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.