The Finite Element Embedded Library in C++
Platform & Compiler |
|
|
Travis/Ubuntu & Clang >= 3.5 |
||
Buildkite Ubuntu 16.10 |
Feel+\+ is a C++ library for arbitrary order Galerkin methods (e.g. finite and spectral element methods) continuous or discontinuous in 1D 2D and 3D. The objectives of this framework is quite ambitious; ambitions which could be express in various ways such as :
-
the creation of a versatile mathematical kernel solving easily problems using different techniques thus allowing testing and comparing methods, e.g. cG versus dG,
-
the creation of a small and manageable library which shall nevertheless encompass a wide range of numerical methods and techniques,
-
build mathematical software that follows closely the mathematical abstractions associated with partial differential equations (PDE),
-
the creation of a library entirely in C++ allowing to create complex and typically multi-physics applications such as fluid-structure interaction or mass transport in haemodynamic.
-
develop branch : Feel++ Online Reference Manual
-
1D 2D and 3D (including high order) geometries and also lower topological dimension 1D(curve) in 2D and 3D or 2D(surface) in 3D
-
continuous and discontinuous arbitrary order Galerkin Methods in 1D, 2D and 3D including finite and spectral element methods
-
domain specific embedded language in C++ for variational formulations
-
interfaced with PETSc for linear and non-linear solvers
-
seamless parallel computations using PETSc
-
interfaced with SLEPc for large-scale sparse standard and generalized eigenvalue solvers
-
supports Gmsh for mesh generation
-
supports Gmsh for post-processing (including on high order geometries)
-
supports Paraview and CEI/Ensight for post-processing and the following file formats: ensight gold, gmsh, xdmf
Here is a full example to solve -\Delta u = f \mbox{ in } \Omega,\quad u=g \mbox{ on } \partial \Omega
#include <feel/feel.hpp>
int main(int argc, char**argv )
{
using namespace Feel;
Environment env( _argc=argc, _argv=argv,
_desc=feel_options(),
_about=about(_name="qs_laplacian",
_author="Feel++ Consortium",
_email="feelpp-devel@feelpp.org"));
auto mesh = unitSquare();
auto Vh = Pch<1>( mesh );
auto u = Vh->element();
auto v = Vh->element();
auto l = form1( _test=Vh );
l = integrate(_range=elements(mesh),
_expr=id(v));
auto a = form2( _trial=Vh, _test=Vh );
a = integrate(_range=elements(mesh),
_expr=gradt(u)*trans(grad(v)) );
a+=on(_range=boundaryfaces(mesh), _rhs=l, _element=u,
_expr=constant(0.) );
a.solve(_rhs=l,_solution=u);
auto e = exporter( _mesh=mesh, _name="qs_laplacian" );
e->add( "u", u );
e->save();
return 0;
}Here is a full non-linear example - the Bratu equation - to solve -\Delta u + e^u = 0 \mbox{ in } \Omega,\quad u=0 \mbox{ on } \partial \Omega.
#include <feel/feel.hpp>
inline
Feel::po::options_description
makeOptions()
{
Feel::po::options_description bratuoptions( "Bratu problem options" );
bratuoptions.add_options()
( "lambda", Feel::po::value<double>()->default_value( 1 ),
"exp() coefficient value for the Bratu problem" )
( "penalbc", Feel::po::value<double>()->default_value( 30 ),
"penalisation parameter for the weak boundary conditions" )
( "hsize", Feel::po::value<double>()->default_value( 0.1 ),
"first h value to start convergence" )
( "export-matlab", "export matrix and vectors in matlab" )
;
return bratuoptions.add( Feel::feel_options() );
}
/**
* Bratu Problem
*
* solve \f$ -\Delta u + \lambda \exp(u) = 0, \quad u_\Gamma = 0\f$ on \f$\Omega\f$
*/
int
main( int argc, char** argv )
{
using namespace Feel;
Environment env( _argc=argc, _argv=argv,
_desc=makeOptions(),
_about=about(_name="bratu",
_author="Christophe Prud'homme",
_email="christophe.prudhomme@feelpp.org"));
auto mesh = unitSquare();
auto Vh = Pch<3>( mesh );
auto u = Vh->element();
auto v = Vh->element();
double penalbc = option(_name="penalbc").as<double>();
double lambda = option(_name="lambda").as<double>();
auto Jacobian = [=](const vector_ptrtype& X, sparse_matrix_ptrtype& J)
{
auto a = form2( _test=Vh, _trial=Vh, _matrix=J );
a = integrate( elements( mesh ), gradt( u )*trans( grad( v ) ) );
a += integrate( elements( mesh ), lambda*( exp( idv( u ) ) )*idt( u )*id( v ) );
a += integrate( boundaryfaces( mesh ),
( - trans( id( v ) )*( gradt( u )*N() ) - trans( idt( u ) )*( grad( v )*N() + penalbc*trans( idt( u ) )*id( v )/hFace() ) );
};
auto Residual = [=](const vector_ptrtype& X, vector_ptrtype& R)
{
auto u = Vh->element();
u = *X;
auto r = form1( _test=Vh, _vector=R );
r = integrate( elements( mesh ), gradv( u )*trans( grad( v ) ) );
r += integrate( elements( mesh ), lambda*exp( idv( u ) )*id( v ) );
r += integrate( boundaryfaces( mesh ),
( - trans( id( v ) )*( gradv( u )*N() ) - trans( idv( u ) )*( grad( v )*N() ) + penalbc*trans( idv( u ) )*id( v )/hFace() ) );
};
u.zero();
backend()->nlSolver()->residual = Residual;
backend()->nlSolver()->jacobian = Jacobian;
backend()->nlSolve( _solution=u );
auto e = exporter( _mesh=mesh );
e->add( "u", u );
e->save();
}