gsFunctionExpr.hpp

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#pragma once
 
#include <gsCore/gsLinearAlgebra.h>
 
/* ExprTk options */
 
//This define will enable printing of debug information to stdout during
//the compilation process.
//#define exprtk_enable_debugging
 
// This define will disable the ability for expressions to have comments.
// Expressions that have comments when parsed with a build that has this
// option, will result in a compilation failure.
#define exprtk_disable_comments
 
// This define will disable the loop-wise 'break' and 'continue'
// capabilities. Any expression that contains those keywords will result
// in a compilation failure.
#define exprtk_disable_break_continue
 
// This define will disable the short-circuit '&' (and) and '|' (or)
// operators
#define exprtk_disable_sc_andor
 
// This define will disable all enhanced features such as strength
// reduction and special function optimisations and expression specific
// type instantiations. This feature will reduce compilation times and
// binary sizes but will also result in massive performance degradation
// of expression evaluations.
#if !defined(NDEBUG) || defined(__MINGW32__)
#define exprtk_disable_enhanced_features
#endif
 
// This define will disable all string processing capabilities. Any
// expression that contains a string or string related syntax will result
// in a compilation failure.
#define exprtk_disable_string_capabilities
 
#define exprtk_disable_rtl_io_file
#define exprtk_disable_rtl_vecops
 
// The order in which header files are included is essential.
//
// It is important that all forward declaration file
// "exprtk_X_forward.hpp"" are included BEFORE the header file
// "exprtk.hpp" so that specializations for is_true(), is_false(),
// etcetera are not known for ALL types that should be supported. All
// adaptor files "exprtk_X_adaptor.hpp" have to be included AFTER the
// file "exprtk.hpp".
 
#if defined(GISMO_WITH_ADIFF)
#define DScalar gismo::ad::DScalar2<real_t,-1>
#include <exprtk_ad_forward.hpp>
#endif
 
#if defined(gsMpfr_ENABLED)
#include <exprtk_mpfr_forward.hpp>
#endif
 
#if defined(gsGmp_ENABLED)
#include <exprtk_gmp_forward.hpp>
#endif
 
#if defined(gsCoDiPack_ENABLED)
#include <gsCoDiPack/exprtk_codi_forward.hpp>
#endif
 
#if defined(gsUniversal_ENABLED)
#include <gsUniversal/exprtk_universal_forward.hpp>
#endif
 
#include <exprtk.hpp>
 
#if defined(GISMO_WITH_ADIFF)
#include <exprtk_ad_adaptor.hpp>
#endif
 
#if defined(gsMpfr_ENABLED)
#include <exprtk_mpfr_adaptor.hpp>
#endif
 
#if defined(gsGmp_ENABLED)
#include <exprtk_gmp_adaptor.hpp>
#endif
 
#if defined(gsCoDiPack_ENABLED)
#include <gsCoDiPack/exprtk_codi_adaptor.hpp>
#endif
 
#if defined(gsUniversal_ENABLED)
#include <gsUniversal/exprtk_universal_adaptor.hpp>
#endif
 
 
#include <gsIO/gsXml.h>
 
namespace
{
 
// addition of mixed derivative for expressions
// see https://en.wikipedia.org/wiki/Finite_difference_coefficient
template <typename T>
T mixed_derivative(const exprtk::expression<T>& e,
                     T& x, T& y,
                     const double& h = 0.00001)
{
    T num = (T)(0.0), tmp;
    T x_init = x;
    T y_init = y;
 
    x = x_init + (T)(2.0) * h;
    y = y_init + (T)(2.0) * h;
    num += e.value();
    y = y_init - (T)(2.0) * h;
    num -= e.value();
    x = x_init - (T)(2.0) * h;
    num += e.value();
    y = y_init + (T)(2.0) * h;
    num -= e.value();
 
    x = x_init + h;
    y = y_init + h;
    tmp = e.value();
    y = y_init - h;
    tmp -= e.value();
    x = x_init - h;
    tmp += e.value();
    y = y_init + h;
    tmp -= e.value();
    num += (T)(64.0) * tmp;
 
    x = x_init + (T)(2.0) * h;
    y = y_init - h;
    tmp = e.value();
    y = y_init + h;
    tmp -= e.value();
    x = x_init - (T)(2.0) * h;
    tmp += e.value();
    y = y_init - h;
    tmp -= e.value();
 
    y = y_init + (T)(2.0) * h;
    x = x_init - h;
    tmp += e.value();
    x = x_init + h;
    tmp -= e.value();
    y = y_init - (T)(2.0) * h;
    tmp += e.value();
    x = x_init - h;
    tmp -= e.value();
    num += (T)(8.0) * tmp;
 
    x = x_init;
    y = y_init;
    return num / ( (T)(144.0)*h*h );
}
 
} //namespace
 
#define N_VARS 7
 
namespace gismo
{
 
template<typename T> class gsFunctionExpr<T>::gsFunctionExprPrivate
{
public:
 
#ifdef GISMO_WITH_ADIFF
    typedef DScalar Numeric_t;
#else
    typedef T Numeric_t;
#endif
 
    typedef exprtk::symbol_table<Numeric_t>  SymbolTable_t;
    typedef exprtk::expression<Numeric_t>    Expression_t;
    typedef exprtk::parser<Numeric_t>        Parser_t;
 
public:
 
    gsFunctionExprPrivate(const short_t _dim)
    : vars(), dim(_dim)
    {
        GISMO_ENSURE( dim <= N_VARS, "The number of variables can be at most 7 (x,y,z,w,u,v,t)." );
        init();
    }
 
    gsFunctionExprPrivate(const gsFunctionExprPrivate & other)
    : vars(), dim(other.dim)
    {
        GISMO_ASSERT ( string.size() == expression.size(), "Corrupted FunctionExpr");
        init();
        //copy_n(other.vars, N_VARS+1, vars);
        string    .reserve(string.size());
        expression.reserve(string.size());
        for (size_t i = 0; i!= other.string.size(); ++i)
            addComponent(other.string[i]);
    }
 
    void addComponent(const std::string & strExpression)
    {
        string.push_back( strExpression );// Keep string data
        std::string & str = string.back();
        str.erase(std::remove(str.begin(), str.end(),' '), str.end() );
        gismo::util::string_replace(str, "**", "^");
 
        // String expression
        expression.push_back(Expression_t());
        Expression_t & expr = expression.back();
        //expr.release();
        expr.register_symbol_table(symbol_table);
 
        // Parser
        Parser_t parser;
        //Collect variable symbols
        //parser.dec().collect_variables() = true;
        bool success = parser.compile(str, expr);
        if ( ! success )
            gsWarn<<"gsFunctionExpr error: " <<parser.error() <<" while parsing "<<str<<"\n";
        /*
           typedef typename exprtk::parser_t::
           dependent_entity_collector::symbol_t symbol_t;
 
           std::deque<symbol_t> symbol_list;
           parser.dec().symbols(symbol_list);
           for (size_t i = 0; i != symbol_list.size(); ++i)
           {
           symbol_t& symbol = symbol_list[i];
           // do something
           }
        */
    }
 
    void init()
    {
        //symbol_table.clear();
        // Identify symbol table
        symbol_table.add_variable("x",vars[0]);
        symbol_table.add_variable("y",vars[1]);
        symbol_table.add_variable("z",vars[2]);
        symbol_table.add_variable("w",vars[3]);
        symbol_table.add_variable("u",vars[4]);
        symbol_table.add_variable("v",vars[5]);
        symbol_table.add_variable("t",vars[6]);
        //symbol_table.remove_variable("w",vars[3]);
        symbol_table.add_pi();
        //symbol_table.add_constant("C", 1);
    }
 
public:
    mutable Numeric_t         vars[N_VARS];
    SymbolTable_t             symbol_table;
    std::vector<Expression_t> expression;
    std::vector<std::string>  string;
    short_t dim;
 
private:
    gsFunctionExprPrivate();
    gsFunctionExprPrivate operator= (const gsFunctionExprPrivate & other);
};
 
/* / /AM: under construction
template<typename T> class gsSymbolListPrivate
{
public:
    exprtk::symbol_table<T> symbol_table;
    std::vector<T> vars  ;
    std::vector<T> params;
};
 
template<typename T>
gsSymbolList<T>::gsSymbolList() : my(new gsSymbolListPrivate<T>) {}
 
template<typename T>
gsSymbolList<T>::~gsSymbolList()
{
delete my;
}
 
template<typename T>
void gsSymbolList<T>::setDefault()
{
    my->symbol_table.clear();
 
    // Identify symbol table
    my->symbol_table.add_variable("x",my->vars[0]);
    my->symbol_table.add_variable("y",my->vars[1]);
    my->symbol_table.add_variable("z",my->vars[2]);
    my->symbol_table.add_variable("w",my->vars[3]);
    my->symbol_table.add_variable("u",my->vars[4]);
    my->symbol_table.add_variable("v",my->vars[5]);
    //my->symbol_table.remove_variable("w",my->vars[3]);
    my->symbol_table.add_pi();
    //my->symbol_table.add_constant("C", 1);
}
 
template<typename T>
bool gsSymbolList<T>::addConstant(const std::string & constant_name, const T& value)
{
return my->symbol_table.add_constant(constant_name, value);
}
 
template<typename T>
bool gsSymbolList<T>::addVariable(const std::string & variable_name)
{
    my->vars.push_back(0.0);
    return my->symbol_table.add_variable(variable_name, my->vars.back() );
}
 
template<typename T>
bool gsSymbolList<T>::addParameter(const std::string & variable_name)
{
    my->params.push_back(0.0);
    return my->symbol_table.add_variable(variable_name, my->params.back() );
}
 
template<typename T>
bool gsSymbolList<T>::hasSymbol(const std::string& symbol_name)
{
return true;
}
*/
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr() : my(new PrivateData_t(0))
{ }
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(const std::string & expression_string, short_t ddim)
: my(new PrivateData_t(ddim))
{
    my->addComponent(expression_string);
}
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(const std::string & expression_string1,
                                  const std::string & expression_string2,
                                  short_t ddim)
: my(new PrivateData_t(ddim))
{
    my->addComponent(expression_string1);
    my->addComponent(expression_string2);
}
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(const std::string & expression_string1,
                                  const std::string & expression_string2,
                                  const std::string & expression_string3,
                                  short_t ddim)
: my(new PrivateData_t(ddim))
{
    my->addComponent(expression_string1);
    my->addComponent(expression_string2);
    my->addComponent(expression_string3);
}
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(const std::string & expression_string1,
                   const std::string & expression_string2,
                   const std::string & expression_string3,
                   const std::string & expression_string4,
                   short_t ddim)
: my(new PrivateData_t(ddim))
{
    my->addComponent(expression_string1);
    my->addComponent(expression_string2);
    my->addComponent(expression_string3);
    my->addComponent(expression_string4);
}
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(const std::string & expression_string1,
                   const std::string & expression_string2,
                   const std::string & expression_string3,
                   const std::string & expression_string4,
                   const std::string & expression_string5,
                   const std::string & expression_string6,
                   const std::string & expression_string7,
                   const std::string & expression_string8,
                   const std::string & expression_string9,
                   short_t ddim)
: my(new PrivateData_t(ddim))
{
    my->addComponent(expression_string1);
    my->addComponent(expression_string2);
    my->addComponent(expression_string3);
    my->addComponent(expression_string4);
    my->addComponent(expression_string5);
    my->addComponent(expression_string6);
    my->addComponent(expression_string7);
    my->addComponent(expression_string8);
    my->addComponent(expression_string9);
}
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(const std::vector<std::string> & expression_string,
                                  short_t ddim)
: my(new PrivateData_t(ddim))
{
    for (size_t i = 0; i!= expression_string.size(); ++i)
        my->addComponent(expression_string[i]);
}
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(const gsFunctionExpr & other)
{
    my = new PrivateData_t(*other.my);
}
#if EIGEN_HAS_RVALUE_REFERENCES
 
template<typename T>
gsFunctionExpr<T>::gsFunctionExpr(gsFunctionExpr && other)
{
    my = other.my; other.my = NULL;
}
 
template<typename T>
gsFunctionExpr<T> & gsFunctionExpr<T>::operator=(const gsFunctionExpr& other)
{
    if (this != &other)
    {
        delete my;
        my = new PrivateData_t(*other.my);
    }
    return *this;
}
 
template<typename T>
gsFunctionExpr<T> & gsFunctionExpr<T>::operator=(gsFunctionExpr&& other)
{
    if (this != &other)
    {
        delete my;
        my = other.my; other.my = NULL;
    }
    return *this;
}
 
#else
template<typename T>
gsFunctionExpr<T> & gsFunctionExpr<T>::operator=(gsFunctionExpr other)
{
    std::swap(my,other.my);
    return *this;
}
#endif
 
template<typename T>
gsFunctionExpr<T>::~gsFunctionExpr()
{
    delete my;
}
 
template<typename T>
short_t gsFunctionExpr<T>::domainDim() const
{
    return my->dim;
}
 
template<typename T>
short_t gsFunctionExpr<T>::targetDim() const
{
    return static_cast<short_t>(my->string.size());
}
 
template<typename T>
void gsFunctionExpr<T>::addComponent(const std::string & strExpression)
{
    my->addComponent(strExpression);
}
 
template<typename T>
const std::string & gsFunctionExpr<T>::expression(int i) const
{
    return my->string[i];
}
 
template<typename T>
void gsFunctionExpr<T>::set_x (T const & v) const { my->vars[0]= v; }
 
template<typename T>
void gsFunctionExpr<T>::set_y (T const & v) const { my->vars[1]= v; }
 
template<typename T>
void gsFunctionExpr<T>::set_z (T const & v) const { my->vars[2]= v; }
 
template<typename T>
void gsFunctionExpr<T>::set_w (T const & v) const { my->vars[3]= v; }
 
template<typename T>
void gsFunctionExpr<T>::set_u (T const & v) const { my->vars[4]= v; }
 
template<typename T>
void gsFunctionExpr<T>::set_v (T const & v) const { my->vars[5]= v; }
 
template<typename T>
void gsFunctionExpr<T>::set_t (T const & t) const { my->vars[6]= t; }
 
template<typename T>
void gsFunctionExpr<T>::eval_into(const gsMatrix<T>& u, gsMatrix<T>& result) const
{
    GISMO_ASSERT ( u.rows() == my->dim, "Inconsistent point dimension (expected: "
                   << my->dim <<", got "<< u.rows() <<")\n"<< *this);
 
    const short_t n = targetDim();
    result.resize(n, u.cols());
 
#pragma omp critical (gsFunctionExpr_run)
    for ( index_t p = 0; p!=u.cols(); p++ ) // for all evaluation points
    {
        copy_n(u.col(p).data(), my->dim, my->vars);
 
        for (short_t c = 0; c!= n; ++c) // for all components
#           ifdef GISMO_WITH_ADIFF
            result(c,p) = my->expression[c].value().getValue();
#           else
            result(c,p) = my->expression[c].value();
#           endif
    }
}
 
template<typename T>
void gsFunctionExpr<T>::eval_component_into(const gsMatrix<T>& u, const index_t comp, gsMatrix<T>& result) const
{
    GISMO_ASSERT ( u.rows() == my->dim, "Inconsistent point dimension (expected: "
                   << my->dim <<", got "<< u.rows() <<")");
 
    GISMO_ASSERT (comp < targetDim(),
                  "Given component number is higher then number of components");
 
    result.resize(1, u.cols());
#   pragma omp critical (gsFunctionExpr_run)
    for ( index_t p = 0; p!=u.cols(); ++p )
    {
        copy_n(u.col(p).data(), my->dim, my->vars);
 
#           ifdef GISMO_WITH_ADIFF
            result(0,p) = my->expression[comp].value().getValue();
#           else
            result(0,p) = my->expression[comp].value();
#           endif
    }
}
 
template<typename T>
void gsFunctionExpr<T>::deriv_into(const gsMatrix<T>& u, gsMatrix<T>& result) const
{
    //gsDebug<< "Using finite differences (gsFunctionExpr::deriv_into) for derivatives.\n";
    const short_t d = domainDim();
    GISMO_ASSERT ( u.rows() == my->dim, "Inconsistent point dimension (expected: "
                   << my->dim <<", got "<< u.rows() <<")");
 
    const short_t n = targetDim();
    result.resize(d*n, u.cols());
#   pragma omp critical (gsFunctionExpr_run)
    for ( index_t p = 0; p!=u.cols(); p++ ) // for all evaluation points
    {
#       ifdef GISMO_WITH_ADIFF
        for (short_t k = 0; k!=d; ++k)
            my->vars[k].setVariable(k,d,u(k,p));
        for (short_t c = 0; c!= n; ++c) // for all components
            my->expression[c].value().gradient_into(result.block(c*d,p,d,1));
            //result.block(c*d,p,d,1) = my->expression[c].value().getGradient(); //fails on constants
#       else
        copy_n(u.col(p).data(), my->dim, my->vars);
        for (short_t c = 0; c!= n; ++c) // for all components
            for ( short_t j = 0; j!=d; j++ ) // for all variables
                result(c*d + j, p) =
                    exprtk::derivative<T>(my->expression[c], my->vars[j], 0.00001 ) ;
#       endif
    }
}
 
template<typename T>
void gsFunctionExpr<T>::deriv2_into(const gsMatrix<T>& u, gsMatrix<T>& result) const
{
    const short_t d = domainDim();
    GISMO_ASSERT ( u.rows() == my->dim, "Inconsistent point dimension (expected: "
                   << my->dim <<", got "<< u.rows() <<")");
 
    const short_t n = targetDim();
    const index_t stride = d + d*(d-1)/2;
    result.resize(stride*n, u.cols() );
#   pragma omp critical (gsFunctionExpr_run)
    for ( index_t p = 0; p!=u.cols(); p++ ) // for all evaluation points
    {
#       ifndef GISMO_WITH_ADIFF
        copy_n(u.col(p).data(), my->dim, my->vars);
#       endif
 
        for (short_t c = 0; c!= n; ++c) // for all components
        {
#           ifdef GISMO_WITH_ADIFF
            for (index_t v = 0; v!=d; ++v)
                my->vars[v].setVariable(v,d,u(v,p));
            const DScalar &            ads  = my->expression[c].value();
            const DScalar::Hessian_t & Hmat = ads.getHessian(); // note: can fail
 
            for ( index_t k=0; k!=d; ++k)
            {
                result(c*stride + k,p) = Hmat(k,k);
                index_t m = d;
                for ( index_t l=k+1; l<d; ++l)
                    result(c*stride + m++,p) = Hmat(k,l);
            }
#           else
            for (short_t k = 0; k!=d; ++k)
            {
                // H_{k,k}
                result(c*stride + k,p) = exprtk::
                    second_derivative<T>(my->expression[c], my->vars[k], 0.00001);
 
                short_t m = d;
                for (short_t l=k+1; l<d; ++l)
                {
                    // H_{k,l}
                    result(c*stride + m++,p) =
                        mixed_derivative<T>( my->expression[c], my->vars[k],
                                             my->vars[l], 0.00001 );
                }
            }
#           endif
        }
    }
}
 
template<typename T>
gsMatrix<T>
gsFunctionExpr<T>::hess(const gsMatrix<T>& u, unsigned coord) const
{
    //gsDebug<< "Using finite differences (gsFunctionExpr::hess) for Hessian.\n";
    GISMO_ENSURE(coord == 0, "Error, function is real");
    GISMO_ASSERT ( u.cols() == 1, "Need a single evaluation point." );
    const index_t d = u.rows();
    GISMO_ASSERT ( u.rows() == my->dim, "Inconsistent point dimension (expected: "
                   << my->dim <<", got "<< u.rows() <<")");
 
    gsMatrix<T> res(d, d);
 
#   pragma omp critical (gsFunctionExpr_run)
{
#   ifdef GISMO_WITH_ADIFF
    for (index_t v = 0; v!=d; ++v)
        my->vars[v].setVariable(v, d, u(v,0) );
    my->expression[coord].value().hessian_into(res);
#   else
    copy_n(u.data(), my->dim, my->vars);
    for( index_t j=0; j!=d; ++j )
    {
        res(j,j) = exprtk::
            second_derivative<T>( my->expression[coord], my->vars[j], 0.00001);
 
        for( index_t k = 0; k!=j; ++k )
            res(k,j) = res(j,k) =
                mixed_derivative<T>( my->expression[coord], my->vars[k],
                                     my->vars[j], 0.00001 );
    }
#   endif
}
    return res;
}
 
template<typename T>
gsMatrix<T> * gsFunctionExpr<T>::mderiv(const gsMatrix<T> & u,
                                        const index_t k,
                                        const index_t j) const
{
    GISMO_ASSERT ( u.rows() == my->dim, "Inconsistent point size.");
    const short_t n = targetDim();
    gsMatrix<T> * res= new gsMatrix<T>(n,u.cols()) ;
 
#   pragma omp critical (gsFunctionExpr_run)
    for( index_t p=0; p!=res->cols(); ++p )
    {
#       ifndef GISMO_WITH_ADIFF
        copy_n(u.col(p).data(), my->dim, my->vars);
#       endif
 
        for (short_t c = 0; c!= n; ++c) // for all components
        {
#           ifdef GISMO_WITH_ADIFF
            for (index_t v = 0; v!=my->dim; ++v)
                my->vars[v].setVariable(v, my->dim, u(v,p) );
            (*res)(c,p) = my->expression[c].value().getHessian()(k,j); //note: can fail
#           else
            (*res)(c,p) =
                mixed_derivative<T>( my->expression[c], my->vars[k], my->vars[j], 0.00001 ) ;
#           endif
        }
    }
    return res;
}
 
template<typename T>
gsMatrix<T> gsFunctionExpr<T>::laplacian(const gsMatrix<T>& u) const
{
    //gsDebug<< "Using finite differences (gsFunction::laplacian) for Laplacian.\n";
    GISMO_ASSERT ( u.rows() == my->dim, "Inconsistent point size.");
    const short_t n = targetDim();
    gsMatrix<T> res(n,u.cols());
 
    #   pragma omp critical (gsFunctionExpr_run)
    for( index_t p = 0; p != res.cols(); ++p )
    {
#       ifndef GISMO_WITH_ADIFF
        copy_n(u.col(p).data(), my->dim, my->vars);
#       endif
 
        for (short_t c = 0; c!= n; ++c) // for all components
        {
#           ifdef GISMO_WITH_ADIFF
            for (index_t v = 0; v!=my->dim; ++v)
                my->vars[v].setVariable(v, my->dim, u(v,p) );
            res(c,p) = my->expression[c].value().getHessian().trace();
#           else
            T & val = res(c,p);
            for ( index_t j = 0; j!=my->dim; ++j )
                val += exprtk::
                    second_derivative<T>( my->expression[c], my->vars[j], 0.00001 );
#           endif
        }
    }
    return  res;
}
 
template<typename T>
std::ostream & gsFunctionExpr<T>::print(std::ostream &os) const
{
    os <<"[ ";
    if( my->string.empty() )
        os << "empty";
    else
    {
        os << my->string[0];
 
        for (short_t k = 1; k<targetDim(); ++k)
            os <<", " << my->string[k];
    }
    os <<" ]";
    return os;
}
 
namespace internal
{
 
template<class T>
class gsXml< gsFunctionExpr<T> >
{
private:
    gsXml() { }
    typedef gsFunctionExpr<T> Object;
public:
    GSXML_COMMON_FUNCTIONS(Object);
    static std::string tag ()  { return "Function"; }
    static std::string type () { return "FunctionExpr"; }
 
    GSXML_GET_POINTER(Object);
 
    static void get_into (gsXmlNode * node, Object & obj)
    {
        GISMO_ASSERT( node->first_attribute("dim"), "Reading gsFunctionExpr XML: No dim found" ) ;
        const int d = atoi( node->first_attribute("dim")->value() );
 
        std::vector< std::string > expr_strings;
 
        gsXmlNode * child = node->first_node("c");
 
        if (child != NULL )
        {
            for (; child; child = child->next_sibling() )
                expr_strings.push_back(  child->value() );
        }
        else
            expr_strings.push_back(  node->value() );
 
        obj = gsFunctionExpr<T>( expr_strings, d );
    }
 
    static gsXmlNode * put (const Object & obj,
                            gsXmlTree & data )
    {
        gsXmlNode * func = makeNode("Function", data);
        func->append_attribute(makeAttribute("dim", obj.domainDim(), data));
        func->append_attribute(makeAttribute("type", "FunctionExpr", data));
 
        const short_t tdim = obj.targetDim();
 
        if ( tdim == 1)
        {
            func->value( makeValue(obj.expression(), data) );
        }
        else
        {
            gsXmlNode * cnode;
            for (short_t c = 0; c!=tdim; ++c)
            {
                cnode = makeNode("c", obj.expression(c), data);
                func->append_node(cnode);
            }
        }
 
        return func;
    }
};
 
} // internal
 
}; // namespace gismo