Provided by: librheolef-dev_7.2-3build5_amd64 bug

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.