Actual source code: ex10.c
petsc-3.6.4 2016-04-12
2: static char help[] = "Linear elastiticty with dimensions using 20 node serendipity elements.\n\
3: This also demonstrates use of block\n\
4: diagonal data structure. Input arguments are:\n\
5: -m : problem size\n\n";
7: #include <petscksp.h>
9: /* This code is not intended as an efficient implementation, it is only
10: here to produce an interesting sparse matrix quickly.
12: PLEASE DO NOT BASE ANY OF YOUR CODES ON CODE LIKE THIS, THERE ARE MUCH
13: BETTER WAYS TO DO THIS. */
15: extern PetscErrorCode GetElasticityMatrix(PetscInt,Mat*);
16: extern PetscErrorCode Elastic20Stiff(PetscReal**);
17: extern PetscErrorCode AddElement(Mat,PetscInt,PetscInt,PetscReal**,PetscInt,PetscInt);
18: extern PetscErrorCode paulsetup20(void);
19: extern PetscErrorCode paulintegrate20(PetscReal K[60][60]);
23: int main(int argc,char **args)
24: {
25: Mat mat;
27: PetscInt i,its,m = 3,rdim,cdim,rstart,rend;
28: PetscMPIInt rank,size;
29: PetscScalar v,neg1 = -1.0;
30: Vec u,x,b;
31: KSP ksp;
32: PetscReal norm;
34: PetscInitialize(&argc,&args,(char*)0,help);
35: PetscOptionsGetInt(NULL,"-m",&m,NULL);
36: MPI_Comm_rank(PETSC_COMM_WORLD,&rank);
37: MPI_Comm_size(PETSC_COMM_WORLD,&size);
39: /* Form matrix */
40: GetElasticityMatrix(m,&mat);
42: /* Generate vectors */
43: MatGetSize(mat,&rdim,&cdim);
44: MatGetOwnershipRange(mat,&rstart,&rend);
45: VecCreate(PETSC_COMM_WORLD,&u);
46: VecSetSizes(u,PETSC_DECIDE,rdim);
47: VecSetFromOptions(u);
48: VecDuplicate(u,&b);
49: VecDuplicate(b,&x);
50: for (i=rstart; i<rend; i++) {
51: v = (PetscScalar)(i-rstart + 100*rank);
52: VecSetValues(u,1,&i,&v,INSERT_VALUES);
53: }
54: VecAssemblyBegin(u);
55: VecAssemblyEnd(u);
57: /* Compute right-hand-side */
58: MatMult(mat,u,b);
60: /* Solve linear system */
61: KSPCreate(PETSC_COMM_WORLD,&ksp);
62: KSPSetOperators(ksp,mat,mat);
63: KSPGMRESSetRestart(ksp,2*m);
64: KSPSetTolerances(ksp,1.e-10,PETSC_DEFAULT,PETSC_DEFAULT,PETSC_DEFAULT);
65: KSPSetType(ksp,KSPCG);
66: KSPSetFromOptions(ksp);
67: KSPSolve(ksp,b,x);
68: KSPGetIterationNumber(ksp,&its);
69: /* Check error */
70: VecAXPY(x,neg1,u);
71: VecNorm(x,NORM_2,&norm);
73: PetscPrintf(PETSC_COMM_WORLD,"Norm of error %g Number of iterations %D\n",(double)norm,its);
75: /* Free work space */
76: KSPDestroy(&ksp);
77: VecDestroy(&u);
78: VecDestroy(&x);
79: VecDestroy(&b);
80: MatDestroy(&mat);
82: PetscFinalize();
83: return 0;
84: }
85: /* -------------------------------------------------------------------- */
88: /*
89: GetElasticityMatrix - Forms 3D linear elasticity matrix.
90: */
91: PetscErrorCode GetElasticityMatrix(PetscInt m,Mat *newmat)
92: {
93: PetscInt i,j,k,i1,i2,j_1,j2,k1,k2,h1,h2,shiftx,shifty,shiftz;
94: PetscInt ict,nz,base,r1,r2,N,*rowkeep,nstart;
96: IS iskeep;
97: PetscReal **K,norm;
98: Mat mat,submat = 0,*submatb;
99: MatType type = MATSEQBAIJ;
101: m /= 2; /* This is done just to be consistent with the old example */
102: N = 3*(2*m+1)*(2*m+1)*(2*m+1);
103: PetscPrintf(PETSC_COMM_SELF,"m = %D, N=%D\n",m,N);
104: MatCreateSeqAIJ(PETSC_COMM_SELF,N,N,80,NULL,&mat);
106: /* Form stiffness for element */
107: PetscMalloc1(81,&K);
108: for (i=0; i<81; i++) {
109: PetscMalloc1(81,&K[i]);
110: }
111: Elastic20Stiff(K);
113: /* Loop over elements and add contribution to stiffness */
114: shiftx = 3; shifty = 3*(2*m+1); shiftz = 3*(2*m+1)*(2*m+1);
115: for (k=0; k<m; k++) {
116: for (j=0; j<m; j++) {
117: for (i=0; i<m; i++) {
118: h1 = 0;
119: base = 2*k*shiftz + 2*j*shifty + 2*i*shiftx;
120: for (k1=0; k1<3; k1++) {
121: for (j_1=0; j_1<3; j_1++) {
122: for (i1=0; i1<3; i1++) {
123: h2 = 0;
124: r1 = base + i1*shiftx + j_1*shifty + k1*shiftz;
125: for (k2=0; k2<3; k2++) {
126: for (j2=0; j2<3; j2++) {
127: for (i2=0; i2<3; i2++) {
128: r2 = base + i2*shiftx + j2*shifty + k2*shiftz;
129: AddElement(mat,r1,r2,K,h1,h2);
130: h2 += 3;
131: }
132: }
133: }
134: h1 += 3;
135: }
136: }
137: }
138: }
139: }
140: }
142: for (i=0; i<81; i++) {
143: PetscFree(K[i]);
144: }
145: PetscFree(K);
147: MatAssemblyBegin(mat,MAT_FINAL_ASSEMBLY);
148: MatAssemblyEnd(mat,MAT_FINAL_ASSEMBLY);
150: /* Exclude any superfluous rows and columns */
151: nstart = 3*(2*m+1)*(2*m+1);
152: ict = 0;
153: PetscMalloc1(N-nstart,&rowkeep);
154: for (i=nstart; i<N; i++) {
155: MatGetRow(mat,i,&nz,0,0);
156: if (nz) rowkeep[ict++] = i;
157: MatRestoreRow(mat,i,&nz,0,0);
158: }
159: ISCreateGeneral(PETSC_COMM_SELF,ict,rowkeep,PETSC_COPY_VALUES,&iskeep);
160: MatGetSubMatrices(mat,1,&iskeep,&iskeep,MAT_INITIAL_MATRIX,&submatb);
161: submat = *submatb;
162: PetscFree(submatb);
163: PetscFree(rowkeep);
164: ISDestroy(&iskeep);
165: MatDestroy(&mat);
167: /* Convert storage formats -- just to demonstrate conversion to various
168: formats (in particular, block diagonal storage). This is NOT the
169: recommended means to solve such a problem. */
170: MatConvert(submat,type,MAT_INITIAL_MATRIX,newmat);
171: MatDestroy(&submat);
173: PetscViewerSetFormat(PETSC_VIEWER_STDOUT_WORLD,PETSC_VIEWER_ASCII_INFO);
174: MatView(*newmat,PETSC_VIEWER_STDOUT_WORLD);
175: MatNorm(*newmat,NORM_1,&norm);
176: PetscPrintf(PETSC_COMM_WORLD,"matrix 1 norm = %g\n",(double)norm);
178: return 0;
179: }
180: /* -------------------------------------------------------------------- */
183: PetscErrorCode AddElement(Mat mat,PetscInt r1,PetscInt r2,PetscReal **K,PetscInt h1,PetscInt h2)
184: {
185: PetscScalar val;
186: PetscInt l1,l2,row,col;
189: for (l1=0; l1<3; l1++) {
190: for (l2=0; l2<3; l2++) {
191: /*
192: NOTE you should never do this! Inserting values 1 at a time is
193: just too expensive!
194: */
195: if (K[h1+l1][h2+l2] != 0.0) {
196: row = r1+l1; col = r2+l2; val = K[h1+l1][h2+l2];
197: MatSetValues(mat,1,&row,1,&col,&val,ADD_VALUES);
198: row = r2+l2; col = r1+l1;
199: MatSetValues(mat,1,&row,1,&col,&val,ADD_VALUES);
200: }
201: }
202: }
203: return 0;
204: }
205: /* -------------------------------------------------------------------- */
206: PetscReal N[20][64]; /* Interpolation function. */
207: PetscReal part_N[3][20][64]; /* Partials of interpolation function. */
208: PetscReal rst[3][64]; /* Location of integration pts in (r,s,t) */
209: PetscReal weight[64]; /* Gaussian quadrature weights. */
210: PetscReal xyz[20][3]; /* (x,y,z) coordinates of nodes */
211: PetscReal E,nu; /* Physcial constants. */
212: PetscInt n_int,N_int; /* N_int = n_int^3, number of int. pts. */
213: /* Ordering of the vertices, (r,s,t) coordinates, of the canonical cell. */
214: PetscReal r2[20] = {-1.0,0.0,1.0,-1.0,1.0,-1.0,0.0,1.0,
215: -1.0,1.0,-1.0,1.0,
216: -1.0,0.0,1.0,-1.0,1.0,-1.0,0.0,1.0};
217: PetscReal s2[20] = {-1.0,-1.0, -1.0,0.0,0.0,1.0, 1.0, 1.0,
218: -1.0,-1.0,1.0,1.0,
219: -1.0,-1.0, -1.0,0.0,0.0,1.0, 1.0, 1.0};
220: PetscReal t2[20] = {-1.0,-1.0,-1.0,-1.0,-1.0,-1.0,-1.0,-1.0,
221: 0.0,0.0,0.0,0.0,
222: 1.0,1.0,1.0,1.0,1.0,1.0,1.0,1.0};
223: PetscInt rmap[20] = {0,1,2,3,5,6,7,8,9,11,15,17,18,19,20,21,23,24,25,26};
224: /* -------------------------------------------------------------------- */
227: /*
228: Elastic20Stiff - Forms 20 node elastic stiffness for element.
229: */
230: PetscErrorCode Elastic20Stiff(PetscReal **Ke)
231: {
232: PetscReal K[60][60],x,y,z,dx,dy,dz,m,v;
233: PetscInt i,j,k,l,Ii,J;
235: paulsetup20();
237: x = -1.0; y = -1.0; z = -1.0; dx = 2.0; dy = 2.0; dz = 2.0;
238: xyz[0][0] = x; xyz[0][1] = y; xyz[0][2] = z;
239: xyz[1][0] = x + dx; xyz[1][1] = y; xyz[1][2] = z;
240: xyz[2][0] = x + 2.*dx; xyz[2][1] = y; xyz[2][2] = z;
241: xyz[3][0] = x; xyz[3][1] = y + dy; xyz[3][2] = z;
242: xyz[4][0] = x + 2.*dx; xyz[4][1] = y + dy; xyz[4][2] = z;
243: xyz[5][0] = x; xyz[5][1] = y + 2.*dy; xyz[5][2] = z;
244: xyz[6][0] = x + dx; xyz[6][1] = y + 2.*dy; xyz[6][2] = z;
245: xyz[7][0] = x + 2.*dx; xyz[7][1] = y + 2.*dy; xyz[7][2] = z;
246: xyz[8][0] = x; xyz[8][1] = y; xyz[8][2] = z + dz;
247: xyz[9][0] = x + 2.*dx; xyz[9][1] = y; xyz[9][2] = z + dz;
248: xyz[10][0] = x; xyz[10][1] = y + 2.*dy; xyz[10][2] = z + dz;
249: xyz[11][0] = x + 2.*dx; xyz[11][1] = y + 2.*dy; xyz[11][2] = z + dz;
250: xyz[12][0] = x; xyz[12][1] = y; xyz[12][2] = z + 2.*dz;
251: xyz[13][0] = x + dx; xyz[13][1] = y; xyz[13][2] = z + 2.*dz;
252: xyz[14][0] = x + 2.*dx; xyz[14][1] = y; xyz[14][2] = z + 2.*dz;
253: xyz[15][0] = x; xyz[15][1] = y + dy; xyz[15][2] = z + 2.*dz;
254: xyz[16][0] = x + 2.*dx; xyz[16][1] = y + dy; xyz[16][2] = z + 2.*dz;
255: xyz[17][0] = x; xyz[17][1] = y + 2.*dy; xyz[17][2] = z + 2.*dz;
256: xyz[18][0] = x + dx; xyz[18][1] = y + 2.*dy; xyz[18][2] = z + 2.*dz;
257: xyz[19][0] = x + 2.*dx; xyz[19][1] = y + 2.*dy; xyz[19][2] = z + 2.*dz;
258: paulintegrate20(K);
260: /* copy the stiffness from K into format used by Ke */
261: for (i=0; i<81; i++) {
262: for (j=0; j<81; j++) {
263: Ke[i][j] = 0.0;
264: }
265: }
266: Ii = 0;
267: m = 0.0;
268: for (i=0; i<20; i++) {
269: J = 0;
270: for (j=0; j<20; j++) {
271: for (k=0; k<3; k++) {
272: for (l=0; l<3; l++) {
273: Ke[3*rmap[i]+k][3*rmap[j]+l] = v = K[Ii+k][J+l];
275: m = PetscMax(m,PetscAbsReal(v));
276: }
277: }
278: J += 3;
279: }
280: Ii += 3;
281: }
282: /* zero out the extremely small values */
283: m = (1.e-8)*m;
284: for (i=0; i<81; i++) {
285: for (j=0; j<81; j++) {
286: if (PetscAbsReal(Ke[i][j]) < m) Ke[i][j] = 0.0;
287: }
288: }
289: /* force the matrix to be exactly symmetric */
290: for (i=0; i<81; i++) {
291: for (j=0; j<i; j++) {
292: Ke[i][j] = (Ke[i][j] + Ke[j][i])/2.0;
293: }
294: }
295: return 0;
296: }
297: /* -------------------------------------------------------------------- */
300: /*
301: paulsetup20 - Sets up data structure for forming local elastic stiffness.
302: */
303: PetscErrorCode paulsetup20(void)
304: {
305: PetscInt i,j,k,cnt;
306: PetscReal x[4],w[4];
307: PetscReal c;
309: n_int = 3;
310: nu = 0.3;
311: E = 1.0;
313: /* Assign integration points and weights for
314: Gaussian quadrature formulae. */
315: if (n_int == 2) {
316: x[0] = (-0.577350269189626);
317: x[1] = (0.577350269189626);
318: w[0] = 1.0000000;
319: w[1] = 1.0000000;
320: } else if (n_int == 3) {
321: x[0] = (-0.774596669241483);
322: x[1] = 0.0000000;
323: x[2] = 0.774596669241483;
324: w[0] = 0.555555555555555;
325: w[1] = 0.888888888888888;
326: w[2] = 0.555555555555555;
327: } else if (n_int == 4) {
328: x[0] = (-0.861136311594053);
329: x[1] = (-0.339981043584856);
330: x[2] = 0.339981043584856;
331: x[3] = 0.861136311594053;
332: w[0] = 0.347854845137454;
333: w[1] = 0.652145154862546;
334: w[2] = 0.652145154862546;
335: w[3] = 0.347854845137454;
336: } else SETERRQ(PETSC_COMM_SELF,1,"Unknown value for n_int");
338: /* rst[][i] contains the location of the i-th integration point
339: in the canonical (r,s,t) coordinate system. weight[i] contains
340: the Gaussian weighting factor. */
342: cnt = 0;
343: for (i=0; i<n_int; i++) {
344: for (j=0; j<n_int; j++) {
345: for (k=0; k<n_int; k++) {
346: rst[0][cnt] =x[i];
347: rst[1][cnt] =x[j];
348: rst[2][cnt] =x[k];
349: weight[cnt] = w[i]*w[j]*w[k];
350: ++cnt;
351: }
352: }
353: }
354: N_int = cnt;
356: /* N[][j] is the interpolation vector, N[][j] .* xyz[] */
357: /* yields the (x,y,z) locations of the integration point. */
358: /* part_N[][][j] is the partials of the N function */
359: /* w.r.t. (r,s,t). */
361: c = 1.0/8.0;
362: for (j=0; j<N_int; j++) {
363: for (i=0; i<20; i++) {
364: if (i==0 || i==2 || i==5 || i==7 || i==12 || i==14 || i== 17 || i==19) {
365: N[i][j] = c*(1.0 + r2[i]*rst[0][j])*
366: (1.0 + s2[i]*rst[1][j])*(1.0 + t2[i]*rst[2][j])*
367: (-2.0 + r2[i]*rst[0][j] + s2[i]*rst[1][j] + t2[i]*rst[2][j]);
368: part_N[0][i][j] = c*r2[i]*(1 + s2[i]*rst[1][j])*(1 + t2[i]*rst[2][j])*
369: (-1.0 + 2.0*r2[i]*rst[0][j] + s2[i]*rst[1][j] +
370: t2[i]*rst[2][j]);
371: part_N[1][i][j] = c*s2[i]*(1 + r2[i]*rst[0][j])*(1 + t2[i]*rst[2][j])*
372: (-1.0 + r2[i]*rst[0][j] + 2.0*s2[i]*rst[1][j] +
373: t2[i]*rst[2][j]);
374: part_N[2][i][j] = c*t2[i]*(1 + r2[i]*rst[0][j])*(1 + s2[i]*rst[1][j])*
375: (-1.0 + r2[i]*rst[0][j] + s2[i]*rst[1][j] +
376: 2.0*t2[i]*rst[2][j]);
377: } else if (i==1 || i==6 || i==13 || i==18) {
378: N[i][j] = .25*(1.0 - rst[0][j]*rst[0][j])*
379: (1.0 + s2[i]*rst[1][j])*(1.0 + t2[i]*rst[2][j]);
380: part_N[0][i][j] = -.5*rst[0][j]*(1 + s2[i]*rst[1][j])*
381: (1 + t2[i]*rst[2][j]);
382: part_N[1][i][j] = .25*s2[i]*(1 + t2[i]*rst[2][j])*
383: (1.0 - rst[0][j]*rst[0][j]);
384: part_N[2][i][j] = .25*t2[i]*(1.0 - rst[0][j]*rst[0][j])*
385: (1 + s2[i]*rst[1][j]);
386: } else if (i==3 || i==4 || i==15 || i==16) {
387: N[i][j] = .25*(1.0 - rst[1][j]*rst[1][j])*
388: (1.0 + r2[i]*rst[0][j])*(1.0 + t2[i]*rst[2][j]);
389: part_N[0][i][j] = .25*r2[i]*(1 + t2[i]*rst[2][j])*
390: (1.0 - rst[1][j]*rst[1][j]);
391: part_N[1][i][j] = -.5*rst[1][j]*(1 + r2[i]*rst[0][j])*
392: (1 + t2[i]*rst[2][j]);
393: part_N[2][i][j] = .25*t2[i]*(1.0 - rst[1][j]*rst[1][j])*
394: (1 + r2[i]*rst[0][j]);
395: } else if (i==8 || i==9 || i==10 || i==11) {
396: N[i][j] = .25*(1.0 - rst[2][j]*rst[2][j])*
397: (1.0 + r2[i]*rst[0][j])*(1.0 + s2[i]*rst[1][j]);
398: part_N[0][i][j] = .25*r2[i]*(1 + s2[i]*rst[1][j])*
399: (1.0 - rst[2][j]*rst[2][j]);
400: part_N[1][i][j] = .25*s2[i]*(1.0 - rst[2][j]*rst[2][j])*
401: (1 + r2[i]*rst[0][j]);
402: part_N[2][i][j] = -.5*rst[2][j]*(1 + r2[i]*rst[0][j])*
403: (1 + s2[i]*rst[1][j]);
404: }
405: }
406: }
407: return 0;
408: }
409: /* -------------------------------------------------------------------- */
412: /*
413: paulintegrate20 - Does actual numerical integration on 20 node element.
414: */
415: PetscErrorCode paulintegrate20(PetscReal K[60][60])
416: {
417: PetscReal det_jac,jac[3][3],inv_jac[3][3];
418: PetscReal B[6][60],B_temp[6][60],C[6][6];
419: PetscReal temp;
420: PetscInt i,j,k,step;
422: /* Zero out K, since we will accumulate the result here */
423: for (i=0; i<60; i++) {
424: for (j=0; j<60; j++) {
425: K[i][j] = 0.0;
426: }
427: }
429: /* Loop over integration points ... */
430: for (step=0; step<N_int; step++) {
432: /* Compute the Jacobian, its determinant, and inverse. */
433: for (i=0; i<3; i++) {
434: for (j=0; j<3; j++) {
435: jac[i][j] = 0;
436: for (k=0; k<20; k++) {
437: jac[i][j] += part_N[i][k][step]*xyz[k][j];
438: }
439: }
440: }
441: det_jac = jac[0][0]*(jac[1][1]*jac[2][2]-jac[1][2]*jac[2][1])
442: + jac[0][1]*(jac[1][2]*jac[2][0]-jac[1][0]*jac[2][2])
443: + jac[0][2]*(jac[1][0]*jac[2][1]-jac[1][1]*jac[2][0]);
444: inv_jac[0][0] = (jac[1][1]*jac[2][2]-jac[1][2]*jac[2][1])/det_jac;
445: inv_jac[0][1] = (jac[0][2]*jac[2][1]-jac[0][1]*jac[2][2])/det_jac;
446: inv_jac[0][2] = (jac[0][1]*jac[1][2]-jac[1][1]*jac[0][2])/det_jac;
447: inv_jac[1][0] = (jac[1][2]*jac[2][0]-jac[1][0]*jac[2][2])/det_jac;
448: inv_jac[1][1] = (jac[0][0]*jac[2][2]-jac[2][0]*jac[0][2])/det_jac;
449: inv_jac[1][2] = (jac[0][2]*jac[1][0]-jac[0][0]*jac[1][2])/det_jac;
450: inv_jac[2][0] = (jac[1][0]*jac[2][1]-jac[1][1]*jac[2][0])/det_jac;
451: inv_jac[2][1] = (jac[0][1]*jac[2][0]-jac[0][0]*jac[2][1])/det_jac;
452: inv_jac[2][2] = (jac[0][0]*jac[1][1]-jac[1][0]*jac[0][1])/det_jac;
454: /* Compute the B matrix. */
455: for (i=0; i<3; i++) {
456: for (j=0; j<20; j++) {
457: B_temp[i][j] = 0.0;
458: for (k=0; k<3; k++) {
459: B_temp[i][j] += inv_jac[i][k]*part_N[k][j][step];
460: }
461: }
462: }
463: for (i=0; i<6; i++) {
464: for (j=0; j<60; j++) {
465: B[i][j] = 0.0;
466: }
467: }
469: /* Put values in correct places in B. */
470: for (k=0; k<20; k++) {
471: B[0][3*k] = B_temp[0][k];
472: B[1][3*k+1] = B_temp[1][k];
473: B[2][3*k+2] = B_temp[2][k];
474: B[3][3*k] = B_temp[1][k];
475: B[3][3*k+1] = B_temp[0][k];
476: B[4][3*k+1] = B_temp[2][k];
477: B[4][3*k+2] = B_temp[1][k];
478: B[5][3*k] = B_temp[2][k];
479: B[5][3*k+2] = B_temp[0][k];
480: }
482: /* Construct the C matrix, uses the constants "nu" and "E". */
483: for (i=0; i<6; i++) {
484: for (j=0; j<6; j++) {
485: C[i][j] = 0.0;
486: }
487: }
488: temp = (1.0 + nu)*(1.0 - 2.0*nu);
489: temp = E/temp;
490: C[0][0] = temp*(1.0 - nu);
491: C[1][1] = C[0][0];
492: C[2][2] = C[0][0];
493: C[3][3] = temp*(0.5 - nu);
494: C[4][4] = C[3][3];
495: C[5][5] = C[3][3];
496: C[0][1] = temp*nu;
497: C[0][2] = C[0][1];
498: C[1][0] = C[0][1];
499: C[1][2] = C[0][1];
500: C[2][0] = C[0][1];
501: C[2][1] = C[0][1];
503: for (i=0; i<6; i++) {
504: for (j=0; j<60; j++) {
505: B_temp[i][j] = 0.0;
506: for (k=0; k<6; k++) {
507: B_temp[i][j] += C[i][k]*B[k][j];
508: }
509: B_temp[i][j] *= det_jac;
510: }
511: }
513: /* Accumulate B'*C*B*det(J)*weight, as a function of (r,s,t), in K. */
514: for (i=0; i<60; i++) {
515: for (j=0; j<60; j++) {
516: temp = 0.0;
517: for (k=0; k<6; k++) {
518: temp += B[k][i]*B_temp[k][j];
519: }
520: K[i][j] += temp*weight[step];
521: }
522: }
523: } /* end of loop over integration points */
524: return 0;
525: }