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curl(3rheolef) [debian man page]

curl(3rheolef)							    rheolef-6.1 						    curl(3rheolef)

NAME
curl -- curl operator SYNOPSIS
form(const space V, const space& M, "curl"); DESCRIPTION
Assembly the form associated to the curl operator on finite element space. In three dimensions, both V and M are vector-valued: / | b(u,q) = | curl(u).q dx | / Omega In two dimensions, only V is vector-valued. The V space may be a either P2 finite element space, while the M space may be P1d. See also form(2) and space(2). EXAMPLE
The following piece of code build the divergence form associated to the P2 approximation for a three dimensional geometry: geo omega("cube"); space V(omega, "P2", "vector"); space M(omega, "P1d", "vector"); form b(V, M, "curl"); while this code becomes in two dimension: geo omega("square"); space V(omega, "P2", "vector"); space M(omega, "P1d"); form b(V, M, "curl"); SEE ALSO
form(2), space(2) rheolef-6.1 rheolef-6.1 curl(3rheolef)

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s_grad_grad(3rheolef)						    rheolef-6.1 					     s_grad_grad(3rheolef)

NAME
s_grad_grad -- grad_grad-like operator for the Stokes stream function computation SYNOPSIS
form(const space V, const space& V, "s_grad_grad"); DESCRIPTION
Assembly the form associated to the -div(grad) variant operator on a finite element space V. The V space may be a either P1 or P2 finite element space. See also form(2) and space(2). On cartesian coordinate systems, the form coincide with the "grad_grad" one (see grad_grad(3)): / | a(u,v) = | grad(u).grad(v) dx | / Omega The stream function on tri-dimensionnal cartesian coordinate systems is such that u = curl psi div psi = 0 where u is the velocity field. Taking the curl of the first relation, using the identity: curl(curl(psi)) = -div(grad(psi)) + grad(div(psi)) and using the div(psi)=0 relation leads to: -div(grad(psi)) = curl(u) This relation leads to a variational formulation involving the the "grad_grad" and the "curl" forms (see grad_grad(3), curl(3)). In the axisymmetric case, the stream function psi is scalar ans is defined from the velocity field u=(ur,uz) by (see Batchelor, 6th ed., 1967, p 543): d psi d psi uz = (1/r) ----- and ur = - (1/r) ----- d r d r See also http://en.wikipedia.org/wiki/Stokes_stream_function . Multiplying by rot(xi)=(d xi/dr, -d xi/dz), and integrating with r dr dz, we get a well-posed variationnal problem: a(psi,xi) = b(xi,u) with / | (d psi d xi d psi d xi) a(psi,xi) = | (----- ---- + ----- ----) dr dz | ( d r d r d z d z ) / Omega and / | (d xi d xi ) b(xi,u) = | (---- ur - ---- uz) r dr dz | (d z d r ) / Omega Notice that a is symmetric definite positive, but without the 'r' weight as is is usual for axisymmetric standard forms. The b form is named "s_curl", for the Stokes curl variant of the "curl" operator (see s_curl(3)) as it is closely related to the "curl" operator, but differs by the r and 1/r factors, as: ( d (r xi) d xi ) curl(xi) = ( (1/r) -------- ; - -----) ( d r d z ) while ( d xi d xi ) s_curl(xi) = ( ---- ; - ---- ) ( d r d z ) EXAMPLE
The following piece of code build the form associated to the P1 approximation: geo g("square"); space V(g, "P1"); form a(V, V, "s_grad_grad"); SEE ALSO
form(2), space(2), grad_grad(3), grad_grad(3), curl(3), s_curl(3) rheolef-6.1 rheolef-6.1 s_grad_grad(3rheolef)

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