184 @section doc_intro_frame_work_num_app Numerical approximation
187 @subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier/Finite element representation
188 The SFEMaNS code uses a hybrid Fourier/Finite element formulation.
189 The Fourier decomposition allows to approximate the problem’s
191 terms that are made
explicit. The variables are then approximated
192 on a meridian section of the domain with a finite element method.
194 The numerical approximation of a
function \f$f\f$ is written in the following
generic form:
197 \sum_{m=1}^M f_h^{m,\cos} \cos(m\
theta) + f_h^{m,\sin} \sin(m\
theta),
201 the number of Fourier modes considered. The unknown \
f$f_h^{m,\cos}
\f$
and 202 \f$f_h^{m,\sin}
\f$ can be approximated independtly modulo the computation
205 of the finite element space of the meridian section results
in 206 \f$(\phi_j \cos_m)_{j\in J, m \in [|0,M|]} \cup (\phi_j \
sin_m)_{j\in J, m \in [|1,M|]}
\f$
207 being a basis of the space of approximation.
209 @subsection doc_intro_fram_work_num_app_space Space of approximation
216 a family of meshes of the meridian
plane \f$\Omega^{2D}
\f$
219 can be restricted to each sub domain of interests. These sub-meshes are
denoted 220 \f$\mathcal{T}_h^{c,
f}
\f$,
\f$\mathcal{T}_h^{T}
\f$,
\f$\mathcal{T}_h^{c}
\f$
221 and \f$\mathcal{T}_h^{v}
\f$. The approximation of the solutions of the
222 Navier-Stokes, heat and Maxwell equations either
involve \f$\mathbb{P}
_1\f$
or 223 \f$\mathbb{P}
_2\f$ Lagrange finite elements. The following defines
224 the space of approximation used
for each dependent variable.
228 respectively approximated in the following spaces:
233 S_{h}^p :=
\left\{ q_h= \sum\limits_{k=-M}^M q_h^k (r,
z) e^{i
k \theta} ;
234 q_h^k \in S_{h}^{p,2D}
\text{, } \overline{q_h^k}=q_h^{-k}
\text{, }
235 -M \leq k \leq M \right\},
237 where we introduce the following finite element space:
240 \textbf{v}_h|_K \in \mathbb{P}_2^6
\text{ } \forall K \in \mathcal{T}_h^{c,
f}
242 S_{h}^{p,2D} : =
\left\{ q_h \in C^0(\overline{
\Omega_{c,
f}^{2D}}) ;
243 q_h|_K \in \mathbb{P}_1^2
\text{ } \forall K \in \mathcal{T}_h^{c,
f} \right\} .
250 S_h^T :=
\left\{ q_h= \sum\limits_{k=-M}^M q_h^k (r,
z) e^{i
k \theta} ;
251 q_h^k \in S_{h}^{T,2D}
\text{, } \overline{q_h^k}=q_h^{-k}
\text{, }
252 -M \leq k \leq M \right\},
254 where we introduce the following finite element space:
256 S_{h}^{T,2D} : =
\left\{ q_h \in C^0(\overline{
\Omega_{T}^{2D}}) ;
257 q_h|_K \in \mathbb{P}_2^2
\text{ } \forall K \in \mathcal{T}_h^{T} \right\} .
260 are respectively approximated in the following spaces:
271 \sum\limits_{k=-M}^M \varphi_h^k (r,
z) e^{
ik \theta} ;
272 \varphi_h^k \in S_{h}^{\phi,2D}, \; -M \leq k \leq M
275 where we introduce the following finite element spaces:
278 C^0(\overline{
\Omega_{c}^{2D}});
280 \forall K \in \mathcal{T}_h^{c} \right\} ,\\
281 S_{h}^{\phi, 2D} : =
\left\{ \varphi_h \in C^0(\overline{
\Omega_{v}^{2D}}) ;
282 \varphi_h|_K \in \mathbb{P}_{l_\phi}^2
\text{ } \forall K \in \mathcal{T}_h^v , \right\}
287 @subsection doc_intro_fram_work_num_app_time_marching Time marching
291 When approximating the Navier-Stokes, heat and Maxwell equations, the
time marching
292 can be summarized by the four following steps:
301 @section doc_intro_SFEMaNS_weak_form_extensions Weak formulation and extensions
303 This section introduces the weak formulations implemented in SFEMaNS and
304 additional features/extensions of the code. The notations introduced
305 previously, such as the domain of approximation
for each equations or
308 @subsection doc_intro_SFEMaNS_possibilities_nst Hydrodynamic setting
311 @subsubsection doc_intro_SFEMaNS_possibilities_nst_1 Approximation of the Navier-Stokes equations
313 The approximation of the Navier-Stokes equations is based on a
314 rotational
form of the prediction-correction projection method
315 detailed in <a href=
'http://www.ams.org/journals/mcom/2004-73-248/S0025-5718-03-01621-1/'>
316 <code>Guermond and Shen (2004)</code></a>. As the code SFEMaNS
317 approximates the predicted velocity, a penalty method of the
318 divergence of the velocity field is also implemented.
320 The method proceeds as
follows:
322 <li>Initialization of the velocity field, the pressure
323 and the pressure increments.
325 matches the Dirichlet
boundary conditions of the
326 problem, be the solutions of the following formulation
for all 329 \label{eq:SFEMaNS_weak_from_NS_1}
335 + \GRAD ( p^n +\frac{4\psi^n - \psi^{n-1}}{3} ) )
\cdot \textbf{v} \\
340 is a penalty coefficent,
348 \label{eq:SFEMaNS_weak_from_NS_2}
349 \int_{
\Omega_{c,
f}} \GRAD \psi^{n+1} \cdot \GRAD q
353 \label{eq:SFEMaNS_weak_from_NS_3}
357 <li>The pressure is updated as
follows:
359 \label{eq:SFEMaNS_weak_from_NS_4}
360 p^{n+1} = p^n + \psi^{n+1} - \frac{1}{\Re} \delta^{n+1} .
364 @subsubsection doc_intro_SFEMaNS_possibilities_nst_2 Entropy viscosity
for under resolved computation
366 Hydrodynamic problems with large kinetic Reynolds number
367 introduce extremely complex flows. Approximating all of
368 the dynamics
's scales of such problems is not always possible 369 with present computational ressources. To address this problem, 370 a nonlinear stabilization method called entropy viscosity is 371 implemented in SFEMaNS. This method has been introduced by 373 <code>Guermond et al. (2011)</code></a>. It consists of introducing an artifical
374 viscosity, denoted \
f$\nu_{E}
\f$, that is taken proportional
375 to the
default of equilibrium of an energy equation.
377 This implementation of
this method in SFEMaNS can be summarized
378 in the three following steps:
380 <li>Define the residual of the Navier-Stokes at
384 \frac{
\bu^n-
\bu^{n-2}}{ 2 \tau}
386 + \ROT (\
bu^{*,n-1}) \times \
bu^{*,n-1}
389 <li>Compute the entropy viscosity on each mesh cell K as
follows:
391 \label{eq:SFEMaNS_NS_entropy_viscosity}
394 \bu^{n-1}\|_{\bL^\infty(K)}}{\|
\bu^{n-1}\|_{\bL^\infty(K)}^2}\right),
396 with \
f$h\
f$ the local mesh size of the cell K,
400 constant in the
interval \f$(0,1)\
f$. It is generally
set to one.
401 <li>When approximating \
f$\
bu^{n+1}
\f$, the
term 403 is added in the
left handside of the Navier-Stokes equations.
406 Thus defined, the entropy viscosity is expected to be smaller
407 than the consistency error in the smooth regions. In regions
408 with large gradients, the entropy viscosity switches to the first
409 order viscosity \
f$\nu_{\max|K}^n:=
c_\text{max} h_K \|
\bu^{n-1}\|_{\bL^\infty(K)}
\f$.
410 Note
that \f$\nu_\max^
n\f$ corresponds to the artifical viscosity
411 induces by first order up-wind scheme in the finite difference
412 and finite volume litterature.
414 Remark: To facilitate the
explicit treatment of the entropy viscosity,
415 the following term can be added in the
left handside of the Navier-Stokes
418 \label{eq:SFEMaNS_NS_LES_c1}
419 - \DIV( c_1 h \GRAD (\
bu^{n+1}-
\bu^{*,n+1})).
425 @subsubsection doc_intro_SFEMaNS_possibilities_nst_3 Extension to non axisymmetric geometry
427 A penalty method of <a href='http://www.sciencedirect.com/science/article/pii/S0168927407000815'>
428 <code>Pasquetti et al. (2008)</code></a>) is implemented so
429 the code SFEMaNS can report of the presence of non axisymmetric
430 solid domain in \
f$\
Omega_{c,
f}
\f$. Such solid domains can either
431 be driving the fluid or represents an obtacle to the fluid motion
433 the Navier-Stokes equations are approximated, is splitted into a
436 axisymmetric and
time dependent. The penalty method introduces
438 field approximated by the Navier-Stokes equations to
440 This penalty
function is defined as
follows:
442 \label{eq:SFEMaNS_NS_penal_1}
452 The velocity field is updated as
follows:
454 \label{eq:SFEMaNS_NS_penal_2}
455 \frac{3
\bu^{n+1}}{2\tau}
459 + \chi^{n+1}
\left(\frac{4
\bu^n -
\bu^{n-1}}{2\tau}
460 - \GRAD( \frac{4\psi^n-\psi^{n-1}}{3}) \right)
463 - ( \ROT \
bu^{*,n+1} ) \times\
bu^{*,n+1}
465 + (1 - \chi^{n+1}) \frac{3
\bu^{n+1}
_\text{obst}}{2\tau},
470 Note that the original scheme is recovered
where \f$\chi=1
\f$.
472 Remark: the correction and update of the pressure is not modified.
474 @subsubsection doc_intro_SFEMaNS_possibilities_nst_4 Extension to multiphase flow
problem 476 The code SFEMaNS can approximate two phase flow problems.
477 The governing equations are written as
follows:
479 \label{eq:SFEMaNS_NS_multiphase_1}
480 \partial_t \rho + \DIV( \
textbf{m}) = 0,
483 \label{eq:SFEMaNS_NS_multiphase_2}
486 - \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu))
490 \label{eq:SFEMaNS_NS_multiphase_3}
495 \f$\varepsilon(\
bu)=\GRAD^
s \bu = \frac12 (\GRAD \
bu +(\GRAD \
bu)^\sf{T})\
f$.
496 The densities, respestively dynamical viscosities, of the two fluids are denoted
497 \
f$\rho_0\
f$ and \
f$\rho_1\
f$, respectively \f$\eta_0\f$ and \f$\eta_1\f$.
500 The approximation method is based on the following ideas.
502 <li>Use of a level
set method to follow the interface evolution.
503 The method consists of approximating \f$\varphi\f$ that takes
504 value in \f$[0,1]\f$ solution of:
506 \partial_t \varphi + \bu \cdot \GRAD \varphi=0.
508 The level
set is equal to 0 in a fluid and 1 in the other fluid.
509 The
interface between the fluid is represented by \
f$\varphi^{-1}({1/2})\
f$.
510 <li>Use the momentum as dependent variable
for the Navier-Stokes equations.
511 The mass matrix becomes
time independent and can be treated with pseudo-spectral method.
512 <li>Rewritte the diffusive term \
f$- \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu))
\f$ as
follows:
514 - \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu)) =
515 - \frac{2}{\Re} \DIV(\overline{\nu} \varepsilon(\bm))
516 - \
left( \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu))
517 - \frac{2}{\Re} \DIV(\overline{\nu} \varepsilon(\bm)) \right)
520 The first term is made implicit
while the second is treated explicitly.
521 The resulting stiffness matrix is
time independent and does not involve nonlinearity.
522 <li>The level
set and Navier-Stokes equations are stabilized with the same entropy viscosity.
530 }{\|
\bu^{n-1}\|_{\bL^\infty(K)}\|\bm^{n-1}\|_{\bL^\infty(K)}}
536 \frac{\bm^n-\bm^{n-2}}{ 2 \tau}
537 -\frac{1}{\Re} \DIV (\eta^{n-1}
\epsilon(\
bu^{n-1}))
538 + \DIV(\bm^n{\otimes}
\bu^n) + \GRAD p^{n-1} -
\textbf{
f}^{n-1} ,
542 \text{Res}_\rho^n= \frac{\rho^n-\rho^{n-2}}{ 2 \tau}
545 To facilitate the
explicit treatment of the entropy viscosity,
547 \f$-\DIV( c_1 h \GRAD (\varphi^{n+1}-\varphi^n))
\f$, can be added
548 in the
left handside of the Navier-Stokes, respectively of level
set equation.
549 <li>A compression term that allows the level
set to not
get flatten over
time 550 iteration is added. It consists of adding the following term in the right
551 handside of the level
set equation:
553 -
\DIV \left(c_\
text{comp}\nu_E h^{-1} \varphi(1-\varphi)\frac{\GRAD\varphi}{\|\varphi\|}\right).
555 The coefficient \
f$c_\
text{comp}
\f$ a tunable constant
in \f$[0,1]
\f$.
563 \frac{\varphi^{n+1}-\varphi^n}{\tau} = -
\bu^n \cdot \GRAD \varphi^n
565 \nu_E^n\GRAD \varphi^n
566 - c_\
text{comp} \nu_E^n h^{-1} \varphi^n(1-\varphi^n)\frac{\GRAD\varphi^n}{\|\varphi^n\|}
571 \rho^{n+1} = \rho_0 + (\rho_1 - \rho_0) F(\varphi^{n+1}), \qquad
572 \eta = \eta_0 + (\eta_1 - \eta_0) F(\varphi^{n+1}),
574 where \
f$F\
f$ is either equal to the identity,
575 \
f$F(\varphi)=
\varphi\f$, or a piecewise ponylomial
function defined by:
581 & \
text{
if $|\varphi - 0.5| \le c_{
\text{reg}}$}, \\
582 1 &
\text{
if $c_{
\text{reg}} \le \varphi - 0.5$}.
588 \frac{\bm^{n+1}-\bm^n}{\tau} - \frac{2\overline{\nu}}{\Re}\DIV(\
epsilon(\bm^{n+1})-\
epsilon(\bm^n))
589 = \frac{2}{\Re}\DIV( \eta^n\
epsilon(\
bu^n))
590 - \DIV(\bm^n\times\
bu^n)
594 <li>Update the pressure as
follows:
609 <li>This method can be used to approximate problems with
610 a stratification or an inclusion
of \f$n\geq 3
\f$ fluids.
611 One level
set is approximated per
interface between two
612 fluids. The fluids properties are reconstructed with
613 recursive convex combinations.
614 <li>MHD multiphase problems with variable electrical conductivity
615 between the fluids can also be considered. The electrical
616 conductivity in the fluid is reconstructed with the level set
617 the same way the
density and the dynamical viscosity are.
618 The magnetic field \
f$\bH^{n+1}
\f$ is updated as
follows:
620 \frac{3\bH^{n+1}-4\bH^n+\bH^{n-1}}{2\tau}
622 \ROT ( \bH^{n+1}-\bH^{*,n+1}) \right)
624 -
\ROT\left( \frac{1}{\sigma\Rm} \ROT \bH^{*,n+1} \right)
625 + \ROT (\
bu^{n+1}\times \mu^c \bH^{*,n+1})
626 + \ROT \
left( \frac{1}{\sigma\Rm}
\textbf{j}^{n+1} \right)
628 with \
f$\bH^{*,n+1}=2\bH^{n+1}-\bH^
n\f$
and \f$\overline{\sigma}
\f$ a
629 function depending of the radial and vertical
631 \
f$\overline{\sigma}(r,
z)\leq \sigma(r,\
theta,
z,
t)
\f$
for 637 @subsection doc_intro_SFEMaNS_possibilities_temp Heat equation
's weak formulation 639 The heat equations is approximated as follows. 641 <li>Initialization of the temperature. 642 <li>For all \f$n\geq0\f$ let \f$T^{n+1}\f$, that matches the 643 Dirichlet boundary conditions of the problem, be the solution 644 of the following formulation for all \f$v\in S_h^T\f$: 646 \label{eq:SFEMaNS_weak_form_temp} 647 \int_{\Omega_T} \frac{3 C }{2 \tau}T^{n+1} v 648 + \lambda \GRAD T^{n+1} \cdot \GRAD v 649 = - \int_{\Omega_T} \left( \frac{4 T^n -T^{n-1}}{2 \tau} 650 - \DIV (T^{*,n+1} \bu^{*,n+1}) + f_T^{n+1}\right) v, 652 where \f$T^{*,n+1}=2 T^n - T^{n-1}\f$. We remind that \f$C\f$ is 653 the volumetric heat capacity, \f$\lambda\f$ the thermal conductivty 654 and \f$f_T\f$ a source term. 659 @subsection doc_intro_SFEMaNS_possibilities_mxw Magnetic setting 661 The code SFEMaNS uses \f$\bH^1\f$ conforming Lagrange finite element to approximate 662 the magnetic field. As a consequence, the zero divergence condition on the 663 magnetic field cannot be enforced by standard penalty technique for 664 non-smooth and non-convex domains. 665 To overcome this obstacle, a method inspired of 667 <code>Bonito and Guermond (2011)</code></a>
668 has been implemented. This method consists of introducting a
673 if the solution
in \f$\Omega^
c\f$ of:
675 - \DIV( h_\
text{loc}^{2(1-
\alpha)} \GRAD p_m^{c,n+1} ) &=
676 - \DIV( \mu^c \bH^{c,n+1}) ,
678 p_m^{c,n+1}|_{\partial \Omega_c} &= 0,
681 constant parameter
in \f$[0.6,0.8]
\f$.
682 <li>Add the
term \f$ -\DIV(\mu^v \GRAD p_\
text{m}^v)\
f$ in the right handside
683 of the scalar potential \
f$\phi\
f$ equation where \
f$p_\
text{m}^
v\f$
684 is the solution
in \f$\Omega^
v\f$ of:
686 \LAP p_m^{v,n+1} = \LAP \phi^{n+1}, \\
691 We note that the magnetic pressure can be eliminated from the equation
693 <a href=
'http://www.sciencedirect.com/science/article/pii/S0021999111002749'>
694 <code>Guermond et al. (2011)</code></a>
for more details.
695 The approximation space used
696 to approximate \
f$ p_\
text{m}^
c\f$ is the following:
701 \sum\limits_{k=-M}^M \varphi_h^k (r,
z) e^{
ik \theta} ;
702 \varphi_h^k \in S_{h}^{
p_\text{m}^c,2D}, \; -M \leq k \leq M
705 where we introduce the following finite element space:
708 \varphi_h|_K \in \mathbb{P}_1^2
\text{ } \forall K \in \mathcal{T}_h^c , \right\}.
711 In addition, an interior penalty method is used to enforce the continuity conditions
713 <a href=
'http://www.sciencedirect.com/science/article/pii/S0021999106002944'>
714 <code>Guermond et al. (2007)</code></a>
for more details.
716 @subsubsection doc_intro_SFEMaNS_possibilities_mxw_1 Approximation of the Maxwell equations with H
718 The Maxwell equations are approximated as
follows:
720 <li>Initialization of the magnetic field \
f$\bH^c\
f$, the scalar potential \
f$\phi\
f$ and the magnetic pressure \
f$p_\
text{m}^
c\f$.
722 solutions of the following formulation
for all \
f$b\in \bV_h^{\bH^c}
\f$,
723 \f$\varphi\in S_h^{\phi}
\f$
726 & \int_{\Omega_c}\mu^c \frac{D\bH^{c,n+1}}{\Delta
t}\SCAL \bb
727 +\int_{\Omega_c} \frac{1}{\sigma R_m} \ROT \bH ^{c,n+1}\cdot \ROT \bb
728 +\int_{
\Omega_v} \muv\frac{\GRAD D\phi^{n+1}}{\Delta
t}\SCAL \GRAD\varphi
729 +\int_{
\Omega_v} \muv\GRAD\phi^{n+1}\SCAL \GRAD\varphi -
732 - \int_{\Omega_c} \mu^c\bH^{c,n+1}\SCAL\GRAD q +
733 \int_{\Omega_c} h^{2(1-
\alpha)}\GRAD p_\
text{m}^{c,n+1}\SCAL \GRAD q
735 h^{2
\alpha}\DIV (\mu^c \bH^{c,n+1} )\DIV (\mu^c \bb)\right)\\
736 & +\int_{\Sigma_{\mu}}
\left \{ \frac{1}{\sigma R_m} \ROT {\bH ^{c,n+1}} \right \}
737 \cdot \left ( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
738 & +\beta_3 \int_{\Sigma_{\mu}} h^{-1}
\left( { \bH_1^{c,n+1}}\times \bn_1^c
739 + {\bH_2^{c,n+1}}\times \bn_2^c\right ) \SCAL \
left ( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
740 & +\beta_1 \int_{\Sigma_{\mu}} h^{-1}
\left({ \mu^c_1\bH_1^{c,n+1}}\cdot \bn_1^c
741 + {\mu^c_2 \bH_2^{c,n+1}}\cdot \bn_2^c\right ) \SCAL \
left ( {\mu^c_1}{ \bb_1}\cdot \bn_1^c
742 + {\mu^c_2}{ \bb_2}\cdot \bn_2^c\right )\\
743 & +\int_{\Sigma} \frac{1}{\sigma R_m} \ROT {\bH ^{c,n+1}}
\cdot 745 + \beta_2 \int_\Sigma h^{-1}
\left( {\bH^{c,n+1}}\CROSS \bn_1^c
746 + {\GRAD \phi^{n+1}}\CROSS \bn_2^c\right ) \SCAL (\bb\CROSS \bnc +
747 \GRAD\varphi\CROSS \bnv)\\
748 & + \beta_1 \int_\Sigma h^{-1}
\left( { \mu^c\bH ^{c,n+1}}\cdot \bn_1^c
749 + {\GRAD \phi^{n+1}}\cdot \bn_2^c\right ) \SCAL ({\mu^c}\bb\cdot \bnc +
750 \GRAD\varphi \cdot \bnv)\\
751 & + \
int _{\Gamma_c} \frac{1}{\sigma R_m} \ROT \bH ^{c,n+1} \cdot ( \bb \CROSS \bnc)
753 \int_{\Gamma_c} h^{-1}
\left( { \bH^{c,n+1}}
\CROSS \bn^c \right ) \SCAL (\bb\CROSS \bnc)
756 & \int_{\Omega_c}
\left( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1} \times \mu^c \bH^{*,n+1} \right )
758 + \
int _{\Sigma_{\mu}}
\left \{ \frac{1}{\sigma R_m}\bj^s +
759 \bu^{n+1} \times \mu^c \bH^{*,n+1} \right \}
\cdot 760 \left( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
761 & +\int_{\Sigma}
\left ( \frac{1}{\sigma R_m} \bj^s +
\bu^{n+1} \times \mu^c \bH^{*,n+1}
763 +\int_{\Gamma_c}(
\ba \times \bn) \cdot \
left ({\bb} \times \bn \right) + \int_{\Gamma_v}
765 & + \int_{\Gamma_c}
\left ( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1} \times
766 \mu^c \bH^{*,n+1} \right )\cdot ( \bb \CROSS \bnc)
767 +\beta_3 \int_{\Gamma_c} h^{-1}
770 where we
set \f$D\bH^{c,n+1}=\dfrac{3\bH^{c,n+1}-4\bH^{c,n}+\bH^{c,n-1}}{2}
\f$,
771 \f$D\phi^{c,n+1}=\dfrac{3\phi^{c,n+1}-4\phi^{c,n}+\phi^{c,n-1}}{2}
\f$,
775 They are normalized
by \f$(\sigma\Rm)^{-1}
\f$ so their value can be
set to one
786 @subsubsection doc_intro_SFEMaNS_possibilities_mxw_2 Extension to magnetic permeability variable in
time and azimuthal direction
788 The use of a Fourier decomposition in the azimuthal direction leads us to use
789 the magnetic
field \f$\bB^c=\mu\bH^
c\f$ as dependent variable of the Maxwell equations
790 in the conducting domain. The mass matrix becomes
time independent and can be computed with pseudo-spectral methods.
791 To
get a
time independent stiffness matrix that does not involve nonlinearity, the diffusive
term 794 \ROT \left( \frac{1}{\sigma\Rm} \ROT \frac{\bB^c}{\mu} \right) =
795 \ROT \
left( \frac{1}{\sigma\Rm \overline{\mu}} \ROT\frac{\bB^c}{\mu} \right)
796 + \ROT \
left( \frac{1}{\sigma\Rm} \ROT ((\frac{1}{\mu}-\frac{1}{\overline{\mu}})\bB^c) \right)
798 with \
f$\overline{\mu}
\f$ a
function depending of the radial and vertical
800 all \f$(r,
\theta,
z,
t)\
f$ considered. The first term is then made implicit
while 801 the term involving \
f$\frac{1}{\mu}
\f$ is treated explicitly.
804 Under the previous notations and assuming,
810 the Maxwell equations are approximated as
follows.
814 solutions of the following formulation
for all \
f$b\in \bV_h^{\bH^c}
\f$,
815 \f$\varphi\in S_h^{\phi}
\f$
818 & \int_{\Omega_c}\frac{D\bB^{c,n+1}}{\Delta
t}\SCAL \bb
819 + \int _{\Omega_c} \frac{1}{\sigma R_m} \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}}\cdot \ROT \bb
820 + \int_{
\Omega_v} \muv\frac{\GRAD D\phi^{n+1}}{\Delta
t}\SCAL \GRAD\varphi
821 + \int_{
\Omega_v} \muv\GRAD\phi^{n+1}\SCAL \GRAD\varphi
824 - \int_{\Omega_c} \bB^{c,n+1}\SCAL\GRAD q + \int_{\Omega_c} h^{2(1-
\alpha)}
825 \GRAD p_\
text{m}^{c,n+1}\SCAL \GRAD q
826 + \int_{\Omega_c} h^{2
\alpha}
827 \overline{\mu^c} \DIV \bB^{c,n+1} \DIV \bb \right)\\
828 & +\
int _{\Sigma_{\mu}}
\left\{ \frac{1}{\sigma R_m}
829 \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}} \right \}
830 \cdot \left ( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
831 & +\beta_3 \int_{\Sigma_{\mu}} h^{-1}
\left(
832 \frac{\bB_1^{c,n+1}}{\overline{\mu^c}_1}\times \bn_1^c + \frac{\bB_2^{c,n+1}}{\overline{\mu^c_2}}\times \bn_2^c
833 \right) \SCAL \
left ( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
834 & +\beta_1 \int_{\Sigma_{\mu}} h^{-1}
835 \left( {\bB_1^{c,n+1}}\cdot \bn_1^c + {\bB_2^{c,n+1}}\cdot \bn_2^c\right)
836 \SCAL \
left( \overline{\mu^c_1}{ \bb_1}\cdot \bn_1^c + \overline{\mu^c_2}{ \bb_2}\cdot \bn_2^c\right )\\
837 & +\
int _{\Sigma} \frac{1}{\sigma R_m} \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}}
\cdot 839 + \beta_2 \int_{\Sigma} h^{-1}
840 \left( \frac{\bB^{c,n+1}}{\overline{\mu^c}}\CROSS \bn_1^c + {\GRAD \phi ^{n+1}}\CROSS \bn_2^c\right)
841 \SCAL (\bb\CROSS \bnc + \GRAD\varphi\CROSS \bnv)\\
842 & + \beta_1 \int_{\Sigma} h^{-1}
843 \left( {\bB ^{c,n+1}}\cdot \bn_1^c + {\GRAD \phi ^{n+1}} \cdot \bn_2^c\right)
844 \SCAL \
left(\overline{{\mu^c}}\bb\cdot \bnc +
845 \GRAD\varphi \cdot \bnv \right )\\
846 & + \int_{\Gamma_c} \frac{1}{\sigma R_m} \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}}
847 \cdot ( \bb \CROSS \bnc) +
\beta_3\left( \int_{\Gamma_c} h^{-1}
851 & \int_{\Omega_c} \frac{1}{\sigma R_m} \ROT (\langle \overline{\mu^c},{\mu^c}
852 \rangle {\bB^{*,n+1}})\cdot \ROT \bb \\
853 & \int_{\Omega_c}
\left( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1} \times
854 \bB^{*,n+1} \right )\cdot \ROT \bb
855 + \int_{\Sigma_{\mu}}
\left \{ \frac{1}{\sigma R_m}\bj^s
856 +
\bu^{n+1} \times \bB^{*,n+1} \right \}
\cdot 857 \left( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
858 & +\int_{\Sigma}
\left( \frac{1}{\sigma R_m} \bj^s
859 +
\bu^{n+1} \times \bB^{*,n+1} \right )
863 & + \int_{\Gamma_c}
\left ( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1}
864 \times \bB^{*,n+1} \right )\cdot ( \bb \CROSS \bnc)
865 +\beta_3 \int_{\Gamma_c} h^{-1}
\left({\bB}
_\text{bdy}^{c,n+1}
\CROSS \bn^c \right ) \SCAL (\bb\CROSS \bnc),
867 where we
set \f$\bB^{*,n+1}=2\bB^n-\bB^{n-1}
\f$
and 868 \f$\langle \overline{\mu^c},{\mu^c}\rangle=\frac{1}{\overline{\mu^c}}- \frac{1}{\mu^c}
\f$.
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 t
real(kind=8), dimension(:,:,:), allocatable, target bn
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 Omega_
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 independently
type(personalized_data), public user
real(kind=8), dimension(:,:,:), allocatable, target density
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
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 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
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
real(kind=8), private beta
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_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
real(kind=8), parameter, private alpha
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 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 form