1. Integrals

Feel++ provide the integrate() function to define integral expressions which can be used to compute integrals, define linear and bi-linear forms.

1.1. Interface

  integrate( _range, _expr, _quad, _geomap );

Please notice that the order of the parameter is not important, these are boost parameters, so you can enter them in the order you want. To make it clear, there are two required parameters and 2 optional and they of course can be entered in any order provided you give the parameter name. If you don’t provide the parameter name (that is to say _range = or the others) they must be entered in the order they are described below.

Required parameters:

  • _range = domain of integration

  • _expr = integrand expression

Optional parameters:

  • _quad = quadrature to use instead of the default one. Several ways are possible to pass the quadrature order for backward compatibility

API

Example

Explanation

Version

_quad=<n>

_quad=5

Pass the quadrature order as an integer, the default quadrature is used

v0.105

_quad=im<Convex>(<n>)

_quad=im<Triangle>(5)

Pass the default quadrature formula on a triangle to integrate exactly order 5 polynomials

v0.105

_quad=im(mesh,<n>)

_quad=im(mesh,5)

Pass the default quadrature formula on a mesh mesh to integrate exactly order 5 polynomials

v0.105

_quad=_Q<n>()

_quad=_Q<5>()

Pass the quadrature order at compile time to integrate exactly order 5 polynomials. It will be deprecated in a future release

up to v0.105
Starting from v0.105, quadratures are built at runtime and no more at compile time which means that quadrature orders can be adjusted dynamically, e.g from a command line option

  • _geomap = type of geometric mapping to use, that is to say:

Feel Parameter

Description

GEOMAP_HO

High order approximation (same of the mesh)

GEOMAP_OPT

Optimal approximation: high order on boundary elements order 1 in the interior

GEOMAP_01

Order 1 approximation (same of the mesh)

1.2. Example

From doc/manual/tutorial/dar.cpp

form1( ... ) = integrate( _range = elements( mesh ),
                          _expr = f*id( v ) );

From doc/manual/tutorial/myintegrals.cpp

  // compute integral f on boundary
  double intf_3 = integrate( _range = boundaryfaces( mesh ),
                             _expr = f );

From doc/manual/advection/advection.cpp

  // using default quadrature
  form2( _test = Xh, _trial = Xh, _matrix = D ) +=
    integrate( _range = internalfaces( mesh ),
               _quad = 2*Order,
               _expr = ( averaget( trans( beta )*idt( u ) ) * jump( id( v ) ) )
               + penalisation*beta_abs*( trans( jumpt( trans( idt( u ) )) )
               *jump( trans( id( v ) ) ) ),
               _geomap = geomap );
  // use deprecated _Q
  form2( _test = Xh, _trial = Xh, _matrix = D ) +=
    integrate( _range = internalfaces( mesh ),
               _quad = _Q<2*Order>(),
               _expr = ( averaget( trans( beta )*idt( u ) ) * jump( id( v ) ) )
               + penalisation*beta_abs*( trans( jumpt( trans( idt( u ) )) )
               *jump( trans( id( v ) ) ) ),
               _geomap = geomap );

From doc/manual/laplacian/laplacian.cpp

 auto l = form1( _test=Xh, _vector=F );
 l = integrate( _range = elements( mesh ),
                _expr=f*id( v ) ) +
     integrate( _range = markedfaces( mesh, "Neumann" ),
                _expr = nu*gradg*vf::N()*id( v ) );

2. Computing my first Integrals

This part explains how to integrate on a mesh with Feel++ (source doc/manual/tutorial/myintegrals.cpp ).

Let’s consider the domain \(\Omega=[0,1]^d\) and associated meshes. Here, we want to integrate the following function

\[\begin{aligned} f(x,y,z) = x^2 + y^2 + z^2 \end{aligned}\]

on the whole domain \(\Omega\) and on part of the boundary \(\Omega\).

There is the appropriate code:

int
main( int argc, char** argv )
{
    // Initialize Feel++ Environment
    Environment env( _argc=argc, _argv=argv,
                     _desc=feel_options(),
                     _about=about( _name="myintegrals" ,
                                   _author="Feel++ Consortium",
                                   _email="feelpp-devel@feelpp.org" ) );

    // create the mesh (specify the dimension of geometric entity)
    auto mesh = unitHypercube<3>();

    // our function to integrate
    auto f = Px()*Px() + Py()*Py() + Pz()*Pz();

    // compute integral of f (global contribution)
    double intf_1 = integrate( _range = elements( mesh ),
                               _expr = f ).evaluate()( 0,0 );

    // compute integral of f (local contribution)
    double intf_2 = integrate( _range = elements( mesh ),
                               _expr = f ).evaluate(false)( 0,0 );

    // compute integral f on boundary
    double intf_3 = integrate( _range = boundaryfaces( mesh ),
                               _expr = f ).evaluate()( 0,0 );

    std::cout << "int global ; local ; boundary" << std::endl
              << intf_1 << ";" << intf_2 << ";" << intf_3 << std::endl;
}