Beispiel #1
0
void Ortho()
{
    cout << "Basis = Dubiner Orthonormal on Simplex !\n";
    cout << "N = " << N << endl;
    D_Order << N << endl;

    typedef T value_type;

    timer t_tot;
    timer t;

    timeout( t.elapsed() );

    cout << "Basis Construction ... ";
    t.restart();
    const OrthonormalPolynomialSet<2, N, Scalar, value_type, Simplex> Dubiner_Basis;
    //const Dubiner<2, N, Normalized<false>, value_type> Dubiner_Basis;
    timeout( t.elapsed() );

    const int_type Dim_Basis = Dubiner_Basis.basis().coeff().size1();

    cout << "Construction of the Integration Method ... \n";
    const Gauss<Simplex<2,1>, N+N, value_type> Quad;

    /** Matrix construction **/

    /** Stiffness **/
    cout << "\n/********************************/" << endl;
    cout << "Construction of Stiffness Matrix..."  <<endl;

    cout <<"Dubiner Basis Derivation... ";
    t.restart();
    const ublas::vector<ublas::matrix<value_type> > Diff( Dubiner_Basis.derivate( Quad.points() ) );
    timeout( t.elapsed() );


    cout << "Wgts construction ... ";
    t.restart();
    ublas::diagonal_matrix<value_type> Wgts( Quad.weights().size() );

    for ( int_type i=0; i < Wgts.size1() ; ++i )
    {
        Wgts( i,i ) = Quad.weights()( i );
    }

    timeout( t.elapsed() );

    cout << "Stiffness numeric construction ... ";

    t.restart();

#if 1
    cout << "prod ... ";
    ublas::matrix<value_type> A ( ublas::prod( Diff( 0 ), Wgts ) );
    A = ublas::prod( A, ublas::trans( Diff( 0 ) ) );
    ublas::matrix<value_type> B ( ublas::prod( Diff( 1 ), Wgts ) );
    B = ublas::prod( B, ublas::trans( Diff( 1 ) ) );
#else
    cout << "axpy_prod ... ";
    ublas::matrix<value_type> A2( Diff( 0 ).size1(), Wgts.size2() );
    ublas::matrix<value_type> A( A2.size1(), Diff( 0 ).size1() );
    ublas::matrix<value_type> B2( Diff( 1 ).size1(), Wgts.size2() );
    ublas::matrix<value_type> B( B2.size1(), Diff( 1 ).size1() );

    ublas::matrix<value_type> tDiff_0( ublas::trans( Diff( 0 ) ) );
    ublas::matrix<value_type> tDiff_1( ublas::trans( Diff( 1 ) ) );

    ublas::axpy_prod( Diff( 0 ), Wgts, A2 );
    ublas::axpy_prod( A2, tDiff_0, A );
    ublas::axpy_prod( Diff( 1 ), Wgts, B2 );
    ublas::axpy_prod( B2, tDiff_1, B );
#endif

    ublas::matrix<value_type> Stiff( A + B );
    timeout( t.elapsed() );

    /** Mass **/

    cout << "\n/********************************/" << endl;
    cout << "Construction of the Mass Matrix..." << endl;

    cout <<"Dubiner Basis Evaluation... ";
    t.restart();
    ublas::matrix<value_type> Psi = Dubiner_Basis.evaluate( Quad.points() );
    timeout( t.elapsed() );

    cout << "Mass numeric construction ... ";
    t.restart();

#if 1
    cout << "prod ... ";
    ublas::matrix<value_type> Mass ( ublas::prod( Psi, Wgts ) );
    Mass = ublas::prod( Mass, ublas::trans( Psi ) );

#else
    cout << "axpy_prod ... ";
    ublas::matrix<value_type> M0( Psi.size1(), Wgts.size2() );
    ublas::matrix<value_type> Mass( Psi.size1(), Psi.size1() );
    ublas::matrix<value_type> tPsi( ublas::trans( Psi ) );

    ublas::axpy_prod( Psi, Wgts, M0 );
    ublas::axpy_prod( M0, tPsi, Mass );
#endif

    timeout( t.elapsed() );

    /** Global **/

    cout << "\n/********************************/" << endl;
    cout << "Assembly of the Global matrix ..."<<endl;
    ublas::matrix<value_type> Global( Stiff + Mass );

    /** Condition number estimation **/

    cout << "Condition number estimation ... ";
    t.restart();
    value_type condition = cond2<value_type>( Global );
    timeout( t.elapsed() );

    cout << "Condition number = " << condition << endl;
    D_cond << condition << endl;

    /** Right hand side **/

    cout << "Construction of the RHS ... ";
    t.restart();
    ublas::vector<value_type> F( Quad.points().size2() );
    ublas::vector<value_type> G0( Quad.fpoints( 0 ).size2() );
    ublas::vector<value_type> G1( Quad.fpoints( 1 ).size2() );
    ublas::vector<value_type> G2( Quad.fpoints( 2 ).size2() );

    ublas::diagonal_matrix<value_type> Wgts0( Quad.weights( 0 ).size() );
    ublas::diagonal_matrix<value_type> Wgts1( Quad.weights( 1 ).size() );
    ublas::diagonal_matrix<value_type> Wgts2( Quad.weights( 2 ).size() );

    for ( int_type i=0; i < F.size(); ++i )
    {
        F( i ) = f<value_type>( Quad.point( i ) );
    }

    F= ublas::prod( Wgts,F );
    F= ublas::prod( Psi,F );

    for ( int_type i=0; i < Wgts0.size1() ; ++i )
    {
        Wgts0( i,i ) = Quad.weights( 0 )( i );
        G0( i ) = g0<value_type>( ublas::column( Quad.fpoints( 0 ), i ) );
    }

    G0= ublas::prod( Wgts0,G0 );
    G0= ublas::prod( Dubiner_Basis.evaluate( Quad.fpoints( 0 ) ),G0 );

    for ( int_type i=0; i < Wgts1.size1() ; ++i )
    {
        Wgts1( i,i ) = Quad.weights( 1 )( i );
        G1( i ) = g1<value_type>( ublas::column( Quad.fpoints( 1 ), i ) );
    }

    G1= ublas::prod( Wgts1,G1 );
    G1= ublas::prod( Dubiner_Basis.evaluate( Quad.fpoints( 1 ) ),G1 );

    for ( int_type i=0; i < Wgts2.size1() ; ++i )
    {
        Wgts2( i,i ) = Quad.weights( 2 )( i );
        G2( i ) = g2<value_type>( ublas::column( Quad.fpoints( 2 ), i ) );
    }

    G2= ublas::prod( Wgts2,G2 );
    G2= ublas::prod( Dubiner_Basis.evaluate( Quad.fpoints( 2 ) ),G2 );

    /** Global right-hand side assembly **/

    F+=G0+G1+G2;
    timeout( t.elapsed() );

    /** System Resolution **/

    cout << "System Resolution ... ";
    t.restart();
    ublas::vector<value_type> Sol( Dim_Basis );
    LU<ublas::matrix<value_type> > lu( Global );

    Sol = lu.solve( F );

    timeout( t.elapsed() );


    /** L_2 Error Estimation **/

    cout << "L_2 Error estimation ... ";
    t.restart();
    ublas::vector<value_type> Error( Quad.points().size2() );

    for ( int_type i=0; i< Error.size(); ++i )
        Error( i ) = exact_sol<value_type>( ublas::column( Quad.points(), i ) );

    Error = Error - ublas::prod( ublas::trans( Psi ),Sol );
    Error = ublas::element_prod( Error,Error );

    value_type err =  sqrt( ublas::inner_prod( Error, Quad.weights() ) );
    timeout( t.elapsed() );

    cout << "\nErreur L_2= " << err << endl;

    D_error_L2 << err << endl;


    /** H¹ Error Estimation **/

    cout << "H^1 Error estimation ... ";
    t.restart();

    ublas::vector<value_type> Error_H1( Quad.points().size2() );

    ublas::vector<value_type> Approx ( Error_H1.size() );
    ublas::vector<value_type> Approx_x( Error_H1.size() );
    ublas::vector<value_type> Approx_y( Error_H1.size() );

    for ( uint16_type i=0; i< Error_H1.size(); ++i )
    {
        Approx( i )   = exact_sol<value_type>( ublas::column( Quad.points(), i ) );
        Approx_x( i ) = dexact_sol_x<value_type>( ublas::column( Quad.points(), i ) );
        Approx_y( i ) = dexact_sol_y<value_type>( ublas::column( Quad.points(), i ) );
    }

    ublas::vector<value_type> Delta( Approx - ublas::prod( ublas::trans( Psi ),Sol ) );
    ublas::vector<value_type> Delta_x( Approx_x - ublas::prod( ublas::trans( Diff( 0 ) ),Sol ) );
    ublas::vector<value_type> Delta_y( Approx_y - ublas::prod( ublas::trans( Diff( 1 ) ),Sol ) );


    Error_H1 =
        ublas::element_prod( Delta,Delta )
        + ublas::element_prod( Delta_x,Delta_x )
        + ublas::element_prod( Delta_y,Delta_y );


    value_type err_H1 =  sqrt( ublas::inner_prod( Error_H1, Quad.weights() ) );

    timeout( t.elapsed() );

    cout << "\nErreur H^1= " << err_H1 << endl;
    D_error << err_H1 << endl;

    double tol =  10*pow( 1.0/pow( N,N ), 0.2 );
    cout << "\n10*N^{-N/5} = " << tol << endl;

    D_tol << tol << endl;

    cout << "\n-------------------------------\n";
    cout << "Total runtime = ";
    double timing( t_tot.elapsed() );

    timeout( timing );
    cout << "-------------------------------\n";
    cout << "\n\n";
    D_timing << timing << endl;

    BOOST_CHECK( ( err_H1 <= tol ) );
}
Beispiel #2
0
void Boundary()
{
    cout << "Basis = Boundary Adapted on Simplex !\n";
    cout << "N = " << N << endl;
    B_Order << N << endl;

    typedef T value_type;

    cout << "Construction of the Integration Method ... ";

    timer t_tot;
    timer t;

    const Gauss<Simplex<2,1>, N+N, value_type> Quad;
    cout << t.elapsed() << " seconds.\n";

    cout << "Basis Construction ... ";
    t.restart();
    const BoundaryAdaptedPolynomialSet<2, N, Scalar, value_type, Simplex> BA_Basis;
    timeout( t.elapsed() );

    const int_type Dim_Basis = BA_Basis.basis().coeff().size1();


    /** Matrix construction **/

    /** Stiffness **/
    cout << "\n/********************************/" << endl;
    cout << "Construction of Stiffness Matrix..."  <<endl;

    cout <<"Boundary Adapted Basis Derivation... ";
    t.restart();
    const ublas::vector<ublas::matrix<value_type> > Diff( BA_Basis.derivate( Quad.points() ) );
    timeout( t.elapsed() );


    cout << "Wgts construction ... ";
    t.restart();
    ublas::diagonal_matrix<value_type> Wgts( Quad.weights().size() );

    for ( int_type i=0; i < Wgts.size1() ; ++i )
    {
        Wgts( i,i ) = Quad.weights()( i );
    }

    timeout( t.elapsed() );

    cout << "Stiffness numeric construction ... ";

    t.restart();
    ublas::matrix<value_type> A ( ublas::prod( Diff( 0 ), Wgts ) );
    A = ublas::prod( A, ublas::trans( Diff( 0 ) ) );
    ublas::matrix<value_type> B ( ublas::prod( Diff( 1 ), Wgts ) );
    B = ublas::prod( B, ublas::trans( Diff( 1 ) ) );

    ublas::matrix<value_type> Stiff( A + B );
    timeout( t.elapsed() );

    /** Mass **/

    cout << "\n/********************************/" << endl;
    cout << "Construction of the Mass Matrix..." << endl;

    cout <<"Boundary Adapted Basis Evaluation... ";
    t.restart();
    ublas::matrix<value_type> Psi = BA_Basis.evaluate( Quad.points() );
    timeout( t.elapsed() );

    cout << "Mass numeric construction ... ";
    t.restart();
    ublas::matrix<value_type> Mass ( ublas::prod( Psi, Wgts ) );
    Mass = ublas::prod( Mass, ublas::trans( Psi ) );
    timeout( t.elapsed() );

    if ( N == 14 )
    {
#if 0
        PrintMatlab<value_type>( Mass );
#else
        PrintMatlab<value_type>( Stiff );
#endif
    }

    /** Global **/

    cout << "\n/********************************/" << endl;
    cout << "Assembly of the Global matrix ..."<<endl;
    ublas::matrix<value_type> Global( Stiff + Mass );

    /** Right hand side **/

    cout << "Construction of the RHS ... ";
    t.restart();
    ublas::vector<value_type> F( Quad.points().size2() );
    ublas::vector<value_type> G0( Quad.fpoints( 0 ).size2() );
    ublas::vector<value_type> G2( Quad.fpoints( 2 ).size2() );
    ublas::diagonal_matrix<value_type> Wgts0( Quad.weights( 0 ).size() );
    ublas::diagonal_matrix<value_type> Wgts2( Quad.weights( 2 ).size() );

    for ( int_type i=0; i < F.size(); ++i )
    {
        F( i ) = f<value_type>( Quad.point( i ) );
    }

    F= ublas::prod( Wgts,F );
    F= ublas::prod( Psi,F );

    for ( int_type i=0; i < Wgts0.size1() ; ++i )
    {
        Wgts0( i,i ) = Quad.weights( 0 )( i );
        G0( i ) = g0<value_type>( ublas::column( Quad.fpoints( 0 ), i ) );
    }

    G0= ublas::prod( Wgts0,G0 );
    G0= ublas::prod( BA_Basis.evaluate( Quad.fpoints( 0 ) ),G0 );

    for ( int_type i=0; i < Wgts2.size1() ; ++i )
    {
        Wgts2( i,i ) = Quad.weights( 2 )( i );
        G2( i ) = g2<value_type>( ublas::column( Quad.fpoints( 2 ), i ) );
    }

    G2= ublas::prod( Wgts2,G2 );
    G2= ublas::prod( BA_Basis.evaluate( Quad.fpoints( 2 ) ),G2 );

    /** Global right-hand side assembly **/

    F+=G0+G2;
    timeout( t.elapsed() );


    /** Imposing Dirichlet boundary conditions **/

    cout << "Imposing Dirichlet boundary conditions ... ";
    t.restart();
    ublas::matrix<value_type> Global2( Global.size1()-( N-1 ), Global.size2()-( N-1 ) );
    ublas::vector<value_type> F2( F.size()- ( N-1 ) );

    for ( int_type i= 0 ; i < 3+( N-1 ) ; ++i )
    {
        for ( int_type j= 0 ; j < 3+( N-1 ) ; ++j )
        {
            Global2( i,j ) = Global( i,j );
        }

        for ( int_type j=3+2*( N-1 ); j < Global.size1() ; ++j )
        {
            Global2( i,j-( N-1 ) ) = Global( i,j );
        }

        F2( i ) = F( i );
    }

    for ( int_type i=3+2*( N-1 ); i < Global.size1() ; ++i )
    {
        for ( int_type j= 0 ; j < 3+( N-1 ) ; ++j )
        {
            Global2( i-N+1,j ) = Global( i,j );
        }

        for ( int_type j=3+2*( N-1 ); j < Global.size1() ; ++j )
        {
            Global2( i-N+1,j-( N-1 ) ) = Global( i,j );
        }

        F2( i-N+1 ) = F( i );
    }

    timeout( t.elapsed() );

    /** Condition number estimation **/

    cout << "Condition number estimation ... ";
    t.restart();
    value_type condition = cond2<value_type>( Global2 );
    timeout( t.elapsed() );

    cout << "Condition number = " << condition << endl;
    B_cond << condition << endl;

    /** System Resolution **/

    cout << "System Resolution ... ";
    t.restart();
    ublas::vector<value_type> Sol( Dim_Basis );
    LU<ublas::matrix<value_type> > lu( Global2 );

    ublas::vector<value_type> Sol2( Dim_Basis-N+1 );

    Sol2 = lu.solve( F2 );

    /** Expanding Solution on the Dirichlet boundary **/

    for ( int_type i= 0 ; i < 3+( N-1 ) ; ++i )
    {
        Sol( i ) = Sol2( i );
    }

    for ( int_type i= 3+( N-1 ) ; i < 3+2*( N-1 )  ; ++i )
    {
        Sol( i ) = value_type( 0.0 );
    }

    for ( int_type i=3+2*( N-1 ); i < Global.size1() ; ++i )
    {
        Sol( i ) = Sol2( i-N+1 );
    }

    timeout( t.elapsed() );

    /** Error Estimation **/

    cout << "Error estimation ... ";
    t.restart();
    ublas::vector<value_type> Error( Quad.points().size2() );

    for ( int_type i=0; i< Error.size(); ++i )
        Error( i ) = exact_sol<value_type>( ublas::column( Quad.points(), i ) );

    Error = Error - ublas::prod( ublas::trans( Psi ),Sol );
    Error = ublas::element_prod( Error,Error );

    value_type err =  sqrt( ublas::inner_prod( Error, Quad.weights() ) );
    timeout( t.elapsed() );

    cout << "\nErreur L_2 = " << err << endl;
    B_error_L2 << err << endl;

    /** H¹ Error Estimation **/

    cout << "H^1 Error estimation ... ";
    t.restart();
    ublas::vector<value_type> Error_H1( Quad.points().size2() );

    ublas::vector<value_type> Approx ( Error_H1.size() );
    ublas::vector<value_type> Approx_x( Error_H1.size() );
    ublas::vector<value_type> Approx_y( Error_H1.size() );

    for ( uint16_type i=0; i< Error_H1.size(); ++i )
    {
        Approx( i )   = exact_sol<value_type>( ublas::column( Quad.points(), i ) );
        Approx_x( i ) = dexact_sol_x<value_type>( ublas::column( Quad.points(), i ) );
        Approx_y( i ) = dexact_sol_y<value_type>( ublas::column( Quad.points(), i ) );
    }

    ublas::vector<value_type> Delta( Approx - ublas::prod( ublas::trans( Psi ),Sol ) );
    ublas::vector<value_type> Delta_x( Approx_x - ublas::prod( ublas::trans( Diff( 0 ) ),Sol ) );
    ublas::vector<value_type> Delta_y( Approx_y - ublas::prod( ublas::trans( Diff( 1 ) ),Sol ) );


    Error_H1 =
        ublas::element_prod( Delta,Delta )
        + ublas::element_prod( Delta_x,Delta_x )
        + ublas::element_prod( Delta_y,Delta_y );


    value_type err_H1 =  sqrt( ublas::inner_prod( Error_H1, Quad.weights() ) );

    timeout( t.elapsed() );

    cout << "\nErreur H^1= " << err_H1 << endl;
    B_error << err_H1 << endl;

    double tol =  10*pow( 1.0/pow( N,N ), 0.2 );
    cout << "\n10*N^{-N/5} = " << tol << endl;

    B_tol << tol << endl;

    cout << "\n-------------------------------\n";
    cout << "Total runtime = ";
    double timing( t_tot.elapsed() );
    timeout( timing );
    cout << "-------------------------------\n";
    cout << "\n\n";
    B_timing << timing << endl;

    BOOST_CHECK( ( err_H1 <= tol ) );
}