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SyFi
0.3
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SyFi
0.3
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#include <Nedelec2Hdiv.h>
Public Member Functions | |
| Nedelec2Hdiv () | |
| Nedelec2Hdiv (Polygon &p, unsigned int order=1) | |
| virtual | ~Nedelec2Hdiv () |
| virtual void | compute_basis_functions () |
Public Attributes | |
| GiNaC::lst | dof_repr |
Definition at line 26 of file Nedelec2Hdiv.h.
Definition at line 32 of file Nedelec2Hdiv.cpp.
References SyFi::StandardFE::description.
: StandardFE() { description = "Nedelec2Hdiv"; }
| SyFi::Nedelec2Hdiv::Nedelec2Hdiv | ( | Polygon & | p, |
| unsigned int | order = 1 |
||
| ) |
Definition at line 37 of file Nedelec2Hdiv.cpp.
References compute_basis_functions().
: StandardFE(p, order) { compute_basis_functions(); }
| virtual SyFi::Nedelec2Hdiv::~Nedelec2Hdiv | ( | ) | [inline, virtual] |
Definition at line 32 of file Nedelec2Hdiv.h.
{}
| void SyFi::Nedelec2Hdiv::compute_basis_functions | ( | ) | [virtual] |
Reimplemented from SyFi::StandardFE.
Definition at line 42 of file Nedelec2Hdiv.cpp.
References SyFi::bernstein(), SyFi::bernsteinv(), test_syfi::debug::c, SyFi::coeff(), SyFi::collapse(), SyFi::StandardFE::description, dof_repr, SyFi::StandardFE::dofs, SyFi::homogenous_polv(), SyFi::inner(), SyFi::Triangle::integrate(), SyFi::Tetrahedron::integrate(), SyFi::istr(), SyFi::matrix_from_equations(), SyFi::normal(), SyFi::StandardFE::Ns, SyFi::StandardFE::order, SyFi::StandardFE::p, SyFi::pol2basisandcoeff(), SyFi::Polygon::str(), SyFi::t, SyFi::Tetrahedron::triangle(), SyFi::Polygon::vertex(), SyFi::x, SyFi::y, and SyFi::z.
Referenced by SyFi::ArnoldFalkWintherWeakSymSigma::compute_basis_functions(), main(), and Nedelec2Hdiv().
{
// remove previously computed basis functions and dofs
Ns.clear();
dofs.clear();
if ( order < 1 )
{
throw(std::logic_error("Nedelec2Hdiv elements must be of order 1 or higher."));
}
if ( p == NULL )
{
throw(std::logic_error("You need to set a polygon before the basisfunctions can be computed"));
}
if ( p->str().find("Line") != string::npos )
{
cout <<"Can not define the Nedelec2Hdiv element on a line"<<endl;
}
else if ( p->str().find("Triangle") != string::npos )
{
cout <<"Can not define the Nedelec2Hdiv element on a Triangle "<<endl;
}
else if ( p->str().find("Tetrahedron") != string::npos )
{
description = istr( "Nedelec2Hdiv_", order) + "_3D";
int k = order;
Tetrahedron& tetrahedron= (Tetrahedron&)(*p);
GiNaC::lst equations;
GiNaC::lst variables;
// create p
GiNaC::ex P_k = bernsteinv(3,k, tetrahedron, "b");
GiNaC::ex P_k_x = P_k.op(0).op(0);
GiNaC::ex P_k_y = P_k.op(0).op(1);
GiNaC::ex P_k_z = P_k.op(0).op(2);
GiNaC::lst pspace = GiNaC::lst( P_k_x, P_k_y, P_k_z);
variables = collapse(GiNaC::ex_to<GiNaC::lst>(P_k.op(1)));
int counter = 0;
GiNaC::symbol t("t");
GiNaC::ex dofi;
GiNaC::ex bernstein_pol;
// dofs related to edges
for (int i=0; i< 4; i++)
{
Triangle triangle = tetrahedron.triangle(i);
GiNaC::lst normal_vec = normal(tetrahedron, i);
bernstein_pol = bernstein(order, triangle, istr("a",i));
GiNaC::ex basis_space = bernstein_pol.op(2);
GiNaC::ex pspace_n = inner(pspace, normal_vec);
GiNaC::ex basis;
for (unsigned int j=0; j< basis_space.nops(); j++)
{
counter++;
basis = basis_space.op(j);
GiNaC::ex integrand = pspace_n*basis;
dofi = triangle.integrate(integrand);
GiNaC::ex eq = dofi == GiNaC::numeric(0);
equations.append(eq);
// GiNaC::lst d = GiNaC::lst(triangle.integrate(x*basis),
// triangle.integrate(y*basis),
// triangle.integrate(z*basis));
GiNaC::lst d = GiNaC::lst(GiNaC::lst(triangle.vertex(0),
triangle.vertex(1),
triangle.vertex(2)),j);
dofs.insert(dofs.end(), d);
GiNaC::ex u = GiNaC::matrix(3,1,GiNaC::lst(GiNaC::symbol("v[0]"), GiNaC::symbol("v[1]"), GiNaC::symbol("v[2]")));
GiNaC::ex n = GiNaC::matrix(3,1,GiNaC::lst(GiNaC::symbol("normal_vec[0]"), GiNaC::symbol("normal_vec[1]"), GiNaC::symbol("normal_vec[2]")));
dof_repr.append(GiNaC::lst(inner(u,n)*basis.subs( x == GiNaC::symbol("xi[0]"))
.subs( y == GiNaC::symbol("xi[1]"))
.subs( z == GiNaC::symbol("xi[2]")), d));
}
}
// dofs related to tetrahedron
int tetradofs = 0;
if ( order > 1 )
{
GiNaC::ex bernstein_pol = bernsteinv(3,k-2, tetrahedron, istr("c", 0));
GiNaC::ex basis_space = bernstein_pol.op(2);
GiNaC::ex basis;
tetradofs += basis_space.nops();
for (unsigned int j=0; j<basis_space.nops(); j++)
{
basis = basis_space.op(j);
GiNaC::ex integrand = inner(pspace,basis);
dofi = tetrahedron.integrate(integrand);
GiNaC::ex eq = dofi == GiNaC::numeric(0);
equations.append(eq);
// GiNaC::lst d = GiNaC::lst(tetrahedron.integrate(x*basis.op(0)),
// tetrahedron.integrate(y*basis.op(1)),
// tetrahedron.integrate(z*basis.op(2)));
GiNaC::lst d = GiNaC::lst(tetrahedron.vertex(0),
tetrahedron.vertex(1),
tetrahedron.vertex(2),
tetrahedron.vertex(3),j);
dofs.insert(dofs.end(), d);
}
}
// Construction of S_k
//
//
if ( order >= 1 )
{
GiNaC::ex H_k = homogenous_polv(3,k-1, 3, "a");
GiNaC::ex H_k_x = H_k.op(0).op(0);
GiNaC::ex H_k_y = H_k.op(0).op(1);
GiNaC::ex H_k_z = H_k.op(0).op(2);
GiNaC::lst H_variables = collapse(GiNaC::ex_to<GiNaC::lst>(H_k.op(1)));
// Equations that make sure that r*x = 0
GiNaC::ex rx = (H_k_x*x + H_k_y*y + H_k_z*z).expand();
exmap pol_map = pol2basisandcoeff(rx);
exmap::iterator iter;
GiNaC::lst S_k;
GiNaC::lst S_k_equations;
GiNaC::lst null_eqs;
for (unsigned int i=0; i<H_variables.nops(); i++)
{
null_eqs.append( H_variables.op(i) == 0);
}
for (iter = pol_map.begin(); iter != pol_map.end(); iter++)
{
GiNaC::ex coeff = (*iter).second;
GiNaC::ex basis;
if (coeff.nops() > 1 )
{
if (coeff.nops() == 2)
{
S_k_equations.remove_all();
S_k_equations.append(coeff.op(0) == GiNaC::numeric(1));
S_k_equations.append(coeff.op(1) == GiNaC::numeric(-1));
basis = H_k.op(0).subs(S_k_equations).subs(null_eqs);;
S_k.append(basis);
}
else if ( coeff.nops() == 3 )
{
// 2 basis functions is added
// The first:
S_k_equations.remove_all();
S_k_equations.append(coeff.op(0) == GiNaC::numeric(-1,2));
S_k_equations.append(coeff.op(1) == GiNaC::numeric(1));
S_k_equations.append(coeff.op(2) == GiNaC::numeric(-1,2));
basis = H_k.op(0).subs(S_k_equations).subs(null_eqs);;
S_k.append(basis);
// The second:
S_k_equations.remove_all();
S_k_equations.append(coeff.op(0) == GiNaC::numeric(-1,2));
S_k_equations.append(coeff.op(1) == GiNaC::numeric(-1,2));
S_k_equations.append(coeff.op(2) == GiNaC::numeric(1));
basis = H_k.op(0).subs(S_k_equations).subs(null_eqs);;
S_k.append(basis);
}
}
}
std::cout <<"len (S_k) " <<S_k.nops()<<std::endl;
// dofs related to tetrahedron
if ( order >= 1 )
{
GiNaC::ex basis;
for (unsigned int j=0; j<S_k.nops(); j++)
{
basis = S_k.op(j);
GiNaC::ex integrand = inner(pspace,basis);
dofi = tetrahedron.integrate(integrand);
GiNaC::ex eq = dofi == GiNaC::numeric(0);
equations.append(eq);
GiNaC::lst d = GiNaC::lst(tetrahedron.vertex(0),
tetrahedron.vertex(1),
tetrahedron.vertex(2),
tetrahedron.vertex(3), tetradofs + j);
dofs.insert(dofs.end(), d);
}
}
}
// invert the matrix:
// GiNaC has a bit strange way to invert a matrix.
// It solves the system AA^{-1} = Id.
// It seems that this way is the only way to do
// properly with the solve_algo::gauss flag.
//
GiNaC::matrix b; GiNaC::matrix A;
matrix_from_equations(equations, variables, A, b);
unsigned int ncols = A.cols();
GiNaC::matrix vars_sq(ncols, ncols);
// matrix of symbols
for (unsigned r=0; r<ncols; ++r)
for (unsigned c=0; c<ncols; ++c)
vars_sq(r, c) = GiNaC::symbol();
GiNaC::matrix id(ncols, ncols);
// identity
const GiNaC::ex _ex1(1);
for (unsigned i=0; i<ncols; ++i)
id(i, i) = _ex1;
// invert the matrix
GiNaC::matrix m_inv(ncols, ncols);
m_inv = A.solve(vars_sq, id, GiNaC::solve_algo::gauss);
for (unsigned int i=0; i<dofs.size(); i++)
{
b.let_op(i) = GiNaC::numeric(1);
GiNaC::ex xx = m_inv.mul(GiNaC::ex_to<GiNaC::matrix>(b));
GiNaC::lst subs;
for (unsigned int ii=0; ii<xx.nops(); ii++)
{
subs.append(variables.op(ii) == xx.op(ii));
}
GiNaC::ex Nj1 = pspace.op(0).subs(subs);
GiNaC::ex Nj2 = pspace.op(1).subs(subs);
GiNaC::ex Nj3 = pspace.op(2).subs(subs);
Ns.insert(Ns.end(), GiNaC::matrix(3,1,GiNaC::lst(Nj1,Nj2,Nj3)));
b.let_op(i) = GiNaC::numeric(0);
}
}
}
| GiNaC::lst SyFi::Nedelec2Hdiv::dof_repr |
Definition at line 29 of file Nedelec2Hdiv.h.
Referenced by compute_basis_functions().
1.7.6.1