GetFEM  5.4.4
getfem_mesher.h
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4  Copyright (C) 2004-2020 Julien Pommier, Yves Renard
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30 ===========================================================================*/
31 
32 /**@file getfem_mesher.h
33  @author Julien Pommier <[email protected]>, Yves Renard <[email protected]>
34  @date May 1, 2004.
35  @brief An experimental mesher.
36 */
37 
38 #ifndef GETFEM_MESHER_H__
39 #define GETFEM_MESHER_H__
40 
41 
42 #include "getfem_mesh.h"
43 #include "gmm/gmm_condition_number.h"
44 #include "bgeot_comma_init.h"
45 #include "gmm/gmm_solver_bfgs.h"
46 #include "getfem_export.h"
47 #include "bgeot_kdtree.h"
48 #include <typeinfo>
49 
50 namespace getfem {
51 
52  class mesher_virtual_function : virtual public dal::static_stored_object {
53  public:
54  virtual scalar_type operator()(const base_node &P) const = 0;
55  virtual ~mesher_virtual_function() {}
56  };
57 
58  class mvf_constant : public mesher_virtual_function {
59  scalar_type c;
60  public:
61  mvf_constant(scalar_type c_) : c(c_) {}
62  scalar_type operator()(const base_node &) const { return c; }
63  };
64 
65  // Signed distance definition. Should not be a real signed distance but
66  // should satisfy that dist(P) is less than the euclidean distance from
67  // P to the boundary and more than 1/sqrt(N) time this distance.
68 
69 #define SEPS 1e-8
70 
71  class mesher_signed_distance : public mesher_virtual_function {
72  protected:
73  mutable size_type id;
74  public:
75  mesher_signed_distance() : id(size_type(-1)) {}
76  virtual ~mesher_signed_distance() {}
77  virtual bool bounding_box(base_node &bmin, base_node &bmax) const = 0;
78  virtual scalar_type operator()(const base_node &P,
79  dal::bit_vector &bv) const = 0;
80  virtual scalar_type grad(const base_node &P,
81  base_small_vector &G) const = 0;
82  virtual void hess(const base_node &P, base_matrix &H) const = 0;
83  virtual void register_constraints(std::vector<const
84  mesher_signed_distance*>& list) const=0;
85  virtual scalar_type operator()(const base_node &P) const = 0;
86  };
87 
88  typedef std::shared_ptr<const mesher_signed_distance> pmesher_signed_distance;
89 
90  class mesher_half_space : public mesher_signed_distance {
91  base_node x0; base_small_vector n; scalar_type xon;
92  public:
93  mesher_half_space() = default;
94  mesher_half_space(const base_node &x0_, const base_small_vector &n_)
95  : x0(x0_), n(n_)
96  { n /= gmm::vect_norm2(n); xon = gmm::vect_sp(x0, n); }
97  bool bounding_box(base_node &, base_node &) const
98  { return false; }
99  virtual scalar_type operator()(const base_node &P) const
100  { return xon - gmm::vect_sp(P,n); }
101  virtual scalar_type operator()(const base_node &P,
102  dal::bit_vector &bv) const {
103  scalar_type d = xon - gmm::vect_sp(P,n);
104  bv[id] = (gmm::abs(d) < SEPS);
105  return d;
106  }
107  virtual void register_constraints(std::vector<const
108  mesher_signed_distance*>& list) const {
109  id = list.size(); list.push_back(this);
110  }
111  scalar_type grad(const base_node &P, base_small_vector &G) const {
112  G = n; G *= scalar_type(-1);
113  return xon - gmm::vect_sp(P,n);
114  }
115  void hess(const base_node &P, base_matrix &H) const {
116  gmm::resize(H, P.size(), P.size()); gmm::clear(H);
117  }
118 
119  };
120 
121  inline pmesher_signed_distance new_mesher_half_space
122  (const base_node &x0, const base_small_vector &n)
123  { return std::make_shared<mesher_half_space>(x0, n); }
124 
125  class mesher_level_set : public mesher_signed_distance {
126  bgeot::base_poly base;
127  mutable std::vector<base_poly> gradient;
128  mutable std::vector<base_poly> hessian;
129  const fem<base_poly> *pf;
130  mutable int initialized;
131  scalar_type shift_ls; // for the computation of a gap on a level_set.
132  public:
133  bool is_initialized(void) const { return initialized; }
134  mesher_level_set() : initialized(0) {}
135  template <typename VECT>
136  mesher_level_set(pfem pf_, const VECT &coeff_,
137  scalar_type shift_ls_ = scalar_type(0)) {
138  init_base(pf_, coeff_);
139  set_shift(shift_ls_);
140  }
141  void set_shift(scalar_type shift_ls_) {
142  shift_ls = shift_ls_;
143  if (shift_ls != scalar_type(0)) {
144  base_node P(pf->dim()); base_small_vector G(pf->dim());
145  grad(P, G);
146  shift_ls *= gmm::vect_norm2(G);
147  }
148  }
149  template <typename VECT> void init_base(pfem pf_, const VECT &coeff_);
150  void init_grad(void) const;
151  void init_hess(void) const;
152 
153  bool bounding_box(base_node &, base_node &) const
154  { return false; }
155  virtual scalar_type operator()(const base_node &P) const
156  { return bgeot::to_scalar(base.eval(P.begin())) + shift_ls; }
157  virtual scalar_type operator()(const base_node &P,
158  dal::bit_vector &bv) const
159  { scalar_type d = (*this)(P); bv[id] = (gmm::abs(d) < SEPS); return d; }
160  virtual void register_constraints(std::vector<const
161  mesher_signed_distance*>& list) const {
162  id = list.size(); list.push_back(this);
163  }
164  scalar_type grad(const base_node &P, base_small_vector &G) const;
165  void hess(const base_node &P, base_matrix &H) const;
166  };
167 
168  template <typename VECT>
169  void mesher_level_set::init_base(pfem pf_, const VECT &coeff_) {
170  std::vector<scalar_type> coeff(coeff_.begin(), coeff_.end());
171  GMM_ASSERT1(gmm::vect_norm2(coeff) != 0, "level is zero!");
172  pf = dynamic_cast<const fem<base_poly>*> (pf_.get());
173  GMM_ASSERT1(pf, "A polynomial fem is required for level set (got "
174  << typeid(pf_).name() << ")");
175  base = base_poly(pf->base()[0].dim(), pf->base()[0].degree());
176  for (unsigned i=0; i < pf->nb_base(0); ++i) {
177  base += pf->base()[i] * coeff[i];
178  }
179  initialized = 0;
180  }
181 
182  template <typename VECT>
183  inline pmesher_signed_distance new_mesher_level_set
184  (pfem pf, const VECT &coeff, scalar_type shift_ls = scalar_type(0))
185  { return std::make_shared<mesher_level_set>(pf, coeff, shift_ls); }
186 
187 
188  class mesher_ball : public mesher_signed_distance {
189  base_node x0; scalar_type R;
190  public:
191  mesher_ball(base_node x0_, scalar_type R_) : x0(x0_), R(R_) {}
192  bool bounding_box(base_node &bmin, base_node &bmax) const {
193  bmin = bmax = x0;
194  for (size_type i=0; i < x0.size(); ++i) { bmin[i] -= R; bmax[i] += R; }
195  return true;
196  }
197  virtual scalar_type operator()(const base_node &P,
198  dal::bit_vector &bv) const {
199  scalar_type d = gmm::vect_dist2(P,x0)-R;
200  bv[id] = (gmm::abs(d) < SEPS);
201  return d;
202  }
203  virtual scalar_type operator()(const base_node &P) const
204  { return gmm::vect_dist2(P,x0)-R; }
205  virtual void register_constraints(std::vector<const
206  mesher_signed_distance*>& list) const {
207  id = list.size(); list.push_back(this);
208  }
209  scalar_type grad(const base_node &P, base_small_vector &G) const {
210  G = P; G -= x0;
211  scalar_type e= gmm::vect_norm2(G), d = e - R;
212  while (e == scalar_type(0))
213  { gmm::fill_random(G); e = gmm::vect_norm2(G); }
214  G /= e;
215  return d;
216  }
217  void hess(const base_node &, base_matrix &) const {
218  GMM_ASSERT1(false, "Sorry, to be done");
219  }
220  };
221 
222  inline pmesher_signed_distance new_mesher_ball(base_node x0, scalar_type R)
223  { return std::make_shared<mesher_ball>(x0, R); }
224 
225  class mesher_rectangle : public mesher_signed_distance {
226  // ajouter une rotation rigide et translation ..
227  base_node rmin, rmax;
228  std::vector<mesher_half_space> hfs;
229  public:
230  mesher_rectangle(base_node rmin_, base_node rmax_)
231  : rmin(rmin_), rmax(rmax_), hfs(rmin.size()*2) {
232  base_node n(rmin_.size());
233  for (unsigned k = 0; k < rmin.size(); ++k) {
234  n[k] = 1.0;
235  hfs[k*2] = mesher_half_space(rmin, n);
236  n[k] = -1.0;
237  hfs[k*2+1] = mesher_half_space(rmax, n);
238  n[k] = 0.0;
239  }
240  }
241  bool bounding_box(base_node &bmin, base_node &bmax) const {
242  bmin = rmin; bmax = rmax;
243  return true;
244  }
245  virtual scalar_type operator()(const base_node &P) const {
246  size_type N = rmin.size();
247  scalar_type d = rmin[0] - P[0];
248  for (size_type i=0; i < N; ++i) {
249  d = std::max(d, rmin[i] - P[i]);
250  d = std::max(d, P[i] - rmax[i]);
251  }
252  return d;
253  }
254 
255  virtual scalar_type operator()(const base_node &P, dal::bit_vector &bv)
256  const {
257  scalar_type d = this->operator()(P);
258  if (gmm::abs(d) < SEPS)
259  for (int k = 0; k < 2*rmin.size(); ++k) hfs[k](P, bv);
260  return d;
261  }
262  scalar_type grad(const base_node &P, base_small_vector &G) const {
263  unsigned i = 0; scalar_type di = hfs[i](P);
264  for (int k = 1; k < 2*rmin.size(); ++k) {
265  scalar_type dk = hfs[k](P);
266  if (dk > di) { i = k; di = dk; }
267  }
268  return hfs[i].grad(P, G);
269  }
270  void hess(const base_node &P, base_matrix &H) const {
271  gmm::resize(H, P.size(), P.size()); gmm::clear(H);
272  }
273  virtual void register_constraints(std::vector<const
274  mesher_signed_distance*>& list) const {
275  for (int k = 0; k < 2*rmin.size(); ++k)
276  hfs[k].register_constraints(list);
277  }
278  };
279 
280  inline pmesher_signed_distance new_mesher_rectangle(base_node rmin,
281  base_node rmax)
282  { return std::make_shared<mesher_rectangle>(rmin, rmax); }
283 
284  class mesher_simplex_ref : public mesher_signed_distance {
285  // To be added : rigid motion, dilatation ...
286  std::vector<mesher_half_space> hfs;
287  unsigned N;
288  base_node org;
289  public:
290  mesher_simplex_ref(unsigned N_) : hfs(N_+1), N(N_), org(N_) {
291  base_node no(N);
292  for (unsigned k = 0; k < N; ++k) {
293  no[k] = 1.0;
294  hfs[k] = mesher_half_space(org, no);
295  no[k] = 0.0;
296  }
297  std::fill(org.begin(), org.end(), 1.0/N);
298  no = -org;
299  hfs[N] = mesher_half_space(org, no);
300  }
301  bool bounding_box(base_node &bmin, base_node &bmax) const {
302  bmin.resize(N); bmax.resize(N);
303  std::fill(bmin.begin(), bmin.end(), scalar_type(0));
304  std::fill(bmax.begin(), bmax.end(), scalar_type(1));
305  return true;
306  }
307  virtual scalar_type operator()(const base_node &P) const {
308  scalar_type d = - P[0];
309  for (size_type i=1; i < N; ++i) d = std::max(d, - P[i]);
310  d = std::max(d, gmm::vect_sp(P - org, org) / gmm::vect_norm2(org));
311  return d;
312  }
313 
314  virtual scalar_type operator()(const base_node &P, dal::bit_vector &bv)
315  const {
316  scalar_type d = this->operator()(P);
317  if (gmm::abs(d) < SEPS) for (unsigned k = 0; k < N+1; ++k) hfs[k](P, bv);
318  return d;
319  }
320  scalar_type grad(const base_node &P, base_small_vector &G) const {
321  unsigned i = 0; scalar_type di = hfs[i](P);
322  for (unsigned k = 1; k < N+1; ++k) {
323  scalar_type dk = hfs[k](P);
324  if (dk > di) { i = k; di = dk; }
325  }
326  return hfs[i].grad(P, G);
327  }
328  void hess(const base_node &P, base_matrix &H) const {
329  gmm::resize(H, P.size(), P.size()); gmm::clear(H);
330  }
331  virtual void register_constraints(std::vector<const
332  mesher_signed_distance*>& list) const
333  { for (unsigned k = 0; k < N+1; ++k) hfs[k].register_constraints(list); }
334  };
335 
336  inline pmesher_signed_distance new_mesher_simplex_ref(unsigned N)
337  { return std::make_shared<mesher_simplex_ref>(N); }
338 
339 
340  class mesher_prism_ref : public mesher_signed_distance {
341  // To be added : rigid motion, dilatation ...
342  std::vector<mesher_half_space> hfs;
343  unsigned N;
344  base_node org;
345  public:
346  mesher_prism_ref(unsigned N_) : hfs(N_+2), N(N_) {
347  base_node no(N);
348  org = no;
349  for (unsigned k = 0; k < N; ++k) {
350  no[k] = 1.0;
351  hfs[k] = mesher_half_space(org, no);
352  no[k] = 0.0;
353  }
354  no[N-1] = -1.0;
355  org[N-1] = 1.0;
356  hfs[N] = mesher_half_space(org, no);
357  std::fill(org.begin(), org.end(), 1.0/N);
358  org[N-1] = 0.0;
359  no = -org;
360  hfs[N+1] = mesher_half_space(org, no);
361  }
362  bool bounding_box(base_node &bmin, base_node &bmax) const {
363  bmin.resize(N); bmax.resize(N);
364  std::fill(bmin.begin(), bmin.end(), scalar_type(0));
365  std::fill(bmax.begin(), bmax.end(), scalar_type(1));
366  return true;
367  }
368  virtual scalar_type operator()(const base_node &P) const {
369  scalar_type d = - P[0];
370  for (size_type i=1; i < N; ++i) d = std::max(d, - P[i]);
371  d = std::max(d, P[N-1] - scalar_type(1));
372  d = std::max(d, gmm::vect_sp(P - org, org) / gmm::vect_norm2(org));
373  return d;
374  }
375 
376  virtual scalar_type operator()(const base_node &P, dal::bit_vector &bv)
377  const {
378  scalar_type d = this->operator()(P);
379  if (gmm::abs(d) < SEPS) for (unsigned k = 0; k < N+2; ++k) hfs[k](P, bv);
380  return d;
381  }
382  scalar_type grad(const base_node &P, base_small_vector &G) const {
383  unsigned i = 0; scalar_type di = hfs[i](P);
384  for (unsigned k = 1; k < N+2; ++k) {
385  scalar_type dk = hfs[k](P);
386  if (dk > di) { i = k; di = dk; }
387  }
388  return hfs[i].grad(P, G);
389  }
390  void hess(const base_node &P, base_matrix &H) const {
391  gmm::resize(H, P.size(), P.size()); gmm::clear(H);
392  }
393  virtual void register_constraints(std::vector<const
394  mesher_signed_distance*>& list) const
395  { for (unsigned k = 0; k < N+2; ++k) hfs[k].register_constraints(list); }
396  };
397 
398  inline pmesher_signed_distance new_mesher_prism_ref(unsigned N)
399  { return std::make_shared<mesher_prism_ref>(N); }
400 
401 
402 
403  // Rem : It would be better for the convergence of Newton's methods to
404  // take somthing more smooth than min. for instance tthe square root
405  // of positive parts or negative parts depending on the position of
406  // point (in or out the domain).
407 
408  class mesher_union : public mesher_signed_distance {
409  std::vector<pmesher_signed_distance> dists;
410  mutable std::vector<scalar_type> vd;
411  mutable bool isin;
412  bool with_min;
413  public:
414  mesher_union(const std::vector<pmesher_signed_distance>
415  &dists_) : dists(dists_)
416  { vd.resize(dists.size()); with_min = true; }
417 
418  mesher_union
419  (const pmesher_signed_distance &a,
420  const pmesher_signed_distance &b,
421  const pmesher_signed_distance &c = pmesher_signed_distance(),
422  const pmesher_signed_distance &d = pmesher_signed_distance(),
423  const pmesher_signed_distance &e = pmesher_signed_distance(),
424  const pmesher_signed_distance &f = pmesher_signed_distance(),
425  const pmesher_signed_distance &g = pmesher_signed_distance(),
426  const pmesher_signed_distance &h = pmesher_signed_distance(),
427  const pmesher_signed_distance &i = pmesher_signed_distance(),
428  const pmesher_signed_distance &j = pmesher_signed_distance(),
429  const pmesher_signed_distance &k = pmesher_signed_distance(),
430  const pmesher_signed_distance &l = pmesher_signed_distance(),
431  const pmesher_signed_distance &m = pmesher_signed_distance(),
432  const pmesher_signed_distance &n = pmesher_signed_distance(),
433  const pmesher_signed_distance &o = pmesher_signed_distance(),
434  const pmesher_signed_distance &p = pmesher_signed_distance(),
435  const pmesher_signed_distance &q = pmesher_signed_distance(),
436  const pmesher_signed_distance &r = pmesher_signed_distance(),
437  const pmesher_signed_distance &s = pmesher_signed_distance(),
438  const pmesher_signed_distance &t = pmesher_signed_distance()) {
439  dists.push_back(a); dists.push_back(b);
440  with_min = true;
441  if (c) dists.push_back(c);
442  if (d) dists.push_back(d);
443  if (e) dists.push_back(e);
444  if (f) dists.push_back(f);
445  if (g) dists.push_back(g);
446  if (h) dists.push_back(h);
447  if (i) dists.push_back(i);
448  if (j) dists.push_back(j);
449  if (k) dists.push_back(k);
450  if (l) dists.push_back(l);
451  if (m) dists.push_back(m);
452  if (n) dists.push_back(n);
453  if (o) dists.push_back(o);
454  if (p) dists.push_back(p);
455  if (q) dists.push_back(q);
456  if (r) dists.push_back(r);
457  if (s) dists.push_back(s);
458  if (t) dists.push_back(t);
459  vd.resize(dists.size());
460  }
461 
462  bool bounding_box(base_node &bmin, base_node &bmax) const {
463  base_node bmin2, bmax2;
464  bool b = dists[0]->bounding_box(bmin, bmax);
465  if (!b) return false;
466  for (size_type k = 1; k < dists.size(); ++k) {
467  b = dists[k]->bounding_box(bmin2, bmax2);
468  if (!b) return false;
469  for (unsigned i=0; i < bmin.size(); ++i) {
470  bmin[i] = std::min(bmin[i],bmin2[i]);
471  bmax[i] = std::max(bmax[i],bmax2[i]);
472  }
473  }
474  return true;
475  }
476  virtual scalar_type operator()(const base_node &P) const {
477  scalar_type d, f(0), g(1);
478  if (with_min) {
479  d = (*(dists[0]))(P);
480  for (size_type k = 1; k < dists.size(); ++k)
481  d = std::min(d, (*(dists[k]))(P));
482  }
483  else { // essai raté ...
484  isin = false;
485  for (size_type k = 0; k < dists.size(); ++k) {
486  vd[k] = (*(dists[k]))(P);
487  if (vd[k] <= scalar_type(0)) isin = true;
488  f += gmm::sqr(gmm::neg(vd[k]));
489  g *= gmm::pos(vd[k]);
490  }
491  d = isin ? -gmm::sqrt(f)
492  : pow(g, scalar_type(1) / scalar_type(dists.size()));
493  }
494  return d;
495  }
496  scalar_type operator()(const base_node &P, dal::bit_vector &bv) const {
497  if (with_min) {
498  scalar_type d = vd[0] = (*(dists[0]))(P);
499  bool ok = (d > -SEPS);
500  for (size_type k = 1; k < dists.size(); ++k) {
501  vd[k] = (*(dists[k]))(P); if (vd[k] <= -SEPS) ok = false;
502  d = std::min(d,vd[k]);
503  }
504  for (size_type k = 0; ok && k < dists.size(); ++k) {
505  if (vd[k] < SEPS) (*(dists[k]))(P, bv);
506  }
507  return d;
508  }
509  else { // essai raté ...
510  vd[0] = (*(dists[0]))(P);
511  bool ok = (vd[0] > -SEPS);
512  for (size_type k = 1; k < dists.size(); ++k) {
513  vd[k] = (*(dists[k]))(P); if (vd[k] <= -SEPS) ok = false;
514  }
515  for (size_type k = 0; ok && k < dists.size(); ++k) {
516  if (vd[k] < SEPS) (*(dists[k]))(P, bv);
517  }
518  return operator()(P);
519  }
520  }
521  virtual void register_constraints(std::vector<const
522  mesher_signed_distance*>& list) const {
523  for (size_type k = 0; k < dists.size(); ++k)
524  dists[k]->register_constraints(list);
525  }
526  scalar_type grad(const base_node &P, base_small_vector &G) const {
527  scalar_type d;
528  if (with_min /* || gmm::abs(d) < SEPS*/) {
529  d = (*(dists[0]))(P);
530  size_type i = 0;
531  for (size_type k = 1; k < dists.size(); ++k) {
532  scalar_type d2 = (*(dists[k]))(P);
533  if (d2 < d) { d = d2; i = k; }
534  }
535  return dists[i]->grad(P, G);
536  }
537  else { // essai raté ...
538  d = operator()(P);
539  base_small_vector Gloc;
540  for (size_type k = 0; k < dists.size(); ++k) {
541  dists[k]->grad(P, Gloc);
542  if (isin)
543  Gloc *= -gmm::neg(vd[k]);
544  else
545  Gloc *= pow(d, scalar_type(dists.size())) / vd[k];
546  if (!k) G = Gloc; else G += Gloc;
547  }
548  if (isin)
549  G *= scalar_type(1) / d;
550  else
551  G /= pow(d, scalar_type(dists.size()-1)) * scalar_type(dists.size());
552  return d;
553  }
554  }
555  void hess(const base_node &P, base_matrix &H) const {
556  scalar_type d = (*(dists[0]))(P);
557  if (with_min || gmm::abs(d) < SEPS) {
558  size_type i = 0;
559  for (size_type k = 1; k < dists.size(); ++k) {
560  scalar_type d2 = (*(dists[k]))(P);
561  if (d2 < d) { d = d2; i = k; }
562  }
563  dists[i]->hess(P, H);
564  }
565  else {
566  GMM_ASSERT1(false, "Sorry, to be done");
567  }
568  }
569  };
570 
571  inline pmesher_signed_distance new_mesher_union
572  (const std::vector<pmesher_signed_distance> &dists)
573  { return std::make_shared<mesher_union>(dists); }
574 
575  inline pmesher_signed_distance new_mesher_union
576  (const pmesher_signed_distance &a,
577  const pmesher_signed_distance &b,
578  const pmesher_signed_distance &c = pmesher_signed_distance(),
579  const pmesher_signed_distance &d = pmesher_signed_distance(),
580  const pmesher_signed_distance &e = pmesher_signed_distance(),
581  const pmesher_signed_distance &f = pmesher_signed_distance(),
582  const pmesher_signed_distance &g = pmesher_signed_distance(),
583  const pmesher_signed_distance &h = pmesher_signed_distance(),
584  const pmesher_signed_distance &i = pmesher_signed_distance(),
585  const pmesher_signed_distance &j = pmesher_signed_distance(),
586  const pmesher_signed_distance &k = pmesher_signed_distance(),
587  const pmesher_signed_distance &l = pmesher_signed_distance(),
588  const pmesher_signed_distance &m = pmesher_signed_distance(),
589  const pmesher_signed_distance &n = pmesher_signed_distance(),
590  const pmesher_signed_distance &o = pmesher_signed_distance(),
591  const pmesher_signed_distance &p = pmesher_signed_distance(),
592  const pmesher_signed_distance &q = pmesher_signed_distance(),
593  const pmesher_signed_distance &r = pmesher_signed_distance(),
594  const pmesher_signed_distance &s = pmesher_signed_distance(),
595  const pmesher_signed_distance &t = pmesher_signed_distance()) {
596  return std::make_shared<mesher_union>
597  (a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t);
598  }
599 
600 
601 
602  class mesher_intersection : public mesher_signed_distance {
603  std::vector<pmesher_signed_distance> dists;
604  mutable std::vector<scalar_type> vd;
605 
606  // const mesher_signed_distance &a, &b;
607  public:
608 
609  mesher_intersection(const std::vector<pmesher_signed_distance>
610  &dists_) : dists(dists_)
611  { vd.resize(dists.size()); }
612 
613  mesher_intersection
614  (const pmesher_signed_distance &a,
615  const pmesher_signed_distance &b,
616  const pmesher_signed_distance &c = pmesher_signed_distance(),
617  const pmesher_signed_distance &d = pmesher_signed_distance(),
618  const pmesher_signed_distance &e = pmesher_signed_distance(),
619  const pmesher_signed_distance &f = pmesher_signed_distance(),
620  const pmesher_signed_distance &g = pmesher_signed_distance(),
621  const pmesher_signed_distance &h = pmesher_signed_distance(),
622  const pmesher_signed_distance &i = pmesher_signed_distance(),
623  const pmesher_signed_distance &j = pmesher_signed_distance(),
624  const pmesher_signed_distance &k = pmesher_signed_distance(),
625  const pmesher_signed_distance &l = pmesher_signed_distance(),
626  const pmesher_signed_distance &m = pmesher_signed_distance(),
627  const pmesher_signed_distance &n = pmesher_signed_distance(),
628  const pmesher_signed_distance &o = pmesher_signed_distance(),
629  const pmesher_signed_distance &p = pmesher_signed_distance(),
630  const pmesher_signed_distance &q = pmesher_signed_distance(),
631  const pmesher_signed_distance &r = pmesher_signed_distance(),
632  const pmesher_signed_distance &s = pmesher_signed_distance(),
633  const pmesher_signed_distance &t = pmesher_signed_distance()) {
634  dists.push_back(a); dists.push_back(b);
635  if (c) dists.push_back(c);
636  if (d) dists.push_back(d);
637  if (e) dists.push_back(e);
638  if (f) dists.push_back(f);
639  if (g) dists.push_back(g);
640  if (h) dists.push_back(h);
641  if (i) dists.push_back(i);
642  if (j) dists.push_back(j);
643  if (k) dists.push_back(k);
644  if (l) dists.push_back(l);
645  if (m) dists.push_back(m);
646  if (n) dists.push_back(n);
647  if (o) dists.push_back(o);
648  if (p) dists.push_back(p);
649  if (q) dists.push_back(q);
650  if (r) dists.push_back(r);
651  if (s) dists.push_back(s);
652  if (t) dists.push_back(t);
653  vd.resize(dists.size());
654  }
655  bool bounding_box(base_node &bmin, base_node &bmax) const {
656  base_node bmin2, bmax2;
657  bool first;
658  bool b = dists[0]->bounding_box(bmin, bmax); first = !b;
659  for (size_type k = 1; k < dists.size(); ++k) {
660  bool bb = dists[k]->bounding_box(bmin2, bmax2);
661  for (unsigned i=0; i < bmin.size() && bb && !first; ++i) {
662  bmin[i] = std::max(bmin[i],bmin2[i]);
663  bmax[i] = std::max(std::min(bmax[i],bmax2[i]), bmin[i]);
664  }
665  if (first && bb) { bmin = bmin2; bmax = bmax2; first = false; }
666  b = b || bb;
667  }
668  return b;
669  }
670  virtual scalar_type operator()(const base_node &P) const {
671  scalar_type d = (*(dists[0]))(P);
672  for (size_type k = 1; k < dists.size(); ++k)
673  d = std::max(d, (*(dists[k]))(P));
674  return d;
675 
676  }
677  scalar_type operator()(const base_node &P, dal::bit_vector &bv) const {
678  scalar_type d = vd[0] = (*(dists[0]))(P);
679  bool ok = (d < SEPS);
680  for (size_type k = 1; k < dists.size(); ++k) {
681  vd[k] = (*(dists[k]))(P); if (vd[k] >= SEPS) ok = false;
682  d = std::min(d,vd[k]);
683  }
684  for (size_type k = 0; ok && k < dists.size(); ++k) {
685  if (vd[k] > -SEPS) (*(dists[k]))(P, bv);
686  }
687  return d;
688  }
689  virtual void register_constraints(std::vector<const
690  mesher_signed_distance*>& list) const {
691  for (size_type k = 0; k < dists.size(); ++k) {
692  dists[k]->register_constraints(list);
693  }
694  }
695  scalar_type grad(const base_node &P, base_small_vector &G) const {
696  scalar_type d = (*(dists[0]))(P);
697  size_type i = 0;
698  for (size_type k = 1; k < dists.size(); ++k) {
699  scalar_type d2 = (*(dists[k]))(P);
700  if (d2 > d) { d = d2; i = k; }
701  }
702  return dists[i]->grad(P, G);
703  }
704  void hess(const base_node &P, base_matrix &H) const {
705  scalar_type d = (*(dists[0]))(P);
706  size_type i = 0;
707  for (size_type k = 1; k < dists.size(); ++k) {
708  scalar_type d2 = (*(dists[k]))(P);
709  if (d2 > d) { d = d2; i = k; }
710  }
711  dists[i]->hess(P, H);
712  }
713  };
714 
715  inline pmesher_signed_distance new_mesher_intersection
716  (const std::vector<pmesher_signed_distance> &dists)
717  { return std::make_shared<mesher_intersection>(dists); }
718 
719  inline pmesher_signed_distance new_mesher_intersection
720  (const pmesher_signed_distance &a,
721  const pmesher_signed_distance &b,
722  const pmesher_signed_distance &c = pmesher_signed_distance(),
723  const pmesher_signed_distance &d = pmesher_signed_distance(),
724  const pmesher_signed_distance &e = pmesher_signed_distance(),
725  const pmesher_signed_distance &f = pmesher_signed_distance(),
726  const pmesher_signed_distance &g = pmesher_signed_distance(),
727  const pmesher_signed_distance &h = pmesher_signed_distance(),
728  const pmesher_signed_distance &i = pmesher_signed_distance(),
729  const pmesher_signed_distance &j = pmesher_signed_distance(),
730  const pmesher_signed_distance &k = pmesher_signed_distance(),
731  const pmesher_signed_distance &l = pmesher_signed_distance(),
732  const pmesher_signed_distance &m = pmesher_signed_distance(),
733  const pmesher_signed_distance &n = pmesher_signed_distance(),
734  const pmesher_signed_distance &o = pmesher_signed_distance(),
735  const pmesher_signed_distance &p = pmesher_signed_distance(),
736  const pmesher_signed_distance &q = pmesher_signed_distance(),
737  const pmesher_signed_distance &r = pmesher_signed_distance(),
738  const pmesher_signed_distance &s = pmesher_signed_distance(),
739  const pmesher_signed_distance &t = pmesher_signed_distance()) {
740  return std::make_shared<mesher_intersection>
741  (a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t);
742  }
743 
744 
745 
746  class mesher_setminus : public mesher_signed_distance {
747  pmesher_signed_distance a, b;
748  public:
749  mesher_setminus(const pmesher_signed_distance& a_,
750  const pmesher_signed_distance &b_) : a(a_), b(b_) {}
751  bool bounding_box(base_node &bmin, base_node &bmax) const
752  { return a->bounding_box(bmin,bmax); }
753  scalar_type operator()(const base_node &P, dal::bit_vector &bv) const {
754  scalar_type da = (*a)(P), db = -(*b)(P);
755  if (da < SEPS && db < SEPS) {
756  if (da > -SEPS) (*a)(P, bv);
757  if (db > -SEPS) (*b)(P, bv);
758  }
759  return std::max(da, db);
760  }
761  scalar_type operator()(const base_node &P) const
762  { return std::max((*a)(P),-(*b)(P)); }
763  virtual void register_constraints(std::vector<const
764  mesher_signed_distance*>& list) const {
765  a->register_constraints(list); b->register_constraints(list);
766  }
767  scalar_type grad(const base_node &P, base_small_vector &G) const {
768  scalar_type da = (*a)(P), db = -(*b)(P);
769  if (da > db) return a->grad(P, G);
770  else { b->grad(P, G); G *= scalar_type(-1); return db; }
771  }
772  void hess(const base_node &P, base_matrix &H) const {
773  scalar_type da = (*a)(P), db = -(*b)(P);
774  if (da > db) a->hess(P, H);
775  else { b->hess(P, H); gmm::scale(H, scalar_type(-1)); }
776  }
777 
778  };
779 
780  inline pmesher_signed_distance new_mesher_setminus
781  (const pmesher_signed_distance &a, const pmesher_signed_distance &b)
782  { return std::make_shared<mesher_setminus>(a, b); }
783 
784 
785  class mesher_tube : public mesher_signed_distance {
786  base_node x0; base_node n; scalar_type R;
787  public:
788  mesher_tube(base_node x0_, base_node n_, scalar_type R_)
789  : x0(x0_), n(n_), R(R_)
790  { n /= gmm::vect_norm2(n); }
791  bool bounding_box(base_node &, base_node &) const
792  { return false; }
793  virtual scalar_type operator()(const base_node &P) const {
794  base_node v(P); v -= x0;
795  gmm::add(gmm::scaled(n, -gmm::vect_sp(v, n)), v);
796  return gmm::vect_norm2(v) - R;
797  }
798  virtual scalar_type operator()(const base_node &P,
799  dal::bit_vector &bv) const {
800  scalar_type d = (*this)(P);
801  bv[id] = (gmm::abs(d) < SEPS);
802  return d;
803  }
804  virtual void register_constraints(std::vector<const
805  mesher_signed_distance*>& list) const {
806  id = list.size(); list.push_back(this);
807  }
808  scalar_type grad(const base_node &P, base_small_vector &G) const {
809  G = P; G -= x0;
810  gmm::add(gmm::scaled(n, -gmm::vect_sp(G, n)), G);
811  scalar_type e = gmm::vect_norm2(G), d = e - R;
812  while (e == scalar_type(0)) {
813  gmm::fill_random(G);
814  gmm::add(gmm::scaled(n, -gmm::vect_sp(G, n)), G);
815  e = gmm::vect_norm2(G);
816  }
817  G /= e;
818  return d;
819  }
820  void hess(const base_node &, base_matrix &) const {
821  GMM_ASSERT1(false, "Sorry, to be done");
822  }
823  };
824 
825  inline pmesher_signed_distance new_mesher_tube(base_node x0, base_node n,
826  scalar_type R)
827  { return std::make_shared<mesher_tube>(x0,n,R); }
828 
829 
830  class mesher_cylinder : public mesher_signed_distance {
831  base_node x0; base_small_vector n;
832  scalar_type L, R;
833  pmesher_signed_distance t, p1, p2, i1;
834  public:
835  mesher_cylinder(const base_node &c, const base_small_vector &no,
836  scalar_type L_, scalar_type R_)
837  : x0(c), n(no/gmm::vect_norm2(no)), L(L_), R(R_),
838  t(new_mesher_tube(x0, n, R)),
839  p1(new_mesher_half_space(x0, n)),
840  p2(new_mesher_half_space(x0+n*L, -1.0 * n)),
841  i1(new_mesher_intersection(p1, p2, t)) {}
842  bool bounding_box(base_node &bmin, base_node &bmax) const {
843  base_node x1(x0+n*L);
844  bmin = bmax = x0;
845  for (unsigned i = 0; i < gmm::vect_size(x0); ++i) {
846  bmin[i] = std::min(x0[i], x1[i]) - R;
847  bmax[i] = std::max(x0[i], x1[i]) + R;
848  }
849  return true;
850  }
851  virtual scalar_type operator()(const base_node &P) const {return (*i1)(P); }
852  virtual scalar_type operator()(const base_node &P,
853  dal::bit_vector& bv) const
854  { return (*i1)(P, bv); }
855  scalar_type grad(const base_node &P, base_small_vector &G) const
856  { return i1->grad(P, G); }
857  void hess(const base_node &, base_matrix &) const {
858  GMM_ASSERT1(false, "Sorry, to be done");
859  }
860  virtual void register_constraints(std::vector<const
861  mesher_signed_distance*>& list) const
862  { i1->register_constraints(list); }
863  };
864 
865  inline pmesher_signed_distance new_mesher_cylinder
866  (const base_node &c, const base_small_vector &no,
867  scalar_type L, scalar_type R)
868  { return std::make_shared<mesher_cylinder>(c,no,L,R); }
869 
870 
871  class mesher_infinite_cone : public mesher_signed_distance {
872  // uses the true distance to a cone.
873  base_node x0; base_node n; scalar_type alpha;
874  public:
875  mesher_infinite_cone(base_node x0_, base_node n_, scalar_type alpha_)
876  : x0(x0_), n(n_), alpha(alpha_)
877  { n /= gmm::vect_norm2(n); }
878  bool bounding_box(base_node &, base_node &) const
879  { return false; }
880  virtual scalar_type operator()(const base_node &P) const {
881  base_node v(P); v -= x0;
882  scalar_type v_n = gmm::vect_sp(v, n);
883  gmm::add(gmm::scaled(n, -v_n), v);
884  return gmm::vect_norm2(v) * cos(alpha) - gmm::abs(v_n) * sin(alpha);
885  }
886  virtual scalar_type operator()(const base_node &P,
887  dal::bit_vector &bv) const {
888  scalar_type d = (*this)(P);
889  bv[id] = (gmm::abs(d) < SEPS);
890  return d;
891  }
892  virtual void register_constraints(std::vector<const
893  mesher_signed_distance*>& list) const {
894  id = list.size(); list.push_back(this);
895  }
896  scalar_type grad(const base_node &P, base_small_vector &v) const {
897  v = P; v -= x0;
898  scalar_type v_n = gmm::vect_sp(v, n);
899  gmm::add(gmm::scaled(n, -v_n), v);
900  scalar_type no = gmm::vect_norm2(v);
901  scalar_type d = no * cos(alpha) - gmm::abs(v_n) * sin(alpha);
902  while (no == scalar_type(0)) {
903  gmm::fill_random(v);
904  gmm::add(gmm::scaled(n, -gmm::vect_sp(v, n)), v);
905  no = gmm::vect_norm2(v);
906  }
907  v *= cos(alpha) / no;
908  v -= (sin(alpha) * gmm::sgn(v_n)) * n;
909  return d;
910  }
911  void hess(const base_node &, base_matrix &) const {
912  GMM_ASSERT1(false, "Sorry, to be done");
913  }
914  };
915 
916  inline pmesher_signed_distance new_mesher_infinite_cone
917  (base_node x0, base_node n, scalar_type alpha)
918  { return std::make_shared<mesher_infinite_cone>(x0,n,alpha); }
919 
920 
921  class mesher_cone : public mesher_signed_distance {
922  base_node x0; base_small_vector n;
923  scalar_type L, alpha;
924  pmesher_signed_distance t, p1, p2, i1;
925  public:
926  mesher_cone(const base_node &c, const base_small_vector &no,
927  scalar_type L_, scalar_type alpha_)
928  : x0(c), n(no/gmm::vect_norm2(no)), L(L_), alpha(alpha_),
929  t(new_mesher_infinite_cone(x0, n, alpha)),
930  p1(new_mesher_half_space(x0, n)),
931  p2(new_mesher_half_space(x0+n*L, -1.0 * n)),
932  i1(new_mesher_intersection(p1, p2, t)) {}
933  bool bounding_box(base_node &bmin, base_node &bmax) const {
934  base_node x1(x0+n*L);
935  scalar_type R = L * sin(alpha);
936  bmin = bmax = x0;
937  for (unsigned i = 0; i < gmm::vect_size(x0); ++i) {
938  bmin[i] = std::min(x0[i], x1[i]) - R;
939  bmax[i] = std::max(x0[i], x1[i]) + R;
940  }
941  return true;
942  }
943  virtual scalar_type operator()(const base_node &P) const {return (*i1)(P); }
944  virtual scalar_type operator()(const base_node &P,
945  dal::bit_vector& bv) const
946  { return (*i1)(P, bv); }
947  scalar_type grad(const base_node &P, base_small_vector &G) const
948  { return i1->grad(P, G); }
949  void hess(const base_node &, base_matrix &) const {
950  GMM_ASSERT1(false, "Sorry, to be done");
951  }
952  virtual void register_constraints(std::vector<const
953  mesher_signed_distance*>& list) const
954  { i1->register_constraints(list); }
955  };
956 
957  inline pmesher_signed_distance new_mesher_cone
958  (const base_node &c, const base_small_vector &no,
959  scalar_type L, scalar_type alpha)
960  { return std::make_shared<mesher_cone>(c,no,L,alpha); }
961 
962 
963  class mesher_ellipse : public mesher_signed_distance {
964  base_node x0; base_small_vector n, t;
965  scalar_type r, R, a;
966  public:
967  mesher_ellipse(const base_node &center, const base_small_vector &no,
968  scalar_type r_, scalar_type R_)
969  : x0(center), n(no/gmm::vect_norm2(no)), r(r_), R(R_) {
970  t[0] = -n[1]; t[1] = n[0];
971  if (R < r) { std::swap(r, R); std::swap(n, t); }
972  a = sqrt(R*R - r*r);
973  }
974  bool bounding_box(base_node &bmin, base_node &bmax) const {
975  bmin = bmax = x0;
976  for (unsigned i = 0; i < 2; ++i) { bmin[i] -= R; bmax[i] += R; }
977  return true;
978  }
979  virtual scalar_type operator()(const base_node &P) const {
980  base_small_vector v(P); v -= x0;
981  scalar_type vt = gmm::vect_sp(v, t);
982  vt = std::max(-a, std::min(a, vt));
983  base_node x1 = x0 + t*vt;
984  base_small_vector v1(P); v1 -= x1;
985  scalar_type v1n = gmm::vect_sp(v1, n), v1t = gmm::vect_sp(v1, t);
986  scalar_type x1n = gmm::vect_sp(x1, n), x1t = gmm::vect_sp(x1, t);
987  scalar_type ea = v1n*v1n / (r*r) + v1t * v1t / (R*R);
988  scalar_type eb = 2. * (x1n*v1n / (r*r) + x1t*v1t / (R*R));
989  scalar_type ec = x1n*x1n / (r*r) + x1t * x1t / (R*R);
990 
991  scalar_type delta = eb*eb - 4 * ea * ec;
992  assert(delta >= 0);
993  scalar_type lambda = (-eb + sqrt(delta)) / (2. * ea);
994  return (1.-lambda)*gmm::vect_norm2(v1);
995  }
996  virtual scalar_type operator()(const base_node &P,
997  dal::bit_vector& bv) const {
998  scalar_type d = this->operator()(P);
999  bv[id] = (gmm::abs(d) < SEPS);
1000  return d;
1001  }
1002  scalar_type grad(const base_node &, base_small_vector &) const
1003  { GMM_ASSERT1(false, "Sorry, to be done"); return 0.; }
1004  void hess(const base_node &, base_matrix &) const {
1005  GMM_ASSERT1(false, "Sorry, to be done");
1006  }
1007  virtual void register_constraints(std::vector<const
1008  mesher_signed_distance*>& list) const
1009  { id = list.size(); list.push_back(this); }
1010  };
1011 
1012  inline pmesher_signed_distance new_mesher_ellipse
1013  (const base_node &center, const base_small_vector &no,
1014  scalar_type r, scalar_type R)
1015  { return std::make_shared<mesher_ellipse>(center,no,r,R); }
1016 
1017 
1018 
1019  class mesher_torus : public mesher_signed_distance { // to be done
1020  // ajouter une rotation rigide et translation ..
1021  scalar_type R, r;
1022  public:
1023  mesher_torus(scalar_type RR, scalar_type rr) : R(RR), r(rr) {}
1024  bool bounding_box(base_node &bmin, base_node &bmax) const {
1025  bmin = base_node(3); bmax = base_node(3);
1026  bmin[0] = bmin[1] = -R-r; bmin[2] = -r;
1027  bmax[0] = bmax[1] = +R+r; bmax[2] = +r;
1028  return true;
1029  }
1030  virtual scalar_type operator()(const base_node &P) const {
1031  scalar_type x = P[0], y = P[1], z = P[2], c = sqrt(x*x + y*y);
1032  return (c == 0.) ? R - r : sqrt(gmm::sqr(c-R) + z*z) - r;
1033  }
1034  virtual scalar_type operator()(const base_node &P, dal::bit_vector&bv)
1035  const {
1036  scalar_type d = this->operator()(P);
1037  bv[id] = (gmm::abs(d) < SEPS);
1038  return d;
1039  }
1040  scalar_type grad(const base_node &P, base_small_vector &G) const {
1041  G.resize(3);
1042  scalar_type x = P[0], y = P[1], z = P[2], c = sqrt(x*x + y*y), d(0);
1043  if (c == 0.) {
1044  d = R - r;
1045  gmm::fill_random(G); G[2] = 0.0; G /= gmm::vect_norm2(G);
1046  }
1047  else {
1048  scalar_type w = 1. - R / c, e = sqrt(gmm::sqr(c-R) + z*z);
1049  d = e - r;
1050  if (e == 0.) {
1051  gmm::fill_random(G); G[0] = x; G[1] = y; G /= gmm::vect_norm2(G);
1052  }
1053  else {
1054  G[0] = x * w / e; G[1] = y * w / e; G[2] = z / e;
1055  }
1056  }
1057  return d;
1058  }
1059  void hess(const base_node &, base_matrix &) const {
1060  GMM_ASSERT1(false, "Sorry, to be done");
1061  }
1062  virtual void register_constraints(std::vector<const
1063  mesher_signed_distance*>&list) const
1064  { id = list.size(); list.push_back(this); }
1065  };
1066 
1067 
1068  inline pmesher_signed_distance new_mesher_torus(scalar_type R, scalar_type r)
1069  { return std::make_shared<mesher_torus>(R,r); }
1070 
1071  // mesher
1072  void build_mesh(mesh &m, const pmesher_signed_distance& dist_,
1073  scalar_type h0, const std::vector<base_node> &fixed_points
1074  = std::vector<base_node>(), size_type K = 1, int noise = -1,
1075  size_type iter_max = 500, int prefind = 1,
1076  scalar_type dist_point_hull = 4,
1077  scalar_type boundary_threshold_flatness = 0.11);
1078 
1079  // exported functions
1080  bool try_projection(const mesher_signed_distance& dist, base_node &X,
1081  bool on_surface = false);
1082  bool pure_multi_constraint_projection
1083  (const std::vector<const mesher_signed_distance*> &list_constraints,
1084  base_node &X, const dal::bit_vector &cts);
1085  scalar_type curvature_radius_estimate(const mesher_signed_distance &dist,
1086  base_node X, bool proj = false);
1087  scalar_type min_curvature_radius_estimate
1088  (const std::vector<const mesher_signed_distance*> &list_constraints,
1089  const base_node &X, const dal::bit_vector &cts, size_type hide_first = 0);
1090 
1091 }
1092 
1093 #endif /* GETFEM_MESHER_H__ */
bgeot_comma_init.h
convenient initialization of vectors via overload of "operator,".
bgeot_kdtree.h
Simple implementation of a KD-tree.
bgeot::polynomial::eval
T eval(const ITER &it) const
Evaluate the polynomial.
Definition: bgeot_poly.h:453
dal::static_stored_object
base class for static stored objects
Definition: dal_static_stored_objects.h:206
getfem_export.h
Export solutions to various formats.
getfem_mesh.h
Define a getfem::getfem_mesh object.
gmm::vect_norm2
number_traits< typename linalg_traits< V >::value_type >::magnitude_type vect_norm2(const V &v)
Euclidean norm of a vector.
Definition: gmm_blas.h:557
gmm::fill_random
void fill_random(L &l)
fill a vector or matrix with random value (uniform [-1,1]).
Definition: gmm_blas.h:130
gmm::clear
void clear(L &l)
clear (fill with zeros) a vector or matrix.
Definition: gmm_blas.h:59
gmm::vect_dist2
number_traits< typename linalg_traits< V1 >::value_type >::magnitude_type vect_dist2(const V1 &v1, const V2 &v2)
Euclidean distance between two vectors.
Definition: gmm_blas.h:597
gmm::resize
void resize(V &v, size_type n)
*‍/
Definition: gmm_blas.h:210
gmm::vect_sp
strongest_value_type< V1, V2 >::value_type vect_sp(const V1 &v1, const V2 &v2)
*‍/
Definition: gmm_blas.h:264
gmm::add
void add(const L1 &l1, L2 &l2)
*‍/
Definition: gmm_blas.h:1276
gmm_condition_number.h
computation of the condition number of dense matrices.
gmm_solver_bfgs.h
Implements BFGS (Broyden, Fletcher, Goldfarb, Shanno) algorithm.
getfem::pfem
std::shared_ptr< const getfem::virtual_fem > pfem
type of pointer on a fem description
Definition: getfem_fem.h:244
bgeot::size_type
size_t size_type
used as the common size type in the library
Definition: bgeot_poly.h:49
bgeot::alpha
size_type alpha(short_type n, short_type d)
Return the value of which is the number of monomials of a polynomial of variables and degree .
Definition: bgeot_poly.cc:47
getfem
GEneric Tool for Finite Element Methods.
Definition: getfem_accumulated_distro.h:47