Rheolef  7.2
an efficient C++ finite element environment
basis_fem_RTk.cc
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1 #include "basis_fem_RTk.h"
22 #include "equispaced.icc"
23 #include "warburton.icc"
24 #include "basis_ordering.icc"
25 #include "reference_element_aux.icc"
26 #include "eigen_util.h"
28 #include "piola_fem_hdiv.h"
29 
30 namespace rheolef {
31 using namespace std;
32 
33 // =========================================================================
34 // basis members
35 // =========================================================================
36 // allocators
37 // -------------------------------------------------------------------------
38 template<class T>
40 {
41 }
42 template<class T>
44  : basis_rep<T> (sopt),
45  _b_pre_kp1(),
46  _hat_node(),
47  _vdm(),
48  _inv_vdm(),
49  _quad(),
50  _bar_a(),
51  _bkm1_node_internal_d()
52 {
54  // requieres a hierarchical pre basis => Dubiner or monomial
55  // - monomial is ill conditionned, but easier to decompose
56  // - dubiner: TODO project the tilde_psi basis on it
57  // note: the internal dof basis is independent
58  _b_pre_kp1 = basis_raw_basic<T> ("M"+std::to_string(k+1));
60 
61  // piola FEM transformation:
62  typedef piola_fem_hdiv<T> piola_fem_type;
63  base::_piola_fem.piola_fem<T>::base::operator= (new_macro(piola_fem_type));
64 }
65 template<class T>
66 const Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1>&
68 {
69  base::_initialize_data_guard (hat_K);
70  return _hat_node [hat_K.variant()];
71 }
72 template<class T>
73 const Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>&
75 {
76  base::_initialize_data_guard (hat_K);
77  return _vdm [hat_K.variant()];
78 }
79 template<class T>
80 const Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>&
82 {
83  base::_initialize_data_guard (hat_K);
84  return _inv_vdm [hat_K.variant()];
85 }
86 template<class T>
87 void
89 {
90  size_type k = degree()-1;
91  for (size_type map_d = 0; map_d < 4; ++map_d) {
92  base::_ndof_on_subgeo_internal [map_d].fill (0);
93  }
94  // --------------------------------------------------------------------------
95  // 1. compute ndof and dof pointers:
96  // --------------------------------------------------------------------------
97  // see BreFor-1991 p. 116 for K=t,T & p. 119 for K=q,H ; BerDur-2013 for K=P
98  base::_ndof_on_subgeo_internal [0][reference_element::p] = 1;
99  base::_ndof_on_subgeo_internal [1][reference_element::p] = 1;
100  base::_ndof_on_subgeo_internal [1][reference_element::e] = k; // RT0 is P1 on 1d with K=e
101  base::_ndof_on_subgeo_internal [2][reference_element::e] = k+1;
102  base::_ndof_on_subgeo_internal [2][reference_element::t] = k*(k+1);
103  base::_ndof_on_subgeo_internal [2][reference_element::q] = 2*k*(k+1); // Rav-1981, p. 23
104  base::_ndof_on_subgeo_internal [3][reference_element::t] = (k+1)*(k+2)/2;
105  base::_ndof_on_subgeo_internal [3][reference_element::q] = sqr(k+1);
106  base::_ndof_on_subgeo_internal [3][reference_element::T] = 3*k*(k+1)*(k+2)/6;
107  base::_ndof_on_subgeo_internal [3][reference_element::P] = sqr(k)*(k+1)/2; // see BerDur-2013
108  base::_ndof_on_subgeo_internal [3][reference_element::H] = 3*k*sqr(k+1);
109  base::_helper_make_discontinuous_ndof_on_subgeo (base::is_continuous(), base::_ndof_on_subgeo_internal, base::_ndof_on_subgeo);
110  base::_helper_initialize_first_ixxx_by_dimension_from_nxxx_on_subgeo (base::_ndof_on_subgeo_internal, base::_first_idof_by_dimension_internal);
111  base::_helper_initialize_first_ixxx_by_dimension_from_nxxx_on_subgeo (base::_ndof_on_subgeo, base::_first_idof_by_dimension);
112  // --------------------------------------------------------------------------
113  // 2. nodes for dofs : interpolate v.n on sides + internal quadrature nodes
114  // --------------------------------------------------------------------------
115  // nodes coincides with dofs on sides : Lagrange nodes
116  // interior nodes are quadrature node
117  quadrature_option qopt;
118  qopt.set_family (quadrature_option::gauss);
119  // order = 2*(k-1)) for P_{k-1} internal modes for simplicial elts hat_K=etT
120  // = 2*k for P_{k-1,k,k} internal modes for non-simplicial elts hat_K=qHP
121  qopt.set_order (2*(k+2)); // takes the max to have an uniform quadrature for any cases TODO: too much ? min-adjust=2*k to converge
122  _quad = quadrature<T> (qopt);
123  for (size_type map_d = 0; map_d < 4; ++map_d) {
124  base::_nnod_on_subgeo_internal [map_d].fill (0);
125  }
126  base::_nnod_on_subgeo_internal [0][reference_element::p] = 1;
127  base::_nnod_on_subgeo_internal [1][reference_element::p] = 1;
128  base::_nnod_on_subgeo_internal [1][reference_element::e] = k == 0 ? 0 : _quad.size (reference_element::e);
129  base::_nnod_on_subgeo_internal [2][reference_element::e] = k+1;
130  base::_nnod_on_subgeo_internal [2][reference_element::t] = k*(k+1)/2; // k == 0 ? 0 : _quad.size (reference_element::t);
131  base::_nnod_on_subgeo_internal [2][reference_element::q] = k == 0 ? 0 : _quad.size (reference_element::q);
132  base::_nnod_on_subgeo_internal [3][reference_element::t] = (k+1)*(k+2)/2;
133  base::_nnod_on_subgeo_internal [3][reference_element::q] = sqr(k+1);
134  base::_nnod_on_subgeo_internal [3][reference_element::T] = k == 0 ? 0 : _quad.size (reference_element::T);
135  base::_nnod_on_subgeo_internal [3][reference_element::P] = k == 0 ? 0 : _quad.size (reference_element::P);
136  base::_nnod_on_subgeo_internal [3][reference_element::H] = k == 0 ? 0 : _quad.size (reference_element::H);
137  base::_helper_make_discontinuous_ndof_on_subgeo (base::is_continuous(), base::_nnod_on_subgeo_internal, base::_nnod_on_subgeo);
138  base::_helper_initialize_first_ixxx_by_dimension_from_nxxx_on_subgeo (base::_nnod_on_subgeo_internal, base::_first_inod_by_dimension_internal);
139  base::_helper_initialize_first_ixxx_by_dimension_from_nxxx_on_subgeo (base::_nnod_on_subgeo, base::_first_inod_by_dimension);
140 }
141 template<class T>
142 void
144 {
145  if (hat_K.dimension() == 0) return; // nothing to do
146 
147  // abbrevs
148  size_type d = hat_K.dimension();
149  size_type k = degree()-1;
150  size_type variant = hat_K.variant();
151 
152  // --------------------------------------------------------------------------
153  // 1. nodes for dofs : interpolate v.n on sides + internal quadrature nodes
154  // --------------------------------------------------------------------------
155  // 1.1. insert nodes
156  Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1>& hat_node = _hat_node [hat_K.variant()];
157  hat_node.resize (base::_first_inod_by_dimension [variant][d+1]);
158  size_type loc_nnod = hat_node.size();
159  // 1.2. nodes on sides
160  size_type loc_inod = 0;
161  for (size_type loc_isid = 0, loc_nsid = hat_K.n_subgeo(d-1); loc_isid < loc_nsid; ++loc_isid) {
162  reference_element hat_S = hat_K.subgeo (d-1, loc_isid);
163  size_type sid_variant = hat_S.variant();
164  size_type loc_nsidvert = hat_S.n_vertex();
165  Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1> hat_node_sid;
166  const size_type internal = 1;
167  switch (base::_sopt.get_node()) {
169 #ifdef TODO
170  // TODO: pointset warburton could support the "internal=1" option
171  pointset_lagrange_warburton (sid_variant, k, hat_node_sid, internal); break;
172 #endif // TODO
174  pointset_lagrange_equispaced (sid_variant, k, hat_node_sid, internal); break;
175  default:
176  error_macro ("unsupported node set: " << base::_sopt.get_node_name());
177  }
178  point loc_vertex [4]; // restricted to point_basic<Float> in reference_element::vertex()
179  for (size_type loc_jsidvert = 0; loc_jsidvert < loc_nsidvert; ++loc_jsidvert) {
180  size_type loc_jvertex = hat_K.subgeo_local_vertex (d-1, loc_isid, loc_jsidvert);
181  loc_vertex[loc_jsidvert] = hat_K.vertex (loc_jvertex);
182  }
183  for (size_type loc_inod_sid = 0, loc_nnod_sid = hat_node_sid.size(); loc_inod_sid < loc_nnod_sid; ++loc_inod_sid) {
184  check_macro (loc_inod < size_t(hat_node.size()), "invalid loc_inod");
185  T xi0 = hat_node_sid [loc_inod_sid][0],
186  xi1 = hat_node_sid [loc_inod_sid][1];
187  if (loc_nsidvert == 4) { // map from [-1,1]^2 to [0,1]^2
188  xi0 = (1+xi0)/2;
189  xi1 = (1+xi1)/2;
190  }
191  for (size_type alpha = 0; alpha < d; ++alpha) {
192  hat_node[loc_inod][alpha] = loc_vertex [0][alpha];
193  if (d >= 2) {
194  hat_node[loc_inod][alpha] += xi0*(loc_vertex[1][alpha] - loc_vertex[0][alpha]);
195  }
196  if (d == 3) {
197  hat_node[loc_inod][alpha] += xi1*(loc_vertex[loc_nsidvert-1][alpha] - loc_vertex[0][alpha]);
198  }
199  }
200  ++loc_inod;
201  }
202  }
203  // 1.3. insert internal quadrature nodes for integrating exactly P(k-1) in hat_K, when k>0
204  if (k > 0) {
205  if (variant == reference_element::t) {
206  // experimental for triangle: internal interpolation without quadrature
207  Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1> hat_node_internal;
208  const size_type internal = 1;
209  switch (base::_sopt.get_node()) {
211 #ifdef TODO
212  // TODO: pointset warburton could support the "internal=1" option
213  pointset_lagrange_warburton (variant, k-1, hat_node_internal, internal); break;
214 #endif // TODO
216  pointset_lagrange_equispaced (variant, k-1, hat_node_internal, internal); break;
217  default:
218  error_macro ("unsupported node set: " << base::_sopt.get_node_name());
219  }
220  trace_macro ("hat_node_internal.size="<<hat_node_internal.size());
221  for (size_type loc_inod_int = 0; loc_inod_int < size_type(hat_node_internal.size()); ++loc_inod_int) {
222  trace_macro ("hat_node_internal["<<loc_inod_int<<"]="<<hat_node_internal[loc_inod_int]);
223  hat_node[loc_inod] = hat_node_internal [loc_inod_int];
224  ++loc_inod;
225  }
226  } else { // variant == TPH : still quadrature
227  for (typename quadrature<T>::const_iterator iter_q = _quad.begin(hat_K),
228  last_q = _quad.end(hat_K); iter_q != last_q; iter_q++) {
229  hat_node[loc_inod] = (*iter_q).x;
230  ++loc_inod;
231  }
232  }
233  }
234  check_macro (loc_inod == loc_nnod, "invalid node count: loc_inod="<<loc_inod<<" and loc_nnod="<<loc_nnod);
235 
236  // --------------------------------------------------------------------------
237  // 2. build a transformation tilde_A for evaluating polynomials from (b_pre_{k+1})^d
238  // from a pre-basis:
239  // tilde_psi = {(p,0), (0,p), p in Bkm1} cup {(x0*p,x1*p), p in bar_Bk}
240  // --------------------------------------------------------------------------
241  // 2.1. compute ndof per side, for each side (prism have different sides)
242  size_type dim_Pkm1 = (k == 0) ? 0 : reference_element::n_node(hat_K.variant(), k-1);
243  size_type dim_Pk = reference_element::n_node (hat_K.variant(), k);
244  size_type dim_Pkp1 = _b_pre_kp1.ndof (hat_K);
245  size_type loc_ndof_sid_tot = 0;
246  for (size_type loc_isid = 0, loc_nsid = hat_K.n_subgeo(d-1); loc_isid < loc_nsid; ++loc_isid) {
247  reference_element hat_S = hat_K.subgeo (d-1, loc_isid);
248  // dofs on sides are Lagrange dofs, associated to nodes:
249  loc_ndof_sid_tot += base::_nnod_on_subgeo_internal [d][hat_S.variant()];
250  }
251  size_type loc_ndof = base::_first_idof_by_dimension [variant][d+1];
252  size_type loc_nnod_sid_tot = loc_ndof_sid_tot; // dofs on sides are Lagrange one, associated to nodes
253 
254 #ifdef TO_CLEAN
255  // check size
256  warning_macro ("first_inod(hat_K,d) ="<<base::first_inod(hat_K,d));
257  warning_macro ("first_inod(hat_K,d+1)="<<base::first_inod(hat_K,d+1));
258  warning_macro ("dim(P"<<int(k-1)<<")="<<dim_Pkm1);
259  warning_macro ("dim(P"<<k<<")="<<dim_Pk);
260  warning_macro ("dim(P"<<k+1<<")="<<dim_Pkp1);
261  warning_macro ("loc_ndof_sid_tot="<<loc_ndof_sid_tot);
262  warning_macro ("size(hat_K)="<<base::ndof (hat_K));
263  warning_macro ("first_idof(hat_K,d-1)="<<base::first_idof(hat_K,d-1));
264  warning_macro ("first_idof(hat_K,d) ="<<base::first_idof(hat_K,d));
265  warning_macro ("first_idof(hat_K,d+1)="<<base::first_idof(hat_K,d+1));
266  warning_macro ("dim(RT"<<k<<")="<<loc_ndof);
267  warning_macro ("dim((P"<<k+1<<")^"<<d<<")="<<d*dim_Pkp1);
268  warning_macro ("a_tilde(loc_ndof)="<<loc_ndof<<",d*dim(Pkp1)="<<d*dim_Pkp1<<")");
269 #endif // TO_CLEAN
270 
271  check_macro (loc_ndof == base::ndof (hat_K), "invalid ndof="<<loc_ndof<<" != "<< base::ndof (hat_K));
272  //
273  // 2.2. decompose the tilde_psi RTk basis on the basis of (P_{k+1})^d
274  // note: explicit easy decomposition when using monomial basis for (P_{k+1})^d
275  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> tilde_a (loc_ndof, d*dim_Pkp1);
276  tilde_a.fill (0);
277  // 2.2.1 homogeneous polynoms:
278  // (q,0) and (0,r), q,r in basis(Pk)
279  // leads to a diag matrix block
280  // since basis(Pk) subset basis(P_{k+1}) and the basis is hierarchical
281  trace_macro ("basis(1): (q,0) and (0,r), q,r in basis(Pk)...");
282  for (size_type loc_comp_idof = 0; loc_comp_idof < dim_Pk; ++loc_comp_idof) {
283  for (size_type alpha = 0; alpha < d; ++alpha) {
284  size_type loc_idof = d*loc_comp_idof + alpha;
285  size_type loc_jpkp1 = d*loc_comp_idof + alpha;
286  trace_macro ("loc_idof="<<loc_idof<<", loc_jpkp1="<<loc_jpkp1);
287  tilde_a (loc_idof, loc_jpkp1) = 1;
288  }
289  }
290  // 2.2.2 non-homogeneous polynoms
291  trace_macro ("basis(2): (x0*p,x1*p), p in basis(P_k)\basis(P_{k-1}) [exactly k degree]");
292  std::vector<size_type> inod2ideg_kp1;
293  build_inod2ideg (hat_K, k+1, inod2ideg_kp1);
294  switch (hat_K.variant()) {
295  case reference_element::e:
296  case reference_element::t:
297  case reference_element::T: {
298  // triangle : (x0*p,x1*p), p in basis(P_k)\basis(P_{k-1}) [exactly k degree]
299  // =(b_iO,b_i1) in P_{k+1} when b is the hierarchical monomial basis
300  // and i0=ilat2ideg([i+1,k-i ]), i=0..k
301  // i1=ilat2ideg([i ,k-i+1])
302  std::vector<point_basic<size_type> > power_index;
303  make_power_indexes_sorted_by_degrees (hat_K, k, power_index);
304  for (size_type loc_ideg = dim_Pkm1, loc_ndeg = power_index.size(); loc_ideg < loc_ndeg; ++loc_ideg) {
305  for (size_type alpha = 0; alpha < d; ++alpha) {
306  point_basic<size_type> ilat = power_index [loc_ideg];
307  size_type loc_idof = d*dim_Pk + (loc_ideg - dim_Pkm1);
308  ilat [alpha]++; // x_alpha*monomial(i,k-i) = x^{i+1}*y^{k-i} when alpha=0, etc
309  size_type loc_inod_kp1 = ilat2loc_inod (hat_K, k+1, ilat);
310  size_type loc_ideg_kp1 = inod2ideg_kp1 [loc_inod_kp1];
311  size_type loc_jpkp1d = d*loc_ideg_kp1 + alpha;
312  trace_macro ("loc_idof="<<loc_idof<<", loc_jpkp1d="<<loc_jpkp1d);
313  tilde_a (loc_idof, loc_jpkp1d) = 1;
314  }
315  }
316  break;
317  }
319  case reference_element::H: {
320  // quadrangle : (x0*p, 0)
321  // ( ,x1*q), p,q in basis(P_k)\basis(P_{k-1}) [exactly k degree]
322  // =(b_i0, 0)
323  // ( 0,b_i1), b_i0, b_i1 in P_{k+1} when b is the hierarchical monomial basis
324  // and i0=ilat2ideg([i+1,j ]), i=0..k
325  // i1=ilat2ideg([i ,j+1])
326  size_type sub_variant = (d == 2) ? reference_element::e : reference_element::q;
327  reference_element hat_sub (sub_variant); // q=e*e ; H = e*q
328  std::vector<point_basic<size_type> > power_index_sub;
329  make_power_indexes_sorted_by_degrees (hat_sub, k, power_index_sub);
330  for (size_type loc_ideg = 0, loc_ndeg = power_index_sub.size(); loc_ideg < loc_ndeg; ++loc_ideg) {
331  for (size_type alpha = 0; alpha < d; ++alpha) {
332  point_basic<size_type> ilat_sub = power_index_sub [loc_ideg];
333  size_type loc_idof = d*dim_Pk + d*loc_ideg + alpha;
334  point_basic<size_type> ilat (0,0,0);
335  ilat [alpha] = k+1; // (x^{k+1}*p(y), 0) & (0, p(x)*y^{k+1}) with p in Pk(sub)
336  ilat [(alpha+1)%d] = ilat_sub[0];
337  if (d == 3) ilat [(alpha+2)%d] = ilat_sub[1];
338  size_type loc_inod_kp1 = ilat2loc_inod (hat_K, k+1, ilat);
339  size_type loc_ideg_kp1 = inod2ideg_kp1 [loc_inod_kp1];
340  size_type loc_jpkp1d = d*loc_ideg_kp1 + alpha;
341  trace_macro ("loc_idof="<<loc_idof<<", loc_jpkp1d="<<loc_jpkp1d);
342  tilde_a (loc_idof, loc_jpkp1d) = 1;
343  }
344  }
345  break;
346  }
348  default: error_macro ("unexpected element `"<<hat_K.name()<<"' (HINT: see BerDur-2013)");
349  }
350 #undef DEBUG_RTK // print matrix
351 #ifdef DEBUG_RTK
352  cout << setprecision(std::numeric_limits<T>::digits10)
353  << "tilde_a=[" << endl << tilde_a <<"]"<<endl
354  << "[u,s,vt]=svd(tilde_a)" << endl
355  << "id=eye(size(s))" << endl
356  << "a=u*id*vt" << endl
357  ;
358 #endif // DEBUG_RTK
359  // --------------------------------------------------------------------------
360  // 3. build a transformation A for evaluating polynomials from (b_pre_{k+1})^d
361  // for a raw-basis:
362  // psi = A*b_pre
363  // --------------------------------------------------------------------------
364  // get psi raw RTk basis by othogonalizing the tilde_psi RTk basis
365  // note: use SVD
366  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>
367  u (tilde_a.rows(), tilde_a.rows()),
368  vt (tilde_a.cols(), tilde_a.cols());
369  Eigen::Matrix<T,Eigen::Dynamic,1> s (tilde_a.rows());
370  Eigen::JacobiSVD<Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> >
371  svd (tilde_a, Eigen::ComputeFullU | Eigen::ComputeFullV);
372  // SVD: tilde_a = u*s1*trans(vt) TODO check with eigen u/ut v/vt !
373  s = svd.singularValues();
374  u = svd.matrixU();
375  vt = svd.matrixV();
376  size_type rank_s = 0;
378  for (size_type loc_idof = 0; loc_idof < size_type(s.size()); ++loc_idof) {
379  rank_s += (abs(s[loc_idof]) > eps) ? 1 : 0;
380  }
381  check_macro (rank_s == loc_ndof,
382  "invalid polynomial space dimension = " << rank_s << " < loc_ndof = " << loc_ndof);
383  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> id (tilde_a.rows(), tilde_a.cols());
384  id.setIdentity();
385  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> a = id*vt.transpose();
386 
387 #ifdef DEBUG_RTK
388  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> s1 (tilde_a.rows(), tilde_a.cols());
389  s1.fill(0);
390  for (size_type iloc = 0; iloc < size_t(tilde_a.rows()); iloc++) {
391  s1(iloc,iloc) = s(iloc);
392  }
393  cout << "s1 = ["<< endl << s1 <<"];" << endl
394  << "u1 = ["<< endl << u <<"];"<<endl
395  << "vt1 = ["<< endl << vt <<"];"<<endl
396  << "id1 = ["<< endl << id <<"];"<<endl
397  << "a1 = ["<< endl << a <<"]"<<endl
398  << "err=norm(tilde_a-u*s*vt')"<<endl
399  << "err1=norm(tilde_a-u1*s1*vt1')"<<endl
400  << "err_a=abs(a-a1)"<<endl
401  << "err_svd=max(max(abs(a-a1)))"<<endl
402  << "err_u=max(max(abs(u-u1)))"<<endl
403  << "err_v=max(max(abs(vt-vt1)))"<<endl
404  << "err_s=max(diag(s1)-diag(s))"<<endl
405  ;
406  T err_svd = (tilde_a - u*s1*vt.transpose()).norm();
407  cout << "err_svd = " << err_svd << endl;
408 #endif // DEBUG_RTK
409 
410  // -------------------------------------------------------------------------------
411  // 4. build a transformation bar_A for evaluating polynomials from (b_pre_{k+1})^d
412  // for the dof-basis:
413  // phi = A*b_pre
414  // -------------------------------------------------------------------------------
415  // 4.1. defines the quadrature nodes, for integral dofs
416  // and evaluate the basis of P_{k-1} on it
417  std::array<
418  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>
419  ,3>&
420  bkm1_node_internal_d = _bkm1_node_internal_d [variant];
421  if (k > 0) {
422  size_type loc_nnod_int = base::nnod_on_subgeo (d,hat_K.variant());
423  Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1> hat_node_internal (loc_nnod_int);
424  for (size_type loc_inod_int = 0; loc_inod_int < loc_nnod_int; ++loc_inod_int) {
425  size_type loc_inod = loc_nnod_sid_tot + loc_inod_int;
426  hat_node_internal [loc_inod_int] = hat_node [loc_inod];
427  }
428  string basis_dof_name;
429  switch (base::_sopt.get_raw_polynomial()) {
430  case basis_option::monomial: basis_dof_name = "M"; break;
431  case basis_option::dubiner: basis_dof_name = "D"; break;
432  case basis_option::bernstein: basis_dof_name = "B"; break;
433  default: error_macro ("unsupported raw polynomial basis `"<<base::_sopt.get_raw_polynomial_name()<<"'");
434  }
435 trace_macro("basis_dof_name="<<basis_dof_name);
436  // bkm1_node_internal(i,j) = bj(xi)
437  switch (variant) {
438  case reference_element::e:
439  case reference_element::t:
440  case reference_element::T: {
441  // int_K b*v dx with v=[p,0],[0,p] and p in P_{k-1}
442  basis_raw_basic<T> b_dof_km1 (basis_dof_name+std::to_string(k-1));
443  details::basis_on_pointset_evaluate (b_dof_km1, hat_K, hat_node_internal, bkm1_node_internal_d[0]);
444  for (size_type alpha = 1; alpha < d; ++alpha) {
445  bkm1_node_internal_d [alpha] = bkm1_node_internal_d [0];
446  }
447  break;
448  }
450  case reference_element::H: {
451  // int_K b*v dx with v=[p,0],[0,q] and p in P_{k-1,k}, q in P_{k,k-1}
452  basis_raw_basic<T> b_dof_km1 (basis_dof_name+std::to_string(k-1)),
453  b_dof_k (basis_dof_name+std::to_string(k));
454  std::array<Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1>,3> hat_node_internal_comp;
455  std::array<
456  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>
457  ,3>
458  bkm1_node_internal_comp,
459  bk_node_internal_comp;
461  for (size_type alpha = 0; alpha < d; ++alpha) {
462  hat_node_internal_comp [alpha].resize (hat_node_internal.size());
463  for (size_type loc_inod_int = 0, loc_nnod_int = hat_node_internal.size(); loc_inod_int < loc_nnod_int; ++loc_inod_int) {
464  // resize from q=[-1,1]^2 to e=[0,1]
465  hat_node_internal_comp [alpha][loc_inod_int][0] = (1+hat_node_internal [loc_inod_int][alpha])/2;
466  }
467  details::basis_on_pointset_evaluate (b_dof_km1, hat_e, hat_node_internal_comp[alpha], bkm1_node_internal_comp[alpha]);
468  details::basis_on_pointset_evaluate (b_dof_k, hat_e, hat_node_internal_comp[alpha], bk_node_internal_comp[alpha]);
469  }
470  size_type loc_ndof_int = bkm1_node_internal_comp[0].cols()*pow(bk_node_internal_comp[0].cols(),d-1);
471  for (size_type alpha = 0; alpha < d; ++alpha) {
472  size_type alpha2 = (alpha+1)%d;
473  size_type alpha3 = (alpha+2)%d;
474  bkm1_node_internal_d [alpha].resize (hat_node_internal.size(), loc_ndof_int);
475  for (size_type loc_inod_int = 0, loc_nnod_int = hat_node_internal.size(); loc_inod_int < loc_nnod_int; ++loc_inod_int) {
476  size_type loc_idof_int = 0;
477  if (variant == reference_element::q) {
478  for (size_type i = 0; i < size_type(bkm1_node_internal_comp [alpha].cols()); ++i) {
479  for (size_type j = 0; j < size_type( bk_node_internal_comp [alpha].cols()); ++j) {
480  bkm1_node_internal_d [alpha ] (loc_inod_int,loc_idof_int)
481  = bkm1_node_internal_comp [alpha ] (loc_inod_int,i)
482  * bk_node_internal_comp [alpha2] (loc_inod_int,j);
483  ++loc_idof_int;
484  }}
485  } else {
486  for (size_type i = 0; i < size_type(bkm1_node_internal_comp [alpha].cols()); ++i) {
487  for (size_type j = 0; j < size_type( bk_node_internal_comp [alpha].cols()); ++j) {
488  for (size_type k = 0; k < size_type( bk_node_internal_comp [alpha].cols()); ++k) {
489  bkm1_node_internal_d [alpha ] (loc_inod_int,loc_idof_int)
490  = bkm1_node_internal_comp [alpha ] (loc_inod_int,i)
491  * bk_node_internal_comp [alpha2] (loc_inod_int,j)
492  * bk_node_internal_comp [alpha3] (loc_inod_int,k);
493  ++loc_idof_int;
494  }}}
495  }
496  }
497  }
498  break;
499  }
501  default: error_macro ("unexpected element `"<<hat_K.name()<<"'");
502  }
503  }
504  // -----------------------------------------
505  // 4.2. compute basis of (P_{k+1})^d at all nodes
506  // -----------------------------------------
507  size_type loc_n_bkp1 = _b_pre_kp1.ndof (hat_K);
508  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> bkp1_node (loc_nnod, loc_n_bkp1);
509  details::basis_on_pointset_evaluate (_b_pre_kp1, hat_K, hat_node, bkp1_node); // bkp1_node(i,j) = bj(xi)
510  // vector expansion: bkp1d_node(i,d*j+alpha) = [bj(xi),0,0], [0,bj(xi),0], etc
511  // -> compute the dofs for all this expansion
512  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic> bkp1d_dof (loc_ndof, d*loc_n_bkp1); // dof_i(bjd)
513  bkp1d_dof.fill (std::numeric_limits<T>::max());
514  Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1> bkp1d_j_node (loc_nnod);
515  Eigen::Matrix<T,Eigen::Dynamic,1> bkp1d_j_dof (loc_ndof);
516  for (size_type loc_j_bkp1 = 0; loc_j_bkp1 < loc_n_bkp1; ++loc_j_bkp1) {
517  for (size_type alpha = 0; alpha < d; ++alpha) {
518  for (size_type loc_inod = 0; loc_inod < loc_nnod; ++loc_inod) {
519  bkp1d_j_node [loc_inod] = point_basic<T>(0,0,0);
520  bkp1d_j_node [loc_inod][alpha] = bkp1_node(loc_inod,loc_j_bkp1);
521  }
522  bkp1d_j_dof.fill (std::numeric_limits<T>::max());
523  _compute_dofs (hat_K, bkp1d_j_node, bkp1d_j_dof);
524  size_type loc_j_bkp1d = d*loc_j_bkp1 + alpha;
525  check_macro (bkp1d_dof.rows() == bkp1d_j_dof.size(), "invalid sizes");
526  bkp1d_dof.col (loc_j_bkp1d) = bkp1d_j_dof;
527  }
528  }
529  // -----------------------------------------
530  // 4.3. vdm
531  // -----------------------------------------
532  // VDM(i,j) = dof_i(phi_j)
533  // = phi_j.n(xi) for side dofs
534  // int phi_j(x)*p(x)*dx, for all p in P(k-1)
535  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>& vdm = _vdm [hat_K.variant()];
536  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>& inv_vdm = _inv_vdm[hat_K.variant()];
537  vdm = bkp1d_dof*a.transpose(); // = trans(a*trans(bkp1d_dof));
538  bool invert_ok = invert(vdm, inv_vdm);
539  check_macro (invert_ok,
540  "unisolvence failed for " << base::name() <<"(" << hat_K.name() << ") basis");
541  // -----------------------------------------
542  // 4.4. final composition matrix: bar_a = trans(inv_vdm)*a
543  // -----------------------------------------
544  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>& bar_a = _bar_a [hat_K.variant()];
545  bar_a = inv_vdm.transpose()*a;
546 #ifdef DEBUG_RTK
547  cout << "bkp1d_dof=[" << endl << bkp1d_dof <<"]"<<endl
548  << "vdm=[" << endl << vdm <<"]"<<endl
549  << "det_vdm=" << vdm.determinant() <<endl
550  << "cond_vdm=" << cond(vdm) <<endl
551  << "bar_a1=inv(vdm)'*a;"<<endl
552  << "bar_a=[" << endl << bar_a <<"]"<<endl;
553 #endif // DEBUG_RTK
554 }
555 // ----------------------------------------------------------------------------
556 // evaluation of all basis functions at hat_x:
557 // ----------------------------------------------------------------------------
558 template<class T>
559 void
561  reference_element hat_K,
562  const point_basic<T>& hat_x,
563  Eigen::Matrix<value_type,Eigen::Dynamic,1>& value) const
564 {
565  base::_initialize_data_guard (hat_K);
566  size_type d = hat_K.dimension();
567  size_type loc_ndof = base::ndof (hat_K);
568  //
569  // 1) evaluate the basis of P_{k+1}(hat_K) at hat_x : bkp1(x)
570  //
571  const Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>& bar_a = _bar_a [hat_K.variant()];
572  Eigen::Matrix<T,Eigen::Dynamic,1> bkp1;
573  _b_pre_kp1.evaluate (hat_K, hat_x, bkp1);
574  value.resize (loc_ndof);
575  //
576  // 2) evaluate the basis of RTk at hat_x: phi(x)
577  // [phi(x)] = [A2]*[bkp1d(x)]
578  // where bkp1d = basis of (P_{k+1}(hat_K))^d by vectorization
579  // nb
580  // ---
581  // \.
582  // phi_{i,a}(hat_x) = / A(i,[j,a])*b_j(hat_x)
583  // ---
584  // j=0
585  for (size_type loc_idof = 0; loc_idof < loc_ndof; ++loc_idof) {
586  value[loc_idof] = point_basic<T>(0,0,0);
587  for (size_type loc_jdof_bkp1 = 0, loc_ndof_bkp1 = bkp1.size(); loc_jdof_bkp1 < loc_ndof_bkp1; ++loc_jdof_bkp1) {
588  for (size_type alpha = 0; alpha < d; ++alpha) {
589  size_type loc_jdof_bkp1d = d*loc_jdof_bkp1 + alpha;
590  value[loc_idof][alpha] += bar_a(loc_idof,loc_jdof_bkp1d)*bkp1[loc_jdof_bkp1];
591  }
592  }
593  }
594 }
595 template<class T>
596 void
598  reference_element hat_K,
599  const point_basic<T>& hat_x,
600  Eigen::Matrix<tensor_basic<T>,Eigen::Dynamic,1>& value) const
601 {
602  base::_initialize_data_guard (hat_K);
603  size_type d = hat_K.dimension();
604  size_type loc_ndof = base::ndof (hat_K);
605  //
606  // 1) evaluate the grad basis of P_{k+1}(hat_K) at hat_x : grad bkp1(x)
607  //
608  const Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>& bar_a = _bar_a [hat_K.variant()];
609  Eigen::Matrix<point_basic<T>,Eigen::Dynamic,1> grad_bkp1;
610  _b_pre_kp1.grad_evaluate (hat_K, hat_x, grad_bkp1);
611  value.resize (loc_ndof);
612  //
613  // 2) evaluate the grad basis of RTk at hat_x: grad phi(x)
614  // [grad phi(x)] = [A2]*[grad bkp1d(x)]
615  // where grad bkp1d = grad basis of (P_{k+1}(hat_K))^d by vectorization
616  //
617  // nb
618  // ---
619  // d phi_{i,a} \ d b_j
620  // ----------(hat_x) = / A(i,[j,a])*--------(hat_x)
621  // d hat_x_b --- d hat_x_b
622  // j=0
623  for (size_type loc_idof = 0; loc_idof < loc_ndof; ++loc_idof) {
624  value[loc_idof] = tensor_basic<T>();
625  for (size_type loc_jdof_bkp1 = 0, loc_ndof_bkp1 = grad_bkp1.size(); loc_jdof_bkp1 < loc_ndof_bkp1; ++loc_jdof_bkp1) {
626  for (size_type alpha = 0; alpha < d; ++alpha) {
627  size_type loc_jdof_bkp1d = d*loc_jdof_bkp1 + alpha;
628  for (size_type beta = 0; beta < d; ++beta) {
629  value[loc_idof](alpha,beta) += bar_a(loc_idof,loc_jdof_bkp1d)*grad_bkp1[loc_jdof_bkp1][beta];
630  }
631  }
632  }
633  }
634 }
635 // ----------------------------------------------------------------------------
636 // dofs for a scalar-valued function
637 // ----------------------------------------------------------------------------
638 // note: as virtual and template members are not available,
639 // the function "f" has been already evaluated on the hat_node[] set
640 // note2: moments on sides are scalar products with normals (not integrals on sides)
641 // for RT0 on the triangle we have the basis psi associated with momentum as:
642 // psi: matrix([x,y-1],[sqrt(2)*x,sqrt(2)*y],[x-1,y]);
643 // n : matrix([0,-1],[1/sqrt(2),1/sqrt(2)],[-1,0]);
644 // xm : matrix([1/2,0],[1/2,1/2],[0,1/2]);
645 // moment(i,f) := subst(xm[i][1],x,subst(xm[i][2],y,f.n[i]));
646 //
647 template<class T>
648 void
650  reference_element hat_K,
651  const Eigen::Matrix<value_type,Eigen::Dynamic,1>& f_xnod,
652  Eigen::Matrix<T,Eigen::Dynamic,1>& dof) const
653 {
654 trace_macro ("_compute_dofs (hat_K="<<hat_K.name()<<")...");
655  base::_initialize_data_guard (hat_K);
656  size_type k = degree()-1;
657  size_type d = hat_K.dimension();
658  size_type loc_ndof = base::ndof (hat_K);
659 
660  dof.resize (loc_ndof);
661  if (d == 0) return;
662 
663  // side dofs are Lagrange ones, associated to nodes
664  size_type loc_inod = 0;
665  size_type loc_idof = 0;
666  for (size_type loc_isid = 0, loc_nsid = hat_K.n_subgeo(d-1); loc_isid < loc_nsid; ++loc_isid) {
667  reference_element hat_S = hat_K.subgeo (d-1, loc_isid);
668  point_basic<Float> hat_n;
669  Float hat_S_meas = hat_K.side_measure (loc_isid);
670  hat_K.side_normal (loc_isid, hat_n);
671  size_type loc_ndof_sid = base::_nnod_on_subgeo_internal [d][hat_S.variant()];
672  for (size_type loc_idof_sid = 0; loc_idof_sid < loc_ndof_sid; ++loc_idof_sid) {
673  dof[loc_idof] = hat_S_meas*dot(f_xnod[loc_inod],hat_n);
674  loc_idof++;
675  loc_inod++;
676  }
677  }
678  // internal dofs
679  if (k == 0) return; // no internal dofs when k==0
680  const std::array<
681  Eigen::Matrix<T,Eigen::Dynamic,Eigen::Dynamic>
682  ,3>&
683  bkm1_node_internal_d = _bkm1_node_internal_d [hat_K.variant()];
684 
685  size_type loc_ndof_boundary = loc_idof;
686  size_type loc_ndof_int_d = d*bkm1_node_internal_d[0].cols();
687  size_type first_loc_inod_int = loc_inod;
688  check_macro (loc_ndof == loc_ndof_boundary + loc_ndof_int_d,
689  "invalid internal dof count: loc_ndof="<<loc_ndof
690  << ", loc_ndof_boundary="<<loc_ndof_boundary
691  << ", loc_ndof_int_d="<<loc_ndof_int_d);
692 
693  size_type variant = hat_K.variant();
694  if (variant == reference_element::t) {
695  // interpolate all vector components on a lattice of internal nodes
696  for (size_type loc_idof_int = 0, loc_ndof_int = bkm1_node_internal_d[0].cols(); loc_idof_int < loc_ndof_int; ++loc_idof_int) {
697  size_type loc_inod = first_loc_inod_int + loc_idof_int;
698  for (size_type alpha = 0; alpha < d; ++alpha) {
699  dof [loc_idof] = f_xnod [loc_inod][alpha];
700  loc_idof++;
701  }
702  }
703  } else { // variant == qTPH : quadrature
704  for (size_type loc_idof_int = 0, loc_ndof_int = bkm1_node_internal_d[0].cols(); loc_idof_int < loc_ndof_int; ++loc_idof_int) {
705  for (size_type alpha = 0; alpha < d; ++alpha) {
706  loc_inod = first_loc_inod_int;
707  T sum = 0;
708  size_type inod_q = 0;
709  for (typename quadrature<T>::const_iterator iter_q = _quad.begin(hat_K),
710  last_q = _quad.end(hat_K); iter_q != last_q; iter_q++, inod_q++, ++loc_inod) {
711  sum += f_xnod [loc_inod][alpha]
712  *bkm1_node_internal_d[alpha] (inod_q, loc_idof_int)
713  *(*iter_q).w;
714  }
715  check_macro (loc_idof < loc_ndof, "invalid size");
716  dof [loc_idof] = sum;
717  loc_idof++;
718  }
719  }
720  }
721  check_macro (loc_idof == loc_ndof, "invalid dof count");
722 trace_macro ("_compute_dofs (hat_K="<<hat_K.name()<<") done");
723 }
724 // ----------------------------------------------------------------------------
725 // instanciation in library
726 // ----------------------------------------------------------------------------
727 #define _RHEOLEF_instanciation(T) \
728 template class basis_fem_RTk<T>;
729 
731 
732 }// namespace rheolef
see the Float page for the full documentation
see the point page for the full documentation
void grad_evaluate(reference_element hat_K, const point_basic< T > &hat_x, Eigen::Matrix< tensor_basic< T >, Eigen::Dynamic, 1 > &value) const
void _compute_dofs(reference_element hat_K, const Eigen::Matrix< value_type, Eigen::Dynamic, 1 > &f_xnod, Eigen::Matrix< T, Eigen::Dynamic, 1 > &dof) const
void evaluate(reference_element hat_K, const point_basic< T > &hat_x, Eigen::Matrix< value_type, Eigen::Dynamic, 1 > &value) const
const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > & vdm(reference_element hat_K) const
void _initialize_cstor_sizes() const
basis_raw_basic< T > _b_pre_kp1
basis_fem_RTk(size_type k, const basis_option &sopt)
const Eigen::Matrix< point_basic< T >, Eigen::Dynamic, 1 > & hat_node(reference_element hat_K) const
std::string family_name() const
Definition: basis_fem_RTk.h:71
const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > & inv_vdm(reference_element hat_K) const
void _initialize_data(reference_element hat_K) const
see the basis_option page for the full documentation
Definition: basis_option.h:93
reference_element::size_type size_type
Definition: basis.h:214
piola_fem< T > _piola_fem
Definition: basis.h:424
basis_option _sopt
Definition: basis.h:423
static std::string standard_naming(std::string family_name, size_t degree, const basis_option &sopt)
Definition: basis_rep.cc:44
std::string _name
Definition: basis.h:422
void set_family(family_type type)
rep::const_iterator const_iterator
Definition: quadrature.h:195
see the reference_element page for the full documentation
void side_normal(size_type loc_isid, point_basic< Float > &hat_n) const
const point_basic< Float > & vertex(size_type iloc) const
static const variant_type H
static const variant_type q
static const variant_type e
reference_element subgeo(size_type subgeo_dim, size_type loc_isid) const
static const variant_type p
Float side_measure(size_type loc_isid) const
variant_type variant() const
size_type subgeo_local_vertex(size_type subgeo_dim, size_type loc_isid, size_type loc_jsidvert) const
static size_type n_node(variant_type variant, size_type order)
size_type n_subgeo(size_type subgeo_dim) const
static const variant_type T
static const variant_type P
static const variant_type t
integrate_option quadrature_option
size_t size_type
Definition: basis_get.cc:76
point_basic< T >
Definition: piola_fem.h:135
rheolef::std value
point u(const point &x)
#define trace_macro(message)
Definition: dis_macros.h:111
#define error_macro(message)
Definition: dis_macros.h:49
#define warning_macro(message)
Definition: dis_macros.h:53
Expr1::float_type T
Definition: field_expr.h:230
check_macro(expr1.have_homogeneous_space(Xh1), "dual(expr1,expr2); expr1 should have homogeneous space. HINT: use dual(interpolate(Xh, expr1),expr2)")
Float alpha[pmax+1][pmax+1]
Definition: bdf.icc:28
void basis_on_pointset_evaluate(const Basis &b, const reference_element &hat_K, const Eigen::Matrix< point_basic< T >, Eigen::Dynamic, 1 > &hat_x, Eigen::Matrix< Value, Eigen::Dynamic, Eigen::Dynamic > &vdm)
size_type ndof(const basis_basic< T > &b, const geo_size &gs, size_type map_dim)
This file is part of Rheolef.
void pointset_lagrange_equispaced(reference_element hat_K, size_t order_in, Eigen::Matrix< point_basic< T >, Eigen::Dynamic, 1 > &hat_xnod, size_t internal=0)
Definition: equispaced.icc:44
T norm(const vec< T, M > &x)
norm(x): see the expression page for the full documentation
Definition: vec.h:387
rheolef::std enable_if ::type dot const Expr1 expr1, const Expr2 expr2 dot(const Expr1 &expr1, const Expr2 &expr2)
dot(x,y): see the expression page for the full documentation
Definition: vec_expr_v2.h:415
void pointset_lagrange_warburton(reference_element hat_K, size_t degree, Eigen::Matrix< point_basic< T >, Eigen::Dynamic, 1 > &hat_xnod, bool map_on_reference_element=true)
Definition: warburton.icc:574
T cond(const Eigen::Matrix< T, Eigen::Dynamic, Eigen::Dynamic > &a)
Definition: eigen_util.h:60
space_mult_list< T, M > pow(const space_basic< T, M > &X, size_t n)
Definition: space_mult.h:120
_RHEOLEF_instanciation(Float, sequential, std::allocator< Float >) _RHEOLEF_instanciation(Float
void invert(tiny_matrix< T > &a, tiny_matrix< T > &inv_a)
Definition: tiny_lu.h:127
Float beta[][pmax+1]
Definition: leveque.h:25
Float epsilon