187 @section doc_intro_frame_work_num_app Numerical approximation
190 @subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier/Finite element representation
191 The SFEMaNS code uses a hybrid Fourier/Finite element formulation.
192 The Fourier decomposition allows to approximate the problem’s
194 terms that are made
explicit. The variables are then approximated
195 on a meridian section of the domain with a finite element method.
197 The numerical approximation of a
function \f$f\f$ is written in the following
generic form:
200 \sum_{m=1}^M f_h^{m,\cos} \cos(m\
theta) + f_h^{m,\sin} \sin(m\
theta),
204 the number of Fourier modes considered. The unknown \
f$f_h^{m,\cos}
\f$
and
205 \f$f_h^{m,\sin}
\f$ can be approximated independtly modulo the computation
208 of the finite element space of the meridian section results
in
209 \f$(\phi_j \cos_m)_{j\in J, m \in [|0,M|]} \cup (\phi_j \
sin_m)_{j\in J, m \in [|1,M|]}
\f$
210 being a basis of the space of approximation.
212 @subsection doc_intro_fram_work_num_app_space Space of approximation
219 a family of meshes of the meridian
plane \f$\Omega^{2D}
\f$
222 can be restricted to each sub domain of interests. These sub-meshes are
denoted
223 \f$\mathcal{T}_h^{c,
f}
\f$,
\f$\mathcal{T}_h^{T}
\f$,
\f$\mathcal{T}_h^{c}
\f$
224 and \f$\mathcal{T}_h^{v}
\f$. The approximation of the solutions of the
225 Navier-Stokes, heat and Maxwell equations either
involve \f$\mathbb{P}
_1\f$
or
226 \f$\mathbb{P}
_2\f$ Lagrange finite elements. The following defines
227 the space of approximation used
for each dependent variable.
231 respectively approximated in the following spaces:
236 S_{h}^p :=
\left\{ q_h= \sum\limits_{k=-M}^M q_h^k (r,
z) e^{i
k \theta} ;
237 q_h^k \in S_{h}^{p,2D}
\text{, } \overline{q_h^k}=q_h^{-k}
\text{, }
238 -M \leq k \leq M \right\},
240 where we introduce the following finite element space:
243 \textbf{v}_h|_K \in \mathbb{P}_2^6
\text{ } \forall K \in \mathcal{T}_h^{c,
f}
245 S_{h}^{p,2D} : =
\left\{ q_h \in C^0(\overline{
\Omega_{c,
f}^{2D}}) ;
246 q_h|_K \in \mathbb{P}_1^2
\text{ } \forall K \in \mathcal{T}_h^{c,
f} \right\} .
253 S_h^T :=
\left\{ q_h= \sum\limits_{k=-M}^M q_h^k (r,
z) e^{i
k \theta} ;
254 q_h^k \in S_{h}^{T,2D}
\text{, } \overline{q_h^k}=q_h^{-k}
\text{, }
255 -M \leq k \leq M \right\},
257 where we introduce the following finite element space:
259 S_{h}^{T,2D} : =
\left\{ q_h \in C^0(\overline{
\Omega_{T}^{2D}}) ;
260 q_h|_K \in \mathbb{P}_2^2
\text{ } \forall K \in \mathcal{T}_h^{T} \right\} .
263 are respectively approximated in the following spaces:
274 \sum\limits_{k=-M}^M \varphi_h^k (r,
z) e^{
ik \theta} ;
275 \varphi_h^k \in S_{h}^{\phi,2D}, \; -M \leq k \leq M
278 where we introduce the following finite element spaces:
281 C^0(\overline{
\Omega_{c}^{2D}});
283 \forall K \in \mathcal{T}_h^{c} \right\} ,\\
284 S_{h}^{\phi, 2D} : =
\left\{ \varphi_h \in C^0(\overline{
\Omega_{v}^{2D}}) ;
285 \varphi_h|_K \in \mathbb{P}_{l_\phi}^2
\text{ } \forall K \in \mathcal{T}_h^v , \right\}
290 @subsection doc_intro_fram_work_num_app_time_marching Time marching
292 To present the time marching, we introduce a time
step \f$
\tau\f$ and denote
294 When approximating the Navier-Stokes, heat and Maxwell equations, the time marching
295 can be summarized by the four following steps:
304 @section doc_intro_SFEMaNS_weak_form_extensions Weak formulation and extensions
306 This section introduces the weak formulations implemented in SFEMaNS and
307 additional features/extensions of the code. The notations introduced
308 previously, such as the domain of approximation
for each equations or
311 @subsection doc_intro_SFEMaNS_possibilities_nst Hydrodynamic setting
314 @subsubsection doc_intro_SFEMaNS_possibilities_nst_1 Approximation of the Navier-Stokes equations
316 The approximation of the Navier-Stokes equations is based on a
317 rotational
form of the prediction-correction projection method
318 detailed in <a href=
'http://www.ams.org/journals/mcom/2004-73-248/S0025-5718-03-01621-1/'>
319 <code>Guermond and Shen (2004)</code></a>. As the code SFEMaNS
320 approximates the predicted velocity, a penalty method of the
321 divergence of the velocity field is also implemented.
323 The method proceeds as
follows:
325 <li>Initialization of the velocity field, the pressure
326 and the pressure increments.
328 matches the Dirichlet
boundary conditions of the
329 problem, be the solutions of the following formulation
for all
332 \label{eq:SFEMaNS_weak_from_NS_1}
338 + \GRAD ( p^n +\frac{4\psi^n - \psi^{n-1}}{3} ) )
\cdot \textbf{v} \\
343 is a penalty coefficent,
351 \label{eq:SFEMaNS_weak_from_NS_2}
352 \int_{
\Omega_{c,
f}} \GRAD \psi^{n+1} \cdot \GRAD q
356 \label{eq:SFEMaNS_weak_from_NS_3}
360 <li>The pressure is updated as
follows:
362 \label{eq:SFEMaNS_weak_from_NS_4}
363 p^{n+1} = p^n + \psi^{n+1} - \frac{1}{\Re} \delta^{n+1} .
367 @subsubsection doc_intro_SFEMaNS_possibilities_nst_2 Entropy viscosity
for under resolved computation
369 Hydrodynamic problems with large kinetic Reynolds number
370 introduce extremely complex flows. Approximating all of
371 the dynamics
's scales of such problems is not always possible
372 with present computational ressources. To address this problem,
373 a nonlinear stabilization method called entropy viscosity is
374 implemented in SFEMaNS. This method has been introduced by
376 <code>Guermond et al. (2011)</code></a>. It consists in introducing an artifical
377 viscosity, denoted \
f$\nu_{E}
\f$, that is taken proportional
378 to the
default of equilibrium of an energy equation.
380 This implementation of
this method in SFEMaNS can be summarized
381 in the three following steps:
383 <li>Define the residual of the Navier-Stokes at
387 \frac{
\bu^n-
\bu^{n-2}}{ 2 \tau}
389 + \ROT (\
bu^{*,n-1}) \times \
bu^{*,n-1}
392 <li>Compute the entropy viscosity on each mesh cell K as
follows:
394 \label{eq:SFEMaNS_NS_entropy_viscosity}
397 \bu^{n-1}\|_{\bL^\infty(K)}}{\|
\bu^{n-1}\|_{\bL^\infty(K)}^2}\right),
399 with \
f$h\
f$ the local mesh size of the cell K,
403 constant in the
interval \f$(0,1)\
f$. It is generally set to one.
404 <li>When approximating \
f$\
bu^{n+1}
\f$, the
term
406 is added in the
left handside of the Navier-Stokes equations.
409 Thus defined, the entropy viscosity is expected to be smaller
410 than the consistency error in the smooth regions. In regions
411 with large gradients, the entropy viscosity switches to the first
412 order viscosity \
f$\nu_{\max|K}^n:=
c_\text{max} h_K \|
\bu^{n-1}\|_{\bL^\infty(K)}
\f$.
413 Note
that \f$\nu_\max^
n\f$ corresponds to the artifical viscosity
414 induces by first order up-wind scheme in the finite difference
415 and finite volume litterature.
417 Remark: To facilitate the
explicit treatment of the entropy viscosity,
418 the following term can be added in the
left handside of the Navier-Stokes
421 \label{eq:SFEMaNS_NS_LES_c1}
422 - \DIV( c_1 h \GRAD (\
bu^{n+1}-
\bu^{*,n+1})).
428 @subsubsection doc_intro_SFEMaNS_possibilities_nst_3 Extension to non axisymmetric geometry
430 A penalty method of <a href='http://www.sciencedirect.com/science/article/pii/S0168927407000815'>
431 <code>Pasquetti et al. (2008)</code></a>) is implemented so
432 the code SFEMaNS can report of the presence of non axisymmetric
433 solid domain in \
f$\
Omega_{c,
f}
\f$. Such solid domains can either
434 be driving the fluid or represents an obtacle to the fluid motion
436 the Navier-Stokes equations are approximated, is splitted into a
439 axisymmetric and time dependent. The penalty method introduces
441 field approximated by the Navier-Stokes equations to
443 This penalty
function is defined as
follows:
445 \label{eq:SFEMaNS_NS_penal_1}
455 The velocity field is updated as
follows:
457 \label{eq:SFEMaNS_NS_penal_2}
458 \frac{3
\bu^{n+1}}{2\tau}
462 + \chi^{n+1}
\left(\frac{4
\bu^n -
\bu^{n-1}}{2\tau}
463 - \GRAD( \frac{4\psi^n-\psi^{n-1}}{3}) \right)
466 - ( \ROT \
bu^{*,n+1} ) \times\
bu^{*,n+1}
468 + (1 - \chi^{n+1}) \frac{3
\bu^{n+1}
_\text{obst}}{2\tau},
473 Note that the original scheme is recovered
where \f$\chi=1
\f$.
475 Remark: the correction and update of the pressure is not modified.
477 @subsubsection doc_intro_SFEMaNS_possibilities_nst_4 Extension to multiphase flow
problem
479 The code SFEMaNS can approximate two phase flow problems.
480 The governing equations can be written as
follows:
482 \label{eq:SFEMaNS_NS_multiphase_1}
483 \partial_t \rho + \DIV( \
textbf{m}) = 0,
486 \label{eq:SFEMaNS_NS_multiphase_2}
489 - \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu))
493 \label{eq:SFEMaNS_NS_multiphase_3}
498 \f$\varepsilon(\
bu)=\GRAD^
s \bu = \frac12 (\GRAD \
bu +(\GRAD \
bu)^\sf{T})\
f$.
499 The densities, respestively dynamical viscosities, of the two fluids are denoted
500 \
f$\rho_0\
f$ and \
f$\rho_1\
f$, respectively \f$\eta_0\f$ and \f$\eta_1\f$.
503 The approximation method is based on the following ideas.
505 <li>Use of a level set method to follow the interface evolution.
506 The method consists of approximating \f$\varphi\f$ that takes
507 value in \f$[0,1]\f$ solution of:
509 \partial_t \varphi + \bu \cdot \GRAD \varphi=0.
511 The level set is equal to 0 in a fluid and 1 in the other fluid.
512 The
interface between the fluid is represented by \
f$\varphi^{-1}({1/2})\
f$.
513 <li>Use the momentum as dependent variable
for the Navier-Stokes equations.
514 The mass matrix becomes time independent and can be treated with pseudo-spectral method.
515 <li>Rewritte the diffusive term \
f$- \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu))
\f$ as
follows:
517 - \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu)) =
518 - \frac{2}{\Re} \DIV(\overline{\nu} \varepsilon(\bm))
519 - \
left( \frac{2}{\Re} \DIV(\eta \varepsilon(\
bu))
520 - \frac{2}{\Re} \DIV(\overline{\nu} \varepsilon(\bm)) \right)
523 The first term is made implicit
while the second is treated explicitly.
524 The resulting stiffness matrix is time independent and does not involve nonlinearity.
525 <li>The level set and Navier-Stokes equations are stabilized with the same entropy viscosity.
533 }{\|
\bu^{n-1}\|_{\bL^\infty(K)}\|\bm^{n-1}\|_{\bL^\infty(K)}}
539 \frac{\bm^n-\bm^{n-2}}{ 2 \tau}
540 -\frac{1}{\Re} \DIV (\eta^{n-1}\epsilon(\
bu^{n-1}))
541 + \DIV(\bm^n{\otimes}
\bu^n) + \GRAD p^{n-1} -
\textbf{
f}^{n-1} ,
545 \text{Res}_\rho^n= \frac{\bm^n-\bm^{n-2}}{ 2 \tau}
548 To facilitate the
explicit treatment of the entropy viscosity,
550 \f$-\DIV( c_1 h \GRAD (\varphi^{n+1}-\varphi^n))
\f$, can be added
551 in the
left handside of the Navier-Stokes, respectively of level set equation.
552 <li>A compression term that allows the level set to not
get flatten over time
553 iteration is added. It consists of adding the following term in the right
554 handside of the level set equation:
556 -
\DIV \left(c_\
text{comp}\nu_E h^{-1} \varphi(1-\varphi)\frac{\GRAD\varphi}{\|\varphi\|}\right).
558 The coefficient \
f$c_\
text{comp}\f$ a tunable constant
in \f$[0,1]\f$.
564 <li>Compute \
f$\varphi^{n+1}\f$ solution
of
566 \frac{\varphi^{n+1}-\varphi^n}{\tau} = -
\bu^n \cdot \GRAD \varphi^n
568 \nu_E^n\GRAD \varphi^n
569 - c_\
text{comp} \nu_E^n h^{-1} \varphi^n(1-\varphi^n)\frac{\GRAD\varphi^n}{\|\varphi^n\|}
574 \rho^{n+1} = \rho_0 + (\rho_1 - \rho_0) F(\varphi^{n+1}), \qquad
575 \eta = \eta_0 + (\eta_1 - \eta_0) F(\varphi^{n+1}),
577 where \
f$F\
f$ is either equal to the identity,
578 \
f$F(\varphi)=
\varphi\f$, or a piecewise ponylomial
function defined by:
584 & \
text{
if $|\varphi - 0.5| \le c_{
\text{reg}}$}, \\
585 1 &
\text{
if $c_{
\text{reg}} \le \varphi - 0.5$}.
591 \frac{\bm^{n+1}-\bm^n}{\tau} - \frac{2\overline{\nu}}{\Re}\DIV(\epsilon(\bm^{n+1})-\epsilon(\bm^n))
592 = \frac{2}{\Re}\DIV( \eta^n\epsilon(\
bu^n))
593 - \DIV(\bm^n\times\
bu^n)
597 <li>Update the pressure as
follows:
612 <li>This method can be used to approximate problems with
613 a stratification or an inclusion
of \f$n\geq 3\f$ fluids.
614 One level set is approximated per
interface between two
615 fluids. The fluids properties are reconstructed with
616 recursive convex combinations.
617 <li>MHD multiphase problems with variable electrical conductivity
618 between the fluids can also be considered. The electrical
619 conductivity in the fluid is reconstructed with the level set
620 the same way the density and the dynamical viscosity are.
621 The magnetic field \
f$\bH^{n+1}\f$ is updated as
follows:
623 \frac{3\bH^{n+1}-4\bH^n+\bH^{n-1}}{2\tau}
625 \ROT ( \bH^{n+1}-\bH^{*,n+1}) \right)
627 -
\ROT\left( \frac{1}{\sigma\Rm} \ROT \bH^{*,n+1} \right)
628 + \ROT (\
bu^{n+1}\times \mu^c \bH^{*,n+1})
629 + \ROT \
left( \frac{1}{\sigma\Rm}
\textbf{j}^{n+1} \right)
631 with \
f$\bH^{*,n+1}=2\bH^{n+1}-\bH^
n\f$
and \f$\overline{\sigma}\f$ a
632 function depending of the radial and vertical
634 \
f$\overline{\sigma}(r,
z)\leq \sigma(r,\
theta,
z,
t)\f$
for
640 @subsection doc_intro_SFEMaNS_possibilities_temp Heat equation
's weak formulation
642 The heat equations is approximated as follows.
644 <li>Initialization of the temperature.
645 <li>For all \f$n\geq0\f$ let \f$T^{n+1}\f$, that matches the
646 Dirichlet boundary conditions of the problem, be the solution
647 of the following formulation for all \f$v\in S_h^T\f$:
649 \label{eq:SFEMaNS_weak_form_temp}
650 \int_{\Omega_T} \frac{3 C }{2 \tau}T^{n+1} v
651 + \lambda \GRAD T^{n+1} \cdot \GRAD v
652 = - \int_{\Omega_T} \left( \frac{4 T^n -T^{n-1}}{2 \tau}
653 - \DIV (T^{*,n+1} \bu^{*,n+1}) + f_T^{n+1}\right) v,
655 where \f$T^{*,n+1}=2 T^n - T^{n-1}\f$. We remind that \f$C\f$ is
656 the volumetric heat capacity, \f$\lambda\f$ the thermal conductivty
657 and \f$f_T\f$ a source term.
662 @subsection doc_intro_SFEMaNS_possibilities_mxw Magnetic setting
664 The code SFEMaNS uses \f$\bH^1\f$ conforming Lagrange finite element to approximate
665 the magnetic field. As a consequence, the zero divergence condition on the
666 magnetic field cannot be enforced by standard penalty technique for
667 non-smooth and non-convex domains.
668 To overcome this obstacle, a method inspired of
670 <code>Bonito and Guermond (2011)</code></a>
671 has been implemented. This method consists of introducting a
676 if the solution in \f$\Omega^
c\f$ of:
678 - \DIV( h_\
text{loc}^{2(1-\alpha)} \GRAD p_m^{c,n+1} ) &=
679 - \DIV( \mu^c \GRAD \bH^{c,n+1}) ,
681 p_m^{c,n+1}|_{\partial \Omega_c} &= 0,
684 constant parameter in \f$[0.6,0.8]\f$.
685 <li>Add the
term \f$ -\DIV(\mu^v \GRAD p_\
text{m}^v)\
f$ in the right handside
686 of the scalar potential \
f$\phi\
f$ equation where \
f$p_\
text{m}^
v\f$
687 is the solution in \f$\Omega^
v\f$ of:
689 \LAP p_m^{v,n+1} = \LAP \phi^{n+1}, \\
694 We note that the magnetic pressure can be eliminated from the equation
696 <a href=
'http://www.sciencedirect.com/science/article/pii/S0021999111002749'>
697 <code>Guermond et al. (2011)</code></a>
for more details.
698 The approximation space used
699 to approximate \
f$ p_\
text{m}^
c\f$ is the following:
704 \sum\limits_{k=-M}^M \varphi_h^k (r,z) e^{
ik \theta} ;
705 \varphi_h^k \in S_{h}^{
p_\text{m}^c,2D}, \; -M \leq k \leq M
708 where we introduce the following finite element space:
711 \varphi_h|_K \in \mathbb{P}_1^2
\text{ } \forall K \in \mathcal{T}_h^c , \right\}.
714 In addition, an interior penalty method is used to enforce the continuity conditions
716 <a href=
'http://www.sciencedirect.com/science/article/pii/S0021999106002944'>
717 <code>Guermond et al. (2007)</code></a>
for more details.
719 @subsubsection doc_intro_SFEMaNS_possibilities_mxw_1 Approximation of the Maxwell equations with H
721 The Maxwell equations are approximated as
follows:
723 <li>Initialization of the magnetic field \
f$\bH^c\
f$, the scalar potential \f$\phi\f$ and the magnetic pressure \f$p_\
text{m}^
c\f$.
724 <li>For
all \f$n\geq 1\f$, computation
of \f$(\bH^{c,n+1},\phi^{n+1},
p_\text{m}^{c,n+1})\
f$
725 solutions of the following formulation
for all \
f$b\in \bV_h^{\bH^c} \f$,
726 \f$\varphi\in S_h^{\phi}\f$
729 & \int_{\Omega_c}\mu^c \frac{D\bH^{c,n+1}}{\Delta
t}\SCAL \bb
730 +\int_{\Omega_c} \frac{1}{\sigma R_m} \ROT \bH ^{c,n+1}\cdot \ROT \bb
731 +\int_{
\Omega_v} \muv\frac{\GRAD D\phi^{n+1}}{\Delta
t}\SCAL \GRAD\varphi
732 +\int_{
\Omega_v} \muv\GRAD\phi^{n+1}\SCAL \GRAD\varphi -
735 - \int_{\Omega_c} \mu^c\bH^{c,n+1}\SCAL\GRAD q +
736 \int_{\Omega_c} h^{2(1-\alpha)}\GRAD p_\
text{m}^{c,n+1}\SCAL \GRAD q
738 h^{2\alpha}\DIV (\mu^c \bH^{c,n+1} )\DIV (\mu^c \bb)\right)\\
739 & +\int_{\Sigma_{\mu}}
\left \{ \frac{1}{\sigma R_m} \ROT {\bH ^{c,n+1}} \right \}
740 \cdot \left ( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
741 & +\beta_3 \int_{\Sigma_{\mu}} h^{-1}
\left( { \bH_1^{c,n+1}}\times \bn_1^c
742 + {\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 )\\
743 & +\beta_1 \int_{\Sigma_{\mu}} h^{-1}
\left({ \mu^c_1\bH_1^{c,n+1}}\cdot \bn_1^c
744 + {\mu^c_2 \bH_2^{c,n+1}}\cdot \bn_2^c\right ) \SCAL \
left ( {\mu^c_1}{ \bb_1}\cdot \bn_1^c
745 + {\mu^c_2}{ \bb_2}\cdot \bn_2^c\right )\\
746 & +\int_{\Sigma} \frac{1}{\sigma R_m} \ROT {\bH ^{c,n+1}}
\cdot
747 \left( { \bb }\times \bn^c + \nabla \varphi ^{n+1}\times \bn^v\right)
748 + \beta_2 \int_\Sigma h^{-1}
\left( {\bH^{c,n+1}}\CROSS \bn_1^c
749 + {\GRAD \phi^{n+1}}\CROSS \bn_2^c\right ) \SCAL (\bb\CROSS \bnc +
750 \GRAD\varphi\CROSS \bnv)\\
751 & + \beta_1 \int_\Sigma h^{-1}
\left( { \mu^c\bH ^{c,n+1}}\cdot \bn_1^c
752 + {\GRAD \phi^{n+1}}\cdot \bn_2^c\right ) \SCAL ({\mu^c}\bb\cdot \bnc +
753 \GRAD\varphi \cdot \bnv)\\
754 & + \
int _{\Gamma_c} \frac{1}{\sigma R_m} \ROT \bH ^{c,n+1} \cdot ( \bb \CROSS \bnc)
756 \int_{\Gamma_c} h^{-1}
\left( { \bH^{c,n+1}}\CROSS \bn^c \right ) \SCAL (\bb\CROSS \bnc)
759 & \int_{\Omega_c}
\left( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1} \times \mu^c \bH^{*,n+1} \right )
761 + \
int _{\Sigma_{\mu}}
\left \{ \frac{1}{\sigma R_m}\bj^s +
762 \bu^{n+1} \times \mu^c \bH^{*,n+1} \right \}
\cdot
763 \left( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
764 & +\int_{\Sigma}
\left ( \frac{1}{\sigma R_m} \bj^s +
\bu^{n+1} \times \mu^c \bH^{*,n+1}
765 \right)\cdot \
left ( { \bb }\times \bn^c + \nabla \varphi \times \bn^v\right)
766 +\int_{\Gamma_c}(\ba \times \bn) \cdot \
left ({\bb} \times \bn \right) + \int_{\Gamma_v}
767 (\ba \times \bn) \cdot (\nabla \varphi \times \bn)\\
768 & + \int_{\Gamma_c}
\left ( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1} \times
769 \mu^c \bH^{*,n+1} \right )\cdot ( \bb \CROSS \bnc)
770 +\beta_3 \int_{\Gamma_c} h^{-1}
771 \left( {\bH}
_\text{bdy}^{c,n+1}\CROSS \bn^c \right) \SCAL (\bb\CROSS \bnc) ,
773 where we
set \f$D\bH^{c,n+1}=\dfrac{3\bH^{c,n+1}-4\bH^{c,n}+\bH^{c,n-1}}{2}\f$,
774 \f$D\phi^{c,n+1}=\dfrac{3\phi^{c,n+1}-4\phi^{c,n}+\phi^{c,n-1}}{2}\f$,
775 \f$\bH^{*,n+1}=2\bH^{c,n}-\bH^{c,n-1}\f$. We use the
operator \f$\{.\}\f$ defined
by
778 They are normalized
by \f$(\sigma\Rm)^{-1}\f$ so their value can be set to one
789 @subsubsection doc_intro_SFEMaNS_possibilities_mxw_2 Extension to magnetic permeability variable in time and azimuthal direction
791 The use of a Fourier decomposition in the azimuthal direction leads us to use
792 the magnetic
field \f$\bB^c=\mu\bH^
c\f$ as dependent variable of the Maxwell equations
793 in the conducting domain. The mass matrix becomes time independent and can be computed with pseudo-spectral methods.
794 To
get a time independent stiffness matrix that does not involve nonlinearity, the diffusive
term
797 \ROT \left( \frac{1}{\sigma\Rm} \ROT \frac{\bB^c}{\mu} \right) =
798 \ROT \
left( \frac{1}{\sigma\Rm \overline{\mu}} \ROT\frac{\bB^c}{\mu} \right)
799 + \ROT \
left( \frac{1}{\sigma\Rm} \ROT ((\frac{1}{\mu}-\frac{1}{\overline{\mu}})\bB^c) \right)
801 with \
f$\overline{\mu}\f$ a
function depending of the radial and vertical
803 all \f$(r,
\theta,
z,
t)\
f$ considered. The first term is then made implicit
while
804 the term involving \
f$\frac{1}{\mu}\f$ is treated explicitly.
807 Under the previous notations and assuming,
813 the Maxwell equations are approximated as
follows.
816 <li>For
all \f$n\geq 1\f$, computation
of \f$(\bB^{c,n+1},\phi^{n+1},
p_\text{m}^{c,n+1})\
f$
817 solutions of the following formulation
for all \
f$b\in \bV_h^{\bH^c} \f$,
818 \f$\varphi\in S_h^{\phi}\f$
821 & \int_{\Omega_c}\frac{D\bB^{c,n+1}}{\Delta
t}\SCAL \bb
822 + \int _{\Omega_c} \frac{1}{\sigma R_m} \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}}\cdot \ROT \bb
823 + \int_{
\Omega_v} \muv\frac{\GRAD D\phi^{n+1}}{\Delta
t}\SCAL \GRAD\varphi
824 + \int_{
\Omega_v} \muv\GRAD\phi^{n+1}\SCAL \GRAD\varphi
827 - \int_{\Omega_c} \bB^{c,n+1}\SCAL\GRAD q + \int_{\Omega_c} h^{2(1-\alpha)}
828 \GRAD p_\
text{m}^{c,n+1}\SCAL \GRAD q
829 + \int_{\Omega_c} h^{2\alpha}
830 \overline{\mu^c} \DIV \bB^{c,n+1} \DIV \bb \right)\\
831 & +\
int _{\Sigma_{\mu}}
\left\{ \frac{1}{\sigma R_m}
832 \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}} \right \}
833 \cdot \left ( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
834 & +\beta_3 \int_{\Sigma_{\mu}} h^{-1}
\left(
835 \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
836 \right) \SCAL \
left ( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
837 & +\beta_1 \int_{\Sigma_{\mu}} h^{-1}
838 \left( {\bB_1^{c,n+1}}\cdot \bn_1^c + {\bB_2^{c,n+1}}\cdot \bn_2^c\right)
839 \SCAL \
left( \overline{\mu^c_1}{ \bb_1}\cdot \bn_1^c + \overline{\mu^c_2}{ \bb_2}\cdot \bn_2^c\right )\\
840 & +\
int _{\Sigma} \frac{1}{\sigma R_m} \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}}
\cdot
841 \left( {\bb }\times \bn^c + \nabla \varphi \times \bn^v\right)
842 + \beta_2 \int_{\Sigma} h^{-1}
843 \left( \frac{\bB^{c,n+1}}{\overline{\mu^c}}\CROSS \bn_1^c + {\GRAD \phi ^{n+1}}\CROSS \bn_2^c\right)
844 \SCAL (\bb\CROSS \bnc + \GRAD\varphi\CROSS \bnv)\\
845 & + \beta_1 \int_{\Sigma} h^{-1}
846 \left( {\bB ^{c,n+1}}\cdot \bn_1^c + {\GRAD \phi ^{n+1}} \cdot \bn_2^c\right)
847 \SCAL \
left(\overline{{\mu^c}}\bb\cdot \bnc +
848 \GRAD\varphi \cdot \bnv \right )\\
849 & + \int_{\Gamma_c} \frac{1}{\sigma R_m} \ROT \frac{\bB ^{c,n+1}}{\overline{\mu^c}}
850 \cdot ( \bb \CROSS \bnc) +
\beta_3\left( \int_{\Gamma_c} h^{-1}
851 \left( \frac{
\bB_\text{bdy}^{c,n+1}}{\overline{\mu^c}}\CROSS \bn^c \right ) \SCAL (\bb\CROSS \bnc)
854 & \int_{\Omega_c} \frac{1}{\sigma R_m} \ROT (\langle \overline{\mu^c},{\mu^c}
855 \rangle {\bB^{*,n+1}})\cdot \ROT \bb \\
856 & \int_{\Omega_c}
\left( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1} \times
857 \bB^{*,n+1} \right )\cdot \ROT \bb
858 + \int_{\Sigma_{\mu}}
\left \{ \frac{1}{\sigma R_m}\bj^s
859 +
\bu^{n+1} \times \bB^{*,n+1} \right \}
\cdot
860 \left( { \bb_1}\times \bn_1^c + { \bb_2}\times \bn_2^c\right )\\
861 & +\int_{\Sigma}
\left( \frac{1}{\sigma R_m} \bj^s
862 +
\bu^{n+1} \times \bB^{*,n+1} \right )
863 \cdot \
left ( { \bb }\times \bn^c + \nabla \varphi \times \bn^v\right)
864 +\int_{\Gamma_c}(\ba \times \bn) \cdot \
left ({\bb} \times \bn \right)
865 + \int_{\Gamma_v}(\ba \times \bn) \cdot (\nabla \varphi \times \bn)\\
866 & + \int_{\Gamma_c}
\left ( \frac{1}{\sigma R_m}\bj^s +
\bu^{n+1}
867 \times \bB^{*,n+1} \right )\cdot ( \bb \CROSS \bnc)
868 +\beta_3 \int_{\Gamma_c} h^{-1}
\left({\bB}
_\text{bdy}^{c,n+1}\CROSS \bn^c \right ) \SCAL (\bb\CROSS \bnc),
870 where we
set \f$\bB^{*,n+1}=2\bB^n-\bB^{n-1}\f$
and
871 \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
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
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
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
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