Provided by: librheolef-dev_7.2-2_amd64 
NAME
expression - involved by interpolate and integrate (rheolef-7.2)
SYNOPSIS
template <class Expression>
field interpolate (const space& Xh, const Expression& expr);
template <class Expression>
Result integrate (const Expression& expr);
DESCRIPTION
The expressions involved by interpolate(3) and integrate(3) functions base on several
predefined operators and functions that are reviewed here. Let phi, psi be two scalar-
valued field(2) or test(2) objects. Similarly, let u, v be two vector-valued objects and
sigma and tau be two tensor-valued objects. In the following table, the indexes i and j
are summed from 0 to d-1 where d is the dimension of the physical geometry, as described
by the geo(2) class.
c++ code | mathematics
------------------------+----------------------------------------------
dot(u,v) | u.v = sum_i u_i v_i
ddot(sigma,tau) | sigma:tau = sum_ij sigma_ij tau_ij
dddot(sigma,tau) | A:.B = sum_ijk A_ijk B_ijk
tr(sigma) | sum_i sigma_ii
trans(sigma) | sigma^T the transpose
sqr(phi), norm2(phi) | phi^2
norm2(u) | sum_i u_i^2
norm2(sigma) | sum_ij sigma_ij^2
abs(phi), norm(phi) | absolute value
norm(u) | sqrt(norm2(u))
norm(sigma) | sqrt(norm2(sigma))
------------------------+----------------------------------------------
grad(phi) | (d phi/d x_i)_i
grad(u) | (d u_i/d x_j)_ij
D(u) | (grad(u) + grad(u)^T)/2
div(u) | sum_i d u_i/d x_i = tr(D(u))
curl(phi) | (d phi/d x_1, - d phi/d x_0) when d=2
curl(u) | grad^u when d=3
------------------------+----------------------------------------------
normal() | n, the outward unit normal
grad_s(phi) | P*grad(phi) where P = I - n otimes n
grad_s(u) | grad(u)*P
Ds(u) | P*D(u)*P
div_s(u) | tr(Ds(u))
------------------------+----------------------------------------------
grad_h(phi) | grad(phi|K) broken gradient, piecewise
div_h(u) | div(u|K)
Dh(u) | D(u|K)
------------------------+----------------------------------------------
jump(phi) | [phi] = phi|K0 - phi|K1
| on an oriented side S = dK0 inter dK1
average(phi) | {phi} = (phi|K0 + phi|K1)/2
inner(phi) | phi|K0
outer(phi) | phi|K1
h_local() | meas(K)^(1/d)
penalty() | max(meas(dK0)/meas(K0), meas(dK1)/meas(K1))
------------------------+----------------------------------------------
sin, cos, tan, acos, | standard math function extended to field
asin, atan, cosh, sinh, | and test functions
tanh, log, log10, floor |
ceil |
pow(phi,psi) |
atan2(phi,psi) | atan(psi/phi)
fmod(phi,psi) | phi - floor(phi/psi+0.5)*psi
min(phi,psi) |
max(phi,psi) |
------------------------+----------------------------------------------
compose(f,phi) | f(phi), for a given function f
compose(f,phi1,..,phin) | f(phi0,...,phin)
compose(phi,X) | phi(X) for a characteristic X
See also compose(3) and characteristic(2) for details.
IMPLEMENTATION
The practical implementation of expressions involved by interpolate(3) and integrate(3)
functions bases on two very classical C++ idioms: the expression template and the
Substitution failure is not an error (SFINAE). These C++ idioms are described in the
following book:
D. Vandevoorde and N. M. Josuttis, C++ templates: the complete guide, Addison-Wesley, 2002
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.