpu_x,cpu_y,prop_dof,prop_stencil_width,lx,ly,&da_prop);
823:   PetscFree(lx);
824:   PetscFree(ly);

826:   /* define centroid positions */
827:   DMDAGetInfo(da_prop,0,&M,&N,0,0,0,0,0,0,0,0,0,0);
828:   dx   = 1.0/((PetscReal)(M));
829:   dy   = 1.0/((PetscReal)(N));

831:   DMDASetUniformCoordinates(da_prop,0.0+0.5*dx,1.0-0.5*dx,0.0+0.5*dy,1.0-0.5*dy,0.0,1.0);

833:   /* define coefficients */
834:   PetscOptionsGetInt(NULL,"-c_str",&coefficient_structure,NULL);

836:   DMCreateGlobalVector(da_prop,&properties);
837:   DMCreateLocalVector(da_prop,&l_properties);
838:   DMDAVecGetArray(da_prop,l_properties,&element_props);

840:   DMGetCoordinateDM(da_prop,&prop_cda);
841:   DMGetCoordinatesLocal(da_prop,&prop_coords);
842:   DMDAVecGetArray(prop_cda,prop_coords,&_prop_coords);

844:   DMDAGetGhostCorners(prop_cda,&si,&sj,0,&nx,&ny,0);

846:   DMGetCoordinateDM(elas_da,&vel_cda);
847:   DMGetCoordinatesLocal(elas_da,&vel_coords);
848:   DMDAVecGetArray(vel_cda,vel_coords,&_vel_coords);


851:   /* interpolate the coordinates */
852:   for (j = sj; j < sj+ny; j++) {
853:     for (i = si; i < si+nx; i++) {
854:       PetscInt    ngp;
855:       PetscScalar gp_xi[GAUSS_POINTS][2],gp_weight[GAUSS_POINTS];
856:       PetscScalar el_coords[8];

858:       GetElementCoords(_vel_coords,i,j,el_coords);
859:       ConstructGaussQuadrature(&ngp,gp_xi,gp_weight);

861:       for (p = 0; p < GAUSS_POINTS; p++) {
862:         PetscScalar gp_x,gp_y;
863:         PetscInt    n;
864:         PetscScalar xi_p[2],Ni_p[4];

866:         xi_p[0] = gp_xi[p][0];
867:         xi_p[1] = gp_xi[p][1];
868:         ConstructQ12D_Ni(xi_p,Ni_p);

870:         gp_x = 0.0;
871:         gp_y = 0.0;
872:         for (n = 0; n < NODES_PER_EL; n++) {
873:           gp_x = gp_x+Ni_p[n]*el_coords[2*n];
874:           gp_y = gp_y+Ni_p[n]*el_coords[2*n+1];
875:         }
876:         element_props[j][i].gp_coords[2*p]   = gp_x;
877:         element_props[j][i].gp_coords[2*p+1] = gp_y;
878:       }
879:     }
880:   }

882:   /* define the coefficients */
883:   PetscOptionsGetBool(NULL,"-use_gp_coords",&use_gp_coords,&flg);

885:   for (j = sj; j < sj+ny; j++) {
886:     for (i = si; i < si+nx; i++) {
887:       PetscScalar              centroid_x = _prop_coords[j][i].x; /* centroids of cell */
888:       PetscScalar              centroid_y = _prop_coords[j][i].y;
889:       PETSC_UNUSED PetscScalar coord_x,coord_y;


892:       if (coefficient_structure == 0) { /* isotropic */
893:         PetscScalar opts_E,opts_nu;

895:         opts_E  = 1.0;
896:         opts_nu = 0.33;
897:         PetscOptionsGetScalar(NULL,"-iso_E",&opts_E,&flg);
898:         PetscOptionsGetScalar(NULL,"-iso_nu",&opts_nu,&flg);

900:         for (p = 0; p < GAUSS_POINTS; p++) {
901:           element_props[j][i].E[p]  = opts_E;
902:           element_props[j][i].nu[p] = opts_nu;

904:           element_props[j][i].fx[p] = 0.0;
905:           element_props[j][i].fy[p] = 0.0;
906:         }
907:       } else if (coefficient_structure == 1) { /* step */
908:         PetscScalar opts_E0,opts_nu0,opts_xc;
909:         PetscScalar opts_E1,opts_nu1;

911:         opts_E0  = opts_E1  = 1.0;
912:         opts_nu0 = opts_nu1 = 0.333;
913:         opts_xc  = 0.5;
914:         PetscOptionsGetScalar(NULL,"-step_E0",&opts_E0,&flg);
915:         PetscOptionsGetScalar(NULL,"-step_nu0",&opts_nu0,&flg);
916:         PetscOptionsGetScalar(NULL,"-step_E1",&opts_E1,&flg);
917:         PetscOptionsGetScalar(NULL,"-step_nu1",&opts_nu1,&flg);
918:         PetscOptionsGetScalar(NULL,"-step_xc",&opts_xc,&flg);

920:         for (p = 0; p < GAUSS_POINTS; p++) {
921:           coord_x = centroid_x;
922:           coord_y = centroid_y;
923:           if (use_gp_coords) {
924:             coord_x = element_props[j][i].gp_coords[2*p];
925:             coord_y = element_props[j][i].gp_coords[2*p+1];
926:           }

928:           element_props[j][i].E[p]  = opts_E0;
929:           element_props[j][i].nu[p] = opts_nu0;
930:           if (PetscRealPart(coord_x) > PetscRealPart(opts_xc)) {
931:             element_props[j][i].E[p]  = opts_E1;
932:             element_props[j][i].nu[p] = opts_nu1;
933:           }

935:           element_props[j][i].fx[p] = 0.0;
936:           element_props[j][i].fy[p] = 0.0;
937:         }
938:       } else if (coefficient_structure == 2) { /* brick */
939:         PetscReal values_E[10];
940:         PetscReal values_nu[10];
941:         PetscInt  nbricks,maxnbricks;
942:         PetscInt  index,span;
943:         PetscInt  jj;

945:         flg        = PETSC_FALSE;
946:         maxnbricks = 10;
947:         PetscOptionsGetRealArray(NULL, "-brick_E",values_E,&maxnbricks,&flg);
948:         nbricks    = maxnbricks;
949:         if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER,"User must supply a list of E values for each brick");

951:         flg        = PETSC_FALSE;
952:         maxnbricks = 10;
953:         PetscOptionsGetRealArray(NULL, "-brick_nu",values_nu,&maxnbricks,&flg);
954:         if (!flg) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER,"User must supply a list of nu values for each brick");
955:         if (maxnbricks != nbricks) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER,"User must supply equal numbers of values for E and nu");

957:         span = 1;
958:         PetscOptionsGetInt(NULL,"-brick_span",&span,&flg);

960:         /* cycle through the indices so that no two material properties are repeated in lines of x or y */
961:         jj    = (j/span)%nbricks;
962:         index = (jj+i/span)%nbricks;
963:         /*printf("j=%d: index = %d \n", j,index); */

965:         for (p = 0; p < GAUSS_POINTS; p++) {
966:           element_props[j][i].E[p]  = values_E[index];
967:           element_props[j][i].nu[p] = values_nu[index];
968:         }
969:       } else if (coefficient_structure == 3) { /* sponge */
970:         PetscScalar opts_E0,opts_nu0;
971:         PetscScalar opts_E1,opts_nu1;
972:         PetscInt    opts_t,opts_w;
973:         PetscInt    ii,jj,ci,cj;

975:         opts_E0  = opts_E1  = 1.0;
976:         opts_nu0 = opts_nu1 = 0.333;
977:         PetscOptionsGetScalar(NULL,"-sponge_E0",&opts_E0,&flg);
978:         PetscOptionsGetScalar(NULL,"-sponge_nu0",&opts_nu0,&flg);
979:         PetscOptionsGetScalar(NULL,"-sponge_E1",&opts_E1,&flg);
980:         PetscOptionsGetScalar(NULL,"-sponge_nu1",&opts_nu1,&flg);

982:         opts_t = opts_w = 1;
983:         PetscOptionsGetInt(NULL,"-sponge_t",&opts_t,&flg);
984:         PetscOptionsGetInt(NULL,"-sponge_w",&opts_w,&flg);

986:         ii = (i)/(opts_t+opts_w+opts_t);
987:         jj = (j)/(opts_t+opts_w+opts_t);

989:         ci = i - ii*(opts_t+opts_w+opts_t);
990:         cj = j - jj*(opts_t+opts_w+opts_t);

992:         for (p = 0; p < GAUSS_POINTS; p++) {
993:           element_props[j][i].E[p]  = opts_E0;
994:           element_props[j][i].nu[p] = opts_nu0;
995:         }
996:         if ((ci >= opts_t) && (ci < opts_t+opts_w)) {
997:           if ((cj >= opts_t) && (cj < opts_t+opts_w)) {
998:             for (p = 0; p < GAUSS_POINTS; p++) {
999:               element_props[j][i].E[p]  = opts_E1;
1000:               element_props[j][i].nu[p] = opts_nu1;
1001:             }
1002:           }
1003:         }

1005:       } else SETERRQ(PETSC_COMM_SELF,PETSC_ERR_USER,"Unknown coefficient_structure");
1006:     }
1007:   }
1008:   DMDAVecRestoreArray(prop_cda,prop_coords,&_prop_coords);

1010:   DMDAVecRestoreArray(vel_cda,vel_coords,&_vel_coords);

1012:   DMDAVecRestoreArray(da_prop,l_properties,&element_props);
1013:   DMLocalToGlobalBegin(da_prop,l_properties,ADD_VALUES,properties);
1014:   DMLocalToGlobalEnd(da_prop,l_properties,ADD_VALUES,properties);

1016:   PetscOptionsGetBool(NULL,"-no_view",&no_view,NULL);
1017:   if (!no_view) {
1018:     DMDAViewCoefficientsGnuplot2d(da_prop,properties,"Coeffcients for elasticity eqn.","properties");
1019:     DMDACoordViewGnuplot2d(elas_da,"mesh");
1020:   }

1022:   /* Generate a matrix with the correct non-zero pattern of type AIJ. This will work in parallel and serial */
1023:   DMSetMatType(elas_da,MATAIJ);
1024:   DMCreateMatrix(elas_da,&A);
1025:   MatSetBlockSize(A,2);
1026:   DMGetCoordinates(elas_da,&vel_coords);
1027:   MatNullSpaceCreateRigidBody(vel_coords,&matnull);
1028:   MatSetNearNullSpace(A,matnull);
1029:   MatNullSpaceDestroy(&matnull);
1030:   MatCreateVecs(A,&f,&X);

1032:   /* assemble A11 */
1033:   MatZeroEntries(A);
1034:   VecZeroEntries(f);

1036:   AssembleA_Elasticity(A,elas_da,da_prop,properties);
1037:   /* build force vector */
1038:   AssembleF_Elasticity(f,elas_da,da_prop,properties);


1041:   KSPCreate(PETSC_COMM_WORLD,&ksp_E);
1042:   KSPSetOptionsPrefix(ksp_E,"elas_");  /* elasticity */

1044:   PetscOptionsGetBool(NULL,"-use_nonsymbc",&use_nonsymbc,&flg);
1045:   /* solve */
1046:   if (!use_nonsymbc) {
1047:     Mat        AA;
1048:     Vec        ff,XX;
1049:     IS         is;
1050:     VecScatter scat;

1052:     DMDABCApplySymmetricCompression(elas_da,A,f,&is,&AA,&ff);
1053:     VecDuplicate(ff,&XX);

1055:     KSPSetOperators(ksp_E,AA,AA);
1056:     KSPSetFromOptions(ksp_E);

1058:     KSPSolve(ksp_E,ff,XX);

1060:     /* push XX back into X */
1061:     DMDABCApplyCompression(elas_da,NULL,X);

1063:     VecScatterCreate(XX,NULL,X,is,&scat);
1064:     VecScatterBegin(scat,XX,X,INSERT_VALUES,SCATTER_FORWARD);
1065:     VecScatterEnd(scat,XX,X,INSERT_VALUES,SCATTER_FORWARD);
1066:     VecScatterDestroy(&scat);

1068:     MatDestroy(&AA);
1069:     VecDestroy(&ff);
1070:     VecDestroy(&XX);
1071:     ISDestroy(&is);
1072:   } else {
1073:     DMDABCApplyCompression(elas_da,A,f);

1075:     KSPSetOperators(ksp_E,A,A);
1076:     KSPSetFromOptions(ksp_E);

1078:     KSPSolve(ksp_E,f,X);
1079:   }

1081:   if (!no_view) {DMDAViewGnuplot2d(elas_da,X,"Displacement solution for elasticity eqn.","X");}
1082:   KSPDestroy(&ksp_E);

1084:   VecDestroy(&X);
1085:   VecDestroy(&f);
1086:   MatDestroy(&A);

1088:   DMDestroy(&elas_da);
1089:   DMDestroy(&da_prop);

1091:   VecDestroy(&properties);
1092:   VecDestroy(&l_properties);
1093:   return(0);
1094: }

1098: int main(int argc,char **args)
1099: {
1101:   PetscInt       mx,my;

1103:   PetscInitialize(&argc,&args,(char*)0,help);

1105:   mx   = my = 10;
1106:   PetscOptionsGetInt(NULL,"-mx",&mx,NULL);
1107:   PetscOptionsGetInt(NULL,"-my",&my,NULL);

1109:   solve_elasticity_2d(mx,my);

1111:   PetscFinalize();
1112:   return 0;
1113: }

1115: /* -------------------------- helpers for boundary conditions -------------------------------- */

1119: static PetscErrorCode BCApply_EAST(DM da,PetscInt d_idx,PetscScalar bc_val,Mat A,Vec b)
1120: {
1121:   DM                     cda;
1122:   Vec                    coords;
1123:   PetscInt               si,sj,nx,ny,i,j;
1124:   PetscInt               M,N;
1125:   DMDACoor2d             **_coords;
1126:   const PetscInt         *g_idx;
1127:   PetscInt               *bc_global_ids;
1128:   PetscScalar            *bc_vals;
1129:   PetscInt               nbcs;
1130:   PetscInt               n_dofs;
1131:   PetscErrorCode         ierr;
1132:   ISLocalToGlobalMapping ltogm;

1135:   /* enforce bc's */
1136:   DMGetLocalToGlobalMapping(da,&ltogm);
1137:   ISLocalToGlobalMappingGetIndices(ltogm,&g_idx);

1139:   DMGetCoordinateDM(da,&cda);
1140:   DMGetCoordinatesLocal(da,&coords);
1141:   DMDAVecGetArray(cda,coords,&_coords);
1142:   DMDAGetGhostCorners(cda,&si,&sj,0,&nx,&ny,0);
1143:   DMDAGetInfo(da,0,&M,&N,0,0,0,0,&n_dofs,0,0,0,0,0);

1145:   /* --- */

1147:   PetscMalloc1(ny*n_dofs,&bc_global_ids);
1148:   PetscMalloc1(ny*n_dofs,&bc_vals);

1150:   /* init the entries to -1 so VecSetValues will ignore them */
1151:   for (i = 0; i < ny*n_dofs; i++) bc_global_ids[i] = -1;

1153:   i = nx-1;
1154:   for (j = 0; j < ny; j++) {
1155:     PetscInt                 local_id;
1156:     PETSC_UNUSED PetscScalar coordx,coordy;

1158:     local_id = i+j*nx;

1160:     bc_global_ids[j] = g_idx[n_dofs*local_id+d_idx];

1162:     coordx = _coords[j+sj][i+si].x;
1163:     coordy = _coords[j+sj][i+si].y;

1165:     bc_vals[j] =  bc_val;
1166:   }
1167:   ISLocalToGlobalMappingRestoreIndices(ltogm,&g_idx);
1168:   nbcs = 0;
1169:   if ((si+nx) == (M)) nbcs = ny;

1171:   if (b) {
1172:     VecSetValues(b,nbcs,bc_global_ids,bc_vals,INSERT_VALUES);
1173:     VecAssemblyBegin(b);
1174:     VecAssemblyEnd(b);
1175:   }
1176:   if (A) {
1177:     MatZeroRows(A,nbcs,bc_global_ids,1.0,0,0);
1178:   }

1180:   PetscFree(bc_vals);
1181:   PetscFree(bc_global_ids);

1183:   DMDAVecRestoreArray(cda,coords,&_coords);
1184:   return(0);
1185: }

1189: static PetscErrorCode BCApply_WEST(DM da,PetscInt d_idx,PetscScalar bc_val,Mat A,Vec b)
1190: {
1191:   DM                     cda;
1192:   Vec                    coords;
1193:   PetscInt               si,sj,nx,ny,i,j;
1194:   PetscInt               M,N;
1195:   DMDACoor2d             **_coords;
1196:   const PetscInt         *g_idx;
1197:   PetscInt               *bc_global_ids;
1198:   PetscScalar            *bc_vals;
1199:   PetscInt               nbcs;
1200:   PetscInt               n_dofs;
1201:   PetscErrorCode         ierr;
1202:   ISLocalToGlobalMapping ltogm;

1205:   /* enforce bc's */
1206:   DMGetLocalToGlobalMapping(da,&ltogm);
1207:   ISLocalToGlobalMappingGetIndices(ltogm,&g_idx);

1209:   DMGetCoordinateDM(da,&cda);
1210:   DMGetCoordinatesLocal(da,&coords);
1211:   DMDAVecGetArray(cda,coords,&_coords);
1212:   DMDAGetGhostCorners(cda,&si,&sj,0,&nx,&ny,0);
1213:   DMDAGetInfo(da,0,&M,&N,0,0,0,0,&n_dofs,0,0,0,0,0);

1215:   /* --- */

1217:   PetscMalloc1(ny*n_dofs,&bc_global_ids);
1218:   PetscMalloc1(ny*n_dofs,&bc_vals);

1220:   /* init the entries to -1 so VecSetValues will ignore them */
1221:   for (i = 0; i < ny*n_dofs; i++) bc_global_ids[i] = -1;

1223:   i = 0;
1224:   for (j = 0; j < ny; j++) {
1225:     PetscInt                 local_id;
1226:     PETSC_UNUSED PetscScalar coordx,coordy;

1228:     local_id = i+j*nx;

1230:     bc_global_ids[j] = g_idx[n_dofs*local_id+d_idx];

1232:     coordx = _coords[j+sj][i+si].x;
1233:     coordy = _coords[j+sj][i+si].y;

1235:     bc_vals[j] =  bc_val;
1236:   }
1237:   ISLocalToGlobalMappingRestoreIndices(ltogm,&g_idx);
1238:   nbcs = 0;
1239:   if (si == 0) nbcs = ny;

1241:   if (b) {
1242:     VecSetValues(b,nbcs,bc_global_ids,bc_vals,INSERT_VALUES);
1243:     VecAssemblyBegin(b);
1244:     VecAssemblyEnd(b);
1245:   }
1246:   if (A) {
1247:     MatZeroRows(A,nbcs,bc_global_ids,1.0,0,0);
1248:   }

1250:   PetscFree(bc_vals);
1251:   PetscFree(bc_global_ids);

1253:   DMDAVecRestoreArray(cda,coords,&_coords);
1254:   return(0);
1255: }

1259: static PetscErrorCode DMDABCApplyCompression(DM elas_da,Mat A,Vec f)
1260: {

1264:   BCApply_EAST(elas_da,0,-1.0,A,f);
1265:   BCApply_EAST(elas_da,1, 0.0,A,f);
1266:   BCApply_WEST(elas_da,0,1.0,A,f);
1267:   BCApply_WEST(elas_da,1,0.0,A,f);
1268:   return(0);
1269: }

1273: static PetscErrorCode DMDABCApplySymmetricCompression(DM elas_da,Mat A,Vec f,IS *dofs,Mat *AA,Vec *ff)
1274: {
1276:   PetscInt       start,end,m;
1277:   PetscInt       *unconstrained;
1278:   PetscInt       cnt,i;
1279:   Vec            x;
1280:   PetscScalar    *_x;
1281:   IS             is;
1282:   VecScatter     scat;

1285:   /* push bc's into f and A */
1286:   VecDuplicate(f,&x);
1287:   BCApply_EAST(elas_da,0,-1.0,A,x);
1288:   BCApply_EAST(elas_da,1, 0.0,A,x);
1289:   BCApply_WEST(elas_da,0,1.0,A,x);
1290:   BCApply_WEST(elas_da,1,0.0,A,x);

1292:   /* define which dofs are not constrained */
1293:   VecGetLocalSize(x,&m);
1294:   PetscMalloc1(m,&unconstrained);
1295:   VecGetOwnershipRange(x,&start,&end);
1296:   VecGetArray(x,&_x);
1297:   cnt  = 0;
1298:   for (i = 0; i < m; i+=2) {
1299:     PetscReal val1,val2;

1301:     val1 = PetscRealPart(_x[i]);
1302:     val2 = PetscRealPart(_x[i+1]);
1303:     if (PetscAbs(val1) < 0.1 && PetscAbs(val2) < 0.1) {
1304:       unconstrained[cnt] = start + i;
1305:       cnt++;
1306:       unconstrained[cnt] = start + i + 1;
1307:       cnt++;
1308:     }
1309:   }
1310:   VecRestoreArray(x,&_x);

1312:   ISCreateGeneral(PETSC_COMM_WORLD,cnt,unconstrained,PETSC_COPY_VALUES,&is);
1313:   PetscFree(unconstrained);
1314:   ISSetBlockSize(is,2);

1316:   /* define correction for dirichlet in the rhs */
1317:   MatMult(A,x,f);
1318:   VecScale(f,-1.0);

1320:   /* get new matrix */
1321:   MatGetSubMatrix(A,is,is,MAT_INITIAL_MATRIX,AA);
1322:   /* get new vector */
1323:   MatCreateVecs(*AA,NULL,ff);

1325:   VecScatterCreate(f,is,*ff,NULL,&scat);
1326:   VecScatterBegin(scat,f,*ff,INSERT_VALUES,SCATTER_FORWARD);
1327:   VecScatterEnd(scat,f,*ff,INSERT_VALUES,SCATTER_FORWARD);

1329:   {                             /* Constrain near-null space */
1330:     PetscInt nvecs;
1331:     const Vec *vecs;
1332:     Vec *uvecs;
1333:     PetscBool has_const;
1334:     MatNullSpace mnull,unull;
1335:     MatGetNearNullSpace(A,&mnull);
1336:     MatNullSpaceGetVecs(mnull,&has_const,&nvecs,&vecs);
1337:     VecDuplicateVecs(*ff,nvecs,&uvecs);
1338:     for (i=0; i<nvecs; i++) {
1339:       VecScatterBegin(scat,vecs[i],uvecs[i],INSERT_VALUES,SCATTER_FORWARD);
1340:       VecScatterEnd(scat,vecs[i],uvecs[i],INSERT_VALUES,SCATTER_FORWARD);
1341:     }
1342:     MatNullSpaceCreate(PetscObjectComm((PetscObject)A),has_const,nvecs,uvecs,&unull);
1343:     MatSetNearNullSpace(*AA,unull);
1344:     MatNullSpaceDestroy(&unull);
1345:     for (i=0; i<nvecs; i++) {
1346:       VecDestroy(&uvecs[i]);
1347:     }
1348:     PetscFree(uvecs);
1349:   }

1351:   VecScatterDestroy(&scat);

1353:   *dofs = is;
1354:   VecDestroy(&x);
1355:   return(0);
1356: }