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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).

Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito [AT] 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.

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