.TH "integrate" 3rheolef "Version 7.2" "rheolef" \" -*- nroff -*-
.ad l
.nh
.SH NAME
integrate \- expression integration (rheolef-7\&.2)
.SH "SYNOPSIS"
.PP
.PP
.nf
    template <typename Expression>
    Value integrate (geo domain, Expression, integrate_option iopt);
.fi
.PP
 
.SH "DESCRIPTION"
.PP
This overloaded function is able to return either a scalar constant, a \fBfield(2)\fP or a bilinear \fBform(2)\fP, depending upon its arguments\&.
.PP
.IP "1." 4
When the expression involves both \fCtrial\fP and \fBtest(2)\fP functions, the result is a bilinear \fBform(2)\fP
.IP "2." 4
When the expression involves either a \fCtrial\fP or a \fBtest(2)\fP function, the result is a linear form, represented by the \fBfield(2)\fP class
.IP "3." 4
When the expression involves neither a \fCtrial\fP nor a \fBtest(2)\fP function, the result is a scalar constant
.PP
.PP
The general call involves three arguments:
.PP
.IP "1." 4
the \fBgeo(2)\fP domain of integration
.IP "2." 4
the expression to integrate
.IP "3." 4
the \fBintegrate_option(3)\fP
.PP
.PP
Here is the overloaded synopsis: 
.PP
.nf
    Float integrate (geo domain, Expression, integrate_option iopt);
    field integrate (geo domain, Expression, integrate_option iopt);
    form  integrate (geo domain, Expression, integrate_option iopt);

.fi
.PP
 
.SH "OMITTED ARGUMENTS"
.PP
Some argument could be omitted when the expression involves a \fBtest(2)\fP function:
.PP
.IP "\(bu" 2
when the domain of integration is omitted, then it is taken as those of the \fBtest(2)\fP function
.PP
.PP
The reduced synopsis is: 
.PP
.nf
    field integrate (Expression, integrate_option iopt);
    form  integrate (Expression, integrate_option iopt);

.fi
.PP
.IP "\(bu" 2
when the \fBintegrate_option(3)\fP is omitted, then a Gauss quadrature formula is considered such that it integrates exactly \fC2*k+1\fP polynomials where \fCk\fP is the polynomial degree of the \fBtest(2)\fP function\&. When a \fCtrial\fP function is also involved, then this degree is \fCk1+k2+1\fP where \fCk1\fP and \fCk2\fP are the polynomial degree of the \fBtest(2)\fP and \fCtrial\fP functions\&.
.PP
.PP
The reduced synopsis is: 
.PP
.nf
    field integrate (geo domain, Expression);
    form  integrate (geo domain, Expression);

.fi
.PP
 Both arguments could be omitted an the synopsis becomes: 
.PP
.nf
    field integrate (Expression);
    form  integrate (Expression);

.fi
.PP
 
.SH "INTEGRATION OVER A SUBDOMAIN"
.PP
Let \fComega\fP be a finite element mesh of a geometric domain, as described by the \fBgeo(2)\fP class\&. A subdomain is defined by indexation, e\&.g\&. \fComega['left']\fP and, when a \fBtest(2)\fP function is involved, the \fComega\fP could be omitted, and only the string \fC'left'\fP has to be present e\&.g\&. 
.PP
.nf
    test v (Xh);
    field lh = integrate ('left', 2*v);

.fi
.PP
 is equivalent to 
.PP
.nf
    field lh = integrate (omega['left'], 2*v);

.fi
.PP
 
.SH "MEASURE OF A DOMAIN"
.PP
Finally, when only the domain argument is provided, the \fCintegrate\fP function returns its measure: 
.PP
.nf
    Float integrate (geo domain);

.fi
.PP
 
.SH "EXAMPLES"
.PP
The computation of the measure of a domain: 
.PP
.nf
    Float meas_omega = integrate (omega);
    Float meas_left  = integrate (omega['left']);

.fi
.PP
 The integral of a function: 
.PP
.nf
    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);

.fi
.PP
 The function can be replaced by any expression combining functions, class-functions and \fBfield(2)\fP\&.
.PP
The right-hand-side involved by the variational formulation 
.PP
.nf
    space Xh (omega, 'P1');
    test v (Xh);
    field lh = integrate (f*v);

.fi
.PP
 For a bilinear form: 
.PP
.nf
    trial u (Xh);
    form m = integrate (u*v);
    form a = integrate (dot(grad(u),grad(v)));

.fi
.PP
 The expression can also combine functions, class-functions and \fBfield(2)\fP\&.
.SH "IMPLEMENTATION"
.PP
This documentation has been generated from file main/lib/integrate\&.h 
.SH AUTHOR
Pierre  Saramito  <Pierre.Saramito@imag.fr>
.SH 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.