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13    
14    
15  \chapter{Models}  \chapter{Models}
16    \label{MODELS CHAPTER}
17    
18  The following sections give a breif overview of the model classes and their corresponding methods.  The following sections give a breif overview of the model classes and their corresponding methods.
19    
20  \section{Stokes Cartesian (Saddle Point Problem)}  \section{Stokes Problem}
21    The velocity \index{velocity} field $v$ and pressure $p$ of an incompressible fluid \index{incompressible fluid} is given as the solution of the Stokes problem\index{Stokes problem}
22  \subsection{Description}  \begin{equation}\label{Stokes 1}
23    -\left(\eta(v\hackscore{i,j}+ v\hackscore{i,j})\right)\hackscore{,j}+p\hackscore{,i}=f\hackscore{i}-\sigma\hackscore{ij,j}
24  Saddle point type problems emerge in a number of applications throughout physics and engineering. Finite element discretisation of the Navier-Stokes (momentum) equations for incompressible flow leads to equations of a saddle point type, which can be formulated as a solution of the following operator problem for $u \in V$ and $p \in Q$ with suitable Hilbert spaces $V$ and $Q$:  \end{equation}
25    where $\eta$ is the viscosity, $F\hackscore{i}$ defines an internal force \index{force, internal} and $\sigma\hackscore{ij}$ is an intial stress \index{stress, initial}. We assume an incompressible media:
26    \begin{equation}\label{Stokes 2}
27    -v\hackscore{i,i}=0
28    \end{equation}
29    Natural boundary conditions are taken in the form
30    \begin{equation}\label{Stokes Boundary}
31    \left(\eta(v\hackscore{i,j}+ v\hackscore{i,j})\right)n\hackscore{j}-n\hackscore{i}p=s\hackscore{i}+\sigma\hackscore{ij} n\hackscore{i}
32    \end{equation}
33    which can be overwritten by constraints of the form
34    \begin{equation}\label{Stokes Boundary0}
35    v\hackscore{i}(x)=v^D\hackscore{i}(x)
36    \end{equation}
37    at some locations $x$ at the boundary of the domain. The index $i$ may depend on the location $x$ on the boundary.
38    $v^D$ is a given function on the domain.
39    
40    \subsection{Solution Method \label{STOKES SOLVE}}
41    In block form equation equations~\ref{Stokes 1} and~\ref{Stokes 2} takes the form of a saddle point problem
42    \index{saddle point problem}
43  \begin{equation}  \begin{equation}
44  \left[ \begin{array}{cc}  \left[ \begin{array}{cc}
45  A     & B \\  A     & B^{*} \\
46  b^{*} & 0 \\  B & 0 \\
47  \end{array} \right]  \end{array} \right]
48  \left[ \begin{array}{c}  \left[ \begin{array}{c}
49  u \\  v \\
50  p \\  p \\
51  \end{array} \right]  \end{array} \right]
52  =\left[ \begin{array}{c}  =\left[ \begin{array}{c}
53  f \\  G \\
54  g \\  0 \\
55  \end{array} \right]  \end{array} \right]
56  \label{SADDLEPOINT}  \label{SADDLEPOINT}
57  \end{equation}  \end{equation}
58    where $A$ is coercive, self-adjoint linear operator in a suitable Hilbert space, $B$ is the $(-1) \cdot$ divergence operator and $B^{*}$ is it adjoint operator (=gradient operator). For more details on the mathematics see references \cite{AAMIRBERKYAN2008,MBENZI2005}.
59    We use iterative techniques to solve this problem. To make sure that the incomressibilty condition holds
60    with sufficient accuracy we check for
61    \begin{equation}
62    \|v\hackscore{k,k}\| \hackscore \le  \epsilon
63    \|\sqrt{v\hackscore{j,k}v\hackscore{j,k}}\|
64    \label{STOKES STOP}
65    \end{equation}
66    where $\epsilon$ is the desired relative accuracy and
67    \begin{equation}
68    \|p\|^2= \int\hackscore{\Omega} p^2 \; dx
69    \label{PRESSURE NORM}
70    \end{equation}
71    defines the $L^2$-norm. We use the Uzawa scheme \index{Uzawa scheme} to solve the problem.
72    
73    In fact the first equation in~\ref{SADDLEPOINT} gives for a known pressure
74    \begin{equation}
75    v=A^{-1}(G-B^{*}p)
76    \label{V CALC}
77    \end{equation}
78    which is inserted into the second equation leading to
79    \begin{equation}
80    S p =  B A^{-1} G
81    \end{equation}
82    with the Schur complement \index{Schur complement} $S=BA^{-1}B^{*}$. This problem can be solved iteratively
83    with the preconditioner $\hat{S}$ defined as $q=\hat{S}^{-1}p$ by solving
84    \begin{equation}
85    \frac{1}{\eta}q = p
86    \end{equation}
87    see~\cite{ELMAN} for more details. Note that the residual for the current approximation $p$ is given as
88    \begin{equation}
89    r=B A^{-1} (G - B^* p) = Bv
90    \end{equation}
91    where $v$ is given by~\ref{V CALC}.
92    
93  where $A$ is coercive, self-adjoint linear operator in $V$, $B$ is a linear operator from $Q$ into $V$ and $B^{*}$ is the adjoint operator of $B$. $f$ and $g$ are given elements from $V$ and $Q$ respectivitly. For more details on the mathematics see references \cite{AAMIRBERKYAN2008,MBENZI2005}.  If one uses the generalized minimal residual method (GMRES) \index{generalized minimal residual method!GMRES}
94    the method is directly applied to the preconditioned system
95    \begin{equation}
96    \hat{S}^{-1} S p =  \hat{S}^{-1} B A^{-1} G
97    \end{equation}
98    We use the norm
99    \begin{equation}
100    \|p\|\hackscore{GMRES} = \|\hat{S} p \|
101    \end{equation}
102    Notice that for the residual $\hat{r}=\hat{S}^{-1} r$ one has
103    \begin{equation}
104    \
105    \end{equation}
106    If $p^{0}$ provides an initial guess for the pressure we use~\ref{V CALC} to get a first initial guess for the
107    velocity $v^{0}$ which we use to set an absolute tolerance $ATOL =\epsilon \|\sqrt{v^{0}\hackscore{j,k}v^{0}\hackscore{j,k}}\|$.
108    The GMRES is terminated when
109    \begin{equation}
110    \|\hat{r}\|\hackscore{GMRES} \le ATOL
111    \end{equation}
112    Notice that $\|\hat{r}\|\hackscore{GMRES}= \|r \| = \|Bv\|  = \|v\hackscore{k,k}\|$ so we we can expect that
113    the target stopping criterion~\ref{STOKES STOP} is fullfilled. However, if $v$ is very different from the
114    initial choice of $v^{0}$ the value of $ATOL$ is corrected and GMRES is restarted with a new tolerance. For time dependend problems this apprach works well as value for $p$ form a previous time step provides a good initial guess.
115    
116    Alternatively, as $S$ is symmetric and positive definite one can apply the preconditioned conjugate gradient method (PCG) \index{preconditioned conjugate gradient method!PCG}. PCG use the norm
117    \begin{equation}
118    \|r\|\hackscore{PCG}^2 = \int\hackscore{\Omega} r \hat{S}^{-1}r \; dx = \int\hackscore{\Omega}  \eta r^2 \; dx
119    \end{equation}
120    To take the extra factor $\eta$ into consideration when checking the stopping criterion we use the following
121    definition for $ATOL$:
122    \begin{equation}
123    ATOL = \epsilon \frac{\|\sqrt{v^{0}\hackscore{j,k}v^{0}\hackscore{j,k}}\|  }{\|v^{0}\hackscore{k,k}\|}
124    \|v^{0}\hackscore{k,k}\|\hackscore{PCG}
125    \end{equation}
126    
 The Uzawa scheme scheme is used to solve the momentum equation with the secondary condition of incompressibility \cite{GROSS2006,AAMIRBERKYAN2008}.  
127    
 \begin{classdesc}{StokesProblemCartesian}{domain,debug}  
 opens the stokes equations on the \Domain domain. Setting debug=True switches the debug mode to on.  
 \end{classdesc}  
128    
129  example usage:  \subsection{Functions}
130    
131  solution=StokesProblemCartesian(mesh) \\  \begin{classdesc}{StokesProblemCartesian}{domain}
132  solution.setTolerance(TOL) \\  opens the Stokes problem\index{Stokes problem} on the \Domain domain. The approximation
133  solution.initialize(fixed\_u\_mask=b\_c,eta=eta,f=Y) \\  order needs to be two.
134  velocity,pressure=solution.solve(velocity,pressure,max\_iter=max\_iter,solver=solver) \\  \end{classdesc}
135    
136  \subsection{Benchmark Problem}  \begin{methoddesc}[StokesProblemCartesian]{initialize}{\optional{f=Data(), \optional{fixed_u_mask=Data(), \optional{eta=1, \optional{surface_stress=Data(), \optional{stress=Data()}}}}}}
137    assigns values to the model parameters. In any call all values must be set.
138    \var{f} defines the external force $f$, \var{eta} the viscosity $\eta$,
139    \var{surface_stress} the surface stress $s$ and \var{stress} the initial stress $\sigma$.
140    The locations and compontents where the velocity is fixed are set by
141    the values of \var{fixed_u_mask}. The method will try to cast the given values to appropriate
142    \Data class objects.
143    \end{methoddesc}
144    
145  Convection problem  \begin{methoddesc}[StokesProblemCartesian]{solve}{v,p,
146    \optional{max_iter=20, \optional{verbose=False, \optional{usePCG=True}}}}
147    solves the problem and return approximations for velocity and pressure.
148    The arguments \var{v} and \var{p} define initial guess. The values of \var{v} marked
149    by \var{fixed_u_mask} remain unchanged.
150    If \var{usePCG} is set to \True
151    reconditioned conjugate gradient method (PCG) \index{preconditioned conjugate gradient method!PCG}  scheme is used. Otherwise the problem is solved generalized minimal residual method (GMRES) \index{generalized minimal residual method!GMRES}. In most cases
152    the PCG scheme is more efficient.
153    \var{max_iter} defines the maximum number of iteration steps.
154    If \var{verbose} is set to \True informations on the progress of of the solver are printed.
155    \end{methoddesc}
156    
157    
158  \section{Temperature Cartesian}  \begin{methoddesc}[StokesProblemCartesian]{setTolerance}{\optional{tolerance=1.e-4}}
159    sets the tolerance in an appropriate norm relative to the right hand side. The tolerance must be non-negative and less than 1.
160    \end{methoddesc}
161    \begin{methoddesc}[StokesProblemCartesian]{getTolerance}{}
162    returns the current relative tolerance.
163    \end{methoddesc}
164    \begin{methoddesc}[StokesProblemCartesian]{setAbsoluteTolerance}{\optional{tolerance=0.}}
165    sets the absolute tolerance for the error in the relevant norm. The tolerance must be non-negative. Typically the
166    absolute talerance is set to 0.
167    \end{methoddesc}
168    \begin{methoddesc}[StokesProblemCartesian]{getAbsoluteTolerance}{}
169    sreturns the current absolute tolerance.
170    \end{methoddesc}
171    \begin{methoddesc}[StokesProblemCartesian]{setSubProblemTolerance}{\optional{rtol=None}}
172    sets the tolerance to solve the involved PDEs. The subtolerance \var{rtol} should not be choosen to large
173    in order to avoid feed back of errors in the subproblem solution into the outer iteration.
174    On the otherhand is choosen to small compute time is wasted.
175    If \var{rtol} is set to \var{None} the sub-tolerance is set automatically depending on the
176    tolerance choosen for the oter iteration.
177    \end{methoddesc}
178    \begin{methoddesc}[StokesProblemCartesian]{getSubProblemTolerance}{}
179    return the tolerance for the involved PDEs.
180    \end{methoddesc}
181    
182    \subsection{Example: Lit Driven Cavity}
183     The following script \file{lit\hackscore driven\hackscore cavity.py}
184    \index{scripts!\file{helmholtz.py}} which is available in the \ExampleDirectory
185    illustrates the usage of the \class{StokesProblemCartesian} class to solve
186    the lit driven cavity problem~\cite{LITDRIVENCAVITY}:
187    \begin{python}
188    from esys.escript import *
189    from esys.finley import Rectangle
190    from esys.escript.models import StokesProblemCartesian
191    NE=25
192    dom = Rectangle(NE,NE,order=2)
193    x = dom.getX()
194    sc=StokesProblemCartesian(dom)
195    mask= (whereZero(x[0])*[1.,0]+whereZero(x[0]-1))*[1.,0] + \
196          (whereZero(x[1])*[0.,1.]+whereZero(x[1]-1))*[1.,1]
197    sc.initialize(eta=.1, fixed_u_mask= mask)
198    v=Vector(0.,Solution(dom))
199    v[0]+=whereZero(x[1]-1.)
200    p=Scalar(0.,ReducedSolution(dom))
201    v,p=sc.solve(v,p, verbose=True)
202    saveVTK("u.xml",velocity=v,pressure=p)
203    \end{python}
204    
205    \section{Darcy Flux}
206    We want to calculate the velocity $u$ and pressure $p$ on a domain $\Omega$ solving
207    the Darcy flux problem \index{Darcy flux}\index{Darcy flow}
208    \begin{equation}\label{DARCY PROBLEM}
209    \begin{array}{rcl}
210    u\hackscore{i} + \kappa\hackscore{ij} p\hackscore{,j} & = & g\hackscore{i} \\
211    u\hackscore{k,k} & = & f
212    \end{array}
213    \end{equation}
214    with the boundary conditions
215    \begin{equation}\label{DARCY BOUNDARY}
216    \begin{array}{rcl}
217    u\hackscore{i} \; n\hackscore{i}  = u^{N}\hackscore{i}  \; n\hackscore{i} & \mbox{ on } & \Gamma\hackscore{N} \\
218    p = p^{D} &  \mbox{ on } & \Gamma\hackscore{D} \\
219    \end{array}
220    \end{equation}
221    where $\Gamma\hackscore{N}$ and $\Gamma\hackscore{D}$ are a partition of the boundary of $\Omega$ with $\Gamma\hackscore{D}$ non empty, $n\hackscore{i}$ is the outer normal field of the boundary of $\Omega$, $u^{N}\hackscore{i}$ and $p^{D}$ are given functions on $\Omega$, $g\hackscore{i}$ and $f$ are given source terms and $\kappa\hackscore{ij}$ is the given permability. We assume that $\kappa\hackscore{ij}$ is symmetric (which is not really required) and positive definite, i.e there are positive constants $\alpha\hackscore{0}$ and $\alpha\hackscore{1}$ wich are independent from the location in $\Omega$ such that
222  \begin{equation}  \begin{equation}
223  \rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T  \alpha\hackscore{0} \; x\hackscore{i} x\hackscore{i} \le \kappa\hackscore{ij} x\hackscore{i} x\hackscore{j} \le \alpha\hackscore{1} \; x\hackscore{i} x\hackscore{i}
 \label{HEAT EQUATION}  
224  \end{equation}  \end{equation}
225    for all $x\hackscore{i}$.
226    
 where $\vec{v}$ is the velocity vector, $T$ is the temperature, $\rho$ is the density, $\eta$ is the viscosity, $c\hackscore{p}$ is the specific heat at constant pressure and $k$ is the thermal conductivity.  
227    
228  \subsection{Description}  \subsection{Solution Method \label{DARCY SOLVE}}
229    In practical applications it is an advantage to calculate the pressure $p$ as a correction of a 'static' pressure $p^{ref}$ which is the solution of
230    \begin{equation}
231    -(\kappa\hackscore{ki}\kappa\hackscore{kj} p\hackscore{,j}^{ref})\hackscore{,i} =  - (\kappa\hackscore{ki} (g\hackscore{k}- u^{N}\hackscore{k}))\hackscore{,i}
232    \mbox{ with }
233    p^{ref} = p^{D} \mbox{ on } \Gamma\hackscore{D}
234    \end{equation}
235    With setting $u \leftarrow u-u^{N}$ and $p \leftarrow p-p^{ref}$ and
236    \begin{equation}
237    \begin{array}{rcl}
238    g\hackscore{i} & \leftarrow & g\hackscore{i} - u^{N}\hackscore{i} -  \kappa\hackscore{ij} p^{ref}\hackscore{,j }\\
239    f & \leftarrow & f - u^{N}\hackscore{k,k}
240    \end{array}
241    \end{equation}
242    we can assume that $u^{N}\hackscore{i}  \; n\hackscore{i}=0$ and
243    $p^{D}=0$.
244    We set
245    \begin{equation}
246    V = \{ q \in H^{1}(\Omega) : q=0 \mbox{ on } \Gamma\hackscore{D} \}
247    \end{equation}
248    and
249    \begin{equation}
250    W = \{ v \in (L^2(\Omega))^{d} : v\hackscore{k,k} \in L^2(\Omega) \mbox{ and } u\hackscore{i} \; n\hackscore{i} =0  \mbox{ on } \Gamma\hackscore{N} \}
251    \end{equation}
252    and define the operator $Q: V \rightarrow (L^2(\Omega))^{d}$ defined by
253    \begin{equation}
254    (Qp)\hackscore{i} = \kappa\hackscore{ij} p\hackscore{,j}
255    \end{equation}
256    and the operator $D: W \rightarrow L^2(\Omega)$ defined by
257    \begin{equation}
258    Dv = v\hackscore{k,k}
259    \end{equation}
260    In operator notation the Darcy problem~\ref{DARCY PROBLEM} is written in the form
261    \begin{equation}
262    \begin{array}{rcl}
263    u + Qp & = & g \\
264    Du & = & f
265    \end{array}
266    \end{equation}
267    We solve this equation by minimising the functional
268    \begin{equation}
269    J(u,p):=\|u + Qp - g\|^2\hackscore{0} + \|Du-f\|\hackscore{0}^2
270    \end{equation}
271    over $W \times V$ where $\|.\|\hackscore{0}$ denotes the norm in $L^2(\Omega)$. A simple calculation shows that
272    one has to solve
273    \begin{equation}
274    ( v + Qq , u + Qp - g) + (Dv,Du-f) =0
275    \end{equation}
276    for all $v\in W$ and $q \in V$.which translates back into operator notation
277    \begin{equation}
278    \begin{array}{rcl}
279    (I+D^*D)u + Qp & = & D^*f + g \\
280    Q^*u  + Q^*Q p & = & Q^*g \\
281    \end{array}
282    \end{equation}
283    where $D^*$ and $Q^*$ denote the adjoint operators.
284    In~\cite{XXX} it has been shown that this problem is continuous and coercive in $W \times V$ and therefore has a unique solution. Also standart FEM methods can be used for discretization. It is also possible
285    to solve the problem is coupled form, however this approach leads in some cases to a very ill-conditioned stiffness matrix in particular in the case of a very small or large permability ($\alpha\hackscore{1} \ll 1$ or $\alpha\hackscore{0} \gg 1$)  
286    
287  \subsection{Method}  The approach we are taking is to eliminate the velocity $u$ from the problem. Assuming that $p$ is known we have
288    \begin{equation}
289    v= (I+D^*D)^{-1} (D^*f + g - Qp)
290    \end{equation}
291    (notice that $(I+D^*D)$ is coercive in $W$) which is inserted into the second equation
292    \begin{equation}
293    Q^* (I+D^*D)^{-1} (D^*f + g - Qp) + Q^* Q p = Q^*g
294    \end{equation}
295    which is
296    \begin{equation}
297    Q^* ( I - (I+D^*D)^{-1} ) Q p = Q^* (g-(I+D^*D)^{-1} (D^*f + g) )
298    \end{equation}
299    We use the PCG \index{linear solver!PCG}\index{PCG} method to solve this. The residual $r$ ($\in V^*$) is given as
300    \begin{equation}
301    \begin{array}{rcl}
302    r & = & Q^*  \left( g -(I+D^*D)^{-1} (D^*f + g) - Qp + (I+D^*D)^{-1}Q p \right)\\
303    & =&  Q^* \left( - Qp - (I+D^*D)^{-1} (D^*f + g - Qp) \right) \\
304    & =&  Q^* \left( g - Qp - v \right)
305    \end{array}
306    \end{equation}
307    So in a partical implementation we use the pair $(Qp,v)$ to represent the residual. This will save the
308    reconstruction of the velocity $v$. In this notation the right hand side is given as
309    $(0,(I+D^*D)^{-1} (D^*f + g))$. The evaluation of the iteration operator for a given $p$ is then
310    returning $(Qp,w)$ where $w$ is the solution of
311    \begin{equation}\label{UPDATE W}
312    (I+D^*D)w = Qp
313    \end{equation}
314    We use $Q^*Q$ as a a preconditioner for the iteration operator $Q^* ( I - (I+D^*D)^{-1} ) Q$.
315    
316  \begin{classdesc}{TemperatureCartesian}{dom,theta=THETA,useSUPG=SUPG}  The iteration PCG \index{linear solver!PCG}\index{PCG} is terminated if
317    \begin{equation}\label{DARCY STOP}
318    \int r \cdot (Q^*Q)^{-1} r \; dx \le \mbox{ATOL}^2
319    \end{equation}
320    where ATOL is a given absolute tolerance.
321    The initial residual $r\hackscore{0}$ is
322    \begin{equation}\label{DARCY STOP 2}
323    r\hackscore{0}=Q^* \left( g - v\hackscore{ref} \right) \mbox{ with } v\hackscore{ref} = (I+D^*D)^{-1} (D^*f + g)
324    \end{equation}
325    so the
326    \begin{equation}\label{DARCY NORM 0}
327    \int r\hackscore0 \cdot (Q^*Q)^{-1} r\hackscore0 \; dx = \int \left( g - v\hackscore{ref} \right)  \cdot  Q  p\hackscore{ref} \; dx \mbox{ with }p\hackscore{ref} = (Q^*Q)^{-1} Q^* \left( g - v\hackscore{ref} \right)
328    \end{equation}
329    So we set
330    \begin{equation}\label{DARCY NORM 1}
331    ATOL = atol + rtol \cdot \max(|g - v\hackscore{ref}|\hackscore{0}, |Q p\hackscore{ref} |\hackscore{0} )
332    \end{equation}
333    where atol and rtol a given absolute and relative tolerances, respectively. The reference flux $v\hackscore{ref}$
334    and reference pressure $p\hackscore{ref}$ may be calcualated from their definition which would require to solve to
335    PDEs but in a practical application estimates can be used for instance solutions from previous time steps or for simplified scenarious (e.g. constant permability).
336    
337    \subsection{Functions}
338    \begin{classdesc}{DarcyFlow}{domain}
339    opens the Darcy flux problem\index{Darcy flux} on the \Domain domain.
340  \end{classdesc}  \end{classdesc}
341    \begin{methoddesc}[DarcyFlow]{setValue}{\optional{f=None, \optional{g=None, \optional{location_of_fixed_pressure=None, \optional{location_of_fixed_flux=None, \optional{permeability=None}}}}}}
342    assigns values to the model parameters. Values can be assigned using various calls - in particular
343    in a time dependend problem only values that change over time needs to be reset. The permability can be defined as scalar (isotropic), a vector (orthotropic) or a matrix (anisotropic).
344    \var{f} and \var{g} are the corresponding parameters in~\ref{DARCY PROBLEM}.
345    The locations and compontents where the flux is prescribed are set by positive values in
346    \var{location_of_fixed_flux}.
347    The locations where the pressure is prescribed are set by
348    by positive values of \var{location_of_fixed_pressure}.
349    The values of the pressure and flux are defined by the initial guess.
350    Notice that at any point on the boundary of the domain the pressure or the normal component of
351    the flux must be defined. There must be at least one point where the pressure is prescribed.
352    The method will try to cast the given values to appropriate
353    \Data class objects.
354    \end{methoddesc}
355    
356  \subsection{Benchmark Problem}  \begin{methoddesc}[DarcyFlow]{setTolerance}{\optional{atol=0,\optional{rtol=1e-8,\optional{p_ref=None,\optional{v_ref=None}}}}}
357    sets the absolute tolerance ATOL according to~\ref{DARCY NORM 1}. If \var{p_ref} is not present $0$ is used.
358    If \var{v_ref} is not present $0$ is used. If the final result ATOL is not positive an exception is thrown.
359    \end{methoddesc}
360    
361    
 \section{Level Set Method}  
362    
363  \subsection{Description}  \begin{methoddesc}[DarcyFlow]{solve}{u0,p0, \optional{max_iter=100, \optional{verbose=False \optional{sub_rtol=1.e-8}}}}
364    solves the problem. and returns approximations for the flux $v$ and the pressure $p$.
365    \var{u0} and \var{p0} define initial guess for flux and pressure. Values marked
366    by positive values \var{location_of_fixed_flux} and \var{location_of_fixed_pressure}, respectively, are kept unchanged.
367    \var{sub_rtol} defines the tolerance used to solve the involved PDEs. \var{sub_rtol} needs to be choosen sufficiently small to ensure convergence but users need to keep in mind that a very small value for \var{sub_rtol} will result in a long compute time. Typically  $\var{sub_rtol}=\var{rtol}^2$ is a good choice if $\var{rtol}$ is not choosen too small.
368    \end{methoddesc}
369    
 \subsection{Method}  
370    
371  Advection and Reinitialisation  \subsection{Example: Gravity Flow}
372    later
373    
374  \begin{classdesc}{LevelSet}{mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth}  %================================================
375  \end{classdesc}  % \section{Temperature Advection Diffusion\label{TEMP ADV DIFF}}
376    
377  %example usage:  %\begin{equation}
378    % \rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T
379    % \label{HEAT EQUATION}
380    % \end{equation}
381    
382  %levelset = LevelSet(mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth)  % where $\vec{v}$ is the velocity vector, $T$ is the temperature, $\rho$ is the density, $\eta$ is the viscosity, % % $c\hackscore{p}$ is the specific heat at constant pressure and $k$ is the thermal conductivity.
383    
384  \begin{methoddesc}[LevelSet]{update\_parameter}{parameter}  % \subsection{Description}
 Update the parameter.  
 \end{methoddesc}  
385    
386  \begin{methoddesc}[LevelSet]{update\_phi}{paramter}{velocity}{dt}{t\_step}  % \subsection{Method}
387  Update level set function; advection and reinitialization  %
388  \end{methoddesc}  % \begin{classdesc}{TemperatureCartesian}{dom,theta=THETA,useSUPG=SUPG}
389    % \end{classdesc}
390    
391    % \subsection{Benchmark Problem}
392    %===============================================================================================================
393    
394    %=========================================================
395    \input{levelsetmodel}
396    
397    % \section{Drucker Prager Model}
398    
399  \subsection{Benchmark Problem}  \section{Isotropic Kelvin Material \label{IKM}}
400    As proposed by Kelvin~\ref{KELVN} material strain $D\hackscore{ij}=v\hackscore{i,j}+v\hackscore{j,i}$ can be decomposed into
401    an elastic part $D\hackscore{ij}^{el}$ and visco-plastic part $D\hackscore{ij}^{vp}$:
402    \begin{equation}\label{IKM-EQU-2}
403    D\hackscore{ij}=D\hackscore{ij}^{el}+D\hackscore{ij}^{vp}
404    \end{equation}
405    with the elastic strain given as
406    \begin{equation}\label{IKM-EQU-3}
407    D\hackscore{ij}^{el'}=\frac{1}{2 \mu} \dot{\sigma}\hackscore{ij}'
408    \end{equation}
409    where $\sigma'\hackscore{ij}$ is the deviatoric stress (Notice that $\sigma'\hackscore{ii}=0$).
410    If the material is composed by materials $q$ the visco-plastic strain can be decomposed as
411    \begin{equation}\label{IKM-EQU-4}
412    D\hackscore{ij}^{vp'}=\sum\hackscore{q} D\hackscore{ij}^{q'}
413    \end{equation}
414    where $D\hackscore{ij}^{q}$ is the strain in material $q$ given as
415    \begin{equation}\label{IKM-EQU-5}
416    D\hackscore{ij}^{q'}=\frac{1}{2 \eta^{q}} \sigma'\hackscore{ij}
417    \end{equation}
418    where $\eta^{q}$ is the viscosity of material $q$. We assume the following
419    betwee the the strain in material $q$
420    \begin{equation}\label{IKM-EQU-5b}
421    \eta^{q}=\eta^{q}\hackscore{N} \left(\frac{\tau}{\tau\hackscore{t}^q}\right)^{\frac{1}{n^{q}}-1}
422    \mbox{ with } \tau=\sqrt{\frac{1}{2}\sigma'\hackscore{ij} \sigma'\hackscore{ij}}
423    \end{equation}
424    for a given power law coefficients $n^{q}\ge1$ and transition stresses $\tau\hackscore{t}^q$, see~\ref{POERLAW}.
425    Notice that $n^{q}=1$ gives a constant viscosity.
426    After inserting equation~\ref{IKM-EQU-5} into equation \ref{IKM-EQU-4} one gets:
427    \begin{equation}\label{IKM-EQU-6}
428    D\hackscore{ij}'^{vp}=\frac{1}{2 \eta^{vp}} \sigma'\hackscore{ij} \mbox{ with } \frac{1}{\eta^{vp}} = \sum\hackscore{q} \frac{1}{\eta^{q}} \;.
429    \end{equation}
430    With
431    \begin{equation}\label{IKM-EQU-8}
432    \dot{\gamma}=\sqrt{2 D\hackscore{ij} D\hackscore{ij}}
433    \end{equation}
434    one gets
435    \begin{equation}\label{IKM-EQU-8b}
436    \tau = \eta^{vp} \dot{\gamma}^{vp} \;.
437    \end{equation}
438    With the Drucker-Prager cohesion factor $\tau\hackscore{Y}$, Drucker-Prager friction $\beta$ and total pressure $p$ we want to achieve
439    \begin{equation}\label{IKM-EQU-8c}
440    \tau \le \tau\hackscore{Y} + \beta \; p
441    \end{equation}
442    which leads to the condition
443    \begin{equation}\label{IKM-EQU-8d}
444    \eta^{vp} \le \frac{\tau\hackscore{Y} + \beta \; p}{ \dot{\gamma}^{vp}} \; .
445    \end{equation}
446    Therefore we modify the definition of $\eta^{vp}$ to the form
447    \begin{equation}\label{IKM-EQU-6b}
448    \frac{1}{\eta^{vp}}=\max(\sum\hackscore{q} \frac{1}{\eta^{q}}, \frac{\dot{\gamma}^{vp}} {\tau\hackscore{Y} + \beta \; p})
449    \end{equation}
450    The deviatoric stress needs to fullfill the equilibrion equation
451    \begin{equation}\label{IKM-EQU-1}
452    -\sigma'\hackscore{ij,j}+p\hackscore{,i}=F\hackscore{i}
453    \end{equation}
454    where $F\hackscore{j}$ is a given external fource. We assume an incompressible media:
455    \begin{equation}\label{IKM-EQU-2}
456    -v\hackscore{i,i}=0
457    \end{equation}
458    Natural boundary conditions are taken in the form
459    \begin{equation}\label{IKM-EQU-Boundary}
460    \sigma'\hackscore{ij}n\hackscore{j}-n\hackscore{i}p=f
461    \end{equation}
462    which can be overwritten by a constraint
463    \begin{equation}\label{IKM-EQU-Boundary0}
464    v\hackscore{i}(x)=0
465    \end{equation}
466    where the index $i$ may depend on the location $x$ on the bondary.
467    
468  Rayleigh-Taylor instability problem  \subsection{Solution Method \label{IKM-SOLVE}}
469    By using a first order finite difference approximation wit step size $dt>0$~\ref{IKM-EQU-3} get the form
470    \begin{equation}\label{IKM-EQU-3b}
471    D\hackscore{ij}'^{el}=\frac{1}{2 \mu dt } \left( \sigma\hackscore{ij}' - \sigma\hackscore{ij}^{'-} \right)
472    \end{equation}
473    where $\sigma\hackscore{ij}^{'-}$ is the deviatoric stress at the precious time step.
474    Now we can combine equations~\ref{IKM-EQU-2}, \ref{IKM-EQU-3b} and~\ref{IKM-EQU-6b} to get
475    \begin{equation}\label{IKM-EQU-10}
476    \sigma\hackscore{ij}' =  2 \eta\hackscore{eff}  \left( D\hackscore{ij}' +
477    \frac{1}{  2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right)  \mbox{ with }
478    \frac{1}{\eta\hackscore{eff}}=\frac{1}{\mu \; dt}+\frac{1}{\eta^{vp}}
479    \end{equation}
480    Notice that $\eta\hackscore{eff}$ is a function of diatoric stress $\sigma\hackscore{ij}'$.
481    After inserting~\ref{IKM-EQU-10} into~\ref{IKM-EQU-1} we get
482    \begin{equation}\label{IKM-EQU-1ib}
483    -\left(\eta\hackscore{eff} (v\hackscore{i,j}+ v\hackscore{i,j})
484    \right)\hackscore{,j}+p\hackscore{,i}=F\hackscore{i}+
485    \frac{\eta\hackscore{eff}}{\mu dt } \sigma\hackscore{ij,j}^{'-}
486    \end{equation}
487    Combining this with the incomressibilty condition~\ref{IKM-EQU-2} we need to solve a
488    Stokes problem as discussed in section~\ref{STOKES SOLVE} in each time step.
489    In oder to perform step~\ref{IKM iteration 2} we need to calculate the $\eta\hackscore{eff}$ which
490    is a function of $\sigma\hackscore{ij}$ via $\tau$.  To get $\tau$ and $\eta\hackscore{eff}$ we need to solve the
491    non-linear equation
492    \begin{equation}
493    \tau = \eta\hackscore{eff} \cdot \dot{\gamma}\hackscore{total} \mbox{ with }
494    \dot{\gamma}\hackscore{total} = \sqrt{ 2 \left( D\hackscore{ij}' +
495    \frac{1}{  2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right)^2}
496    \label{IKM iteration 5}
497    \end{equation}
498    The Newton scheme takes the form
499    \begin{equation}
500    \tau\hackscore{n+1} = \min(\tau\hackscore{n} - \frac{\tau\hackscore{n} - \eta\hackscore{eff}  \cdot \dot{\gamma}\hackscore{total}}{1 - \eta\hackscore{eff}'  \cdot  \dot{\gamma}\hackscore{total}}, \tau\hackscore{Y} + \beta \; p)
501    \label{IKM iteration 6}
502    \end{equation}
503    where $\eta\hackscore{eff}'$ denotes the derivative of $\eta\hackscore{eff}$ with respect of $\tau$. The second term in $\min$ is droped of $\tau\hackscore{Y} + \beta \; p<0$ (?)). We have
504    \begin{equation}
505    \eta\hackscore{eff}' = - \eta\hackscore{eff}^2 \left(\frac{1}{\eta\hackscore{eff}}\right)'
506    \mbox{ with }
507    \left(\frac{1}{\eta\hackscore{eff}}\right)' = \sum\hackscore{q} \left(\frac{1}{\eta^{q}} \right)'
508    \label{IKM iteration 7}
509    \end{equation}
510    \begin{equation}\label{IKM-EQU-5XX}
511    \left(\frac{1}{\eta^{q}} \right)'
512    = \frac{1-\frac{1}{n^{q}}}{\eta^{q}\hackscore{N}} \frac{\tau^{-\frac{1}{n^{q}}}}{(\tau\hackscore{t}^q)^{1-\frac{1}{n^{q}}}}
513    = \frac{1-\frac{1}{n^{q}}}{\tau}\frac{1}{\eta^{q}}
514    \end{equation}
515    Notice that allways $\eta\hackscore{eff}'\le 0$ which makes the denomionator in~\ref{IKM iteration 6}
516    positive.
517    
518    
 \section{Drucker Prager Model}  
519    
 \section{Plate Mantel}  

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