SFEMaNS  version 4.1 (work in progress) Reference documentation for SFEMaNS
doc_intro.h
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187 @section doc_intro_frame_work_num_app Numerical approximation
188
189
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
193  solutions for each Fourier mode independently, modulo nonlinear
194  terms that are made explicit. The variables are then approximated
195  on a meridian section of the domain with a finite element method.
196
197 The numerical approximation of a function \f$f\f$ is written in the following generic form:
198 @f{align*}
199 f(r,\theta,z,t)=f_h^{0,\cos}(r,z,t) +
200  \sum_{m=1}^M f_h^{m,\cos} \cos(m\theta) + f_h^{m,\sin} \sin(m\theta),
201
202 @f}
203 with \f$(r,\theta,z)\f$ the cylindrical coordinates, \f$t\f$ the time and \f$M\f$
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
206  of nonlinear terms. Introducing the functions \f$\cos_m = \cos(m\theta)\f$,
207  \f$\sin_m = \sin(m\theta)\f$ and a basis functions \f$(\phi_j)_{j \in J}\f$
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.
211
212 @subsection doc_intro_fram_work_num_app_space Space of approximation
213  We set the number of Fourier mode to \f$M\f$. We define the meridian sections
214  \f$\Omega_{c,f}^{2D}\f$, \f$\Omega_{T}^{2D}\f$,
215  \f$\Omega_{c}^{2D}\f$, \f$\Omega_v^{2D}\f$ and \f$\Omega^{2D}\f$ of
216  the domains \f$\Omega_{c,f}\f$, \f$\Omega_{T}\f$, \f$\Omega_{c}\f$,
218  We also consider \f$\left\{ \mathcal{T}_h \right\}_{h > 0}\f$
219  a family of meshes of the meridian plane \f$\Omega^{2D}\f$
220  composed of disjoint triangular cells \f$K\f$ of diameters at most \f$h\f$.
221  For a given \f$h>0\f$, we assume that the mesh \f$\mathcal{T}_h\f$
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.
228
229 <ol>
230 <li>The velocity field \f$\bu\f$ and the pressure \f$p\f$ are
231  respectively approximated in the following spaces:
232 \f{align*}{
233 \textbf{V}_{h}^\bu := \left\{ \textbf{v} =\sum\limits_{k=-M}^M \textbf{v}_h^k (r,z)
234 e^{ik \theta} ; \textbf{v}_h^k \in \textbf{V}_{h}^{\bu,2D} \text{, }
235  \overline{\textbf{v}_h^k}=\textbf{v}_h^{-k} \text{, } -M \leq k \leq M \right\} ,\\
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\},
239 \f}
240 where we introduce the following finite element space:
241 \f{align*}{
242 \textbf{V}_{h}^{\bu,2D} : = \left\{ \textbf{v}_h \in C^0(\overline{\Omega_{c,f}^{2D}}) ;
243  \textbf{v}_h|_K \in \mathbb{P}_2^6 \text{ } \forall K \in \mathcal{T}_h^{c,f}
244  \right\} ,\\
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\} .
247 \f}
248 We also introduce the subspace \f$\textbf{V}_{h,0}^{2D}\f$
249  of \f$\textbf{V}_{h}^\bu\f$ composed of vector field that are
250  zero on the boundaries of \f$\Omega_{c,f}\f$.
251 <li>The temperature \f$T\f$ is approximated in the following space:
252 \f{align*}{
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\},
256 \f}
257 where we introduce the following finite element space:
258 \f{align*}{
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\} .
261 \f}
262 <li>The magnetic field \f$\bH^c\f$ and the scalar potentiel \f$\phi\f$
263  are respectively approximated in the following spaces:
264 \f{align*}{
265 \textbf{V}_h^{\bH^c} :=
266 \left\{
267 \textbf{b}=
268 \sum\limits_{k=-M}^M \textbf{b}_h^k (r,z) e^{ik \theta} ;
269  \textbf{b}_h^k \in \textbf{V}_h^{\bH^c,2D}, \; -M \leq k \leq M \right\},
270 \\
271 S_h^{\phi} :=
272 \left\{
273 \varphi=
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
276 \right\},
277 \f}
278 where we introduce the following finite element spaces:
279 \f{align*}{
280 \textbf{V}_{h}^{\bH^c, 2D} : = \left\{ \textbf{b}_h \in
281 C^0(\overline{\Omega_{c}^{2D}});
282  \textbf{b}_h|_K \in \mathbb{P}_{l_\bH}^6 \text{ }
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\}
286 \f}
287 with \f$l_\bH\in\{1,2\}\f$ and \f$l_\phi\in\{1,2\}\f$ such that \f$l_\phi \geq l_\bH\f$.
288 </ol>
289
290 @subsection doc_intro_fram_work_num_app_time_marching Time marching
291
292 To present the time marching, we introduce a time step \f$\tau\f$ and denote
293  by \f$f^n\f$ the approximation of a function at the time \f$t_n=n \tau\f$.
294  When approximating the Navier-Stokes, heat and Maxwell equations, the time marching
295  can be summarized by the four following steps:
296 <ol>
297 <li>Initialization of the temperature \f$T^0, T^1\f$, velocity fields \f$\bu^0, \bu^1\f$ , the dynamical pressure \f$p^0, p^1\f$ , the magnetic field \f$\bH^0, \bH^1\f$ and the scalar potential \f$\phi^0, \phi^1\f$.
298 <li>Approximation of \f$T^{n+1}\f$ after the nonlinear terms are computed with extroplation involving \f$T^n, T^{n-1}, \bu^n\f$ and \f$\bu^{n-1}\f$.
299 <li>Approximation of \f$\bu^{n+1}\f$ and \f$p^{n+1}\f$ after the nonlinear terms are computed with extroplation involving \f$\bu^n, \bu^{n-1}, \bH^n\f$ and \f$\bH^{n-1}\f$.
300 <li>Approximation of \f$\bH^{n+1}\f$ and \f$\phi^{n+1}\f$ after the nonlinear terms are computed with extroplation involving \f$\bu^{n+1}, \bH^n\f$ and \f$\bH^{n-1}\f$.
301 </ol>
302
303
304 @section doc_intro_SFEMaNS_weak_form_extensions Weak formulation and extensions
305
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
309  the time step \f$\tau\f$, are unchanged.
310
311 @subsection doc_intro_SFEMaNS_possibilities_nst Hydrodynamic setting
312
313
314 @subsubsection doc_intro_SFEMaNS_possibilities_nst_1 Approximation of the Navier-Stokes equations
315
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.
322
323  The method proceeds as follows:
324 <ol>
325 <li>Initialization of the velocity field, the pressure
326  and the pressure increments.
327 <li>For \f$n\geq0\f$ let \f$\bu^{n+1}\f$, that
328  matches the Dirichlet boundary conditions of the
329  problem, be the solutions of the following formulation for all
331 \f{equation}{
332 \label{eq:SFEMaNS_weak_from_NS_1}
333 \int_{\Omega_{c,f}} \frac{3}{2 \tau} \textbf{u}^{n+1} \cdot \textbf{v}
335  + \text{c}_\text{div} h^{-1} \DIV\bu^{n+1} \DIV \bv=
336  - \int_{\Omega_{c,f}} ( \frac{-4 \textbf{u}^n
337  + \textbf{u}^{n-1}}{2 \tau}
338  + \GRAD ( p^n +\frac{4\psi^n - \psi^{n-1}}{3} ) ) \cdot \textbf{v} \\
339  + \int_{\Omega_{c,f}} ( \textbf{f}^{n+1} - (\ROT \textbf{u}^{*,n+1} )
340  \times \textbf{u}^{*,n+1} ) \cdot \textbf{v} ,
341 \f}
342 where \f$h\f$ is the local mesh size, \f$\text{c}_\text{div}\f$
343  is a penalty coefficent,
344  \f$\textbf{u}^{*,n+1}=2\textbf{u}^n-\textbf{u}^{n-1}\f$ and
345 \f$\GRAD \textbf{u} : \GRAD \textbf{v} = \sum_{i,j} \partial_i u_j \partial_i v_j\f$.
346 We remind that \f$\Re\f$ is the kinetic Reynolds number
347  and \f$\textbf{f}\f$ a source term.
348 <li>Computation of \f$\psi^{n+1}\f$ and \f$\delta^{n+1}\f$
349  solutions in \f$S_h^p\f$ of:
350 \f{equation}{
351 \label{eq:SFEMaNS_weak_from_NS_2}
353 = \frac{3}{2 \tau} \int_{\Omega_{c,f}} \textbf{u}^{n+1} \cdot \GRAD q,
354 \f}
355 \f{equation}{
356 \label{eq:SFEMaNS_weak_from_NS_3}
357 \int_{\Omega_{c,f}} q \delta^{n+1} = \int_{\Omega_{c,f}} q \DIV \textbf{u} ^{n+1},
358 \f}
359  for all \f$q\f$ in \f$S_h^p\f$.
360 <li>The pressure is updated as follows:
361 \f{equation}{
362 \label{eq:SFEMaNS_weak_from_NS_4}
363  p^{n+1} = p^n + \psi^{n+1} - \frac{1}{\Re} \delta^{n+1} .
364 \f}
365 </ol>
366
367 @subsubsection doc_intro_SFEMaNS_possibilities_nst_2 Entropy viscosity for under resolved computation
368
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.
379
380 This implementation of this method in SFEMaNS can be summarized
381  in the three following steps:
382 <ol>
383 <li>Define the residual of the Navier-Stokes at
384  the time \f$t_n\f$ as follows:
385 \f{equation}{
386 \textbf{Res}_\text{NS}^n:=
387 \frac{\bu^n-\bu^{n-2}}{ 2 \tau}
388 -\frac{1}{\Re} \LAP \bu^{n-1}
389 + \ROT (\bu^{*,n-1}) \times \bu^{*,n-1}
390  + \GRAD p^{n-1} -\textbf{f}^{n-1} ,
391 \f}
392 <li>Compute the entropy viscosity on each mesh cell K as follows:
393 \f{equation}{
394 \label{eq:SFEMaNS_NS_entropy_viscosity}
395 \nu_{E|K}^{n}:=\min\left(c_\text{max} h \|\bu^{n-1}\|_{\bL^\infty(K)},
396  c_\text{e} h^2 \frac{\|\textbf{Res}_\text{NS}^n \cdot
397  \bu^{n-1}\|_{\bL^\infty(K)}}{\|\bu^{n-1}\|_{\bL^\infty(K)}^2}\right),
398 \f}
399 with \f$h\f$ the local mesh size of the cell K,
400  \f$c_\text{max}=\frac{1}{2}\f$ for \f${\mathbb P}_1\f$
401 finite elements and \f$c_{\max}=\frac{1}{8}\f$ for \f${\mathbb P}_2\f$
402  finite elements. The coefficient \f$c_\text{e}\f$ is a tunable
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
405  \f$-\DIV(\nu_{E}^n \GRAD \bu^n)\f$
406  is added in the left handside of the Navier-Stokes equations.
407 </ol>
408
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.
416
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
419 equations:
420 \f{equation}{
421 \label{eq:SFEMaNS_NS_LES_c1}
422 - \DIV( c_1 h \GRAD (\bu^{n+1}-\bu^{*,n+1})).
423 \f}
424 with \f$h\f$ the local mesh size and \f$c_1\f$ is a tunable constant.
425  The coefficient \f$c_1\f$ should be of the same order of
426  \f$c_\text{max} \|\bu\|_{\bL^\infty(\Omega_{c,f})}\f$.
427
428 @subsubsection doc_intro_SFEMaNS_possibilities_nst_3 Extension to non axisymmetric geometry
429
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
435  when their velocity is zero. The domain \f$\Omega_{c,f}\f$, where
436  the Navier-Stokes equations are approximated, is splitted into a
437  fluid domain \f$\Omega_\text{fluid}\f$ and a solid
438  domain \f$\Omega_\text{obs}\f$. These sub domains can be non
439  axisymmetric and time dependent. The penalty method introduces
440  a penalty function \f$\chi\f$. It is used to force the velocity
441  field approximated by the Navier-Stokes equations to
442  match the given velocity field of the solid in \f$\Omega_\text{obs}\f$.
443  This penalty function is defined as follows:
444 \f{equation}{
445 \label{eq:SFEMaNS_NS_penal_1}
446 \chi =
447 \left\{
448 \begin{array}{c}
449  1 \text{ in } \Omega_\text{fluid}, \\
450  0 \text{ in } \Omega_\text{obs}.
451 \end{array}
452 \right.
453 \f}
454
455 The velocity field is updated as follows:
456 \f{equation}{
457 \label{eq:SFEMaNS_NS_penal_2}
458 \frac{3\bu^{n+1}}{2\tau}
459 - \frac{1}{\Re} \LAP \bu^{n+1}
460 =
462 + \chi^{n+1} \left(\frac{4\bu^n - \bu^{n-1}}{2\tau}
464 \\
465 + \chi^{n+1} \left(
466 - ( \ROT \bu^{*,n+1} ) \times\bu^{*,n+1}
467  + \textbf{f}^{n+1} \right)
468  + (1 - \chi^{n+1}) \frac{3\bu^{n+1}_\text{obst}}{2\tau},
469 \f}
470  with \f$\bu_\text{obs}\f$ the given velocity of the solid obstacle.
472  can be time dependent so is the penalty function \f$\chi\f$.
473  Note that the original scheme is recovered where \f$\chi=1\f$.
474
475 Remark: the correction and update of the pressure is not modified.
476
477 @subsubsection doc_intro_SFEMaNS_possibilities_nst_4 Extension to multiphase flow problem
478
479 The code SFEMaNS can approximate two phase flow problems.
480  The governing equations can be written as follows:
481 \f{equation}{
482 \label{eq:SFEMaNS_NS_multiphase_1}
483 \partial_t \rho + \DIV( \textbf{m}) = 0,
484 \f}
485 \f{equation}{
486 \label{eq:SFEMaNS_NS_multiphase_2}
487 \partial_t(\textbf{m})
488 + \DIV(\textbf{m}{\otimes}\bu)
489 - \frac{2}{\Re} \DIV(\eta \varepsilon(\bu))
490 = -\GRAD p + \textbf{f},
491 \f}
492 \f{equation}{
493 \label{eq:SFEMaNS_NS_multiphase_3}
494 \DIV \bu = 0,
495 \f}
496 where \f$\rho\f$ is the density, \f$\eta\f$ the dynamical viscosity,
497  \f$\bm=\rho\bu\f$ the momentum, \f$\textbf{f}\f$ a forcing term and
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$\eta_1\f$.
501
502
503 The approximation method is based on the following ideas.
504 <ol>
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:
508 \f{align*}{
509 \partial_t \varphi + \bu \cdot \GRAD \varphi=0.
510 \f}
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:
516 \f{align*}{
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)
521 \f}
522 with \f$\overline{\nu}\f$ a constant satisfying \f$\overline{\nu}\geq \frac{\eta}{\rho}\f$.
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.
526  For each mesh cell \f$K\f$ and each time iteration \f$n\f$,
527  the entropy viscosity \f$\nu_E\f$ is defined as follows:
528 \f{align*}{
529 \nu_{E|K}^{n}:=\min\left(c_\text{max} h \|\bu^{n-1}\|_{\bL^\infty(K)},
530  c_\text{e} h^2 \frac{\ \max(
531  |\textbf{Res}_\text{NS}^n \cdot \bu^{n-1}\|_{\bL^\infty(K)},
532 |\text{Res}_\rho^n \|\bu{n-1}\|^2|
533 }{\|\bu^{n-1}\|_{\bL^\infty(K)}\|\bm^{n-1}\|_{\bL^\infty(K)}}
534 \right),
535 \f}
536 where
537 \f{align*}{
538 \textbf{Res}_\text{NS}^n=
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} ,
542 \f}
543 and
544 \f{align*}{
545 \text{Res}_\rho^n= \frac{\bm^n-\bm^{n-2}}{ 2 \tau}
546 + \DIV (\bm^{m-1}).
547 \f}
548 To facilitate the explicit treatment of the entropy viscosity,
549 the term \f$- \DIV( c_1 h \GRAD (\bu^{n+1}-\bu^{n}))\f$, respectively
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:
555 \f{align*}{
556  - \DIV \left(c_\text{comp}\nu_E h^{-1} \varphi(1-\varphi)\frac{\GRAD\varphi}{\|\varphi\|}\right).
557 \f}
558 The coefficient \f$c_\text{comp}\f$ a tunable constant in \f$[0,1]\f$.
559  We generally set \f$c_\text{comp}=1\f$.
560 </ol>
561
562 After initialization, the first time order algorithm (BDF1) proceeds as follows:
563 <ol>
564 <li>Compute \f$\varphi^{n+1}\f$ solution of
565 \f{align}{
566 \frac{\varphi^{n+1}-\varphi^n}{\tau} = - \bu^n \cdot \GRAD \varphi^n
567  + \DIV \left(
569  - c_\text{comp} \nu_E^n h^{-1} \varphi^n(1-\varphi^n)\frac{\GRAD\varphi^n}{\|\varphi^n\|}
570  \right).
571 \f}
572 <li>Reconstruct \f$\rho^{n+1}\f$ and \f$\eta^{n+1}\f$ as follows:
573 \f{align*}{
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}),
576 \f}
577 where \f$F\f$ is either equal to the identity,
578  \f$F(\varphi)=\varphi\f$, or a piecewise ponylomial function defined by:
579 \f{align*}{
580 F(\varphi) =
581 \begin{cases}
582 0 & \text{if $\varphi - 0.5\le -c_{\text{reg}}$}, \\
583 \frac12 \left(1+\frac{(\varphi-0.5)((\varphi-0.5)^2 - 3 c_{\text{reg}}^2)}{-2c^3_{\text{reg}}}\right)
584 & \text{if $|\varphi - 0.5| \le c_{\text{reg}}$}, \\
585 1 & \text{if $c_{\text{reg}} \le \varphi - 0.5$}.
586 \end{cases}
587 \f}
588 The tunable coefficient \f$c_\text{reg}\f$ lives in \f$[0,0.5]\f$. We generally set \f$c_\text{reg}=0.5\f$.
589 <li>Compute \f$\bm^{n+1}\f$ solution of:
590 \f{align}{
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)
595 +\textbf{f}^{n+1}.
596 \f}
597 <li>Update the pressure as follows:
598 <ol>
599 <li>Solve \f$\phi^{n+1}\f$ solution of
600 \f{align*}{
601 - \LAP \phi^{n+1} = \frac{\rho_\text{min}}{\tau} \DIV \bu^{n+1},
602 \f}
603 with \f$\rho_\text{min}=\min(\rho_0,\rho_1)\f$ and \f$\bu^{n+1}=\frac{1}{\rho^{n+1}}\bm^{n+1}\f$.
604 <li>Set \f$p^{n+1}=p^n + \phi^{n+1} - \frac{\eta_\text{min}}{\Re} \DIV \bu^{n+1}\f$
605  with \f$\eta_\text{min}=\min(\eta_0,\eta_1)\f$.
606 </ol>
607 </ol>
608
609
610 Remarks:
611 <ol>
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:
622 \f{align}{
623 \frac{3\bH^{n+1}-4\bH^n+\bH^{n-1}}{2\tau}
624  + \ROT \left( \frac{1}{\overline{\sigma}\Rm}
625  \ROT ( \bH^{n+1}-\bH^{*,n+1}) \right)
626 =
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)
630 \f}
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
633  coordinates \f$(r,z)\f$ such that
634  \f$\overline{\sigma}(r,z)\leq \sigma(r,\theta,z,t)\f$ for
635  all \f$(r,\theta,z,t)\f$ considered.
636 </ol>
637
638
639
640 @subsection doc_intro_SFEMaNS_possibilities_temp Heat equation's weak formulation
641
642 The heat equations is approximated as follows.
643 <ol>
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$v\in S_h^T\f$:
648 \f{equation}{
649 \label{eq:SFEMaNS_weak_form_temp}
650 \int_{\Omega_T} \frac{3 C }{2 \tau}T^{n+1} 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,
654 \f}
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.
658 </ol>
659
660
661
662 @subsection doc_intro_SFEMaNS_possibilities_mxw Magnetic setting
663
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
669  <a href='http://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02464-6/'>
670 <code>Bonito and Guermond (2011)</code></a>
671  has been implemented. This method consists of introducting a
672  magnetic pressure denoted \f$p_\text{m}\f$ and proceeds as follows.
673 <ol>
674 <li>Add the term \f$-\mu^c \GRAD p_\text{m}^c\f$ in the right handside
675  of the magnetic field \f$\bH^c\f$ equation where \f$p_\text{m}^c\f$
676  if the solution in \f$\Omega^c\f$ of:
677 \f{align*}{
678 - \DIV( h_\text{loc}^{2(1-\alpha)} \GRAD p_m^{c,n+1} ) &=
679  - \DIV( \mu^c \GRAD \bH^{c,n+1}) ,
680  \\
681 p_m^{c,n+1}|_{\partial \Omega_c} &= 0,
682 \f}
683 where \f$h\f$ is the local mesh size and \f$\alpha\f$ a
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:
688 \f{align*}{
689 \LAP p_m^{v,n+1} = \LAP \phi^{n+1}, \\
690 \GRAD p_m^{v,n+1} \cdot \textbf{n} |_{\partial \Omega_v} = 0.
691 \f}
692 </ol>
693
694 We note that the magnetic pressure can be eliminated from the equation
695  of the scalar potential \f$\phi\f$. We refer to
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:
700 \f{align*}{
701 S_h^{p_\text{m}^c} :=
702 \left\{
703 \varphi=
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
706 \right\},
707 \f}
708 where we introduce the following finite element space:
709 \f{align*}{
710 S_{h}^{p_\text{m}^c, 2D} : = \left\{ \varphi_h \in C^0(\overline{\Omega_{c}^{2D}}) ;
711  \varphi_h|_K \in \mathbb{P}_1^2 \text{ } \forall K \in \mathcal{T}_h^c , \right\}.
712 \f}
713
714 In addition, an interior penalty method is used to enforce the continuity conditions
715  across the interfaces \f$\Sigma_\mu\f$ and \f$\Sigma\f$. We refer to
716  <a href='http://www.sciencedirect.com/science/article/pii/S0021999106002944'>
717  <code>Guermond et al. (2007)</code></a> for more details.
718
719 @subsubsection doc_intro_SFEMaNS_possibilities_mxw_1 Approximation of the Maxwell equations with H
720
721 The Maxwell equations are approximated as follows:
722 <ol>
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$\varphi\in S_h^{\phi}\f$
727  and \f$q\in S_h^{p_\text{m}^c} \f$.
728 \f{align*}{
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
733  \int_{\partial\Omega_v} \muv\varphi \bn\SCAL \GRAD \phi^{n+1}\\
734 & + \beta_1\left(\int_{\Omega_c} \mu^c\GRAD p_\text{m}^{c,n+1}\SCAL\bb
735  - \int_{\Omega_c} \mu^c\bH^{c,n+1}\SCAL\GRAD q +
737  + \int_{\Omega_c}
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 +
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 +
754 & + \int _{\Gamma_c} \frac{1}{\sigma R_m} \ROT \bH ^{c,n+1} \cdot ( \bb \CROSS \bnc)
755  + \beta _3\left(
756  \int_{\Gamma_c} h^{-1} \left( { \bH^{c,n+1}}\CROSS \bn^c \right ) \SCAL (\bb\CROSS \bnc)
757  \right )\\
758 & =\\
759 & \int_{\Omega_c} \left( \frac{1}{\sigma R_m}\bj^s + \bu^{n+1} \times \mu^c \bH^{*,n+1} \right )
760  \cdot \ROT \bb
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) ,
772 \f}
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
776  \f$\{f\}=\frac{f_1+f_2}{2}\f$ on the interface \f$\Sigma_\mu\f$.
777  The constants \f$\beta_1, \beta_2\f$ and \f$\beta_3\f$ are penalty coefficients.
778  They are normalized by \f$(\sigma\Rm)^{-1}\f$ so their value can be set to one
779  in the data file. The function \f$\bH_\text{bdy}^{c}\f$ is a user function
780  used to impose Dirichlet boundary conditions on the surface
782
783 </ol>
784
785
786
787
788
789 @subsubsection doc_intro_SFEMaNS_possibilities_mxw_2 Extension to magnetic permeability variable in time and azimuthal direction
790
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
795  \f$\ROT \left(\frac{1}{\sigma\Rm} \ROT\frac{\bB^c}{\mu} \right)\f$ is rewritten as follows:
796 \f{align*}{
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)
800 \f}
801 with \f$\overline{\mu}\f$ a function depending of the radial and vertical
802  coordinates \f$(r,z)\f$ such that \f$\overline{\mu}(r,z)\leq \mu(r,\theta,z,t)\f$ for
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.
805
806
807 Under the previous notations and assuming,
808 \f{align*}{
812 \f}
813 the Maxwell equations are approximated as follows.
814 <ol>
815 <li>Initialization of the magnetic field \f$\bB^c\f$, the scalar potential \f$\phi\f$ and the magnetic pressure \f$p_\text{m}^c\f$.
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$\varphi\in S_h^{\phi}\f$
819  and \f$q\in S_h^{p_\text{m}^c} \f$.
820 \f{align*}{
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
825  - \int_{\partial\Omega_v} \muv\varphi \bn\SCAL \GRAD \phi^{n+1}\\
826 & + \beta_1\left(\int_{\Omega_c} \overline{\mu^c}\GRAD p_\text{m}^{c,n+1}\SCAL\bb
827  - \int_{\Omega_c} \bB^{c,n+1}\SCAL\GRAD q + \int_{\Omega_c} h^{2(1-\alpha)}
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)
852  \right)\\
853 & =\\
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),
869 \f}
870 where we set \f$\bB^{*,n+1}=2\bB^n-\bB^{n-1}\f$ and
871  \f$\langle \overline{\mu^c},{\mu^c}\rangle=\frac{1}{\overline{\mu^c}}- \frac{1}{\mu^c}\f$.
872  The function \f$\bB_\text{bdy}^{c}\f$ is a user function used to impose Dirichlet
873  boundary conditions on the surface \f$\Gamma_c=\partial\Omega_c \setminus \Sigma\f$.
874
875
876
877  */
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 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
Definition: doc_intro.h:215
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 independently
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 $f f is written in the following generic f with f$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 $M f the number of Fourier modes considered The unknown f f f f f f f f Omega_v f and f Omega f We also consider f a penalty method of the divergence of the velocity field is also implemented The method proceeds as follows Definition: doc_intro.h:218 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$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 left
Definition: doc_intro.h:218
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: 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_h
Definition: doc_intro.h:204
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 $M f the number of Fourier modes considered The unknown f f sin_m Definition: doc_intro.h:207 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$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 $n geq0 f let f that matches the Dirichlet boundary conditions of the be the solutions of the following formulation for all f textbf Definition: doc_intro.h:330 section doc_intro_frame_work_num_app Numerical approximation subsection doc_intro_fram_work_num_app_Fourier_FEM Fourier Finite element representation The SFEMaNS code uses a hybrid Fourier Finite element formulation The Fourier decomposition allows to approximate the problem’s solutions for each Fourier mode modulo nonlinear terms that are made explicit The variables are then approximated on a meridian section of the domain with a finite element method The numerical approximation of a function f$f f is written in the following generic z
Definition: doc_intro.h:193
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: doc_intro.h:193