Actual source code: ex51.c

petsc-master 2017-09-25
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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*);

 28: int main(int argc,char **args)
 29: {
 30:   PetscInt       p = 2, m = 5;
 31:   PetscInt       num1Dnodes, num2Dnodes;
 32:   PetscScalar    *Ke1D,*Ke2D,*Me1D,*Me2D;
 33:   PetscScalar    *r,*ue,val;
 34:   Vec            u,ustar,b,q;
 35:   Mat            A,Mass;
 36:   KSP            ksp;
 37:   PetscInt       M,N;
 38:   PetscMPIInt    rank,size;
 39:   PetscReal      x,y,h,norm;
 40:   PetscInt       *idx,indx,count,*rows,i,j,k,start,end,its;
 41:   PetscReal      *rowsx,*rowsy;
 42:   PetscReal      *gllNode, *gllWgts;

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

 56:   /* Create stiffness matrix */
 57:   MatCreate(PETSC_COMM_WORLD,&A);
 58:   MatSetSizes(A,PETSC_DECIDE,PETSC_DECIDE,N,N);
 59:   MatSetFromOptions(A);
 60:   MatSetUp(A);

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

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

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

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

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

107:   PetscFree(Me1D);
108:   PetscFree(Ke1D);
109:   PetscFree(Me2D);
110:   PetscFree(Ke2D);

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

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

146:   PetscFree(idx);
147:   PetscFree(r);
148:   PetscFree(ue);

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

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

193:   VecAssemblyBegin(u);
194:   VecAssemblyEnd(u);
195:   VecAssemblyBegin(b);
196:   VecAssemblyEnd(b);

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

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

211:   PetscFree(gllNode);
212:   PetscFree(gllWgts);

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

222:   PetscFinalize();
223:   return ierr;
224: }

226: /* --------------------------------------------------------------------- */

228: /* 1d element stiffness mass matrix  */
229: static PetscErrorCode Form1DElementMass(PetscReal H,PetscInt P,PetscReal *gqn,PetscReal *gqw,PetscScalar *Me1D)
230: {
231:   PetscInt i,j,k;
232:   PetscInt indx;

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

247: /* --------------------------------------------------------------------- */

249: /* 1d element stiffness matrix for derivative */
250: static PetscErrorCode Form1DElementStiffness(PetscReal H,PetscInt P,PetscReal *gqn,PetscReal *gqw,PetscScalar *Ke1D)
251: {
252:   PetscInt i,j,k;
253:   PetscInt indx;

256:   for (j=0;j<(P+1);++j) {
257:     for (i=0;i<(P+1);++i) {
258:       indx = j*(P+1)+i;
259:       Ke1D[indx] = 0.0;
260:       for (k=0; k<(P+1);++k) {
261:         Ke1D[indx] += (1./H)*gqw[k]*derivPolyBasisFunc(P,i,gqn,gqn[k])*derivPolyBasisFunc(P,j,gqn,gqn[k]);
262:       }
263:     }
264:   }
265:   return(0);
266: }

268: /* --------------------------------------------------------------------- */

270:    /* element mass matrix */
271: static PetscErrorCode Form2DElementMass(PetscInt P,PetscScalar *Me1D,PetscScalar *Me2D)
272: {
273:   PetscInt i1,j1,i2,j2;
274:   PetscInt indx1,indx2,indx3;

277:   for (j2=0;j2<(P+1);++j2) {
278:     for (i2=0; i2<(P+1);++i2) {
279:       for (j1=0;j1<(P+1);++j1) {
280:         for (i1=0;i1<(P+1);++i1) {
281:           indx1 = j1*(P+1)+i1;
282:           indx2 = j2*(P+1)+i2;
283:           indx3 = (j2*(P+1)+j1)*(P+1)*(P+1)+(i2*(P+1)+i1);
284:           Me2D[indx3] = Me1D[indx1]*Me1D[indx2];
285:         }
286:       }
287:     }
288:   }
289:   return(0);
290: }

292: /* --------------------------------------------------------------------- */

294: /* element stiffness for Laplacian */
295: static PetscErrorCode Form2DElementStiffness(PetscInt P,PetscScalar *Ke1D,PetscScalar *Me1D,PetscScalar *Ke2D)
296: {
297:   PetscInt i1,j1,i2,j2;
298:   PetscInt indx1,indx2,indx3;

301:   for (j2=0;j2<(P+1);++j2) {
302:     for (i2=0; i2<(P+1);++i2) {
303:       for (j1=0;j1<(P+1);++j1) {
304:         for (i1=0;i1<(P+1);++i1) {
305:           indx1 = j1*(P+1)+i1;
306:           indx2 = j2*(P+1)+i2;
307:           indx3 = (j2*(P+1)+j1)*(P+1)*(P+1)+(i2*(P+1)+i1);
308:           Ke2D[indx3] = Ke1D[indx1]*Me1D[indx2] + Me1D[indx1]*Ke1D[indx2];
309:         }
310:       }
311:     }
312:   }
313:   return(0);
314: }

316: /* --------------------------------------------------------------------- */

318: static PetscErrorCode FormNodalRhs(PetscInt P,PetscReal x,PetscReal y,PetscReal H,PetscReal* nds,PetscScalar *r)
319: {
320:   PetscInt i,j,indx;

323:   indx=0;
324:   for (j=0;j<(P+1);++j) {
325:     for (i=0;i<(P+1);++i) {
326:       r[indx] = src(x+H*nds[i],y+H*nds[j]);
327:       indx++;
328:     }
329:   }
330:   return(0);
331: }

333: /* --------------------------------------------------------------------- */

335: static PetscErrorCode FormNodalSoln(PetscInt P,PetscReal x,PetscReal y,PetscReal H,PetscReal* nds,PetscScalar *u)
336: {
337:   PetscInt i,j,indx;

340:   indx=0;
341:   for (j=0;j<(P+1);++j) {
342:     for (i=0;i<(P+1);++i) {
343:       u[indx] = ubdy(x+H*nds[i],y+H*nds[j]);
344:       indx++;
345:     }
346:   }
347:   return(0);
348: }

350: /* --------------------------------------------------------------------- */

352: static PetscReal polyBasisFunc(PetscInt order, PetscInt basis, PetscReal *xLocVal, PetscReal xval)
353: {
354:   PetscReal denominator = 1.;
355:   PetscReal numerator   = 1.;
356:   PetscInt  i           =0;

358:   for (i=0; i<(order+1); i++) {
359:     if (i!=basis) {
360:       numerator   *= (xval-xLocVal[i]);
361:       denominator *= (xLocVal[basis]-xLocVal[i]);
362:     }
363:   }
364:   return (numerator/denominator);
365: }

367: /* --------------------------------------------------------------------- */

369: static PetscReal derivPolyBasisFunc(PetscInt order, PetscInt basis, PetscReal *xLocVal, PetscReal xval)
370: {
371:   PetscReal denominator;
372:   PetscReal numerator;
373:   PetscReal numtmp;
374:   PetscInt  i=0,j=0;

376:   denominator=1.;
377:   for (i=0; i<(order+1); i++) {
378:     if (i!=basis) denominator *= (xLocVal[basis]-xLocVal[i]);
379:   }
380:   numerator = 0.;
381:   for (j=0;j<(order+1);++j) {
382:     if (j != basis) {
383:       numtmp = 1.;
384:       for (i=0;i<(order+1);++i) {
385:         if (i!=basis && j!=i) numtmp *= (xval-xLocVal[i]);
386:       }
387:       numerator += numtmp;
388:     }
389:   }

391:   return (numerator/denominator);
392: }

394: /* --------------------------------------------------------------------- */

396: static PetscReal ubdy(PetscReal x,PetscReal y)
397: {
398:   return x*x*y*y;
399: }

401: static PetscReal src(PetscReal x,PetscReal y)
402: {
403:   return -2.*y*y - 2.*x*x;
404: }
405: /* --------------------------------------------------------------------- */

407: static void leggaulob(PetscReal x1, PetscReal x2, PetscReal x[], PetscReal w[], int n)
408: /*******************************************************************************
409: Given the lower and upper limits of integration x1 and x2, and given n, this
410: routine returns arrays x[0..n-1] and w[0..n-1] of length n, containing the abscissas
411: and weights of the Gauss-Lobatto-Legendre n-point quadrature formula.
412: *******************************************************************************/
413: {
414:   PetscInt    j,m;
415:   PetscReal z1,z,xm,xl,q,qp,Ln,scale;
416:   if (n==1) {
417:     x[0] = x1;   /* Scale the root to the desired interval, */
418:     x[1] = x2;   /* and put in its symmetric counterpart.   */
419:     w[0] = 1.;   /* Compute the weight */
420:     w[1] = 1.;   /* and its symmetric counterpart. */
421:   } else {
422:     x[0] = x1;   /* Scale the root to the desired interval, */
423:     x[n] = x2;   /* and put in its symmetric counterpart.   */
424:     w[0] = 2./(n*(n+1));;   /* Compute the weight */
425:     w[n] = 2./(n*(n+1));   /* and its symmetric counterpart. */
426:     m    = (n+1)/2; /* The roots are symmetric, so we only find half of them. */
427:     xm   = 0.5*(x2+x1);
428:     xl   = 0.5*(x2-x1);
429:     for (j=1; j<=(m-1); j++) { /* Loop over the desired roots. */
430:       z=-1.0*PetscCosReal((PETSC_PI*(j+0.25)/(n))-(3.0/(8.0*n*PETSC_PI))*(1.0/(j+0.25)));
431:       /* Starting with the above approximation to the ith root, we enter */
432:       /* the main loop of refinement by Newton's method.                 */
433:       do {
434:         qAndLEvaluation(n,z,&q,&qp,&Ln);
435:         z1 = z;
436:         z  = z1-q/qp; /* Newton's method. */
437:       } while (PetscAbsReal(z-z1) > 3.0e-11);
438:       qAndLEvaluation(n,z,&q,&qp,&Ln);
439:       x[j]   = xm+xl*z;      /* Scale the root to the desired interval, */
440:       x[n-j] = xm-xl*z;      /* and put in its symmetric counterpart.   */
441:       w[j]   = 2.0/(n*(n+1)*Ln*Ln);  /* Compute the weight */
442:       w[n-j] = w[j];                 /* and its symmetric counterpart. */
443:     }
444:   }
445:   if (n%2==0) {
446:     qAndLEvaluation(n,0.0,&q,&qp,&Ln);
447:     x[n/2]=(x2-x1)/2.0;
448:     w[n/2]=2.0/(n*(n+1)*Ln*Ln);
449:   }
450:   /* scale the weights according to mapping from [-1,1] to [0,1] */
451:   scale = (x2-x1)/2.0;
452:   for (j=0; j<=n; ++j) w[j] = w[j]*scale;
453: }


456: /******************************************************************************/
457: static void qAndLEvaluation(int n, PetscReal x, PetscReal *q, PetscReal *qp, PetscReal *Ln)
458: /*******************************************************************************
459: Compute the polynomial qn(x) = L_{N+1}(x) - L_{n-1}(x) and its derivative in
460: addition to L_N(x) as these are needed for the GLL points.  See the book titled
461: "Implementing Spectral Methods for Partial Differential Equations: Algorithms
462: for Scientists and Engineers" by David A. Kopriva.
463: *******************************************************************************/
464: {
465:   PetscInt k;

467:   PetscReal Lnp;
468:   PetscReal Lnp1, Lnp1p;
469:   PetscReal Lnm1, Lnm1p;
470:   PetscReal Lnm2, Lnm2p;

472:   Lnm1  = 1.0;
473:   *Ln   = x;
474:   Lnm1p = 0.0;
475:   Lnp   = 1.0;

477:   for (k=2; k<=n; ++k) {
478:     Lnm2  = Lnm1;
479:     Lnm1  = *Ln;
480:     Lnm2p = Lnm1p;
481:     Lnm1p = Lnp;
482:     *Ln   = (2.*k-1.)/(1.0*k)*x*Lnm1 - (k-1.)/(1.0*k)*Lnm2;
483:     Lnp   = Lnm2p + (2.0*k-1)*Lnm1;
484:   }
485:   k     = n+1;
486:   Lnp1  = (2.*k-1.)/(1.0*k)*x*(*Ln) - (k-1.)/(1.0*k)*Lnm1;
487:   Lnp1p = Lnm1p + (2.0*k-1)*(*Ln);
488:   *q    = Lnp1 - Lnm1;
489:   *qp   = Lnp1p - Lnm1p;
490: }