integrate(3rheolef)

💖 integrate — expression integration (rheolef-7.2) 💖

okay bestie, pull up a chair, because this one is actually iconic. this is integrate(3rheolef) — yes, the manual page, and yes, we're doing this together 🫶

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✨ NAME ✨

integrate — expression integration, brought to you by rheolef-7.2 🎀

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🌸 SYNOPSIS 🌸

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

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👑 DESCRIPTION 👑

so this function is overloaded, which is giving main character energy because it literally changes what it returns based on the vibes of its arguments. it can hand you back a scalar constant, a field(2), or a bilinear form(2). no cap.

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 takes three arguments:

1. the geo(2) domain of integration
2. the expression to integrate
3. the integrate_option(3)

here's the overloaded synopsis, look at her:

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

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🦋 OMITTED ARGUMENTS 🦋

some arguments can just… not show up, lowkey, 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. 💀 (math girlies stay winning)

the reduced synopsis is:

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

both arguments could be omitted and the synopsis becomes:

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

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🧚 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);

literally the same thing, just shorter. obsessed 😌

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⭐ MEASURE OF A DOMAIN ⭐

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

Float integrate (geo domain);

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🔥 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). periodt 💜

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💅 IMPLEMENTATION 💅

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

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🌈 AUTHOR 🌈

Pierre Saramito <Pierre.Saramito@imag.fr>

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🎀 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.

okay so that last bit is real — there's literally NO WARRANTY, to the extent permitted by law. not a vibe check, a legal fact 🫶

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and that's the whole page, bestie!! this was rheolef, Version 7.2, integrate(3rheolef) — mwah, go integrate something iconic 💖✨👑