SFEMaNS  version 4.1 (work in progress)
Reference documentation for SFEMaNS
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doc_intro.h File Reference

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Functions

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with 
f (r,\theta, z)\f $the cylindrical coordinates
 

Variables

section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode 
independently
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic 
form
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic 
theta
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic 
z
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic 
t
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f 
$f_h
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f 
sin_m
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f 
Omega_ {T}^{2D}\f$
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f f
f 
Omega_v
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f 
left
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as 
follows
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as the pressure and
the pressure increments< li >
For f $n geq0 f let f 
bu {n+1}\f$
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as the pressure and
the pressure increments< li >
For f $n geq0 f let f that
matches the Dirichlet boundary
conditions of the 
problem
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as the pressure and
the pressure increments< li >
For f $n geq0 f let f that
matches the Dirichlet boundary
conditions of the be the
solutions of the following
formulation for all f 
textbf
 
section
doc_intro_frame_work_num_app
Numerical approximation
subsection
doc_intro_fram_work_num_app_Fourier_FEM
Fourier Finite element
representation The SFEMaNS
code uses a hybrid Fourier
Finite element formulation The
Fourier decomposition allows
to approximate the problem’s
solutions for each Fourier
mode modulo nonlinear terms
that are made explicit The
variables are then
approximated on a meridian
section of the domain with a
finite element method The
numerical approximation of a
function f $f f is written in
the following generic f with f
$t f the time and f $M f the
number of Fourier modes
considered The unknown f f f f
f f f f Omega_v f and f Omega
f We also consider f a penalty
method of the divergence of
the velocity field is also
implemented The method
proceeds as the pressure and
the pressure increments< li >
For f $n geq0 f let f that
matches the Dirichlet boundary
conditions of the be the
solutions of the following
formulation for all f f 
text
 

Function Documentation

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f ( ,
theta,
z   
)

Here is the caller graph for this function:

Variable Documentation

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f $f_h
Initial value:
{m,\cos}\f$ and
\f$f_h^{m,\sin}\f$ can be approximated independtly modulo the computation
of nonlinear terms. Introducing the functions \f$\cos_m = \cos(m\theta)\f$
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic theta
Definition: doc_intro.h:193
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f $f_h
Definition: doc_intro.h:204

Definition at line 204 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments<li> For f $n geq0 f let f bu {n+1}\f$

Definition at line 327 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as follows

Definition at line 218 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic form

Definition at line 193 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode independently

Definition at line 193 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f left

Definition at line 218 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f Omega_ {T}^{2D}\f$

Definition at line 214 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f Omega_v
Initial value:
{2D}\f$ and \f$\Omega^{2D}\f$ of
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f Omega_
Definition: doc_intro.h:214
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates

Definition at line 215 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments<li> For f $n geq0 f let f that matches the Dirichlet boundary conditions of the problem

Definition at line 327 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f sin_m
Initial value:
= \sin(m\theta)\f$ and a basis functions \f$(\phi_j)_{j \in J}\f$
of the finite element space of the meridian section results in
\f$(\phi_j \cos_m)_{j\in J, m \in [|0,M|]} \cup (\phi_j \sin_m)_{j\in J, m \in [|1,M|]}\f$
being a basis of the space of approximation.
@subsection doc_intro_fram_work_num_app_space Space of approximation
We set the number of Fourier mode to \f$M\f$. We define the meridian sections
\f$\Omega_{c,f}^{2D}\f$
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f Omega_
Definition: doc_intro.h:214
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic theta
Definition: doc_intro.h:193
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f sin_m
Definition: doc_intro.h:207

Definition at line 207 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic t
Initial value:
=f_h^{0,\cos}(r,z,t) +
\sum_{m=1}^M f_h^{m,\cos} \cos(m\theta) + f_h^{m,\sin} \sin(m\theta)
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic t
Definition: doc_intro.h:199
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic theta
Definition: doc_intro.h:193
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic z
Definition: doc_intro.h:193

Definition at line 199 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments<li> For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f text
Initial value:
{c}_\text{div}\f$
is a penalty coefficent
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f text
Definition: doc_intro.h:342
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates

Definition at line 342 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f f textbf
Initial value:
{v}\f$ in \f$\textbf{V}_{h,0}^\bu\f$:
\f{equation}{
\label{eq:SFEMaNS_weak_from_NS_1}
\int_{\Omega_{c,f}} \frac{3}{2 \tau} \textbf{u}^{n+1} \cdot \textbf{v}
+ \frac{1}{\Re} \GRAD \textbf{u}^{n+1} : \GRAD \textbf{v}
+ \text{c}_\text{div} h^{-1} \DIV\bu^{n+1} \DIV \bv=
- \int_{\Omega_{c,f}} ( \frac{-4 \textbf{u}^n
+ \textbf{u}^{n-1}}{2 \tau}
+ \GRAD ( p^n +\frac{4\psi^n - \psi^{n-1}}{3} ) ) \cdot \textbf{v} \
+ \int_{\Omega_{c,f}} ( \textbf{f}^{n+1} - (\ROT \textbf{u}^{*,n+1} )
\times \textbf{u}^{*,n+1} ) \cdot \textbf{v} ,
where \f$h\f$ is the local mesh size
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f Omega_
Definition: doc_intro.h:214
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f f text
Definition: doc_intro.h:342
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f(r,\theta, z)\f $the cylindrical coordinates
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f bu
Definition: doc_intro.h:327
section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic f with f $t f the time and f $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as the pressure and the pressure increments< li > For f $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f textbf
Definition: doc_intro.h:330

Definition at line 330 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic theta

Definition at line 193 of file doc_intro.h.

section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f $f f is written in the following generic z

Definition at line 193 of file doc_intro.h.