void
NavierStokes::run()
{
this->init();
auto U = Xh->element( "(u,p)" );
auto V = Xh->element( "(u,q)" );
auto u = U.element<0>( "u" );
auto v = V.element<0>( "u" );
auto p = U.element<1>( "p" );
auto q = V.element<1>( "p" );
#if defined( FEELPP_USE_LM )
auto lambda = U.element<2>();
auto nu = V.element<2>();
#endif
//# endmarker4 #
LOG(INFO) << "[dof] number of dof: " << Xh->nDof() << "\n";
LOG(INFO) << "[dof] number of dof/proc: " << Xh->nLocalDof() << "\n";
LOG(INFO) << "[dof] number of dof(U): " << Xh->functionSpace<0>()->nDof() << "\n";
LOG(INFO) << "[dof] number of dof/proc(U): " << Xh->functionSpace<0>()->nLocalDof() << "\n";
LOG(INFO) << "[dof] number of dof(P): " << Xh->functionSpace<1>()->nDof() << "\n";
LOG(INFO) << "[dof] number of dof/proc(P): " << Xh->functionSpace<1>()->nLocalDof() << "\n";
LOG(INFO) << "Data Summary:\n";
LOG(INFO) << " hsize = " << meshSize << "\n";
LOG(INFO) << " export = " << this->vm().count( "export" ) << "\n";
LOG(INFO) << " mu = " << mu << "\n";
LOG(INFO) << " bccoeff = " << penalbc << "\n";
//# marker5 #
auto deft = gradt( u )+trans(gradt(u));
auto def = grad( v )+trans(grad(v));
//# endmarker5 #
//# marker6 #
// total stress tensor (trial)
auto SigmaNt = -idt( p )*N()+mu*deft*N();
// total stress tensor (test)
auto SigmaN = -id( p )*N()+mu*def*N();
//# endmarker6 #
auto F = M_backend->newVector( Xh );
auto D = M_backend->newMatrix( Xh, Xh );
// right hand side
auto ns_rhs = form1( _test=Xh, _vector=F );
LOG(INFO) << "[navier-stokes] vector local assembly done\n";
// construction of the BDF
auto bdfns=bdf(_space=Xh);
/*
* Construction of the left hand side
*/
auto navierstokes = form2( _test=Xh, _trial=Xh, _matrix=D );
mpi::timer chrono;
navierstokes += integrate( elements( mesh ), mu*inner( deft,def )+ trans(idt( u ))*id( v )*bdfns->polyDerivCoefficient( 0 ) );
LOG(INFO) << "mu*inner(deft,def)+(bdf(u),v): " << chrono.elapsed() << "\n";
chrono.restart();
navierstokes +=integrate( elements( mesh ), - div( v )*idt( p ) + divt( u )*id( q ) );
LOG(INFO) << "(u,p): " << chrono.elapsed() << "\n";
chrono.restart();
#if defined( FEELPP_USE_LM )
navierstokes +=integrate( elements( mesh ), id( q )*idt( lambda ) + idt( p )*id( nu ) );
LOG(INFO) << "(lambda,p): " << chrono.elapsed() << "\n";
chrono.restart();
#endif
std::for_each( inflow_conditions.begin(), inflow_conditions.end(),
[&]( BoundaryCondition const& bc )
{
// right hand side
ns_rhs += integrate( markedfaces( mesh, bc.marker() ), inner( idf(&bc,BoundaryCondition::operator()),-SigmaN+penalbc*id( v )/hFace() ) );
navierstokes +=integrate( boundaryfaces( mesh ), -inner( SigmaNt,id( v ) ) );
navierstokes +=integrate( boundaryfaces( mesh ), -inner( SigmaN,idt( u ) ) );
navierstokes +=integrate( boundaryfaces( mesh ), +penalbc*inner( idt( u ),id( v ) )/hFace() );
});
std::for_each( wall_conditions.begin(), wall_conditions.end(),
[&]( BoundaryCondition const& bc )
{
navierstokes +=integrate( boundaryfaces( mesh ), -inner( SigmaNt,id( v ) ) );
navierstokes +=integrate( boundaryfaces( mesh ), -inner( SigmaN,idt( u ) ) );
navierstokes +=integrate( boundaryfaces( mesh ), +penalbc*inner( idt( u ),id( v ) )/hFace() );
});
std::for_each( outflow_conditions.begin(), outflow_conditions.end(),
[&]( BoundaryCondition const& bc )
{
ns_rhs += integrate( markedfaces( mesh, bc.marker() ), inner( idf(&bc,BoundaryCondition::operator()),N() ) );
});
LOG(INFO) << "bc: " << chrono.elapsed() << "\n";
chrono.restart();
u = vf::project( _space=Xh->functionSpace<0>(), _expr=cst(0.) );
p = vf::project( _space=Xh->functionSpace<1>(), _expr=cst(0.) );
M_bdf->initialize( U );
for( bdfns->start(); bdfns->isFinished(); bdfns->next() )
{
// add time dependent terms
auto bdf_poly = bdfns->polyDeriv();
form1( _test=Xh, _vector=Ft ) =
integrate( _range=elements(mesh), _expr=trans(idv( bdf_poly ))*id( v ) );
// add convective terms
form1( _test=Xh, _vector=Ft ) +=
integrate( _range=elements(mesh), _expr=trans(gradv(u)*idv( u ))*id(v) );
// add contrib from time independent terms
Ft->add( 1., F );
// add time stepping terms from BDF to right hand side
backend()->solve( _matrix=D, _solution=U, _rhs=Ft );
this->exportResults( bdfns->time(), U );
}
} // NavierNavierstokes::run
/**
* @name derivativeMatrix
* @brief This is the main working function of the file. Every work of the file is done in this function.
* @param[in] vector<double> Points
* This would denote the inteprolating points for which the Lagrange polynomials have to be written.
*
* Example usage
* @code
* double integral = lobattoIntegration(unsigned n);//Storing the coefficient of the nth degree Legendre roots.
* @endcode
*/
vector< vector<double> > derivativeMatrix(vector<double> Points)
{
unsigned n = Points.size();
sort(Points.begin(),Points.end());
double start = Points[0];///The first element.
double end = Points[n-1];///The last element.
vector< vector<double> > DerivativeMatrix;
DerivativeMatrix=zeros(n,n);
vector< vector<double> > LagrangePolynomials = lagrangePolynomials(Points);
unsigned i,j;///Counters for the loop.
function<double(double)> eval;
for(i=0;i<n;i++)
{
for(j=0;j<n;j++)
{
eval = [&LagrangePolynomials,&i,&j](double x){ return ((polyEval(LagrangePolynomials[i],x)*polyEval(polyDeriv(LagrangePolynomials[j]),x)));};
DerivativeMatrix[i][j] = lobattoIntegration(start,end,n,eval);
}
}
return DerivativeMatrix;
}
