/usr/include/rheolef/form_vf_assembly.h is in librheolef-dev 6.6-1build2.
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#define _RHEO_FORM_VF_ASSEMBLY_H
///
/// This file is part of Rheolef.
///
/// Copyright (C) 2000-2009 Pierre Saramito <Pierre.Saramito@imag.fr>
///
/// Rheolef is free software; you can redistribute it and/or modify
/// it under the terms of the GNU General Public License as published by
/// the Free Software Foundation; either version 2 of the License, or
/// (at your option) any later version.
///
/// Rheolef is distributed in the hope that it will be useful,
/// but WITHOUT ANY WARRANTY; without even the implied warranty of
/// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
/// GNU General Public License for more details.
///
/// You should have received a copy of the GNU General Public License
/// along with Rheolef; if not, write to the Free Software
/// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
///
/// =========================================================================
#include "rheolef/form.h"
#include "rheolef/test.h"
#include "rheolef/quadrature.h"
#include "rheolef/field_vf_assembly.h" // for dg_dis_idof()
#include "rheolef/ublas-io.h" // for debug
namespace rheolef {
/*
let:
a(u,v) = int_domain expr(u,v) dx
The integrals are evaluated over each element K of the domain
by using a quadrature formulae given by fopt
expr(u,v) is a bilinear expression with respect to the
trial and test functions u and v
The trial function u is replaced by each of the basis function of
the corresponding finite element space Xh: (phi_j), j=0..dim(Xh)-1
The test function v is replaced by each of the basis function of
the corresponding finite element space Yh: (psi_i), i=0..dim(Yh)-1
The integrals over the domain omega is the sum of integrals over K.
The integrals over K are transformed on the reference element with
the piola transformation:
F : hat_K ---> K
hat_x |--> x = F(hat_x)
exemples:
1) expr(v) = u*v
int_K phi_j(x)*psi_i(x) dx
= int_{hat_K} hat_phi_j(hat_x)*hat_psi_i(hat_x) det(DF(hat_x)) d hat_x
= sum_q hat_phi_j(hat_xq)*hat_psi_i(hat_xq) det(DF(hat_xq)) hat_wq
The value(q,i,j) = (hat_phi_j(hat_xq)*hat_psi_i(hat_xq))
refers to basis values on the reference element.
There are evaluated on time for all over the reference element hat_K
and for the given quadrature formulae by:
expr.initialize (dom, quad);
This expression is represented by the 'test' class (see test.h)
3) expr(v) = dot(grad(u),grad(v)) dx
The 'grad' node returns
value(q,i) = trans(inv(DF(hat_wq))*grad_phi_i(hat_xq) that is vector-valued
The grad_phi values are obtained by a grad_value(q,i) method on the 'test' class.
The 'dot' performs on the fly the product
value(q,i,j) = dot (value1(q,i), value2(q,j))
This approch generalize for an expression tree.
*/
// external utilities:
template <class T> bool is_symmetric (ublas::matrix<T>& m, const T& tol);
template <class T> void local_lump (ublas::matrix<T>& m);
template <class T> void local_invert (ublas::matrix<T>& m, bool is_diag);
// ====================================================================
// common implementation for integration on a band or an usual domain
// ====================================================================
template <class T, class M>
template <class Expr>
void
form_basic<T,M>::assembly_internal (
const geo_basic<T,M>& dom,
const geo_basic<T,M>& band,
const band_basic<T,M>& gh,
const Expr& expr,
const form_option_type& fopt,
bool is_on_band)
{
// ----------------------------------------
// 0) init assembly loop
// ----------------------------------------
_X = expr.get_trial_space();
_Y = expr.get_test_space();
check_macro (band.get_background_geo() == _X.get_geo().get_background_geo(),
"assembly: incompatible integration domain "<<band.name() << " and trial function based domain "
<< _X.get_geo().name());
check_macro (band.get_background_geo() == _Y.get_geo().get_background_geo(),
"assembly: incompatible integration domain "<<band.name() << " and test function based domain "
<< _Y.get_geo().name());
quadrature<T> quad;
size_type n_derivative = expr.n_derivative();
if (fopt.get_order() != std::numeric_limits<quadrature_option_type::size_type>::max()) {
quad.set_order (fopt.get_order());
} else {
size_type k1 = _X.degree();
size_type k2 = _Y.degree();
size_type quad_order = k1 + k2 + 1;
if (dom.get_background_geo().sizes().ownership_by_variant[reference_element::q].dis_size() != 0 ||
dom.get_background_geo().sizes().ownership_by_variant[reference_element::P].dis_size() != 0 ||
dom.get_background_geo().sizes().ownership_by_variant[reference_element::H].dis_size() != 0) {
// integrate exactly ??
quad_order += 2;
}
if (dom.coordinate_system() != space_constant::cartesian) quad_order++; // multiplies by a 'r' weight
if (quad_order >= n_derivative) quad_order -= n_derivative;
quad.set_order (quad_order);
}
quad.set_family (fopt.get_family());
trace_macro ("quadrature : " << quad.get_family_name() << " order " << quad.get_order());
if (fopt.invert) check_macro (
_X.get_numbering().has_compact_support_inside_element() &&
_Y.get_numbering().has_compact_support_inside_element(),
"local inversion requires compact support inside elements (e.g. discontinuous or bubble)");
if (fopt.lump) check_macro (n_derivative == 0,
"local mass lumping requires no derivative operators");
if (!is_on_band) {
expr.initialize (dom, quad, fopt.ignore_sys_coord);
} else {
expr.initialize (gh, quad, fopt.ignore_sys_coord);
}
expr.template valued_check<T>();
basis_on_pointset<T> piola_on_quad;
piola_on_quad.set (quad, _X.get_geo().get_piola_basis());
asr<T,M> auu (_Y.iu_ownership(), _X.iu_ownership()),
aub (_Y.iu_ownership(), _X.ib_ownership()),
abu (_Y.ib_ownership(), _X.iu_ownership()),
abb (_Y.ib_ownership(), _X.ib_ownership());
std::vector<size_type> dis_idy, dis_jdx, dis_inod_K;
ublas::matrix<T> ak, value;
value.clear();
size_type d = dom.dimension();
size_type map_d = dom.map_dimension();
tensor_basic<T> DF;
bool X_is_dg = (!_X.get_numbering().is_continuous() && map_d < _X.get_geo().map_dimension());
bool Y_is_dg = (!_Y.get_numbering().is_continuous() && map_d < _Y.get_geo().map_dimension());
if (X_is_dg) _X.get_geo().neighbour_guard();
if (Y_is_dg) _Y.get_geo().neighbour_guard();
bool is_sym = true;
T sym_tol = 1e3*std::numeric_limits<T>::epsilon();
for (size_type ie = 0, ne = dom.size(map_d); ie < ne; ie++) {
// ----------------------------------------
// 1) compute local form ak
// ----------------------------------------
const geo_element& K = dom.get_geo_element (map_d, ie);
expr.element_initialize (K);
if (! is_on_band) {
assembly_dis_idof (_X, dom, K, dis_jdx);
assembly_dis_idof (_Y, dom, K, dis_idy);
} else {
size_type L_ie = gh.sid_ie2bnd_ie (ie);
const geo_element& L = band [L_ie];
_X.dis_idof (L, dis_jdx);
_Y.dis_idof (L, dis_idy);
}
ak.resize (dis_idy.size(), dis_jdx.size());
value.resize (dis_idy.size(), dis_jdx.size());
ak.clear();
reference_element hat_K = K.variant();
dom.dis_inod (K, dis_inod_K);
typename quadrature<T>::const_iterator
first_quad = quad.begin(hat_K),
last_quad = quad.end (hat_K);
for (size_type q = 0; first_quad != last_quad; ++first_quad, ++q) {
jacobian_piola_transformation (dom, piola_on_quad, K, dis_inod_K, q, DF);
T det_DF = det_jacobian_piola_transformation (DF, d, map_d);
T wq = 1;
if (! fopt.ignore_sys_coord) {
wq *= weight_coordinate_system (dom, piola_on_quad, K, dis_inod_K, q);
}
wq *= det_DF;
wq *= (*first_quad).w;
expr.basis_evaluate (hat_K, q, value);
for (size_type loc_idof = 0, loc_ndof = value.size1(); loc_idof < loc_ndof; ++loc_idof) {
for (size_type loc_jdof = 0, loc_mdof = value.size2(); loc_jdof < loc_mdof; ++loc_jdof) {
ak (loc_idof,loc_jdof) += value (loc_idof,loc_jdof)*wq;
}}
}
#ifdef TO_CLEAN
std::cerr << "ak=" << ak;
#endif // TO_CLEAN
// ----------------------------------------
// 2) optional local post-traitement
// ----------------------------------------
if (is_sym) is_sym = is_symmetric (ak, sym_tol);
if (fopt.lump ) local_lump (ak);
if (fopt.invert) local_invert (ak, fopt.lump);
// ----------------------------------------
// 3) assembly local ak in global form a
// ----------------------------------------
for (size_type loc_idof = 0, ny = ak.size1(); loc_idof < ny; loc_idof++) {
for (size_type loc_jdof = 0, nx = ak.size2(); loc_jdof < nx; loc_jdof++) {
const T& value = ak (loc_idof, loc_jdof);
// reason to perform:
// - efficient : lumped mass, structured meshes => sparsity increases
// reason to avoid:
// - conserve the sparsity pattern, even with some zero coefs
// usefull when dealing with solver::update_values()
// - also solver_pastix: assume sparsity pattern symmetry
// and failed when a_ij=0 (skipped) and a_ji=1e-15 (conserved) i.e. non-sym pattern
// note: this actual pastix wrapper limitation could be suppressed
if (1+value == 1) continue;
size_type dis_idof = dis_idy [loc_idof];
size_type dis_jdof = dis_jdx [loc_jdof];
size_type dis_iub = _Y.dis_iub (dis_idof);
size_type dis_jub = _X.dis_iub (dis_jdof);
if (_Y.dis_is_blocked(dis_idof))
if (_X.dis_is_blocked(dis_jdof)) abb.dis_entry (dis_iub, dis_jub) += value;
else abu.dis_entry (dis_iub, dis_jub) += value;
else
if (_X.dis_is_blocked(dis_jdof)) aub.dis_entry (dis_iub, dis_jub) += value;
else auu.dis_entry (dis_iub, dis_jub) += value;
}}
}
// ----------------------------------------
// 4) finalize the assembly process
// ----------------------------------------
//
// since all is local, axx.dis_entry_assembly() compute only axx.dis_nnz
//
auu.dis_entry_assembly();
aub.dis_entry_assembly();
abu.dis_entry_assembly();
abb.dis_entry_assembly();
//
// convert dynamic matrix asr to fixed-size one csr
//
_uu = csr<T,M>(auu);
_ub = csr<T,M>(aub);
_bu = csr<T,M>(abu);
_bb = csr<T,M>(abb);
//
// set pattern dimension to uu:
// => used by solvers, for efficiency: direct(d<3) or iterative(d=3)
//
_uu.set_pattern_dimension (map_d);
_ub.set_pattern_dimension (map_d);
_bu.set_pattern_dimension (map_d);
_bb.set_pattern_dimension (map_d);
//
// symmetry is used by solvers, for efficiency: LDL^t or LU, CG or GMRES
//
// Implementation note: cannot be set at compile time
// ex: expression=(eta*u)*v is structurally unsym, but numerical sym
// expression=(eta_h*grad(u))*(nu_h*grad(v)) is structurally sym,
// but numerical unsym when eta and nu are different tensors
// So, test it numerically, at element level:
#ifdef _RHEOLEF_HAVE_MPI
if (dom.comm().size() > 1 && is_distributed<M>::value) {
is_sym = mpi::all_reduce (dom.comm(), size_type(is_sym), mpi::minimum<size_type>());
}
#endif // _RHEOLEF_HAVE_MPI
_uu.set_symmetry (is_sym);
_bb.set_symmetry (is_sym);
// when sym, the main matrix is set definite and positive by default
_uu.set_definite_positive (is_sym);
_bb.set_definite_positive (is_sym);
}
template <class T, class M>
template <class Expr>
inline
void
form_basic<T,M>::assembly (
const geo_basic<T,M>& dom,
const Expr& expr,
const form_option_type& fopt)
{
assembly_internal (dom, dom, band_basic<T,M>(), expr, fopt, false);
}
template <class T, class M>
template <class Expr>
inline
void
form_basic<T,M>::assembly (
const band_basic<T,M>& gh,
const Expr& expr,
const form_option_type& fopt)
{
assembly_internal (gh.level_set(), gh.band(), gh, expr, fopt, true);
}
}// namespace rheolef
#endif // _RHEO_FORM_VF_ASSEMBLY_H
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