Actual source code: ex51.c

petsc-dev 2014-04-23
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  2: static char help[] = "This example solves a linear system in parallel with KSP.  The matrix\n\
  3: uses arbitrary order polynomials for finite elements on the unit square.  To test the parallel\n\
  4: matrix assembly, the matrix is intentionally laid out across processors\n\
  5: differently from the way it is assembled.  Input arguments are:\n\
  6:   -m <size> -p <order> : mesh size and polynomial order\n\n";

  8: /* Contributed by Travis Austin <austin@txcorp.com>, 2010.
  9:    based on src/ksp/ksp/examples/tutorials/ex3.c
 10:  */

 12: #include <petscksp.h>

 14: /* Declare user-defined routines */
 15: static PetscReal      src(PetscReal,PetscReal);
 16: static PetscReal      ubdy(PetscReal,PetscReal);
 17: static PetscReal      polyBasisFunc(PetscInt,PetscInt,PetscReal*,PetscReal);
 18: static PetscReal      derivPolyBasisFunc(PetscInt,PetscInt,PetscReal*,PetscReal);
 19: static PetscErrorCode Form1DElementMass(PetscReal,PetscInt,PetscReal*,PetscReal*,PetscScalar*);
 20: static PetscErrorCode Form1DElementStiffness(PetscReal,PetscInt,PetscReal*,PetscReal*,PetscScalar*);
 21: static PetscErrorCode Form2DElementMass(PetscInt,PetscScalar*,PetscScalar*);
 22: static PetscErrorCode Form2DElementStiffness(PetscInt,PetscScalar*,PetscScalar*,PetscScalar*);
 23: static PetscErrorCode FormNodalRhs(PetscInt,PetscReal,PetscReal,PetscReal,PetscReal*,PetscScalar*);
 24: static PetscErrorCode FormNodalSoln(PetscInt,PetscReal,PetscReal,PetscReal,PetscReal*,PetscScalar*);
 25: static void leggaulob(PetscReal, PetscReal, PetscReal [], PetscReal [], int);
 26: static void qAndLEvaluation(int, PetscReal, PetscReal*, PetscReal*, PetscReal*);

 30: int main(int argc,char **args)
 31: {
 32:   PetscInt       p = 2, m = 5;
 33:   PetscInt       num1Dnodes, num2Dnodes;
 34:   PetscScalar    *Ke1D,*Ke2D,*Me1D,*Me2D;
 35:   PetscScalar    *r,*ue,val;
 36:   Vec            u,ustar,b,q;
 37:   Mat            A,Mass;
 38:   KSP            ksp;
 39:   PetscInt       M,N;
 40:   PetscMPIInt    rank,size;
 41:   PetscReal      x,y,h,norm;
 42:   PetscInt       *idx,indx,count,*rows,i,j,k,start,end,its;
 43:   PetscReal      *rowsx,*rowsy;
 44:   PetscReal      *gllNode, *gllWgts;

 47:   PetscInitialize(&argc,&args,(char*)0,help);
 48:   PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"Options for p-FEM","");
 49:   PetscOptionsInt("-m","Number of elements in each direction","None",m,&m,NULL);
 50:   PetscOptionsInt("-p","Order of each element (tensor product basis)","None",p,&p,NULL);
 51:   PetscOptionsEnd();
 52:   N    = (p*m+1)*(p*m+1); /* dimension of matrix */
 53:   M    = m*m; /* number of elements */
 54:   h    = 1.0/m; /* mesh width */
 55:   MPI_Comm_rank(PETSC_COMM_WORLD,&rank);
 56:   MPI_Comm_size(PETSC_COMM_WORLD,&size);

 58:   /* Create stiffness matrix */
 59:   MatCreate(PETSC_COMM_WORLD,&A);
 60:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
 61:   MatSetFromOptions(A);
 62:   start = rank*(M/size) + ((M%size) < rank ? (M%size) : rank);
 63:   end   = start + M/size + ((M%size) > rank);

 65:   /* Create matrix  */
 66:   MatCreate(PETSC_COMM_WORLD,&Mass);
 67:   MatSetSizes(Mass,PETSC_DECIDE,PETSC_DECIDE,N,N);
 68:   MatSetFromOptions(Mass);
 69:   start = rank*(M/size) + ((M%size) < rank ? (M%size) : rank);
 70:   end   = start + M/size + ((M%size) > rank);

 72:   /* Allocate element stiffness matrices */
 73:   num1Dnodes = (p+1);
 74:   num2Dnodes = num1Dnodes*num1Dnodes;

 76:   PetscMalloc1((num1Dnodes*num1Dnodes),&Me1D);
 77:   PetscMalloc1((num1Dnodes*num1Dnodes),&Ke1D);
 78:   PetscMalloc1((num2Dnodes*num2Dnodes),&Me2D);
 79:   PetscMalloc1((num2Dnodes*num2Dnodes),&Ke2D);
 80:   PetscMalloc1(num2Dnodes,&idx);
 81:   PetscMalloc1(num2Dnodes,&r);
 82:   PetscMalloc1(num2Dnodes,&ue);

 84:   /* Allocate quadrature and create stiffness matrices */
 85:   PetscMalloc1((p+1),&gllNode);
 86:   PetscMalloc1((p+1),&gllWgts);
 87:   leggaulob(0.0,1.0,gllNode,gllWgts,p); /* Get GLL nodes and weights */
 88:   Form1DElementMass(h,p,gllNode,gllWgts,Me1D);
 89:   Form1DElementStiffness(h,p,gllNode,gllWgts,Ke1D);
 90:   Form2DElementMass(p,Me1D,Me2D);
 91:   Form2DElementStiffness(p,Ke1D,Me1D,Ke2D);

 93:   /* Assemble matrix */
 94:   for (i=start; i<end; i++) {
 95:      indx = 0;
 96:      for (k=0;k<(p+1);++k) {
 97:        for (j=0;j<(p+1);++j) {
 98:          idx[indx++] = p*(p*m+1)*(i/m) + p*(i % m) + k*(p*m+1) + j;
 99:        }
100:      }
101:      MatSetValues(A,num2Dnodes,idx,num2Dnodes,idx,Ke2D,ADD_VALUES);
102:      MatSetValues(Mass,num2Dnodes,idx,num2Dnodes,idx,Me2D,ADD_VALUES);
103:   }
104:   MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);
105:   MatAssemblyEnd(A,MAT_FINAL_ASSEMBLY);
106:   MatAssemblyBegin(Mass,MAT_FINAL_ASSEMBLY);
107:   MatAssemblyEnd(Mass,MAT_FINAL_ASSEMBLY);

109:   PetscFree(Me1D);
110:   PetscFree(Ke1D);
111:   PetscFree(Me2D);
112:   PetscFree(Ke2D);

114:   /* Create right-hand-side and solution vectors */
115:   VecCreate(PETSC_COMM_WORLD,&u);
116:   VecSetSizes(u,PETSC_DECIDE,N);
117:   VecSetFromOptions(u);
118:   PetscObjectSetName((PetscObject)u,"Approx. Solution");
119:   VecDuplicate(u,&b);
120:   PetscObjectSetName((PetscObject)b,"Right hand side");
121:   VecDuplicate(u,&q);
122:   PetscObjectSetName((PetscObject)q,"Right hand side 2");
123:   VecDuplicate(b,&ustar);
124:   VecSet(u,0.0);
125:   VecSet(b,0.0);
126:   VecSet(q,0.0);

128:   /* Assemble nodal right-hand-side and soln vector  */
129:   for (i=start; i<end; i++) {
130:     x    = h*(i % m);
131:     y    = h*(i/m);
132:     indx = 0;
133:     for (k=0;k<(p+1);++k) {
134:       for (j=0;j<(p+1);++j) {
135:         idx[indx++] = p*(p*m+1)*(i/m) + p*(i % m) + k*(p*m+1) + j;
136:       }
137:     }
138:     FormNodalRhs(p,x,y,h,gllNode,r);
139:     FormNodalSoln(p,x,y,h,gllNode,ue);
140:     VecSetValues(q,num2Dnodes,idx,r,INSERT_VALUES);
141:     VecSetValues(ustar,num2Dnodes,idx,ue,INSERT_VALUES);
142:   }
143:   VecAssemblyBegin(q);
144:   VecAssemblyEnd(q);
145:   VecAssemblyBegin(ustar);
146:   VecAssemblyEnd(ustar);

148:   PetscFree(idx);
149:   PetscFree(r);
150:   PetscFree(ue);

152:   /* Get FE right-hand side vector */
153:   MatMult(Mass,q,b);

155:   /* Modify matrix and right-hand-side for Dirichlet boundary conditions */
156:   PetscMalloc1(4*p*m,&rows);
157:   PetscMalloc1(4*p*m,&rowsx);
158:   PetscMalloc1(4*p*m,&rowsy);
159:   for (i=0; i<p*m+1; i++) {
160:     rows[i]          = i; /* bottom */
161:     rowsx[i]         = (i/p)*h+gllNode[i%p]*h;
162:     rowsy[i]         = 0.0;
163:     rows[3*p*m-1+i]  = (p*m)*(p*m+1) + i; /* top */
164:     rowsx[3*p*m-1+i] = (i/p)*h+gllNode[i%p]*h;
165:     rowsy[3*p*m-1+i] = 1.0;
166:   }
167:   count = p*m+1; /* left side */
168:   indx  = 1;
169:   for (i=p*m+1; i<(p*m)*(p*m+1); i+= (p*m+1)) {
170:     rows[count]    = i;
171:     rowsx[count]   = 0.0;
172:     rowsy[count++] = (indx/p)*h+gllNode[indx%p]*h;
173:     indx++;
174:   }
175:   count = 2*p*m; /* right side */
176:   indx  = 1;
177:   for (i=2*p*m+1; i<(p*m)*(p*m+1); i+= (p*m+1)) {
178:     rows[count]    = i;
179:     rowsx[count]   = 1.0;
180:     rowsy[count++] = (indx/p)*h+gllNode[indx%p]*h;
181:     indx++;
182:   }
183:   for (i=0; i<4*p*m; i++) {
184:     x    = rowsx[i];
185:     y    = rowsy[i];
186:     val  = ubdy(x,y);
187:     VecSetValues(b,1,&rows[i],&val,INSERT_VALUES);
188:     VecSetValues(u,1,&rows[i],&val,INSERT_VALUES);
189:   }
190:   MatZeroRows(A,4*p*m,rows,1.0,0,0);
191:   PetscFree(rows);
192:   PetscFree(rowsx);
193:   PetscFree(rowsy);

195:   VecAssemblyBegin(u);
196:   VecAssemblyEnd(u);
197:   VecAssemblyBegin(b);
198:   VecAssemblyEnd(b);

200:   /* Solve linear system */
201:   KSPCreate(PETSC_COMM_WORLD,&ksp);
202:   KSPSetOperators(ksp,A,A);
203:   KSPSetInitialGuessNonzero(ksp,PETSC_TRUE);
204:   KSPSetFromOptions(ksp);
205:   KSPSolve(ksp,b,u);

207:   /* Check error */
208:   VecAXPY(u,-1.0,ustar);
209:   VecNorm(u,NORM_2,&norm);
210:   KSPGetIterationNumber(ksp,&its);
211:   PetscPrintf(PETSC_COMM_WORLD,"Norm of error %g Iterations %D\n",(double)(norm*h),its);

213:   PetscFree(gllNode);
214:   PetscFree(gllWgts);

216:   KSPDestroy(&ksp);
217:   VecDestroy(&u);
218:   VecDestroy(&b);
219:   VecDestroy(&q);
220:   VecDestroy(&ustar);
221:   MatDestroy(&A);
222:   MatDestroy(&Mass);

224:   PetscFinalize();
225:   return 0;
226: }

228: /* --------------------------------------------------------------------- */

232: /* 1d element stiffness mass matrix  */
233: static PetscErrorCode Form1DElementMass(PetscReal H,PetscInt P,PetscReal *gqn,PetscReal *gqw,PetscScalar *Me1D)
234: {
235:   PetscInt i,j,k;
236:   PetscInt indx;

239:   for (j=0; j<(P+1); ++j) {
240:     for (i=0; i<(P+1); ++i) {
241:       indx       = j*(P+1)+i;
242:       Me1D[indx] = 0.0;
243:       for (k=0; k<(P+1);++k) {
244:         Me1D[indx] += H*gqw[k]*polyBasisFunc(P,i,gqn,gqn[k])*polyBasisFunc(P,j,gqn,gqn[k]);
245:       }
246:     }
247:   }
248:   return(0);
249: }

251: /* --------------------------------------------------------------------- */

255: /* 1d element stiffness matrix for derivative */
256: static PetscErrorCode Form1DElementStiffness(PetscReal H,PetscInt P,PetscReal *gqn,PetscReal *gqw,PetscScalar *Ke1D)
257: {
258:   PetscInt i,j,k;
259:   PetscInt indx;

262:   for (j=0;j<(P+1);++j) {
263:     for (i=0;i<(P+1);++i) {
264:       indx = j*(P+1)+i;
265:       Ke1D[indx] = 0.0;
266:       for (k=0; k<(P+1);++k) {
267:         Ke1D[indx] += (1./H)*gqw[k]*derivPolyBasisFunc(P,i,gqn,gqn[k])*derivPolyBasisFunc(P,j,gqn,gqn[k]);
268:       }
269:     }
270:   }
271:   return(0);
272: }

274: /* --------------------------------------------------------------------- */

278:    /* element mass matrix */
279: static PetscErrorCode Form2DElementMass(PetscInt P,PetscScalar *Me1D,PetscScalar *Me2D)
280: {
281:   PetscInt i1,j1,i2,j2;
282:   PetscInt indx1,indx2,indx3;

285:   for (j2=0;j2<(P+1);++j2) {
286:     for (i2=0; i2<(P+1);++i2) {
287:       for (j1=0;j1<(P+1);++j1) {
288:         for (i1=0;i1<(P+1);++i1) {
289:           indx1 = j1*(P+1)+i1;
290:           indx2 = j2*(P+1)+i2;
291:           indx3 = (j2*(P+1)+j1)*(P+1)*(P+1)+(i2*(P+1)+i1);
292:           Me2D[indx3] = Me1D[indx1]*Me1D[indx2];
293:         }
294:       }
295:     }
296:   }
297:   return(0);
298: }

300: /* --------------------------------------------------------------------- */

304: /* element stiffness for Laplacian */
305: static PetscErrorCode Form2DElementStiffness(PetscInt P,PetscScalar *Ke1D,PetscScalar *Me1D,PetscScalar *Ke2D)
306: {
307:   PetscInt i1,j1,i2,j2;
308:   PetscInt indx1,indx2,indx3;

311:   for (j2=0;j2<(P+1);++j2) {
312:     for (i2=0; i2<(P+1);++i2) {
313:       for (j1=0;j1<(P+1);++j1) {
314:         for (i1=0;i1<(P+1);++i1) {
315:           indx1 = j1*(P+1)+i1;
316:           indx2 = j2*(P+1)+i2;
317:           indx3 = (j2*(P+1)+j1)*(P+1)*(P+1)+(i2*(P+1)+i1);
318:           Ke2D[indx3] = Ke1D[indx1]*Me1D[indx2] + Me1D[indx1]*Ke1D[indx2];
319:         }
320:       }
321:     }
322:   }
323:   return(0);
324: }

326: /* --------------------------------------------------------------------- */

330: static PetscErrorCode FormNodalRhs(PetscInt P,PetscReal x,PetscReal y,PetscReal H,PetscReal* nds,PetscScalar *r)
331: {
332:   PetscInt i,j,indx;

335:   indx=0;
336:   for (j=0;j<(P+1);++j) {
337:     for (i=0;i<(P+1);++i) {
338:       r[indx] = src(x+H*nds[i],y+H*nds[j]);
339:       indx++;
340:     }
341:   }
342:   return(0);
343: }

345: /* --------------------------------------------------------------------- */

349: static PetscErrorCode FormNodalSoln(PetscInt P,PetscReal x,PetscReal y,PetscReal H,PetscReal* nds,PetscScalar *u)
350: {
351:   PetscInt i,j,indx;

354:   indx=0;
355:   for (j=0;j<(P+1);++j) {
356:     for (i=0;i<(P+1);++i) {
357:       u[indx] = ubdy(x+H*nds[i],y+H*nds[j]);
358:       indx++;
359:     }
360:   }
361:   return(0);
362: }

364: /* --------------------------------------------------------------------- */

368: static PetscReal polyBasisFunc(PetscInt order, PetscInt basis, PetscReal *xLocVal, PetscReal xval)
369: {
370:   PetscReal denominator = 1.;
371:   PetscReal numerator   = 1.;
372:   PetscInt  i           =0;

374:   for (i=0; i<(order+1); i++) {
375:     if (i!=basis) {
376:       numerator   *= (xval-xLocVal[i]);
377:       denominator *= (xLocVal[basis]-xLocVal[i]);
378:     }
379:   }
380:   return (numerator/denominator);
381: }

383: /* --------------------------------------------------------------------- */

387: static PetscReal derivPolyBasisFunc(PetscInt order, PetscInt basis, PetscReal *xLocVal, PetscReal xval)
388: {
389:   PetscReal denominator;
390:   PetscReal numerator;
391:   PetscReal numtmp;
392:   PetscInt  i=0,j=0;

394:   denominator=1.;
395:   for (i=0; i<(order+1); i++) {
396:     if (i!=basis) denominator *= (xLocVal[basis]-xLocVal[i]);
397:   }
398:   numerator = 0.;
399:   for (j=0;j<(order+1);++j) {
400:     if (j != basis) {
401:       numtmp = 1.;
402:       for (i=0;i<(order+1);++i) {
403:         if (i!=basis && j!=i) numtmp *= (xval-xLocVal[i]);
404:       }
405:       numerator += numtmp;
406:     }
407:   }

409:   return (numerator/denominator);
410: }

412: /* --------------------------------------------------------------------- */

414: static PetscReal ubdy(PetscReal x,PetscReal y)
415: {
416:   return x*x*y*y;
417: }

419: static PetscReal src(PetscReal x,PetscReal y)
420: {
421:   return -2.*y*y - 2.*x*x;
422: }
423: /* --------------------------------------------------------------------- */

425: static void leggaulob(PetscReal x1, PetscReal x2, PetscReal x[], PetscReal w[], int n)
426: /*******************************************************************************
427: Given the lower and upper limits of integration x1 and x2, and given n, this
428: routine returns arrays x[0..n-1] and w[0..n-1] of length n, containing the abscissas
429: and weights of the Gauss-Lobatto-Legendre n-point quadrature formula.
430: *******************************************************************************/
431: {
432:   PetscInt    j,m;
433:   PetscReal z1,z,xm,xl,q,qp,Ln,scale;
434:   if (n==1) {
435:     x[0] = x1;   /* Scale the root to the desired interval, */
436:     x[1] = x2;   /* and put in its symmetric counterpart.   */
437:     w[0] = 1.;   /* Compute the weight */
438:     w[1] = 1.;   /* and its symmetric counterpart. */
439:   } else {
440:     x[0] = x1;   /* Scale the root to the desired interval, */
441:     x[n] = x2;   /* and put in its symmetric counterpart.   */
442:     w[0] = 2./(n*(n+1));;   /* Compute the weight */
443:     w[n] = 2./(n*(n+1));   /* and its symmetric counterpart. */
444:     m    = (n+1)/2; /* The roots are symmetric, so we only find half of them. */
445:     xm   = 0.5*(x2+x1);
446:     xl   = 0.5*(x2-x1);
447:     for (j=1; j<=(m-1); j++) { /* Loop over the desired roots. */
448:       z=-1.0*PetscCosReal((PETSC_PI*(j+0.25)/(n))-(3.0/(8.0*n*PETSC_PI))*(1.0/(j+0.25)));
449:       /* Starting with the above approximation to the ith root, we enter */
450:       /* the main loop of refinement by Newton's method.                 */
451:       do {
452:         qAndLEvaluation(n,z,&q,&qp,&Ln);
453:         z1 = z;
454:         z  = z1-q/qp; /* Newton's method. */
455:       } while (fabs(z-z1) > 3.0e-11);
456:       qAndLEvaluation(n,z,&q,&qp,&Ln);
457:       x[j]   = xm+xl*z;      /* Scale the root to the desired interval, */
458:       x[n-j] = xm-xl*z;      /* and put in its symmetric counterpart.   */
459:       w[j]   = 2.0/(n*(n+1)*Ln*Ln);  /* Compute the weight */
460:       w[n-j] = w[j];                 /* and its symmetric counterpart. */
461:     }
462:   }
463:   if (n%2==0) {
464:     qAndLEvaluation(n,0.0,&q,&qp,&Ln);
465:     x[n/2]=(x2-x1)/2.0;
466:     w[n/2]=2.0/(n*(n+1)*Ln*Ln);
467:   }
468:   /* scale the weights according to mapping from [-1,1] to [0,1] */
469:   scale = (x2-x1)/2.0;
470:   for (j=0; j<=n; ++j) w[j] = w[j]*scale;
471: }


474: /******************************************************************************/
475: static void qAndLEvaluation(PetscInt n, PetscReal x, PetscReal *q, PetscReal *qp, PetscReal *Ln)
476: /*******************************************************************************
477: Compute the polynomial qn(x) = L_{N+1}(x) - L_{n-1}(x) and its derivative in
478: addition to L_N(x) as these are needed for the GLL points.  See the book titled
479: "Implementing Spectral Methods for Partial Differential Equations: Algorithms
480: for Scientists and Engineers" by David A. Kopriva.
481: *******************************************************************************/
482: {
483:   PetscInt k;

485:   PetscReal Lnp;
486:   PetscReal Lnp1, Lnp1p;
487:   PetscReal Lnm1, Lnm1p;
488:   PetscReal Lnm2, Lnm2p;

490:   Lnm1  = 1.0;
491:   *Ln   = x;
492:   Lnm1p = 0.0;
493:   Lnp   = 1.0;

495:   for (k=2; k<=n; ++k) {
496:     Lnm2  = Lnm1;
497:     Lnm1  = *Ln;
498:     Lnm2p = Lnm1p;
499:     Lnm1p = Lnp;
500:     *Ln   = (2.*k-1.)/(1.0*k)*x*Lnm1 - (k-1.)/(1.0*k)*Lnm2;
501:     Lnp   = Lnm2p + (2.0*k-1)*Lnm1;
502:   }
503:   k     = n+1;
504:   Lnp1  = (2.*k-1.)/(1.0*k)*x*(*Ln) - (k-1.)/(1.0*k)*Lnm1;
505:   Lnp1p = Lnm1p + (2.0*k-1)*(*Ln);
506:   *q    = Lnp1 - Lnm1;
507:   *qp   = Lnp1p - Lnm1p;
508: }