NAME

       integrate - expression integration (rheolef-7.2)

SYNOPSIS

           template <typename Expression>
           Value integrate (geo domain, Expression, integrate_option iopt);

DESCRIPTION

       This overloaded function is able to return either a scalar constant, a field(2) or a
       bilinear form(2), depending upon its arguments.

       1.  When the expression involves both trial and test(2) functions, the result is a
           bilinear form(2)

       2.  When the expression involves either a trial or a test(2) function, the result is a
           linear form, represented by the field(2) class

       3.  When the expression involves neither a trial nor a test(2) function, the result is a
           scalar constant

       The general call involves three arguments:

       1.  the geo(2) domain of integration

       2.  the expression to integrate

       3.  the integrate_option(3)

       Here is the overloaded synopsis:

           Float integrate (geo domain, Expression, integrate_option iopt);
           field integrate (geo domain, Expression, integrate_option iopt);
           form  integrate (geo domain, Expression, integrate_option iopt);

OMITTED ARGUMENTS

       Some argument could be omitted when the expression involves a test(2) function:

       • when the domain of integration is omitted, then it is taken as those of the test(2)
         function

       The reduced synopsis is:

           field integrate (Expression, integrate_option iopt);
           form  integrate (Expression, integrate_option iopt);

       • when the integrate_option(3) is omitted, then a Gauss quadrature formula is considered
         such that it integrates exactly 2*k+1 polynomials where k is the polynomial degree of
         the test(2) function. When a trial function is also involved, then this degree is
         k1+k2+1 where k1 and k2 are the polynomial degree of the test(2) and trial functions.

       The reduced synopsis is:

           field integrate (geo domain, Expression);
           form  integrate (geo domain, Expression);

        Both arguments could be omitted an the synopsis becomes:

           field integrate (Expression);
           form  integrate (Expression);

INTEGRATION OVER A SUBDOMAIN

       Let omega be a finite element mesh of a geometric domain, as described by the geo(2)
       class. A subdomain is defined by indexation, e.g. omega['left'] and, when a test(2)
       function is involved, the omega could be omitted, and only the string 'left' has to be
       present e.g.

           test v (Xh);
           field lh = integrate ("left", 2*v);

        is equivalent to

           field lh = integrate (omega["left"], 2*v);

MEASURE OF A DOMAIN

       Finally, when only the domain argument is provided, the integrate function returns its
       measure:

           Float integrate (geo domain);

EXAMPLES

       The computation of the measure of a domain:

           Float meas_omega = integrate (omega);
           Float meas_left  = integrate (omega["left"]);

        The integral of a function:

           Float f (const point& x) { return exp(x[0]+x[1]); }
           ...
           integrate_option iopt;
           iopt.set_order (3);
           Float int_f = integrate (omega, f, iopt);

        The function can be replaced by any expression combining functions, class-functions and
       field(2).

       The right-hand-side involved by the variational formulation

           space Xh (omega, "P1");
           test v (Xh);
           field lh = integrate (f*v);

        For a bilinear form:

           trial u (Xh);
           form m = integrate (u*v);
           form a = integrate (dot(grad(u),grad(v)));

        The expression can also combine functions, class-functions and field(2).

IMPLEMENTATION

       This documentation has been generated from file main/lib/integrate.h

AUTHOR

       Pierre  Saramito  <Pierre.Saramito@imag.fr>

COPYRIGHT

       Copyright   (C)  2000-2018  Pierre  Saramito  <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL
       version 3 or later  <http://gnu.org/licenses/gpl.html>.  This  is  free  software:  you
       are free to change and redistribute it.  There is NO WARRANTY, to the extent permitted by
       law.