Mesh Oriented datABase  (version 5.5.1)
An array-based unstructured mesh library
findpt.c
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1 #include <stdio.h>
2 #include <stdlib.h>
3 #include <stdarg.h>
4 #include <math.h> /* for cos, fabs */
5 #include <float.h>
6 #include <string.h> /* for memcpy */
7 
8 #include "moab/FindPtFuncs.h"
9 
10 const unsigned opt_no_constraints_2 = 3 + 1;
11 const unsigned opt_no_constraints_3 = 9 + 3 + 1;
12 
13 /*--------------------------------------------------------------------------
14  Lobatto Polynomial Bounds
15 
16  Needed inputs are the Gauss-Lobatto quadrature nodes and weights:
17  unsigned nr = ..., ns = ...;
18  realType zr[nr], wr[nr];
19  realType zs[ns], ws[ns];
20 
21  lobatto_nodes(zr,nr); lobatto_weights(zr,wr,nr);
22  lobatto_nodes(zs,ns); lobatto_weights(zs,ws,ns);
23 
24  The number of points in the constructed piecewise (bi-)linear bounds
25  is a parameter; more points give tighter bounds
26 
27  unsigned mr = 2*nr, ms = 2*ns;
28 
29  The necessary setup is accomplished via:
30  lob_bnd_base b_data_r;
31  lob_bnd_ext e_data_s;
32 
33  lob_bnd_base_alloc(&b_data_r,nr,mr);
34  lob_bnd_base_setup(&b_data_r,zr,wr);
35  lob_bnd_ext_alloc(&e_data_s,ns,ms);
36  lob_bnd_ext_setup(&e_data_s,zs,ws);
37 
38  Bounds may then be computed via:
39  realType work1r[2*mr], work1s[2*ms], work2[2*mr + 2*mr*ns + 2*mr*ms];
40  realType ur[nr], us[ns]; // 1-d polynomials on the zr[] and zs[] nodes
41  realType u[ns][nr]; // 2-d polynomial on zr[] (x) zs[]
42  realType bound[2]; // = { min, max } (to be computed)
43 
44  lob_bnd_1(&b_data_r ,ur,bound,work1r); // compute bounds on ur
45  lob_bnd_1(&e_data_s.b,us,bound,work1s); // compute bounds on us
46  lob_bnd_2(&b_data_r, &e_data_s,
47  (const double*)&u[0][0],bound,work2); // compute bounds on u
48  The above routines access the zr,zs arrays passed to *_setup
49  (so do not delete them between calls)
50 
51  Memory allocated in *_setup is freed with
52  lob_bnd_base_free(&b_data_r);
53  lob_bnd_ext_free(&e_data_s);
54 
55  --------------------------------------------------------------------------*/
56 
57 typedef struct
58 {
59  unsigned n; /* number of Lobatto nodes in input */
60  unsigned m; /* number of Chebyshev nodes used to calculate bounds */
61  realType *Q0, *Q1; /* Q0[n], Q1[n] -- first two rows of change of basis matrix
62  from Lobatto node Lagrangian to Legendre */
63  const realType* z; /* z[n] -- external; Lobatto nodes */
64  realType* h; /* h[m] -- Chebyshev nodes */
65  realType *uv, *ov; /* uv[n][m], ov[n][m] --
66  uv[j][:] is a piecewise linear function in the nodal
67  basis with nodes h[m] that is everywhere less
68  than or equal to the jth Lagrangian basis
69  function (with the Lobatto nodes z[n])
70  ov[j][:] is everywhere greater than or equal */
71 } lob_bnd_base;
72 
73 typedef struct
74 {
76  realType *uvp, *uvn, *ovp, *ovn; /* [n][m] -- uv and ov split into
77  positive and negative parts */
78 } lob_bnd_ext;
79 
80 static void lob_bnd_base_alloc( lob_bnd_base* p, unsigned n, unsigned m )
81 {
82  p->n = n, p->m = m;
83  p->Q0 = tmalloc( realType, 2 * n + m + 2 * n * m );
84  p->Q1 = p->Q0 + n;
85  p->h = p->Q1 + n;
86  p->uv = p->h + m;
87  p->ov = p->uv + n * m;
88 }
89 
91 {
92  free( p->Q0 );
93 }
94 
95 static void lob_bnd_ext_alloc( lob_bnd_ext* p, unsigned n, unsigned m )
96 {
97  p->b.n = n, p->b.m = m;
98  p->b.Q0 = tmalloc( realType, 2 * n + m + 6 * n * m );
99  p->b.Q1 = p->b.Q0 + n;
100  p->b.h = p->b.Q1 + n;
101  p->b.uv = p->b.h + m;
102  p->b.ov = p->b.uv + n * m;
103  p->uvp = p->b.ov + n * m;
104  p->uvn = p->uvp + n * m;
105  p->ovp = p->uvn + n * m;
106  p->ovn = p->ovp + n * m;
107 }
108 
109 static void lob_bnd_ext_free( lob_bnd_ext* p )
110 {
111  free( p->b.Q0 );
112 }
113 
114 static void lob_bnd_base_setup( lob_bnd_base* p, const realType* z, const realType* w )
115 {
116  unsigned i, j, m = p->m, n = p->n, mm = 2 * m - 1;
117  realType *q = tmalloc( realType, ( 2 * n + 1 ) * mm + 6 * n ), *J = q + mm, *D = J + n * mm, *work = D + n * mm;
118  p->z = z;
119  for( i = 0; i < n; ++i )
120  p->Q0[i] = w[i] / 2, p->Q1[i] = 3 * p->Q0[i] * z[i];
121  p->h[0] = -1, p->h[m - 1] = 1;
122  for( j = 1; j < m - 1; ++j )
123  p->h[j] = mbcos( ( m - j - 1 ) * MOAB_POLY_PI / ( m - 1 ) );
124  for( j = 0; j < m - 1; ++j )
125  q[2 * j] = p->h[j], q[2 * j + 1] = ( p->h[j] + p->h[j + 1] ) / 2;
126  q[mm - 1] = p->h[m - 1];
127  lagrange_weights_deriv( z, n, q, mm, J, D, work );
128  for( i = 0; i < n; ++i )
129  {
130  realType *uv = p->uv + i * m, *ov = p->ov + i * m;
131  ov[0] = uv[0] = J[i];
132  ov[m - 1] = uv[m - 1] = J[( mm - 1 ) * n + i];
133  for( j = 1; j < m - 1; ++j )
134  {
135  unsigned jj = 2 * j;
136  realType c0 = J[( jj - 1 ) * n + i] + ( q[jj] - q[jj - 1] ) * D[( jj - 1 ) * n + i];
137  realType c1 = J[( jj + 1 ) * n + i] + ( q[jj] - q[jj + 1] ) * D[( jj + 1 ) * n + i];
138  realType c2 = J[jj * n + i];
139  rminmax_3( &uv[j], &ov[j], c0, c1, c2 );
140  }
141  }
142  free( q );
143 }
144 
145 static void lob_bnd_ext_setup( lob_bnd_ext* p, const realType* z, const realType* w )
146 {
147  unsigned i, mn = p->b.m * p->b.n;
148  lob_bnd_base_setup( &p->b, z, w );
149  for( i = 0; i < mn; ++i )
150  {
151  realType uvi = p->b.uv[i], ovi = p->b.ov[i];
152  p->uvp[i] = p->uvn[i] = p->ovp[i] = p->ovn[i] = 0;
153  if( uvi > 0 )
154  p->uvp[i] = uvi;
155  else
156  p->uvn[i] = uvi;
157  if( ovi > 0 )
158  p->ovp[i] = ovi;
159  else
160  p->ovn[i] = ovi;
161  }
162 }
163 
164 static void lob_bnd_lines( const lob_bnd_base* p, const realType* u, realType* a, realType* b )
165 {
166  unsigned i, j;
167  realType a0 = 0, a1 = 0;
168  const realType *uv = p->uv, *ov = p->ov;
169  for( i = 0; i < p->n; ++i )
170  a0 += p->Q0[i] * u[i], a1 += p->Q1[i] * u[i];
171  for( j = 0; j < p->m; ++j )
172  b[j] = a[j] = a0 + a1 * p->h[j];
173  for( i = 0; i < p->n; ++i )
174  {
175  realType w = u[i] - ( a0 + a1 * p->z[i] );
176  if( w >= 0 )
177  for( j = 0; j < p->m; ++j )
178  a[j] += w * ( *uv++ ), b[j] += w * ( *ov++ );
179  else
180  for( j = 0; j < p->m; ++j )
181  a[j] += w * ( *ov++ ), b[j] += w * ( *uv++ );
182  }
183 }
184 
185 /* work holds p->m * 2 doubles */
186 static void lob_bnd_1( const lob_bnd_base* p, const realType* u, realType bnd[2], realType* work )
187 {
188  unsigned j;
189  realType *a = work, *b = work + p->m;
190  lob_bnd_lines( p, u, a, b );
191  bnd[0] = a[0], bnd[1] = b[0];
192  for( j = 1; j < p->m; ++j )
193  {
194  if( a[j] < bnd[0] ) bnd[0] = a[j];
195  if( b[j] > bnd[1] ) bnd[1] = b[j];
196  }
197 }
198 
199 /* work holds 2*mr + 2*mr*ns + 2*mr*ms doubles */
200 static void lob_bnd_2( const lob_bnd_base* pr,
201  const lob_bnd_ext* ps,
202  const realType* u,
203  realType bnd[2],
204  realType* work )
205 {
206  unsigned nr = pr->n, mr = pr->m, ns = ps->b.n, ms = ps->b.m;
207  realType *a0 = work, *a1 = a0 + mr, *ar_ = a1 + mr, *ar = ar_, *br_ = ar + mr * ns, *br = br_, *a_ = br + mr * ns,
208  *a = a_, *b_ = a + mr * ms, *b = b_, *uvp, *ovp, *uvn, *ovn;
209  realType b0, b1;
210  unsigned i, j, k;
211  for( i = 0; i < mr; ++i )
212  a0[i] = a1[i] = 0;
213  for( j = 0; j < ns; ++j, u += nr )
214  {
215  realType q0 = ps->b.Q0[j], q1 = ps->b.Q1[j];
216  lob_bnd_lines( pr, u, ar, br );
217  for( i = 0; i < mr; ++i )
218  {
219  realType t = ( *ar++ + *br++ ) / 2;
220  a0[i] += q0 * t, a1[i] += q1 * t;
221  }
222  }
223  for( i = 0; i < mr; ++i )
224  {
225  realType a0i = a0[i], a1i = a1[i];
226  for( k = 0; k < ms; ++k )
227  *b++ = *a++ = a0i + a1i * ps->b.h[k];
228  }
229  ar = ar_, br = br_;
230  uvp = ps->uvp, ovp = ps->ovp, uvn = ps->uvn, ovn = ps->ovn;
231  for( j = 0; j < ns; ++j, uvp += ms, uvn += ms, ovp += ms, ovn += ms )
232  {
233  realType zj = ps->b.z[j];
234  a = a_, b = b_;
235  for( i = 0; i < mr; ++i )
236  {
237  realType t = a0[i] + a1[i] * zj;
238  realType uw = *ar++ - t;
239  realType ow = *br++ - t;
240  if( uw >= 0 ) /* 0 <= uw <= ow */
241  for( k = 0; k < ms; ++k )
242  *a++ += uw * uvp[k] + ow * uvn[k], *b++ += ow * ovp[k] + uw * ovn[k];
243  else if( ow <= 0 ) /* uw <= ow <= 0 */
244  for( k = 0; k < ms; ++k )
245  *a++ += uw * ovp[k] + ow * ovn[k], *b++ += ow * uvp[k] + uw * uvn[k];
246  else /* uw < 0 < ow */
247  for( k = 0; k < ms; ++k )
248  *a++ += uw * ovp[k] + ow * uvn[k], *b++ += ow * ovp[k] + uw * uvn[k];
249  }
250  }
251  b0 = a_[0], b1 = b_[0];
252  for( i = 1; i < mr * ms; ++i )
253  {
254  if( a_[i] < b0 ) b0 = a_[i];
255  if( b_[i] > b1 ) b1 = b_[i];
256  }
257  bnd[0] = b0, bnd[1] = b1;
258 }
259 
260 /*--------------------------------------------------------------------------
261  Small Matrix Inverse
262  --------------------------------------------------------------------------*/
263 
264 static void mat_inv_2( const realType A[4], realType inv[4] )
265 {
266  const realType idet = 1 / ( A[0] * A[3] - A[1] * A[2] );
267  inv[0] = idet * A[3];
268  inv[1] = -( idet * A[1] );
269  inv[2] = -( idet * A[2] );
270  inv[3] = idet * A[0];
271 }
272 
273 static void mat_inv_3( const realType A[9], realType inv[9] )
274 {
275  const realType a = A[4] * A[8] - A[5] * A[7], b = A[5] * A[6] - A[3] * A[8], c = A[3] * A[7] - A[4] * A[6],
276  idet = 1 / ( A[0] * a + A[1] * b + A[2] * c );
277  inv[0] = idet * a;
278  inv[1] = idet * ( A[2] * A[7] - A[1] * A[8] );
279  inv[2] = idet * ( A[1] * A[5] - A[2] * A[4] );
280  inv[3] = idet * b;
281  inv[4] = idet * ( A[0] * A[8] - A[2] * A[6] );
282  inv[5] = idet * ( A[2] * A[3] - A[0] * A[5] );
283  inv[6] = idet * c;
284  inv[7] = idet * ( A[1] * A[6] - A[0] * A[7] );
285  inv[8] = idet * ( A[0] * A[4] - A[1] * A[3] );
286 }
287 
288 static void mat_app_2r( realType y[2], const realType A[4], const realType x[2] )
289 {
290  y[0] = A[0] * x[0] + A[1] * x[1];
291  y[1] = A[2] * x[0] + A[3] * x[1];
292 }
293 
294 static void mat_app_2c( realType y[2], const realType A[4], const realType x[2] )
295 {
296  y[0] = A[0] * x[0] + A[2] * x[1];
297  y[1] = A[1] * x[0] + A[3] * x[1];
298 }
299 
300 static void mat_app_3r( realType y[3], const realType A[9], const realType x[3] )
301 {
302  y[0] = A[0] * x[0] + A[1] * x[1] + A[2] * x[2];
303  y[1] = A[3] * x[0] + A[4] * x[1] + A[5] * x[2];
304  y[2] = A[6] * x[0] + A[7] * x[1] + A[8] * x[2];
305 }
306 
307 static void mat_app_3c( realType y[3], const realType A[9], const realType x[3] )
308 {
309  y[0] = A[0] * x[0] + A[3] * x[1] + A[6] * x[2];
310  y[1] = A[1] * x[0] + A[4] * x[1] + A[7] * x[2];
311  y[2] = A[2] * x[0] + A[5] * x[1] + A[8] * x[2];
312 }
313 
314 static void tinyla_solve_2( realType x[2], const realType A[4], const realType b[2] )
315 {
316  realType inv[4];
317  mat_inv_2( A, inv );
318  mat_app_2r( x, inv, b );
319 }
320 
321 static void tinyla_solve_3( realType x[3], const realType A[9], const realType b[3] )
322 {
323  realType inv[9];
324  mat_inv_3( A, inv );
325  mat_app_3r( x, inv, b );
326 }
327 
328 /* solve
329  A[0] x0 + A[2] x1 = b0,
330  A[2] x0 + A[1] x1 = b1
331 */
332 static void tinyla_solve_sym_2( realType* x0, realType* x1, const realType A[3], realType b0, realType b1 )
333 {
334  const realType idet = 1 / ( A[0] * A[1] - A[2] * A[2] );
335  *x0 = idet * ( A[1] * b0 - A[2] * b1 );
336  *x1 = idet * ( A[0] * b1 - A[2] * b0 );
337 }
338 
339 /*--------------------------------------------------------------------------
340  Oriented Bounding Box
341 
342  Suffixes on names are _2 for 2-d and _3 for 3-d
343 
344  Needed inputs are the Gauss-Lobatto quadrature nodes and weights:
345  unsigned nr = ..., ns = ...;
346  realType zr[nr], wr[nr];
347  realType zs[ns], ws[ns];
348 
349  lobatto_nodes(zr,nr); lobatto_weights(zr,wr,nr);
350  lobatto_nodes(zs,ns); lobatto_weights(zs,ws,ns);
351 
352  The number of points in the constructed piecewise (bi-)linear bounds
353  for the boundaries is a parameter; more points give tighter bounds
354 
355  unsigned mr = 2*nr, ms = 2*ns;
356 
357  Bounding boxes are increased by a relative amount as a parameter
358 
359  realType tol = 0.01;
360 
361  Setup is accomplished via:
362 
363  const realType *z[2] = {zr,zs}, *w[2] = {wr,ws};
364  const unsigned n[2] = {nr,ns}, m[2] = {mr,ms};
365  obbox_data_2 *data = obbox_setup_2(z,w,n,m);
366 
367  Bounding box data may then be computed:
368 
369  obbox_2 box; // will store bounding box information
370  realType xm[ns][nr], ym[ns][nr]; // x, y coordinates of the element nodes
371 
372  obbox_calc_2(data, tol, (const realType *)&xm[0][0],
373  (const realType *)&ym[0][0], &box);
374 
375  A point may be tested:
376 
377  const realType x[2]; // point to test
378  realType r[2];
379 
380  if( obbox_axis_test_2(&box, x) )
381  ... // x failed axis-aligned bounding box test
382 
383  if( obbox_test_2(&box, x, r) )
384  ... // x failed oriented bounding box test
385  else
386  ... // r suitable as initial guess for parametric coords
387 
388  Once all bounding box information has been computed
389 
390  obbox_free_2(data);
391 
392  to free the memory allocated with obbox_setup_2.
393 
394  --------------------------------------------------------------------------*/
395 
396 typedef struct
397 {
399  realType *Jr0, *Dr0, *Js0, *Ds0, *work;
400 } obbox_data_2;
401 
402 typedef struct
403 {
406  realType *Jr0, *Dr0, *Js0, *Ds0, *Jt0, *Dt0, *work;
407 } obbox_data_3;
408 
409 static void obbox_data_alloc_2( obbox_data_2* p, const unsigned n[2], const unsigned m[2] )
410 {
411  const unsigned max_npm = umax_2( n[0] + m[0], n[1] + m[1] );
412  lob_bnd_base_alloc( &p->dr, n[0], m[0] );
413  lob_bnd_base_alloc( &p->ds, n[1], m[1] );
414  p->Jr0 = tmalloc( realType, 2 * n[0] + 2 * n[1] + 2 * max_npm );
415  p->Dr0 = p->Jr0 + n[0];
416  p->Js0 = p->Dr0 + n[0];
417  p->Ds0 = p->Js0 + n[1];
418  p->work = p->Ds0 + n[1];
419 }
420 
422 {
423  lob_bnd_base_free( &p->dr );
424  lob_bnd_base_free( &p->ds );
425  free( p->Jr0 );
426 }
427 
428 static void obbox_data_alloc_3( obbox_data_3* p, const unsigned n[3], const unsigned m[3] )
429 {
430  const unsigned wk1 = 3 * n[0] * n[1] + 2 * m[0] + 2 * m[0] * n[1] + 2 * m[0] * m[1];
431  const unsigned wk2 = 3 * n[0] * n[2] + 2 * m[0] + 2 * m[0] * n[2] + 2 * m[0] * m[2];
432  const unsigned wk3 = 3 * n[1] * n[2] + 2 * m[1] + 2 * m[1] * n[2] + 2 * m[1] * m[2];
433  const unsigned wk_max = umax_3( wk1, wk2, wk3 );
434  lob_bnd_base_alloc( &p->dr, n[0], m[0] );
435  lob_bnd_ext_alloc( &p->ds, n[1], m[1] );
436  lob_bnd_ext_alloc( &p->dt, n[2], m[2] );
437  p->Jr0 = tmalloc( realType, 2 * n[0] + 2 * n[1] + 2 * n[2] + wk_max );
438  p->Dr0 = p->Jr0 + n[0];
439  p->Js0 = p->Dr0 + n[0];
440  p->Ds0 = p->Js0 + n[1];
441  p->Jt0 = p->Ds0 + n[1];
442  p->Dt0 = p->Jt0 + n[2];
443  p->work = p->Dt0 + n[2];
444 }
445 
447 {
448  lob_bnd_base_free( &p->dr );
449  lob_bnd_ext_free( &p->ds );
450  lob_bnd_ext_free( &p->dt );
451  free( p->Jr0 );
452 }
453 
454 static obbox_data_2* obbox_setup_2( const realType* const z[2],
455  const realType* const w[2],
456  const unsigned n[2],
457  const unsigned m[2] )
458 {
459  const realType zero = 0;
460  realType* work;
461  obbox_data_2* p = tmalloc( obbox_data_2, 1 );
462  obbox_data_alloc_2( p, n, m );
463  lob_bnd_base_setup( &p->dr, z[0], w[0] );
464  lob_bnd_base_setup( &p->ds, z[1], w[1] );
465  work = tmalloc( realType, 6 * umax_2( n[0], n[1] ) );
466  lagrange_weights_deriv( z[0], n[0], &zero, 1, p->Jr0, p->Dr0, work );
467  lagrange_weights_deriv( z[1], n[1], &zero, 1, p->Js0, p->Ds0, work );
468  free( work );
469  return p;
470 }
471 
472 static obbox_data_3* obbox_setup_3( const realType* const z[3],
473  const realType* const w[3],
474  const unsigned n[3],
475  const unsigned m[3] )
476 {
477  const realType zero = 0;
478  realType* work;
479  obbox_data_3* p = tmalloc( obbox_data_3, 1 );
480  obbox_data_alloc_3( p, n, m );
481  lob_bnd_base_setup( &p->dr, z[0], w[0] );
482  lob_bnd_ext_setup( &p->ds, z[1], w[1] );
483  lob_bnd_ext_setup( &p->dt, z[2], w[2] );
484  work = tmalloc( realType, 6 * umax_3( n[0], n[1], n[2] ) );
485  lagrange_weights_deriv( z[0], n[0], &zero, 1, p->Jr0, p->Dr0, work );
486  lagrange_weights_deriv( z[1], n[1], &zero, 1, p->Js0, p->Ds0, work );
487  lagrange_weights_deriv( z[2], n[2], &zero, 1, p->Jt0, p->Dt0, work );
488  free( work );
489  return p;
490 }
491 
492 static void obbox_free_2( obbox_data_2* p )
493 {
494  obbox_data_free_2( p );
495  free( p );
496 }
497 
498 static void obbox_free_3( obbox_data_3* p )
499 {
500  obbox_data_free_3( p );
501  free( p );
502 }
503 
504 int obbox_axis_test_2( const obbox_2* p, const realType x[2] )
505 {
506  return ( x[0] < p->axis_bnd[0] || x[0] > p->axis_bnd[1] || x[1] < p->axis_bnd[2] || x[1] > p->axis_bnd[3] );
507 }
508 
509 int obbox_axis_test_3( const obbox_3* p, const realType x[3] )
510 {
511  return ( x[0] < p->axis_bnd[0] || x[0] > p->axis_bnd[1] || x[1] < p->axis_bnd[2] || x[1] > p->axis_bnd[3] ||
512  x[2] < p->axis_bnd[4] || x[2] > p->axis_bnd[5] );
513 }
514 
515 int obbox_test_2( const obbox_2* p, const realType x[2], realType r[2] )
516 {
517  realType xt[2];
518  xt[0] = x[0] - p->x[0];
519  xt[1] = x[1] - p->x[1];
520  r[0] = p->A[0] * xt[0] + p->A[1] * xt[1];
521  if( mbabs( r[0] ) > 1 ) return 1;
522  r[1] = p->A[2] * xt[0] + p->A[3] * xt[1];
523  return mbabs( r[1] ) > 1;
524 }
525 
526 int obbox_test_3( const obbox_3* p, const realType x[3], realType r[3] )
527 {
528  realType xt[3];
529  xt[0] = x[0] - p->x[0];
530  xt[1] = x[1] - p->x[1];
531  xt[2] = x[2] - p->x[2];
532  r[0] = p->A[0] * xt[0] + p->A[1] * xt[1] + p->A[2] * xt[2];
533  if( mbabs( r[0] ) > 1 ) return 1;
534  r[1] = p->A[3] * xt[0] + p->A[4] * xt[1] + p->A[5] * xt[2];
535  if( mbabs( r[1] ) > 1 ) return 1;
536  r[2] = p->A[6] * xt[0] + p->A[7] * xt[1] + p->A[8] * xt[2];
537  return mbabs( r[2] ) > 1;
538 }
539 
540 void obbox_calc_tfm_2( const realType* x,
541  const realType* y,
542  unsigned n,
543  unsigned s,
544  const realType c0[2],
545  const realType A[4],
546  realType* u )
547 {
548  unsigned i;
549  realType* v = u + n;
550  for( i = 0; i < n; ++i, x += s, y += s )
551  {
552  const realType xt = *x - c0[0], yt = *y - c0[1];
553  u[i] = A[0] * xt + A[1] * yt;
554  v[i] = A[2] * xt + A[3] * yt;
555  }
556 }
557 
558 void obbox_calc_tfm_3( const realType* x,
559  const realType* y,
560  const realType* z,
561  unsigned nr,
562  unsigned sr,
563  unsigned ns,
564  unsigned ss,
565  const realType c0[3],
566  const realType A[9],
567  realType* u )
568 {
569  unsigned i, j;
570  realType *v = u + nr * ns, *w = v + nr * ns;
571  for( j = 0; j < ns; ++j, x += ss, y += ss, z += ss )
572  {
573  for( i = 0; i < nr; ++i, x += sr, y += sr, z += sr )
574  {
575  const realType xt = *x - c0[0], yt = *y - c0[1], zt = *z - c0[2];
576  *u++ = A[0] * xt + A[1] * yt + A[2] * zt;
577  *v++ = A[3] * xt + A[4] * yt + A[5] * zt;
578  *w++ = A[6] * xt + A[7] * yt + A[8] * zt;
579  }
580  }
581 }
582 
583 void obbox_merge_2( realType* b, const realType* ob )
584 {
585  if( ob[0] < b[0] ) b[0] = ob[0];
586  if( ob[1] > b[1] ) b[1] = ob[1];
587  if( ob[2] < b[2] ) b[2] = ob[2];
588  if( ob[3] > b[3] ) b[3] = ob[3];
589 }
590 
591 void obbox_merge_3( realType* b, const realType* ob )
592 {
593  if( ob[0] < b[0] ) b[0] = ob[0];
594  if( ob[1] > b[1] ) b[1] = ob[1];
595  if( ob[2] < b[2] ) b[2] = ob[2];
596  if( ob[3] > b[3] ) b[3] = ob[3];
597  if( ob[4] < b[4] ) b[4] = ob[4];
598  if( ob[5] > b[5] ) b[5] = ob[5];
599 }
600 
601 /* work holds 2*n + 2*m realTypes */
602 void obbox_side_2( const realType* x,
603  const realType* y,
604  unsigned n,
605  unsigned s,
606  const realType c0[2],
607  const realType A[4],
608  realType* work,
609  const lob_bnd_base* lbd,
610  realType bnd[4] )
611 {
612  obbox_calc_tfm_2( x, y, n, s, c0, A, work );
613  lob_bnd_1( lbd, work, bnd, work + 2 * n );
614  lob_bnd_1( lbd, work + n, bnd + 2, work + 2 * n );
615 }
616 
617 /* work holds 3*nr*ns + 2*mr + 2*mr*ns + 2*mr*ms realTypes */
618 void obbox_side_3( const realType* x,
619  const realType* y,
620  const realType* z,
621  unsigned nr,
622  unsigned sr,
623  unsigned ns,
624  unsigned ss,
625  const realType c0[3],
626  const realType A[9],
627  realType* work,
628  const lob_bnd_base* dr,
629  const lob_bnd_ext* ds,
630  realType bnd[6] )
631 {
632  obbox_calc_tfm_3( x, y, z, nr, sr, ns, ss, c0, A, work );
633  lob_bnd_2( dr, ds, work, bnd, work + 3 * nr * ns );
634  lob_bnd_2( dr, ds, work + nr * ns, bnd + 2, work + 3 * nr * ns );
635  lob_bnd_2( dr, ds, work + 2 * nr * ns, bnd + 4, work + 3 * nr * ns );
636 }
637 
638 /* return bounds on u = A (x - c0)
639  bnd[0] <= u_0 <= bnd[1]
640  bnd[2] <= u_1 <= bnd[3] */
641 void obbox_bnd_2( const obbox_data_2* p,
642  const realType* x,
643  const realType* y,
644  const realType c0[2],
645  const realType A[4],
646  realType bnd[4] )
647 {
648  unsigned i, nr = p->dr.n, ns = p->ds.n;
649  realType obnd[4];
650 
651  i = nr * ( ns - 1 );
652  obbox_side_2( x, y, nr, 1, c0, A, p->work, &p->dr, bnd );
653  obbox_side_2( x + i, y + i, nr, 1, c0, A, p->work, &p->dr, obnd );
654  obbox_merge_2( bnd, obnd );
655 
656  i = nr - 1;
657  obbox_side_2( x, y, ns, nr, c0, A, p->work, &p->ds, obnd );
658  obbox_merge_2( bnd, obnd );
659  obbox_side_2( x + i, y + i, nr, nr, c0, A, p->work, &p->ds, obnd );
660  obbox_merge_2( bnd, obnd );
661 }
662 
663 /* return bounds on u = A (x - c0)
664  bnd[0] <= u_0 <= bnd[1]
665  bnd[2] <= u_1 <= bnd[3]
666  bnd[4] <= u_2 <= bnd[5] */
667 void obbox_bnd_3( const obbox_data_3* p,
668  const realType* x,
669  const realType* y,
670  const realType* z,
671  const realType c0[3],
672  const realType A[9],
673  realType bnd[6] )
674 {
675  unsigned i, nr = p->dr.n, ns = p->ds.b.n, nt = p->dt.b.n;
676  realType obnd[6];
677 
678  i = nr * ns * ( nt - 1 );
679  obbox_side_3( x, y, z, nr, 1, ns, 0, c0, A, p->work, &p->dr, &p->ds, bnd );
680  obbox_side_3( x + i, y + i, z + i, nr, 1, ns, 0, c0, A, p->work, &p->dr, &p->ds, obnd );
681  obbox_merge_3( bnd, obnd );
682 
683  i = nr * ( ns - 1 );
684  obbox_side_3( x, y, z, nr, 1, nt, i, c0, A, p->work, &p->dr, &p->dt, obnd );
685  obbox_merge_3( bnd, obnd );
686  obbox_side_3( x + i, y + i, z + i, nr, 1, nt, i, c0, A, p->work, &p->dr, &p->dt, obnd );
687  obbox_merge_3( bnd, obnd );
688 
689  i = nr - 1;
690  obbox_side_3( x, y, z, ns, nr, nt, 0, c0, A, p->work, &p->ds.b, &p->dt, obnd );
691  obbox_merge_3( bnd, obnd );
692  obbox_side_3( x + i, y + i, z + i, ns, nr, nt, 0, c0, A, p->work, &p->ds.b, &p->dt, obnd );
693  obbox_merge_3( bnd, obnd );
694 }
695 
696 void obbox_calc_2( const obbox_data_2* p, realType tol, const realType* x, const realType* y, obbox_2* b )
697 {
698  const realType zero[2] = { 0, 0 }, id[4] = { 1, 0, 0, 1 };
699  realType c0[2], jac[4], inv[4], bnd[4], u0[2], d[2];
700 
701  obbox_bnd_2( p, x, y, zero, id, b->axis_bnd );
702  d[0] = b->axis_bnd[1] - b->axis_bnd[0];
703  d[1] = b->axis_bnd[3] - b->axis_bnd[2];
704  b->axis_bnd[0] -= tol * d[0], b->axis_bnd[1] += tol * d[0];
705  b->axis_bnd[2] -= tol * d[1], b->axis_bnd[3] += tol * d[1];
706 
707  c0[0] = tensor_ig2( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.n, x, jac, p->work );
708  c0[1] = tensor_ig2( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.n, y, jac + 2, p->work );
709  mat_inv_2( jac, inv );
710 
711  obbox_bnd_2( p, x, y, c0, inv, bnd );
712 
713  u0[0] = ( bnd[0] + bnd[1] ) / 2;
714  u0[1] = ( bnd[2] + bnd[3] ) / 2;
715  d[0] = 2 / ( ( 1 + tol ) * ( bnd[1] - bnd[0] ) );
716  d[1] = 2 / ( ( 1 + tol ) * ( bnd[3] - bnd[2] ) );
717  b->x[0] = c0[0] + jac[0] * u0[0] + jac[1] * u0[1];
718  b->x[1] = c0[1] + jac[2] * u0[0] + jac[3] * u0[1];
719  b->A[0] = d[0] * inv[0], b->A[1] = d[0] * inv[1];
720  b->A[2] = d[1] * inv[2], b->A[3] = d[1] * inv[3];
721 }
722 
723 void obbox_calc_3( const obbox_data_3* p,
724  realType tol,
725  const realType* x,
726  const realType* y,
727  const realType* z,
728  obbox_3* b )
729 {
730  const realType zero[3] = { 0, 0 }, id[9] = { 1, 0, 0, 0, 1, 0, 0, 0, 1 };
731  realType c0[3], jac[9], inv[9], bnd[6], u0[3], d[3];
732 
733  obbox_bnd_3( p, x, y, z, zero, id, b->axis_bnd );
734  d[0] = b->axis_bnd[1] - b->axis_bnd[0];
735  d[1] = b->axis_bnd[3] - b->axis_bnd[2];
736  d[2] = b->axis_bnd[5] - b->axis_bnd[4];
737  b->axis_bnd[0] -= tol * d[0], b->axis_bnd[1] += tol * d[0];
738  b->axis_bnd[2] -= tol * d[1], b->axis_bnd[3] += tol * d[1];
739  b->axis_bnd[4] -= tol * d[2], b->axis_bnd[5] += tol * d[2];
740 
741  c0[0] =
742  tensor_ig3( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.b.n, p->Jt0, p->Dt0, p->dt.b.n, x, jac, p->work );
743  c0[1] = tensor_ig3( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.b.n, p->Jt0, p->Dt0, p->dt.b.n, y, jac + 3,
744  p->work );
745  c0[2] = tensor_ig3( p->Jr0, p->Dr0, p->dr.n, p->Js0, p->Ds0, p->ds.b.n, p->Jt0, p->Dt0, p->dt.b.n, z, jac + 6,
746  p->work );
747  mat_inv_3( jac, inv );
748 
749  obbox_bnd_3( p, x, y, z, c0, inv, bnd );
750 
751  u0[0] = ( bnd[0] + bnd[1] ) / 2;
752  u0[1] = ( bnd[2] + bnd[3] ) / 2;
753  u0[2] = ( bnd[4] + bnd[5] ) / 2;
754  d[0] = 2 / ( ( 1 + tol ) * ( bnd[1] - bnd[0] ) );
755  d[1] = 2 / ( ( 1 + tol ) * ( bnd[3] - bnd[2] ) );
756  d[2] = 2 / ( ( 1 + tol ) * ( bnd[5] - bnd[4] ) );
757  b->x[0] = c0[0] + jac[0] * u0[0] + jac[1] * u0[1] + jac[2] * u0[2];
758  b->x[1] = c0[1] + jac[3] * u0[0] + jac[4] * u0[1] + jac[5] * u0[2];
759  b->x[2] = c0[2] + jac[6] * u0[0] + jac[7] * u0[1] + jac[8] * u0[2];
760  b->A[0] = d[0] * inv[0], b->A[1] = d[0] * inv[1], b->A[2] = d[0] * inv[2];
761  b->A[3] = d[1] * inv[3], b->A[4] = d[1] * inv[4], b->A[5] = d[1] * inv[5];
762  b->A[6] = d[2] * inv[6], b->A[7] = d[2] * inv[7], b->A[8] = d[2] * inv[8];
763 }
764 
765 /*--------------------------------------------------------------------------
766  Point to Possible Elements Hashing
767 
768  Initializing the data:
769  unsigned nel; // number of elements
770  const unsigned n[3]; // number of nodes in r, s, t directions
771  const realType *xm[3]; // n[0]*n[1]*n[2]*nel x,y,z coordinates
772  realType tol = 0.01; // how far point is allowed to be outside element
773  // relative to element size
774  unsigned max_size = n[0]*n[1]*n[2]*nel; // maximum size of hash table
775 
776  hash_data_3 data;
777  hash_build_3(&data, xm, n, nel, max_size, tol);
778 
779  Using the data:
780  realType x[3]; // point to find
781 
782  unsigned index = hash_index_3(&data, x);
783  unsigned i, b = data.offset[index], e = data.offset[index+1];
784 
785  // point may be in elements
786  // data.offset[b], data.offset[b+1], ... , data.offset[e-1]
787  //
788  // list has maximum size data.max (e.g., e-b <= data.max)
789 
790  for(i=b; i!=e; ++i) {
791  unsigned el = data.offset[i];
792  const obbox_3 *obb = &data.obb[el]; // bounding box data for element el
793  ...
794  }
795 
796  When done:
797  hash_free_3(&data);
798 
799  --------------------------------------------------------------------------*/
800 
801 static int ifloor( realType x )
802 {
803  /*
804  int y = x;
805  return (double)y > x ? y-1 : y;
806  */
807  return mbfloor( x );
808 }
809 
810 static int iceil( realType x )
811 {
812  /*
813  int y = x;
814  return (double)y < x ? y+1 : y;
815  */
816  return mbceil( x );
817 }
818 
819 static unsigned hash_index_helper( realType low, realType fac, unsigned n, realType x )
820 {
821  const int i = ifloor( ( x - low ) * fac );
822  if( i < 0 ) return 0;
823  return umin_2( i, n - 1 );
824 }
825 
826 static uint hash_index_2( const hash_data_2* p, const realType x[2] )
827 {
828  const unsigned n = p->hash_n;
829  return (uint)hash_index_helper( p->bnd[2], p->fac[1], n, x[1] ) * n +
830  hash_index_helper( p->bnd[0], p->fac[0], n, x[0] );
831 }
832 
833 static uint hash_index_3( const hash_data_3* p, const realType x[3] )
834 {
835  const unsigned n = p->hash_n;
836  return ( (uint)hash_index_helper( p->bnd[4], p->fac[2], n, x[2] ) * n +
837  hash_index_helper( p->bnd[2], p->fac[1], n, x[1] ) ) *
838  n +
839  hash_index_helper( p->bnd[0], p->fac[0], n, x[0] );
840 }
841 
842 static void hash_setfac_2( hash_data_2* p, unsigned n )
843 {
844  p->hash_n = n;
845  p->fac[0] = n / ( p->bnd[1] - p->bnd[0] );
846  p->fac[1] = n / ( p->bnd[3] - p->bnd[2] );
847 }
848 
849 static void hash_setfac_3( hash_data_3* p, unsigned n )
850 {
851  p->hash_n = n;
852  p->fac[0] = n / ( p->bnd[1] - p->bnd[0] );
853  p->fac[1] = n / ( p->bnd[3] - p->bnd[2] );
854  p->fac[2] = n / ( p->bnd[5] - p->bnd[4] );
855 }
856 
857 static void hash_range_2( const hash_data_2* p, uint i, unsigned d, unsigned* ia, unsigned* ib )
858 {
859  const realType a = p->obb[i].axis_bnd[d * 2];
860  const realType b = p->obb[i].axis_bnd[d * 2 + 1];
861  const int i0 = ifloor( ( a - p->bnd[d * 2] ) * p->fac[d] );
862  const unsigned i1 = iceil( ( b - p->bnd[d * 2] ) * p->fac[d] );
863  *ia = imax_2( 0, i0 );
864  *ib = imin_2( i1, p->hash_n );
865  if( *ib == *ia ) ++( *ib );
866 }
867 
868 static void hash_range_3( const hash_data_3* p, uint i, unsigned d, unsigned* ia, unsigned* ib )
869 {
870  const realType a = p->obb[i].axis_bnd[d * 2];
871  const realType b = p->obb[i].axis_bnd[d * 2 + 1];
872  const int i0 = ifloor( ( a - p->bnd[d * 2] ) * p->fac[d] );
873  const unsigned i1 = iceil( ( b - p->bnd[d * 2] ) * p->fac[d] );
874  *ia = imax_2( 0, i0 );
875  *ib = imin_2( i1, p->hash_n );
876  if( *ib == *ia ) ++( *ib );
877 }
878 
879 static uint hash_count_2( hash_data_2* p, uint nel, unsigned n )
880 {
881  uint i, count = 0;
882  hash_setfac_2( p, n );
883  for( i = 0; i < nel; ++i )
884  {
885  unsigned ia, ib;
886  uint ci;
887  hash_range_2( p, i, 0, &ia, &ib );
888  ci = ib - ia;
889  hash_range_2( p, i, 1, &ia, &ib );
890  ci *= ib - ia;
891  count += ci;
892  }
893  return count;
894 }
895 
896 static uint hash_count_3( hash_data_3* p, uint nel, unsigned n )
897 {
898  uint i, count = 0;
899  hash_setfac_3( p, n );
900  for( i = 0; i < nel; ++i )
901  {
902  unsigned ia, ib;
903  uint ci;
904  hash_range_3( p, i, 0, &ia, &ib );
905  ci = ib - ia;
906  hash_range_3( p, i, 1, &ia, &ib );
907  ci *= ib - ia;
908  hash_range_3( p, i, 2, &ia, &ib );
909  ci *= ib - ia;
910  count += ci;
911  }
912  return count;
913 }
914 
915 static uint hash_opt_size_2( hash_data_2* p, uint nel, uint max_size )
916 {
917  unsigned nl = 1, nu = ceil( sqrt( max_size - nel ) );
918  uint size_low = 2 + nel;
919  while( nu - nl > 1 )
920  {
921  unsigned nm = nl + ( nu - nl ) / 2;
922  uint size = (uint)nm * nm + 1 + hash_count_2( p, nel, nm );
923  if( size <= max_size )
924  nl = nm, size_low = size;
925  else
926  nu = nm;
927  }
928  hash_setfac_2( p, nl );
929  return size_low;
930 }
931 
932 static uint hash_opt_size_3( hash_data_3* p, uint nel, uint max_size )
933 {
934  unsigned nl = 1, nu = ceil( pow( max_size - nel, 1.0 / 3 ) );
935  uint size_low = 2 + nel;
936  while( nu - nl > 1 )
937  {
938  unsigned nm = nl + ( nu - nl ) / 2;
939  uint size = (uint)nm * nm * nm + 1 + hash_count_3( p, nel, nm );
940  if( size <= max_size )
941  nl = nm, size_low = size;
942  else
943  nu = nm;
944  }
945  hash_setfac_3( p, nl );
946  return size_low;
947 }
948 
949 static void hash_getbb_2( hash_data_2* p, const realType* const elx[2], const unsigned n[2], uint nel, realType tol )
950 {
951  obbox_data_2* data;
952  realType *z[2], *w[2];
953  uint i;
954  unsigned d;
955  const unsigned nn = n[0] * n[1];
956  unsigned int m[2];
957  const realType* x[2];
958 
959  for( i = 0; i < 2; ++i )
960  {
961  x[i] = elx[i];
962  m[i] = 2 * n[i];
963  }
964 
965  z[0] = tmalloc( realType, 2 * ( n[0] + n[1] ) );
966  w[0] = z[0] + n[0];
967  z[1] = w[0] + n[0], w[1] = z[1] + n[1];
968  for( d = 0; d < 2; ++d )
969  lobatto_nodes( z[d], n[d] ), lobatto_weights( z[d], w[d], n[d] );
970  data = obbox_setup_2( (const realType* const*)z, (const realType* const*)w, n, m );
971  obbox_calc_2( data, tol, x[0], x[1], &p->obb[0] );
972  memcpy( &p->bnd[0], (const realType*)&p->obb[0].axis_bnd[0], 4 * sizeof( realType ) );
973  for( i = 0; i < nel; ++i, x[0] += nn, x[1] += nn )
974  {
975  obbox_calc_2( data, tol, x[0], x[1], &p->obb[i] );
976  obbox_merge_2( &p->bnd[0], (const realType*)&p->obb[i].axis_bnd[0] );
977  }
978  obbox_free_2( data );
979  free( z[0] );
980 }
981 
982 static void hash_getbb_3( hash_data_3* p, const realType* const elx[3], const unsigned n[3], uint nel, realType tol )
983 {
984  obbox_data_3* data;
985  const realType* x[3];
986  realType *z[3], *w[3];
987  uint i;
988  unsigned d;
989  const unsigned nn = n[0] * n[1] * n[2];
990  unsigned int m[3];
991 
992  for( i = 0; i < 3; ++i )
993  {
994  x[i] = elx[i];
995  m[i] = 2 * n[i];
996  }
997 
998  z[0] = tmalloc( realType, 2 * ( n[0] + n[1] + n[2] ) );
999  w[0] = z[0] + n[0];
1000  for( d = 1; d < 3; ++d )
1001  z[d] = w[d - 1] + n[d - 1], w[d] = z[d] + n[d];
1002  for( d = 0; d < 3; ++d )
1003  lobatto_nodes( z[d], n[d] ), lobatto_weights( z[d], w[d], n[d] );
1004  data = obbox_setup_3( (const realType* const*)z, (const realType* const*)w, n, m );
1005  obbox_calc_3( data, tol, x[0], x[1], x[2], &p->obb[0] );
1006  memcpy( &p->bnd[0], (const realType*)&p->obb[0].axis_bnd[0], 6 * sizeof( realType ) );
1007  for( i = 0; i < nel; ++i, x[0] += nn, x[1] += nn, x[2] += nn )
1008  {
1009  obbox_calc_3( data, tol, x[0], x[1], x[2], &p->obb[i] );
1010  obbox_merge_3( &p->bnd[0], (const realType*)&p->obb[i].axis_bnd[0] );
1011  }
1012  obbox_free_3( data );
1013  free( z[0] );
1014 }
1015 
1016 static void hash_build_2( hash_data_2* p,
1017  const realType* const x[2],
1018  const unsigned n[2],
1019  uint nel,
1020  uint max_hash_size,
1021  realType tol )
1022 {
1023  uint i, el, size, hn2, sum;
1024  unsigned hn;
1025  unsigned* count;
1026  p->obb = tmalloc( obbox_2, nel );
1027  hash_getbb_2( p, x, n, nel, tol );
1028  size = hash_opt_size_2( p, nel, max_hash_size );
1029  p->offset = tmalloc( uint, size );
1030  hn = p->hash_n;
1031  hn2 = (uint)hn * hn;
1032  count = tcalloc( unsigned, hn2 );
1033  for( el = 0; el < nel; ++el )
1034  {
1035  unsigned ia, ib, j, ja, jb;
1036  hash_range_2( p, el, 0, &ia, &ib );
1037  hash_range_2( p, el, 1, &ja, &jb );
1038  for( j = ja; j < jb; ++j )
1039  for( i = ia; i < ib; ++i )
1040  ++count[(uint)j * hn + i];
1041  }
1042  sum = hn2 + 1, p->max = count[0];
1043  p->offset[0] = sum;
1044  for( i = 0; i < hn2; ++i )
1045  {
1046  if( count[i] > p->max ) p->max = count[i];
1047  sum += count[i];
1048  p->offset[i + 1] = sum;
1049  }
1050  for( el = 0; el < nel; ++el )
1051  {
1052  unsigned ia, ib, j, ja, jb;
1053  hash_range_2( p, el, 0, &ia, &ib );
1054  hash_range_2( p, el, 1, &ja, &jb );
1055  for( j = ja; j < jb; ++j )
1056  for( i = ia; i < ib; ++i )
1057  {
1058  uint arrindex = (uint)j * hn + i;
1059  p->offset[p->offset[arrindex + 1] - count[arrindex]] = el;
1060  --count[arrindex];
1061  }
1062  }
1063  free( count );
1064 }
1065 
1066 static void hash_build_3( hash_data_3* p,
1067  const realType* const x[3],
1068  const unsigned n[3],
1069  uint nel,
1070  uint max_hash_size,
1071  realType tol )
1072 {
1073  uint i, el, size, hn3, sum;
1074  unsigned hn;
1075  unsigned* count;
1076  p->obb = tmalloc( obbox_3, nel );
1077  hash_getbb_3( p, x, n, nel, tol );
1078  size = hash_opt_size_3( p, nel, max_hash_size );
1079  p->offset = tmalloc( uint, size );
1080  hn = p->hash_n;
1081  hn3 = (uint)hn * hn * hn;
1082  count = tcalloc( unsigned, hn3 );
1083  for( el = 0; el < nel; ++el )
1084  {
1085  unsigned ia, ib, j, ja, jb, k, ka, kb;
1086  hash_range_3( p, el, 0, &ia, &ib );
1087  hash_range_3( p, el, 1, &ja, &jb );
1088  hash_range_3( p, el, 2, &ka, &kb );
1089  for( k = ka; k < kb; ++k )
1090  for( j = ja; j < jb; ++j )
1091  for( i = ia; i < ib; ++i )
1092  ++count[( (uint)k * hn + j ) * hn + i];
1093  }
1094  sum = hn3 + 1, p->max = count[0];
1095  p->offset[0] = sum;
1096  for( i = 0; i < hn3; ++i )
1097  {
1098  if( count[i] > p->max ) p->max = count[i];
1099  sum += count[i];
1100  p->offset[i + 1] = sum;
1101  }
1102  for( el = 0; el < nel; ++el )
1103  {
1104  unsigned ia, ib, j, ja, jb, k, ka, kb;
1105  hash_range_3( p, el, 0, &ia, &ib );
1106  hash_range_3( p, el, 1, &ja, &jb );
1107  hash_range_3( p, el, 2, &ka, &kb );
1108  for( k = ka; k < kb; ++k )
1109  for( j = ja; j < jb; ++j )
1110  for( i = ia; i < ib; ++i )
1111  {
1112  uint arrindex = ( (uint)k * hn + j ) * hn + i;
1113  p->offset[p->offset[arrindex + 1] - count[arrindex]] = el;
1114  --count[arrindex];
1115  }
1116  }
1117  free( count );
1118 }
1119 
1120 static void hash_free_2( hash_data_2* p )
1121 {
1122  free( p->obb );
1123  free( p->offset );
1124 }
1125 
1126 static void hash_free_3( hash_data_3* p )
1127 {
1128  free( p->obb );
1129  free( p->offset );
1130 }
1131 
1132 /*--------------------------------------------------------------------------
1133  Optimization algorithm to find a point within an element
1134 
1135  Given x(r) (as values of x,y,z at all Lobatto nodes) and x_star,
1136  find the r that minimizes || x_star - x(r) ||_2
1137 
1138  As a minimization problem, the Newton step is
1139 
1140  __ 3
1141  [ J^T J - >_ d=1 resid_d H_d ] dr = J^t resid
1142 
1143  where resid = x_star - x(r), J = [ dx_i/dr_j ],
1144  and H_d = [ d^2 x_d/dr_i dr_j ].
1145 
1146  This is the appropriate step to take whenever constraints are active,
1147  and the current iterate is on a boundary of the element. When the current
1148  iterate is inside, J is square ( dim r = dim x ), resid will become small,
1149  and the step
1150 
1151  J dr = resid
1152 
1153  may be used instead, still giving quadratic convergence.
1154 
1155 
1156  Names use a _3 suffix for 3-d and _2 for 2-d.
1157  The routines require an initialized lagrange_data array as input:
1158  unsigned d, n[3] = { ... };
1159  realType *z[3] = { tmalloc(realType, n[0]), ... };
1160  for(d=0;d<3;++d) lobatto_nodes(z[d],n[d]);
1161 
1162  lagrange_data ld[3];
1163  for(d=0;d<3;++d) lagrange_setup(&ld[d],z[d],n[d]);
1164 
1165  Initialization:
1166  opt_data_3 data;
1167  opt_alloc_3(&data, ld);
1168 
1169  Use:
1170  const realType *xm[3]; // 3 pointers, each to n[0]*n[1]*n[2] realTypes
1171  // giving the nodal x, y, or z coordinates
1172 
1173  const realType x_star[3] = { ... }; // point to find
1174  realType r[3] = { 0,0,0 }; // initial guess with
1175  unsigned c = opt_no_constraints_3; // these constraints active
1176 
1177  realType dist = opt_findpt_3(&data,xm,x_star,r,&c);
1178  // minimizer is r with constraints c; 2-norm of resid is dist
1179 
1180  Clean-up:
1181  opt_free_3(&data);
1182 
1183  for(d=0;d<3;++d) lagrange_free(&ld[d]);
1184  for(d=0;d<3;++d) free(z[d]);
1185 
1186  The constraint number works as follows. Let cr be the constraints
1187  on the r variable:
1188  cr = 0 r fixed at -1
1189  cr = 1 r not fixed
1190  cr = 2 r fixed at 1
1191  Then the constraint number is (ct*3+cs)*3+cr
1192 
1193  --------------------------------------------------------------------------*/
1194 
1195 /* how many directions are constrained? */
1196 static const char opt_constr_num_2[9] = { 2, 1, 2, 1, 0, 1, 2, 1, 2 };
1197 static const char opt_constr_num_3[27] = { 3, 2, 3, 2, 1, 2, 3, 2, 3, 2, 1, 2, 1, 0,
1198  1, 2, 1, 2, 3, 2, 3, 2, 1, 2, 3, 2, 3 };
1199 
1200 /* which direction is constrained? */
1201 /* static const char opt_constr_dir_2[9] = {-1, 1,-1, 0,-1, 0, -1, 1,-1}; */
1202 static const char opt_constr_dir_3[27] = { -1, -1, -1, -1, 2, -1, -1, -1, -1, -1, 1, -1, 0, -1,
1203  0, -1, 1, -1, -1, -1, -1, -1, 2, -1, -1, -1, -1 };
1204 
1205 /* which direction is not constrained? */
1206 static const char opt_constr_not[27] = { -1, 0, -1, 1, -1, 1, -1, 0, -1, 2, -1, 2, -1, -1,
1207  -1, 2, -1, 2, -1, 0, -1, 1, -1, 1, -1, 0, -1 };
1208 
1209 static const char opt_constr_wide[27] = { 0x00, 0x01, 0x02, 0x04, 0x05, 0x06, 0x08, 0x09, 0x0a,
1210  0x10, 0x11, 0x12, 0x14, 0x15, 0x16, 0x18, 0x19, 0x1a,
1211  0x20, 0x21, 0x22, 0x24, 0x25, 0x26, 0x28, 0x29, 0x2a };
1212 
1213 static const unsigned opt_other1_3[3] = { 1, 0, 0 }, opt_other2_3[3] = { 2, 2, 1 };
1214 
1215 static unsigned opt_constr( unsigned constraints, unsigned d )
1216 {
1217  return ( opt_constr_wide[constraints] >> ( d * 2 ) ) & 3;
1218 }
1219 
1220 static void opt_constr_unpack_2( unsigned constraints, unsigned* c )
1221 {
1222  const char cw = opt_constr_wide[constraints];
1223  c[0] = cw & 3;
1224  c[1] = cw >> 2;
1225 }
1226 
1227 static void opt_constr_unpack_3( unsigned constraints, unsigned* c )
1228 {
1229  const char cw = opt_constr_wide[constraints];
1230  c[0] = cw & 3;
1231  c[1] = ( cw >> 2 ) & 3;
1232  c[2] = cw >> 4;
1233 }
1234 
1235 static unsigned opt_constr_pack_2( const unsigned* c )
1236 {
1237  return c[1] * 3 + c[0];
1238 }
1239 
1240 static unsigned opt_constr_pack_3( const unsigned* c )
1241 {
1242  return ( c[2] * 3 + c[1] ) * 3 + c[0];
1243 }
1244 
1245 /*--------------------------------------------------------------------------
1246 
1247  3 - D
1248 
1249  --------------------------------------------------------------------------*/
1250 
1252 {
1253  const unsigned nr = ld[0].n, ns = ld[1].n, nt = ld[2].n, nf = umax_3( nr * ns, nr * nt, ns * nt ),
1254  ne = umax_3( nr, ns, nt ), nw = 2 * ns * nt + 3 * ns;
1255  p->size[0] = 1;
1256  p->size[1] = nr;
1257  p->size[2] = nr * ns;
1258  p->size[3] = p->size[2] * nt;
1259  p->ld = ld;
1260  p->work = tmalloc( realType, 6 * nf + 9 * ne + nw );
1261  p->fd.x[0] = p->work + nw;
1262  p->fd.x[1] = p->fd.x[0] + nf;
1263  p->fd.x[2] = p->fd.x[1] + nf;
1264  p->fd.fdn[0] = p->fd.x[2] + nf;
1265  p->fd.fdn[1] = p->fd.fdn[0] + nf;
1266  p->fd.fdn[2] = p->fd.fdn[1] + nf;
1267  p->ed.x[0] = p->fd.fdn[2] + nf;
1268  p->ed.x[1] = p->ed.x[0] + ne;
1269  p->ed.x[2] = p->ed.x[1] + ne;
1270  p->ed.fd1[0] = p->ed.x[2] + ne;
1271  p->ed.fd1[1] = p->ed.fd1[0] + ne;
1272  p->ed.fd1[2] = p->ed.fd1[1] + ne;
1273  p->ed.fd2[0] = p->ed.fd1[2] + ne;
1274  p->ed.fd2[1] = p->ed.fd2[0] + ne;
1275  p->ed.fd2[2] = p->ed.fd2[1] + ne;
1276 }
1277 
1279 {
1280  free( p->work );
1281 }
1282 
1283 static void opt_vol_set_3( opt_data_3* p, const realType r[3] )
1284 {
1285  lagrange_1( &p->ld[0], r[0] );
1286  lagrange_1( &p->ld[1], r[1] );
1287  lagrange_1( &p->ld[2], r[2] );
1288 }
1289 
1290 /* work holds 2*ns*nt + 3*ns realTypes */
1291 static void opt_vol_intp_3( opt_data_3* p )
1292 {
1293  unsigned d;
1294  const lagrange_data* ld = p->ld;
1295 
1296  for( d = 0; d < 3; ++d )
1297  p->x[d] = tensor_ig3( ld[0].J, ld[0].D, ld[0].n, ld[1].J, ld[1].D, ld[1].n, ld[2].J, ld[2].D, ld[2].n,
1298  p->elx[d], &p->jac[d * 3], p->work );
1299 }
1300 
1302 {
1303  opt_vol_set_3( p, r );
1304  opt_vol_intp_3( p );
1305 }
1306 
1307 static void opt_face_proj_3( opt_data_3* p )
1308 {
1309  unsigned d, off = 0;
1310  const unsigned dn = p->fd.dn, d1 = p->fd.d1, d2 = p->fd.d2, so = p->size[d2] - p->size[d1 + 1], s1 = p->size[d1],
1311  sn = p->size[dn], n1 = p->ld[d1].n, n2 = p->ld[d2].n, nn = p->ld[dn].n;
1312  const realType* D = p->ld[dn].D_z0;
1313  if( opt_constr( p->fd.constraints, dn ) == 2 ) off = p->size[dn + 1] - p->size[dn], D = p->ld[dn].D_zn;
1314  for( d = 0; d < 3; ++d )
1315  {
1316  unsigned i, j, k, arrindex = 0;
1317  const realType* in = p->elx[d] + off;
1318  for( j = n2; j; --j, in += so )
1319  for( i = n1; i; --i, ++arrindex, in += s1 )
1320  {
1321  const realType* ind = in - off;
1322  realType* fdn = &p->fd.fdn[d][arrindex];
1323  p->fd.x[d][arrindex] = *in;
1324  *fdn = 0;
1325  for( k = 0; k < nn; ++k, ind += sn )
1326  *fdn += *ind * D[k];
1327  }
1328  }
1329 }
1330 
1331 static void opt_face_set_3( opt_data_3* p, const realType r[3], unsigned constr )
1332 {
1333  if( p->fd.constraints != constr )
1334  {
1335  p->fd.constraints = constr;
1336  p->fd.dn = opt_constr_dir_3[constr];
1337  p->fd.d1 = opt_other1_3[p->fd.dn];
1338  p->fd.d2 = opt_other2_3[p->fd.dn];
1339  opt_face_proj_3( p );
1340  }
1341  lagrange_1( &p->ld[p->fd.d1], r[p->fd.d1] );
1342  lagrange_1( &p->ld[p->fd.d2], r[p->fd.d2] );
1343 }
1344 
1345 /* work holds 2*ld[d2].n realTypes */
1346 static void opt_face_intp_3( opt_data_3* p )
1347 {
1348  unsigned d;
1349  const unsigned dn = p->fd.dn, d1 = p->fd.d1, d2 = p->fd.d2, n1 = p->ld[d1].n, n2 = p->ld[d2].n;
1350  const realType *J1 = p->ld[d1].J, *J2 = p->ld[d2].J, *D1 = p->ld[d1].D, *D2 = p->ld[d2].D;
1351 
1352  for( d = 0; d < 3; ++d )
1353  {
1354  realType g[2];
1355  p->x[d] = tensor_ig2( J1, D1, n1, J2, D2, n2, p->fd.x[d], &g[0], p->work );
1356  p->jac[d * 3 + d1] = g[0];
1357  p->jac[d * 3 + d2] = g[1];
1358  p->jac[d * 3 + dn] = tensor_i2( J1, n1, J2, n2, p->fd.fdn[d], p->work );
1359  }
1360 }
1361 
1362 static void opt_face_set_intp_3( opt_data_3* p, const realType r[3], unsigned constr )
1363 {
1364  opt_face_set_3( p, r, constr );
1365  opt_face_intp_3( p );
1366 }
1367 
1368 static void opt_face_hess_3( opt_data_3* p, realType hess[9] )
1369 {
1370  unsigned d;
1371  const unsigned d1 = p->fd.d1, d2 = p->fd.d2, n1 = p->ld[d1].n, n2 = p->ld[d2].n;
1372  const realType *J1 = p->ld[d1].J, *J2 = p->ld[d2].J, *D1 = p->ld[d1].D, *D2 = p->ld[d2].D, *S1 = p->ld[d1].D2,
1373  *S2 = p->ld[d2].D2;
1374 
1375  lagrange_2u( &p->ld[d1] );
1376  lagrange_2u( &p->ld[d2] );
1377 
1378  for( d = 0; d < 3; ++d )
1379  {
1380  (void)tensor_ig2( J1, S1, n1, J2, S2, n2, p->fd.x[d], hess + d * 3, p->work );
1381  hess[d * 3 + 0] = tensor_i2( S1, n1, J2, n2, p->fd.x[d], p->work );
1382  hess[d * 3 + 1] = tensor_i2( J1, n1, S2, n2, p->fd.x[d], p->work );
1383  hess[d * 3 + 2] = tensor_i2( D1, n1, D2, n2, p->fd.x[d], p->work );
1384  }
1385 }
1386 
1387 static void opt_edge_proj_3( opt_data_3* p )
1388 {
1389  unsigned d, off, off1 = 0, off2 = 0;
1390  const unsigned de = p->ed.de, d1 = p->ed.d1, d2 = p->ed.d2, se = p->size[de], s1 = p->size[d1], s2 = p->size[d2],
1391  ne = p->ld[de].n, n1 = p->ld[d1].n, n2 = p->ld[d2].n;
1392  const realType *fD1, *fD2;
1393  if( opt_constr( p->ed.constraints, d1 ) == 0 )
1394  fD1 = p->ld[d1].D_z0;
1395  else
1396  fD1 = p->ld[d1].D_zn, off1 = p->size[d1 + 1] - p->size[d1];
1397  if( opt_constr( p->ed.constraints, d2 ) == 0 )
1398  fD2 = p->ld[d2].D_z0;
1399  else
1400  fD2 = p->ld[d2].D_zn, off2 = p->size[d2 + 1] - p->size[d2];
1401  off = off1 + off2;
1402  for( d = 0; d < 3; ++d )
1403  {
1404  unsigned i, j;
1405  const realType* in = p->elx[d] + off;
1406  for( i = 0; i < ne; ++i, in += se )
1407  {
1408  const realType *in1 = in - off1, *in2 = in - off2;
1409  realType *fd1 = &p->ed.fd1[d][i], *fd2 = &p->ed.fd2[d][i];
1410  p->ed.x[d][i] = *in;
1411  *fd1 = *fd2 = 0;
1412  for( j = 0; j < n1; ++j, in1 += s1 )
1413  *fd1 += *in1 * fD1[j];
1414  for( j = 0; j < n2; ++j, in2 += s2 )
1415  *fd2 += *in2 * fD2[j];
1416  }
1417  }
1418 }
1419 
1420 static void opt_edge_set_3( opt_data_3* p, const realType r[3], unsigned constr )
1421 {
1422  if( p->ed.constraints != constr )
1423  {
1424  p->ed.constraints = constr;
1425  p->ed.de = opt_constr_not[constr];
1426  p->ed.d1 = opt_other1_3[p->ed.de];
1427  p->ed.d2 = opt_other2_3[p->ed.de];
1428  opt_edge_proj_3( p );
1429  }
1430  lagrange_1( &p->ld[p->ed.de], r[p->ed.de] );
1431 }
1432 
1433 static void opt_edge_intp_3( opt_data_3* p )
1434 {
1435  unsigned d;
1436  const unsigned de = p->ed.de, d1 = p->ed.d1, d2 = p->ed.d2, n = p->ld[de].n;
1437  const realType *J = p->ld[de].J, *D = p->ld[de].D;
1438 
1439  for( d = 0; d < 3; ++d )
1440  {
1441  p->x[d] = tensor_ig1( J, D, n, p->ed.x[d], &p->jac[d * 3 + de] );
1442  p->jac[d * 3 + d1] = tensor_i1( J, n, p->ed.fd1[d] );
1443  p->jac[d * 3 + d2] = tensor_i1( J, n, p->ed.fd2[d] );
1444  }
1445 }
1446 
1447 static void opt_edge_set_intp_3( opt_data_3* p, const realType r[3], unsigned constr )
1448 {
1449  opt_edge_set_3( p, r, constr );
1450  opt_edge_intp_3( p );
1451 }
1452 
1453 static void opt_edge_hess_3( opt_data_3* p, realType hess[3] )
1454 {
1455  unsigned d;
1456  const unsigned de = p->ed.de, n = p->ld[de].n;
1457  const realType* D2 = p->ld[de].D2;
1458  lagrange_2u( &p->ld[de] );
1459  for( d = 0; d < 3; ++d )
1460  hess[d] = tensor_i1( D2, n, p->ed.x[d] );
1461 }
1462 
1463 static void opt_point_proj_3( opt_data_3* p )
1464 {
1465  unsigned off[3], offt, d, c[3];
1466  const realType* fD[3];
1468  for( d = 0; d < 3; ++d )
1469  if( c[d] == 0 )
1470  fD[d] = p->ld[d].D_z0, off[d] = 0;
1471  else
1472  fD[d] = p->ld[d].D_zn, off[d] = p->size[d + 1] - p->size[d];
1473  offt = off[0] + off[1] + off[2];
1474  for( d = 0; d < 9; ++d )
1475  p->pd.jac[d] = 0;
1476  for( d = 0; d < 3; ++d )
1477  {
1478  unsigned i, j;
1479  p->pd.x[d] = p->elx[d][offt];
1480  for( i = 0; i < 3; ++i )
1481  {
1482  const realType* in = p->elx[d] + offt - off[i];
1483  for( j = 0; j < p->ld[i].n; ++j, in += p->size[i] )
1484  p->pd.jac[d * 3 + i] += *in * fD[i][j];
1485  }
1486  }
1487 }
1488 
1489 static void opt_point_set_3( opt_data_3* p, unsigned constr )
1490 {
1491  if( p->pd.constraints != constr )
1492  {
1493  p->pd.constraints = constr;
1494  opt_point_proj_3( p );
1495  }
1496 }
1497 
1498 static void opt_point_intp_3( opt_data_3* p )
1499 {
1500  memcpy( p->x, p->pd.x, 3 * sizeof( realType ) );
1501  memcpy( p->jac, p->pd.jac, 9 * sizeof( realType ) );
1502 }
1503 
1504 static void opt_point_set_intp_3( opt_data_3* p, unsigned constr )
1505 {
1506  opt_point_set_3( p, constr );
1507  opt_point_intp_3( p );
1508 }
1509 
1510 #define DIAGNOSTICS 0
1511 
1513  const realType* const elx[3],
1514  const realType xstar[3],
1515  realType r[3],
1516  unsigned* constr )
1517 {
1518  realType dr[3], resid[3], steep[3];
1519 
1520  unsigned c = *constr, ac, d, cc[3], step = 0;
1521 
1522  p->elx[0] = elx[0], p->elx[1] = elx[1], p->elx[2] = elx[2];
1523 
1527 
1528 #if DIAGNOSTICS
1529  printf( "opt_findpt: xstar = %g, %g, %g\n", xstar[0], xstar[1], xstar[2] );
1530 #endif
1531 
1532  do
1533  {
1534  ++step;
1535  if( step == 50 ) /*fail("%s: opt_findpt_3 did not converge\n",__FILE__);*/
1536  return 1.e+30;
1537 #if DIAGNOSTICS
1538  printf( " iteration %u\n", step );
1539  printf( " %d constraint(s) active\n", (int)opt_constr_num_3[c] );
1540 #endif
1541  /* update face/edge/point data if necessary,
1542  and evaluate x(r) as well as the jacobian */
1543  switch( opt_constr_num_3[c] )
1544  {
1545  case 0:
1546  opt_vol_set_intp_3( p, r );
1547  break;
1548  case 1:
1549  opt_face_set_intp_3( p, r, c );
1550  break;
1551  case 2:
1552  opt_edge_set_intp_3( p, r, c );
1553  break;
1554  case 3:
1555  opt_point_set_intp_3( p, c );
1556  break;
1557  }
1558 #if DIAGNOSTICS
1559  printf( " r = %g, %g, %g\n", r[0], r[1], r[2] );
1560  printf( " x = %g, %g, %g\n", p->x[0], p->x[1], p->x[2] );
1561 #endif
1562  /* compute residual */
1563  for( d = 0; d < 3; ++d )
1564  resid[d] = xstar[d] - p->x[d];
1565 #if DIAGNOSTICS
1566  printf( " resid = %g, %g, %g\n", resid[0], resid[1], resid[2] );
1567  printf( " 2-norm = %g\n", r2norm_3( resid[0], resid[1], resid[2] ) );
1568 #endif
1569  /* check constraints against steepest descent direction */
1570  ac = c;
1571  if( opt_constr_num_3[c] )
1572  {
1573  opt_constr_unpack_3( c, cc );
1574  mat_app_3c( steep, p->jac, resid ); /* steepest descent = J^T r */
1575 #if DIAGNOSTICS
1576  printf( " steepest descent = %g, %g, %g\n", steep[0], steep[1], steep[2] );
1577 #endif
1578  for( d = 0; d < 3; ++d )
1579  if( ( cc[d] == 0 && steep[d] > 0 ) || ( cc[d] == 2 && steep[d] < 0 ) ) cc[d] = 1;
1580  ac = opt_constr_pack_3( cc );
1581  }
1582  /* update face/edge/point data if necessary */
1583  if( ac != c )
1584  {
1585  c = ac;
1586 #if DIAGNOSTICS
1587  printf( " relaxed to %d constraints\n", (int)opt_constr_num_3[c] );
1588 #endif
1589  switch( opt_constr_num_3[c] )
1590  {
1591  case 1:
1592  opt_face_set_3( p, r, c );
1593  break;
1594  case 2:
1595  opt_edge_set_3( p, r, c );
1596  break;
1597  case 3:
1598  opt_point_set_3( p, c );
1599  break;
1600  }
1601  }
1602  /* compute Newton step */
1603  switch( opt_constr_num_3[c] )
1604  {
1605  case 0:
1606  tinyla_solve_3( dr, p->jac, resid );
1607  break;
1608  case 1: {
1609  const unsigned dn = p->fd.dn, d1 = p->fd.d1, d2 = p->fd.d2;
1610  realType A[4], H[9];
1611  const realType* J = p->jac;
1612  opt_face_hess_3( p, H );
1613  A[0] = J[d1] * J[d1] + J[3 + d1] * J[3 + d1] + J[6 + d1] * J[6 + d1];
1614  A[1] = J[d2] * J[d2] + J[3 + d2] * J[3 + d2] + J[6 + d2] * J[6 + d2];
1615  A[2] = J[d1] * J[d2] + J[3 + d1] * J[3 + d2] + J[6 + d1] * J[6 + d2];
1616  A[0] -= resid[0] * H[0] + resid[1] * H[3] + resid[2] * H[6];
1617  A[1] -= resid[0] * H[1] + resid[1] * H[4] + resid[2] * H[7];
1618  A[2] -= resid[0] * H[2] + resid[1] * H[5] + resid[2] * H[8];
1619  tinyla_solve_sym_2( &dr[d1], &dr[d2], A, steep[d1], steep[d2] );
1620  dr[dn] = 0;
1621  }
1622  break;
1623  case 2: {
1624  const unsigned de = p->ed.de, d1 = p->ed.d1, d2 = p->ed.d2;
1625  realType fac, H[3];
1626  const realType* J = p->jac + de;
1627  opt_edge_hess_3( p, H );
1628  fac = J[0] * J[0] + J[3] * J[3] + J[6] * J[6] - ( resid[0] * H[0] + resid[1] * H[1] + resid[2] * H[2] );
1629  dr[de] = steep[de] / fac;
1630  dr[d1] = 0, dr[d2] = 0;
1631  }
1632  break;
1633  case 3:
1634  dr[0] = dr[1] = dr[2] = 0;
1635  break;
1636  }
1637 #if DIAGNOSTICS
1638  printf( " dr = %g, %g, %g\n", dr[0], dr[1], dr[2] );
1639 #endif
1640  /* project new iteration onto [-1,1]^3 */
1641  opt_constr_unpack_3( c, cc );
1642  for( d = 0; d < 3; ++d )
1643  {
1644  if( cc[d] != 1 ) continue;
1645  r[d] += dr[d];
1646  if( r[d] <= -1 )
1647  dr[d] -= r[d] + 1, r[d] = -1, cc[d] = 0;
1648  else if( r[d] >= 1 )
1649  dr[d] -= r[d] - 1, r[d] = 1, cc[d] = 2;
1650  }
1651  c = opt_constr_pack_3( cc );
1652  } while( r1norm_3( dr[0], dr[1], dr[2] ) > 3 * MOAB_POLY_EPS );
1653  *constr = c;
1654 #if 0
1655  printf("opt_findpt_3 converged in %u iterations\n", step);
1656 #endif
1657  return r2norm_3( resid[0], resid[1], resid[2] );
1658 }
1659 
1660 #undef DIAGNOSTICS
1661 
1663 {
1664  const unsigned nr = ld[0].n, ns = ld[1].n, ne = umax_2( nr, ns ), nw = 2 * ns;
1665  p->size[0] = 1;
1666  p->size[1] = nr;
1667  p->size[2] = nr * ns;
1668  p->ld = ld;
1669  p->work = tmalloc( realType, 4 * ne + nw );
1670  p->ed.x[0] = p->work + nw;
1671  p->ed.x[1] = p->ed.x[0] + ne;
1672  p->ed.fd1[0] = p->ed.x[1] + ne;
1673  p->ed.fd1[1] = p->ed.fd1[0] + ne;
1674 }
1675 
1677 {
1678  free( p->work );
1679 }
1680 
1681 static void opt_area_set_2( opt_data_2* p, const realType r[2] )
1682 {
1683  lagrange_1( &p->ld[0], r[0] );
1684  lagrange_1( &p->ld[1], r[1] );
1685 }
1686 
1687 /* work holds 2*ns realTypes */
1688 static void opt_area_intp_2( opt_data_2* p )
1689 {
1690  unsigned d;
1691  const lagrange_data* ld = p->ld;
1692 
1693  for( d = 0; d < 2; ++d )
1694  p->x[d] =
1695  tensor_ig2( ld[0].J, ld[0].D, ld[0].n, ld[1].J, ld[1].D, ld[1].n, p->elx[d], &p->jac[d * 2], p->work );
1696 }
1697 
1698 static void opt_area_set_intp_2( opt_data_2* p, const realType r[2] )
1699 {
1700  opt_area_set_2( p, r );
1701  opt_area_intp_2( p );
1702 }
1703 
1704 static void opt_edge_proj_2( opt_data_2* p )
1705 {
1706  unsigned d, off = 0;
1707  const unsigned de = p->ed.de, d1 = p->ed.d1, se = p->size[de], s1 = p->size[d1], ne = p->ld[de].n, n1 = p->ld[d1].n;
1708  const realType* fD1;
1709  if( opt_constr( p->ed.constraints, d1 ) == 0 )
1710  fD1 = p->ld[d1].D_z0;
1711  else
1712  fD1 = p->ld[d1].D_zn, off = p->size[d1 + 1] - p->size[d1];
1713  for( d = 0; d < 2; ++d )
1714  {
1715  unsigned i, j;
1716  const realType* in = p->elx[d] + off;
1717  for( i = 0; i < ne; ++i, in += se )
1718  {
1719  const realType* in1 = in - off;
1720  realType* fd1 = &p->ed.fd1[d][i];
1721  p->ed.x[d][i] = *in;
1722  *fd1 = 0;
1723  for( j = 0; j < n1; ++j, in1 += s1 )
1724  *fd1 += *in1 * fD1[j];
1725  }
1726  }
1727 }
1728 
1729 static void opt_edge_set_2( opt_data_2* p, const realType r[2], unsigned constr )
1730 {
1731  if( p->ed.constraints != constr )
1732  {
1733  p->ed.constraints = constr;
1734  p->ed.de = opt_constr_not[constr];
1735  p->ed.d1 = 1 - p->ed.de;
1736  opt_edge_proj_2( p );
1737  }
1738  lagrange_1( &p->ld[p->ed.de], r[p->ed.de] );
1739 }
1740 
1741 static void opt_edge_intp_2( opt_data_2* p )
1742 {
1743  unsigned d;
1744  const unsigned de = p->ed.de, d1 = p->ed.d1, n = p->ld[de].n;
1745  const realType *J = p->ld[de].J, *D = p->ld[de].D;
1746  for( d = 0; d < 2; ++d )
1747  {
1748  p->x[d] = tensor_ig1( J, D, n, p->ed.x[d], &p->jac[d * 2 + de] );
1749  p->jac[d * 2 + d1] = tensor_i1( J, n, p->ed.fd1[d] );
1750  }
1751 }
1752 
1753 static void opt_edge_set_intp_2( opt_data_2* p, const realType r[2], unsigned constr )
1754 {
1755  opt_edge_set_2( p, r, constr );
1756  opt_edge_intp_2( p );
1757 }
1758 
1759 static void opt_edge_hess_2( opt_data_2* p, realType hess[2] )
1760 {
1761  unsigned d;
1762  const unsigned de = p->ed.de, n = p->ld[de].n;
1763  const realType* D2 = p->ld[de].D2;
1764  lagrange_2u( &p->ld[de] );
1765  for( d = 0; d < 2; ++d )
1766  hess[d] = tensor_i1( D2, n, p->ed.x[d] );
1767 }
1768 
1769 static void opt_point_proj_2( opt_data_2* p )
1770 {
1771  unsigned off[2], offt, d, c[2];
1772  const realType* fD[2];
1774  for( d = 0; d < 2; ++d )
1775  if( c[d] == 0 )
1776  fD[d] = p->ld[d].D_z0, off[d] = 0;
1777  else
1778  fD[d] = p->ld[d].D_zn, off[d] = p->size[d + 1] - p->size[d];
1779  offt = off[0] + off[1];
1780  for( d = 0; d < 4; ++d )
1781  p->pd.jac[d] = 0;
1782  for( d = 0; d < 2; ++d )
1783  {
1784  unsigned i, j;
1785  p->pd.x[d] = p->elx[d][offt];
1786  for( i = 0; i < 2; ++i )
1787  {
1788  const realType* in = p->elx[d] + offt - off[i];
1789  for( j = 0; j < p->ld[i].n; ++j, in += p->size[i] )
1790  p->pd.jac[d * 2 + i] += *in * fD[i][j];
1791  }
1792  }
1793 }
1794 
1795 static void opt_point_set_2( opt_data_2* p, unsigned constr )
1796 {
1797  if( p->pd.constraints != constr )
1798  {
1799  p->pd.constraints = constr;
1800  opt_point_proj_2( p );
1801  }
1802 }
1803 
1804 static void opt_point_intp_2( opt_data_2* p )
1805 {
1806  memcpy( p->x, p->pd.x, 2 * sizeof( realType ) );
1807  memcpy( p->jac, p->pd.jac, 4 * sizeof( realType ) );
1808 }
1809 
1810 static void opt_point_set_intp_2( opt_data_2* p, unsigned constr )
1811 {
1812  opt_point_set_2( p, constr );
1813  opt_point_intp_2( p );
1814 }
1815 
1816 #define DIAGNOSTICS 0
1817 
1819  const realType* const elx[2],
1820  const realType xstar[2],
1821  realType r[2],
1822  unsigned* constr )
1823 {
1824  realType dr[2], resid[2], steep[2];
1825 
1826  unsigned c = *constr, ac, d, cc[2], step = 0;
1827 
1828  p->elx[0] = elx[0], p->elx[1] = elx[1];
1829 
1832 
1833 #if DIAGNOSTICS
1834  printf( "opt_findpt: xstar = %g, %g\n", xstar[0], xstar[1] );
1835 #endif
1836 
1837  do
1838  {
1839  ++step;
1840  if( step == 50 ) /*fail("%s: opt_findpt_2 did not converge\n",__FILE__);*/
1841  return 1.e+30;
1842 #if DIAGNOSTICS
1843  printf( " iteration %u\n", step );
1844  printf( " %d constraint(s) active\n", (int)opt_constr_num_2[c] );
1845 #endif
1846  /* update face/edge/point data if necessary,
1847  and evaluate x(r) as well as the jacobian */
1848  switch( opt_constr_num_2[c] )
1849  {
1850  case 0:
1851  opt_area_set_intp_2( p, r );
1852  break;
1853  case 1:
1854  opt_edge_set_intp_2( p, r, c );
1855  break;
1856  case 2:
1857  opt_point_set_intp_2( p, c );
1858  break;
1859  }
1860 #if DIAGNOSTICS
1861  printf( " r = %g, %g\n", r[0], r[1] );
1862  printf( " x = %g, %g\n", p->x[0], p->x[1] );
1863 #endif
1864  /* compute residual */
1865  for( d = 0; d < 2; ++d )
1866  resid[d] = xstar[d] - p->x[d];
1867 #if DIAGNOSTICS
1868  printf( " resid = %g, %g\n", resid[0], resid[1] );
1869  printf( " 2-norm = %g\n", r2norm_2( resid[0], resid[1] ) );
1870 #endif
1871  /* check constraints against steepest descent direction */
1872  ac = c;
1873  if( opt_constr_num_2[c] )
1874  {
1875  opt_constr_unpack_2( c, cc );
1876  mat_app_2c( steep, p->jac, resid ); /* steepest descent = J^T r */
1877 #if DIAGNOSTICS
1878  printf( " steepest descent = %g, %g\n", steep[0], steep[1] );
1879 #endif
1880  for( d = 0; d < 2; ++d )
1881  if( ( cc[d] == 0 && steep[d] > 0 ) || ( cc[d] == 2 && steep[d] < 0 ) ) cc[d] = 1;
1882  ac = opt_constr_pack_2( cc );
1883  }
1884  /* update face/edge/point data if necessary */
1885  if( ac != c )
1886  {
1887  c = ac;
1888 #if DIAGNOSTICS
1889  printf( " relaxed to %d constraints\n", (int)opt_constr_num_2[c] );
1890 #endif
1891  switch( opt_constr_num_2[c] )
1892  {
1893  case 1:
1894  opt_edge_set_2( p, r, c );
1895  break;
1896  case 2:
1897  opt_point_set_2( p, c );
1898  break;
1899  }
1900  }
1901  /* compute Newton step */
1902  switch( opt_constr_num_2[c] )
1903  {
1904  case 0:
1905  tinyla_solve_2( dr, p->jac, resid );
1906  break;
1907  case 1: {
1908  const unsigned de = p->ed.de, d1 = p->ed.d1;
1909  realType fac, H[2];
1910  const realType* J = p->jac + de;
1911  opt_edge_hess_2( p, H );
1912  fac = J[0] * J[0] + J[2] * J[2] - ( resid[0] * H[0] + resid[1] * H[1] );
1913  dr[de] = steep[de] / fac;
1914  dr[d1] = 0;
1915  }
1916  break;
1917  case 2:
1918  dr[0] = dr[1] = 0;
1919  break;
1920  }
1921 #if DIAGNOSTICS
1922  printf( " dr = %g, %g\n", dr[0], dr[1] );
1923 #endif
1924  /* project new iteration onto [-1,1]^2 */
1925  opt_constr_unpack_2( c, cc );
1926  for( d = 0; d < 2; ++d )
1927  {
1928  if( cc[d] != 1 ) continue;
1929  r[d] += dr[d];
1930  if( r[d] <= -1 )
1931  dr[d] -= r[d] + 1, r[d] = -1, cc[d] = 0;
1932  else if( r[d] >= 1 )
1933  dr[d] -= r[d] - 1, r[d] = 1, cc[d] = 2;
1934  }
1935  c = opt_constr_pack_2( cc );
1936  } while( r1norm_2( dr[0], dr[1] ) > 2 * MOAB_POLY_EPS );
1937  *constr = c;
1938  return r2norm_2( resid[0], resid[1] );
1939 }
1940 
1941 #undef DIAGNOSTICS
1942 
1943 /*--------------------------------------------------------------------------
1944  Point Finding (interface/top-level)
1945 
1946  Initializing the data:
1947  unsigned nel; // number of elements
1948  const unsigned n[3]; // number of nodes in r, s, t directions
1949  const realType *xm[3]; // n[0]*n[1]*n[2]*nel x,y,z coordinates
1950  realType tol = 0.01; // how far point is allowed to be outside element
1951  // relative to element size
1952  unsigned max_size = n[0]*n[1]*n[2]*nel; // maximum size of hash table
1953 
1954  findpt_data_3 *data = findpt_setup_3(xm,n,nel,max_size,tol);
1955 
1956  Using the data:
1957  realType x[3] = { ... }; // point to find
1958  int el; // element number
1959  realType r[3]; // parametric coordinates
1960  int guess = 0; // do we use (el,r,s,t) as an initial guess?
1961  int code; // 0 : normal, -1 : outside all elements,
1962  // 1 : border, or outside but within tolerance
1963  realType dist; // distance in xyz space from returned (el,r,s,t) to given
1964  // (x,y,z)
1965 
1966  code = findpt_3(data, x, guess, &el, r, &dist);
1967 
1968  When done:
1969  findpt_free_3(&data);
1970 
1971  --------------------------------------------------------------------------*/
1972 
1973 /* heap sort on A[0:n-1] with key A[i]->dist
1974  precondition: n!=0 */
1975 static void findpt_list_sort( findpt_listel** A, unsigned n )
1976 {
1977  unsigned i;
1978  --A; /* make A have a base index of 1 */
1979  /* build heap */
1980  for( i = 2; i <= n; ++i )
1981  {
1982  findpt_listel* item = A[i];
1983  unsigned hole = i, parent = hole >> 1;
1984  if( A[parent]->dist >= item->dist ) continue;
1985  do
1986  {
1987  A[hole] = A[parent];
1988  hole = parent;
1989  parent >>= 1;
1990  } while( parent && A[parent]->dist < item->dist );
1991  A[hole] = item;
1992  }
1993  /* extract */
1994  for( i = n - 1; i; --i )
1995  {
1996  findpt_listel* item = A[i + 1];
1997  unsigned hole = 1;
1998  A[i + 1] = A[1];
1999  for( ;; )
2000  {
2001  unsigned ch = hole << 1, r = ch + 1;
2002  if( r <= i && A[ch]->dist < A[r]->dist ) ch = r;
2003  if( ch > i || item->dist >= A[ch]->dist ) break;
2004  A[hole] = A[ch];
2005  hole = ch;
2006  }
2007  A[hole] = item;
2008  }
2009 }
2010 
2012  const unsigned n[2],
2013  uint nel,
2014  uint max_hash_size,
2015  realType bbox_tol )
2016 {
2017  unsigned d;
2018  findpt_data_2* p = tmalloc( findpt_data_2, 1 );
2019 
2020  p->hash = tmalloc( hash_data_2, 1 );
2021  p->od = tmalloc( opt_data_2, 1 );
2022 
2023  for( d = 0; d < 2; ++d )
2024  p->xw[d] = xw[d];
2025  p->nptel = n[0] * n[1];
2026 
2027  hash_build_2( p->hash, xw, n, nel, max_hash_size, bbox_tol );
2028 
2029  for( d = 0; d < 2; ++d )
2030  {
2031  p->z[d] = tmalloc( realType, n[d] );
2032  lobatto_nodes( p->z[d], n[d] );
2033  lagrange_setup( &p->ld[d], p->z[d], n[d] );
2034  }
2035 
2036  p->list = tmalloc( findpt_listel, p->hash->max );
2037  p->sorted = tmalloc( findpt_listel*, p->hash->max );
2038 
2039  opt_alloc_2( p->od, p->ld );
2040  p->od_work = p->od->work;
2041 
2042  return p;
2043 }
2044 
2046  const unsigned n[3],
2047  uint nel,
2048  uint max_hash_size,
2049  realType bbox_tol )
2050 {
2051  unsigned d;
2052  findpt_data_3* p = tmalloc( findpt_data_3, 1 );
2053 
2054  p->hash = tmalloc( hash_data_3, 1 );
2055  p->od = tmalloc( opt_data_3, 1 );
2056 
2057  for( d = 0; d < 3; ++d )
2058  p->xw[d] = xw[d];
2059  p->nptel = n[0] * n[1] * n[2];
2060 
2061  hash_build_3( p->hash, xw, n, nel, max_hash_size, bbox_tol );
2062 
2063  for( d = 0; d < 3; ++d )
2064  {
2065  p->z[d] = tmalloc( realType, n[d] );
2066  lobatto_nodes( p->z[d], n[d] );
2067  lagrange_setup( &p->ld[d], p->z[d], n[d] );
2068  }
2069 
2070  p->list = tmalloc( findpt_listel, p->hash->max );
2071  p->sorted = tmalloc( findpt_listel*, p->hash->max );
2072 
2073  opt_alloc_3( p->od, p->ld );
2074  p->od_work = p->od->work;
2075 
2076  return p;
2077 }
2078 
2080 {
2081  unsigned d;
2082  opt_free_2( p->od );
2083  free( p->od );
2084  hash_free_2( p->hash );
2085  free( p->hash );
2086  free( p->list );
2087  free( p->sorted );
2088  for( d = 0; d < 2; ++d )
2089  free( p->z[d] );
2090  free( p );
2091 }
2092 
2094 {
2095  unsigned d;
2096  opt_free_3( p->od );
2097  free( p->od );
2098  hash_free_3( p->hash );
2099  free( p->hash );
2100  free( p->list );
2101  free( p->sorted );
2102  for( d = 0; d < 3; ++d )
2103  free( p->z[d] );
2104  free( p );
2105 }
2106 
2108 {
2109  return p->hash->bnd;
2110 }
2111 
2113 {
2114  return p->hash->bnd;
2115 }
2116 
2117 static void findpt_hash_2( findpt_data_2* p, const realType x[2] )
2118 {
2119  findpt_listel *list = p->list, **sorted = p->sorted;
2120  const uint hi = hash_index_2( p->hash, x );
2121  const uint* offset = p->hash->offset;
2122  uint i;
2123  const uint b = offset[hi], e = offset[hi + 1];
2124  for( i = b; i != e; ++i )
2125  {
2126  const uint el = offset[i];
2127  realType* r = &list->r[0];
2128  const obbox_2* obb = &p->hash->obb[el];
2129  if( obbox_axis_test_2( obb, x ) ) continue;
2130  if( obbox_test_2( obb, x, r ) ) continue;
2131  list->el = el;
2132  list->dist = r1norm_2( r[0], r[1] );
2133  *sorted++ = list++;
2134  }
2135  p->end = sorted;
2136  if( p->end != p->sorted ) findpt_list_sort( p->sorted, p->end - p->sorted );
2137 }
2138 
2139 static void findpt_hash_3( findpt_data_3* p, const realType x[3] )
2140 {
2141  findpt_listel *list = p->list, **sorted = p->sorted;
2142  const uint hi = hash_index_3( p->hash, x );
2143  const uint* offset = p->hash->offset;
2144  uint i;
2145  const uint b = offset[hi], e = offset[hi + 1];
2146  for( i = b; i != e; ++i )
2147  {
2148  const uint el = offset[i];
2149  realType* r = &list->r[0];
2150  const obbox_3* obb = &p->hash->obb[el];
2151  if( obbox_axis_test_3( obb, x ) ) continue;
2152  if( obbox_test_3( obb, x, r ) ) continue;
2153  list->el = el;
2154  list->dist = r1norm_3( r[0], r[1], r[2] );
2155  *sorted++ = list++;
2156  }
2157  p->end = sorted;
2158  if( p->end != p->sorted ) findpt_list_sort( p->sorted, p->end - p->sorted );
2159 }
2160 
2161 static int findpt_guess_2( findpt_data_2* p, const realType x[2], uint el, realType r[2], realType* dist )
2162 {
2163  const uint arrindex = p->nptel * el;
2164  const realType* elx[2];
2165  realType g[2];
2166  unsigned c = opt_no_constraints_2;
2167  const obbox_2* obb = &p->hash->obb[el];
2168  elx[0] = p->xw[0] + arrindex;
2169  elx[1] = p->xw[1] + arrindex;
2170  if( obbox_axis_test_2( obb, x ) || obbox_test_2( obb, x, g ) ) return 0;
2171  *dist = opt_findpt_2( p->od, elx, x, r, &c );
2172  return c == opt_no_constraints_2;
2173 }
2174 
2175 static int findpt_guess_3( findpt_data_3* p, const realType x[3], uint el, realType r[3], realType* dist )
2176 {
2177  const uint arrindex = p->nptel * el;
2178  const realType* elx[3];
2179  realType g[3];
2180  unsigned c = opt_no_constraints_3;
2181  const obbox_3* obb = &p->hash->obb[el];
2182  elx[0] = p->xw[0] + arrindex;
2183  elx[1] = p->xw[1] + arrindex;
2184  elx[2] = p->xw[2] + arrindex;
2185  if( obbox_axis_test_3( obb, x ) || obbox_test_3( obb, x, g ) ) return 0;
2186  *dist = opt_findpt_3( p->od, elx, x, r, &c );
2187  return c == opt_no_constraints_3;
2188 }
2189 
2190 #define DIAGNOSTICS 0
2191 
2192 static int findpt_pass_2( findpt_data_2* p, const realType x[2], uint* el, realType r[2], realType* dist_min )
2193 {
2194  findpt_listel** qq = p->sorted;
2195  const realType* bnd;
2196  realType dist;
2197  do
2198  {
2199  findpt_listel* q = *qq;
2200  const uint arrindex = p->nptel * q->el;
2201  const realType* elx[2];
2202  unsigned c = opt_no_constraints_2;
2203  elx[0] = p->xw[0] + arrindex;
2204  elx[1] = p->xw[1] + arrindex;
2205  dist = opt_findpt_2( p->od, elx, x, q->r, &c );
2206  if( qq == p->sorted || dist < *dist_min || c == opt_no_constraints_2 )
2207  {
2208  *dist_min = dist;
2209  *el = q->el;
2210  memcpy( r, q->r, 2 * sizeof( realType ) );
2211  if( c == opt_no_constraints_2 ) return 0;
2212  }
2213  } while( ++qq != p->end );
2214  bnd = p->hash->obb[*el].axis_bnd;
2215  return *dist_min > r2norm_2( bnd[1] - bnd[0], bnd[3] - bnd[2] ) ? -1 : 1;
2216 }
2217 
2218 static int findpt_pass_3( findpt_data_3* p, const realType x[3], uint* el, realType r[3], realType* dist_min )
2219 {
2220  findpt_listel** qq = p->sorted;
2221  const realType* bnd;
2222  realType dist;
2223  do
2224  {
2225  findpt_listel* q = *qq;
2226  const uint arrindex = p->nptel * q->el;
2227  const realType* elx[3];
2228  unsigned c = opt_no_constraints_3;
2229  elx[0] = p->xw[0] + arrindex;
2230  elx[1] = p->xw[1] + arrindex;
2231  elx[2] = p->xw[2] + arrindex;
2232  dist = opt_findpt_3( p->od, elx, x, q->r, &c );
2233  if( qq == p->sorted || dist < *dist_min || c == opt_no_constraints_3 )
2234  {
2235  *dist_min = dist;
2236  *el = q->el;
2237  memcpy( r, q->r, 3 * sizeof( realType ) );
2238  if( c == opt_no_constraints_3 )
2239  {
2240 #if DIAGNOSTICS
2241  printf( "point found in element #%d\n", qq - p->sorted );
2242 #endif
2243  return 0;
2244  }
2245  }
2246  } while( ++qq != p->end );
2247  bnd = p->hash->obb[*el].axis_bnd;
2248  return *dist_min > r2norm_3( bnd[1] - bnd[0], bnd[3] - bnd[2], bnd[5] - bnd[4] ) ? -1 : 1;
2249 }
2250 
2251 int findpt_2( findpt_data_2* p, const realType x[2], int guess, uint* el, realType r[2], realType* dist )
2252 {
2253  if( guess && findpt_guess_2( p, x, *el, r, dist ) ) return 0;
2254  findpt_hash_2( p, x );
2255  if( p->sorted == p->end ) return -1;
2256  return findpt_pass_2( p, x, el, r, dist );
2257 }
2258 
2259 int findpt_3( findpt_data_3* p, const realType x[3], int guess, uint* el, realType r[3], realType* dist )
2260 {
2261  if( guess && findpt_guess_3( p, x, *el, r, dist ) ) return 0;
2262  findpt_hash_3( p, x );
2263 #if DIAGNOSTICS
2264  printf( "hashing leaves %d elements to consider\n", p->end - p->sorted );
2265 #endif
2266  if( p->sorted == p->end ) return -1;
2267  return findpt_pass_3( p, x, el, r, dist );
2268 }
2269 
2271 {
2272  lagrange_0( &p->ld[0], r[0] );
2273  lagrange_0( &p->ld[1], r[1] );
2274 }
2275 
2277 {
2278  lagrange_0( &p->ld[0], r[0] );
2279  lagrange_0( &p->ld[1], r[1] );
2280  lagrange_0( &p->ld[2], r[2] );
2281 }
2282 
2283 double findpt_eval_2( findpt_data_2* p, const realType* u )
2284 {
2285  return tensor_i2( p->ld[0].J, p->ld[0].n, p->ld[1].J, p->ld[1].n, u, p->od_work );
2286 }
2287 
2288 double findpt_eval_3( findpt_data_3* p, const realType* u )
2289 {
2290  return tensor_i3( p->ld[0].J, p->ld[0].n, p->ld[1].J, p->ld[1].n, p->ld[2].J, p->ld[2].n, u, p->od_work );
2291 }