# Manpages

## 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

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+x); }
...
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);

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 [AT] imag.fr>