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starting a section for the Level Set class in the Models chapter.

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14 ksteube 1811
15 lgraham 1702 \chapter{Models}
16 lgraham 2128 \label{MODELS CHAPTER}
17 lgraham 1702
18     The following sections give a breif overview of the model classes and their corresponding methods.
19    
20 gross 1878 \section{Stokes Problem}
21 gross 2100 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 gross 1878 \begin{equation}\label{Stokes 1}
23 gross 2100 -\left(\eta(v\hackscore{i,j}+ v\hackscore{i,j})\right)\hackscore{,j}+p\hackscore{,i}=f\hackscore{i}-\sigma\hackscore{ij,j}
24 gross 1878 \end{equation}
25 gross 2100 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 gross 1878 \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 gross 2100 \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 gross 1878 \end{equation}
33 gross 2100 which can be overwritten by constraints of the form
34 gross 1878 \begin{equation}\label{Stokes Boundary0}
35 gross 2100 v\hackscore{i}(x)=v^D\hackscore{i}(x)
36 gross 1878 \end{equation}
37 gross 2100 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 lgraham 1702
40 gross 1878 \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 gross 2100 \index{saddle point problem}
43 lgraham 1702 \begin{equation}
44     \left[ \begin{array}{cc}
45 gross 1878 A & B^{*} \\
46     B & 0 \\
47 lgraham 1702 \end{array} \right]
48     \left[ \begin{array}{c}
49 gross 2100 v \\
50 lgraham 1702 p \\
51     \end{array} \right]
52     =\left[ \begin{array}{c}
53 gross 2100 G \\
54 gross 1878 0 \\
55 lgraham 1702 \end{array} \right]
56     \label{SADDLEPOINT}
57     \end{equation}
58 gross 2100 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 gross 1878 \begin{equation}
62 gross 2100 \|v\hackscore{k,k}\| \hackscore \le \epsilon
63     \|\sqrt{v\hackscore{j,k}v\hackscore{j,k}}\|
64     \end{equation}
65     where $\epsilon$ is the desired relative accuracy and
66     \begin{equation}
67     \|p\|^2= \int\hackscore{\Omega} p^2 \; dx
68     \label{PRESSURE NORM}
69     \end{equation}
70     defines the $L^2$-norm.
71     There are two approaches to solve this problem. The first approach, called the Uzawa scheme \index{Uzawa scheme}
72     eliminates the velocity $v$ from the problem. The second approach solves the equation in coupled form after the application of a preconditioner.
73    
74     \subsubsection{Uzawa scheme}
75     The first eqution in~\ref{SADDLEPOINT} gives $v=A^{-1}(G-B^{*}p)$ assuming $p$ is known. This is inserted into the
76     second eqution which leads to
77     \begin{equation}
78     S p = B A^{-1} G
79     \end{equation}
80     with the Schur complement \index{Schur complement} $S=BA^{-1}B^{*}$. This problem can be solved iteratively using the reconditioned Conjugate Gradient Method (PCG)~\index{PCG!Preconditioned Conjugate Gradient Method}
81     with the preconditioner $\hat{S}$ defined as $q=\hat{S}^{-1}p$ by solving
82     \begin{equation}
83     \frac{1}{\eta}q = p
84     \end{equation}
85     see~\cite{ELMAN} for more details. The evaluation of $w=Sp$ is done in the form
86     \begin{equation}
87     \begin{array}{rcl}
88     A v & = & B^{*}p \\
89     w & = & Bv \\
90     \end{array}
91     \label{EVAL PCG}
92     \end{equation}
93     The residual \index{residual} $r=B A^{-1} G - S p$ is given as
94     \begin{equation}
95     r=B A^{-1} (G - B^* p) = Bv \mbox{ with } v = A^{-1}(G-B^{*}p)
96     \end{equation}
97     Therefore one uses the tuple $(v,Bv)$ to represent the residual of the current pressure $p$. Notice that before the iteration is started the right hand side $B A^{-1} G$ needs to be calculated. The bilinear form $(.,.)$ used is defined as
98     \begin{equation}
99     (p,(v,Bv))=\int\hackscore{\Omega} p \cdot Bv \; dx
100     \end{equation}
101     where $p$ is the pressure increment and $(v,Bv)$ represents an increment in the residual.
102    
103     \subsubsection{Coupled Solver}
104     An alternative approach to solve the saddle point problem~\ref{SADDLEPOINT} directly using an iterative such as
105     the generalized minimal residual method (GMRES) \index{generalized minimal residual method!GMRES} with a suitable
106     preconditioner. Here we use the operator
107     \begin{equation}
108 gross 1878 \left[ \begin{array}{cc}
109     A^{-1} & 0 \\
110     S^{-1} B A^{-1} & -S^{-1} \\
111     \end{array} \right]
112     \label{SADDLEPOINT PRECODITIONER}
113     \end{equation}
114 gross 2100 where again $S$ is the Schur complement~\cite{ELMAN}. In partice we will use an approximation $\hat{S}$ for $S$. The evaluation $(w,q)$ of the iteration operator for a given $(v,p)$ is done as
115 gross 1878 \begin{equation}
116     \begin{array}{rcl}
117 gross 2100 A w & = & Av+B^{*}p \\
118     \hat{S} q & = & B(w-v) \\
119 gross 1878 \end{array}
120 gross 2100 \label{COUPLES SADDLEPOINT iteration}
121 gross 1878 \end{equation}
122 gross 2100 We use the inner product induced by the norm
123 gross 1878 \begin{equation}
124 gross 2100 \|(v,p)\|^2= \int\hackscore{\Omega} v\hackscore{i,j} v\hackscore{i,j} + \left( \frac{p}{\eta}\right)^2\; dx
125     \label{COUPLES NORM}
126     \end{equation}
127     In PDE form~\ref{COUPLES SADDLEPOINT iteration} takes the form
128     \begin{equation}
129 gross 1878 \begin{array}{rcl}
130 gross 2100 -\left(\eta(w\hackscore{i,j}+ w\hackscore{i,j})\right)\hackscore{,j} & = & -\left(\eta(v\hackscore{i,j}+ v\hackscore{i,j})\right)\hackscore{,j}+p\hackscore{,i} \\
131     \frac{1}{\eta} q & = & - (w-v)\hackscore{i,i} \\
132 gross 1878 \end{array}
133     \label{SADDLEPOINT iteration 2}
134     \end{equation}
135 lgraham 1702
136    
137 gross 2100 \subsection{Functions}
138 gross 1878
139 gross 2100 \begin{classdesc}{StokesProblemCartesian}{domain}
140     opens the Stokes problem\index{Stokes problem} on the \Domain domain. The approximation
141     order needs to be two.
142     \end{classdesc}
143 gross 1878
144 gross 2100 \begin{methoddesc}[StokesProblemCartesian]{initialize}{\optional{f=Data(), \optional{fixed_u_mask=Data(), \optional{eta=1, \optional{surface_stress=Data(), \optional{stress=Data()}}}}}}
145     assigns values to the model parameters. In any call all values must be set.
146     \var{f} defines the external force $f$, \var{eta} the viscosity $\eta$,
147     \var{surface_stress} the surface stress $s$ and \var{stress} the initial stress $\sigma$.
148     The locations and compontents where the velocity is fixed are set by
149     the values of \var{fixed_u_mask}. The method will try to cast the given values to appropriate
150     \Data class objects.
151     \end{methoddesc}
152 gross 1878
153 gross 2100 \begin{methoddesc}[StokesProblemCartesian]{solve}{v,p,
154     \optional{max_iter=20, \optional{verbose=False, \optional{useUzawa=True}}}}
155     solves the problem and return approximations for velocity and pressure.
156     The arguments \var{v} and \var{p} define initial guess. The values of \var{v} marked
157     by \var{fixed_u_mask} remain unchanged.
158     If \var{useUzawa} is set to \True
159     the Uzawa\index{Uszwa} scheme is used. Otherwise the problem is solved in coupled form. In most cases
160     the Uzawa scheme is more efficient.
161     \var{max_iter} defines the maximum number of iteration steps.
162     If \var{verbose} is set to \True informations on the progress of of the solver are printed.
163     \end{methoddesc}
164 lgraham 1702
165    
166 gross 2100 \begin{methoddesc}[StokesProblemCartesian]{setTolerance}{\optional{tolerance=1.e-8}}
167     sets the tolerance in an appropriate norm relative to the right hand side. The tolerance must be non-negative and less than 1.
168     \end{methoddesc}
169     \begin{methoddesc}[StokesProblemCartesian]{getTolerance}{}
170     returns the current relative tolerance.
171     \end{methoddesc}
172     \begin{methoddesc}[StokesProblemCartesian]{setAbsoluteTolerance}{\optional{tolerance=0.}}
173     sets the absolute tolerance for the error in the relevant norm. The tolerance must be non-negative. Typically the
174     absolute talerance is set to 0.
175     \end{methoddesc}
176     \begin{methoddesc}[StokesProblemCartesian]{getAbsoluteTolerance}{}
177     sreturns the current absolute tolerance.
178     \end{methoddesc}
179     \begin{methoddesc}[StokesProblemCartesian]{setSubToleranceReductionFactor}{\optional{reduction=None}}
180     sets the reduction factor for the tolerance used to solve the PDEs. A reduction factor
181     in the order of one will minimize compute time per iteration step but my slow down convergence or even lead to
182     divergency. On the other hand a very small value for the PDE tolerance could result in a wast of compute time.
183     If \var{reduction} is set to \var{None} the sub-tolerance is solved adaptively but
184     in cases a very small tolerance is set ($<10^{-6}$) it is recommended to set the
185     reduction factor by hand. This may require some experiments.
186     \end{methoddesc}
187     \begin{methoddesc}[StokesProblemCartesian]{getSubToleranceReductionFactor}{}
188     return the current reduction factor for the sub-problem tolerance.
189     \end{methoddesc}
190 lgraham 1702
191 gross 2100 \subsection{Example: Lit Driven Cavity}
192     The following script \file{lit\hackscore driven\hackscore cavity.py}
193     \index{scripts!\file{helmholtz.py}} which is available in the \ExampleDirectory
194     illustrates the usage of the \class{StokesProblemCartesian} class to solve
195 jfenwick 2104 the lit driven cavity problem~\cite{LITDRIVENCAVITY}:
196 gross 2100 \begin{python}
197     from esys.escript import *
198     from esys.finley import Rectangle
199     from esys.escript.models import StokesProblemCartesian
200     NE=25
201     dom = Rectangle(NE,NE,order=2)
202     x = dom.getX()
203     sc=StokesProblemCartesian(dom)
204     mask= (whereZero(x[0])*[1.,0]+whereZero(x[0]-1))*[1.,0] + \
205     (whereZero(x[1])*[0.,1.]+whereZero(x[1]-1))*[1.,1]
206     sc.initialize(eta=.1, fixed_u_mask= mask)
207     v=Vector(0.,Solution(dom))
208     v[0]+=whereZero(x[1]-1.)
209     p=Scalar(0.,ReducedSolution(dom))
210     v,p=sc.solve(v,p, verbose=True)
211     saveVTK("u.xml",velocity=v,pressure=p)
212     \end{python}
213 lgraham 1702
214 gross 2100 \section{Darcy Flux}
215 gross 2108 We want to calculate the velocity $u$ and pressure $p$ on a domain $\Omega$ solving
216     the Darcy flux problem \index{Darcy flux}\index{Darcy flow}
217 gross 2100 \begin{equation}\label{DARCY PROBLEM}
218     \begin{array}{rcl}
219     u\hackscore{i} + \kappa\hackscore{ij} p\hackscore{,j} & = & g\hackscore{i} \\
220     u\hackscore{k,k} & = & f
221     \end{array}
222     \end{equation}
223     with the boundary conditions
224     \begin{equation}\label{DARCY BOUNDARY}
225     \begin{array}{rcl}
226     u\hackscore{i} \; n\hackscore{i} = u^{N}\hackscore{i} \; n\hackscore{i} & \mbox{ on } & \Gamma\hackscore{N} \\
227     p = p^{D} & \mbox{ on } & \Gamma\hackscore{D} \\
228     \end{array}
229     \end{equation}
230     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
231     \begin{equation}
232     \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}
233     \end{equation}
234     for all $x\hackscore{i}$.
235 lgraham 1702
236    
237 gross 2100 \subsection{Solution Method \label{DARCY SOLVE}}
238     Without loss of generality we can assume that $u^{N}\hackscore{i} \; n\hackscore{i}=0$ and
239     $p^{D}$. Otherewise one solves for $u-u^{N}$ and $p-p^{D}$ and sets
240 lgraham 1702 \begin{equation}
241 gross 2100 \begin{array}{rcl}
242     g\hackscore{i} & \leftarrow & g\hackscore{i} - u^{N}\hackscore{i} - \kappa\hackscore{ij} p^{D}\hackscore{,j }\\
243     f & \leftarrow & f - u^{N}\hackscore{k,k}
244     \end{array}
245     \end{equation}
246     We set
247     \begin{equation}
248     V = \{ q \in H^{1}(\Omega) : q=0 \mbox{ on } \Gamma\hackscore{D} \}
249     \end{equation}
250     and
251     \begin{equation}
252     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} \}
253     \end{equation}
254     and define the operator $Q: V \rightarrow (L^2(\Omega))^{d}$ defined by
255     \begin{equation}
256     (Qp)\hackscore{i} = \kappa\hackscore{ij} p\hackscore{,j}
257     \end{equation}
258     and the operator $D: W \rightarrow L^2(\Omega)$ defined by
259     \begin{equation}
260     Dv = v\hackscore{k,k}
261     \end{equation}
262     In operator notation the Darcy problem~\ref{DARCY PROBLEM} is written in the form
263     \begin{equation}
264     \begin{array}{rcl}
265     u + Qp & = & g \\
266     Du & = & f
267     \end{array}
268     \end{equation}
269     We solve this equation by minimising the functional
270     \begin{equation}
271     J(u,p):=\|u + Qp - g\|^2\hackscore{0} + \|Du-f\|\hackscore{0}^2
272     \end{equation}
273     over $W \times V$ where $\|.\|\hackscore{0}$ denotes the norm in $L^2(\Omega)$. A simple calculation shows that
274     one has to solve
275     \begin{equation}
276     ( v + Qq , u + Qp - g) + (Dv,Du-f) =0
277     \end{equation}
278     for all $v\in W$ and $q \in V$.which translates back into operator notation
279     \begin{equation}
280     \begin{array}{rcl}
281     (I+D^*D)u + Qp & = & D^*f + g \\
282     Q^*u + Q^*Q p & = & Q^* g \\
283     \end{array}
284     \end{equation}
285     where $D^*$ and $Q^*$ denote the adjoint operators.
286     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
287     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 $\alpha\hackscore{0} \gg 1$)
288    
289     The approach we are taking is to eliminate the velocity $u$ from the problem. Assuming that $p$ is known we have
290     \begin{equation}
291     v= (I+D^*D)^{-1} (D^*f + g - Qp)
292     \end{equation}
293     (notice that $(I+D^*D)$ is coercive in $W$) which is inserted into the second equation
294     \begin{equation}
295     Q^* (I+D^*D)^{-1} (D^*f + g - Qp) + Q^* Q p = Q^* g
296     \end{equation}
297     which is
298     \begin{equation}
299     Q^* ( I - (I+D^*D)^{-1} ) Q p = Q^* ( g -(I+D^*D)^{-1} (D^*f + g) )
300     \end{equation}
301     We use the PCG method to solve this. The residual $r$ ($\in V^*$) is given as
302     \begin{equation}
303     \begin{array}{rcl}
304     r & = & Q^* ( g -(I+D^*D)^{-1} (D^*f + g) - Qp + (I+D^*D)^{-1}Q p \\
305     & =& Q^* \left( (g-Qp) - (I+D^*D)^{-1} (D^*f + g - Qp) \right) \\
306     & =& Q^* \left( (g-Qp) - v \right)
307     \end{array}
308     \end{equation}
309     So in a partical implementation we use the pair $(g-Qp,v)$ to represent the residual. This will save the
310     reconstruction of the velocity $v$. In this notation the right hand side is given as
311     $(g,(I+D^*D)^{-1} (D^*f + g))$. The evaluation of the iteration operator for a given $p$ is then
312     returning $(Qp,w)$ where $w$ is the solution of
313     \begin{equation}\label{UPDATE W}
314     (I+D^*D)w = Qp
315     \end{equation}
316     We use $Q^*Q$ as a a preconditioner for the iteration operator $Q^* ( I - (I+D^*D)^{-1} ) Q$.
317    
318     \subsection{Functions}
319 gross 2108 \begin{classdesc}{DarcyFlow}{domain}
320     opens the Darcy flux problem\index{Darcy flux} on the \Domain domain.
321     \end{classdesc}
322 gross 2100
323 gross 2108 \begin{methoddesc}[DarcyFlow]{initialize}{\optional{f=Data(), \optional{fixed_u_mask=Data(), \optional{eta=1, \optional{surface_stress=Data(), \optional{stress=Data()}}}}}}
324     assigns values to the model parameters. In any call all values must be set.
325     \var{f} defines the external force $f$, \var{eta} the viscosity $\eta$,
326     \var{surface_stress} the surface stress $s$ and \var{stress} the initial stress $\sigma$.
327     The locations and compontents where the velocity is fixed are set by
328     the values of \var{fixed_u_mask}. The method will try to cast the given values to appropriate
329     \Data class objects.
330     \end{methoddesc}
331    
332    
333 gross 2100 \subsection{Example: Gravity Flow}
334    
335     %================================================
336     \section{Temperature Advection Diffusion\label{TEMP ADV DIFF}}
337    
338     \begin{equation}
339 lgraham 1709 \rho c\hackscore{p} \left (\frac{\partial T}{\partial t} + \vec{v} \cdot \nabla T \right ) = k \nabla^{2}T
340 lgraham 1702 \label{HEAT EQUATION}
341     \end{equation}
342    
343     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.
344    
345     \subsection{Description}
346    
347     \subsection{Method}
348    
349     \begin{classdesc}{TemperatureCartesian}{dom,theta=THETA,useSUPG=SUPG}
350     \end{classdesc}
351    
352     \subsection{Benchmark Problem}
353 gross 2100 %===============================================================================================================
354 lgraham 1702
355 gross 2100 %=========================================================
356 lgraham 2138 \input{levelsetmodel}
357 gross 2100 % \section{Level Set Method}
358 lgraham 1702
359 gross 2100 %\subsection{Description}
360 lgraham 1702
361 gross 2100 %\subsection{Method}
362 lgraham 1702
363 gross 2100 %Advection and Reinitialisation
364 lgraham 1702
365 gross 2100 %\begin{classdesc}{LevelSet}{mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth}
366     %\end{classdesc}
367 lgraham 1702
368     %example usage:
369    
370     %levelset = LevelSet(mesh, func\_new, reinit\_max, reinit\_each, tolerance, smooth)
371    
372 gross 2100 %\begin{methoddesc}[LevelSet]{update\_parameter}{parameter}
373     %Update the parameter.
374     %\end{methoddesc}
375 lgraham 1702
376 gross 2100 %\begin{methoddesc}[LevelSet]{update\_phi}{paramter}{velocity}{dt}{t\_step}
377     %Update level set function; advection and reinitialization
378     %\end{methoddesc}
379 lgraham 1702
380 gross 2100 %\subsection{Benchmark Problem}
381 lgraham 1702
382 gross 2100 %Rayleigh-Taylor instability problem
383 lgraham 1702
384    
385 gross 1859 % \section{Drucker Prager Model}
386 lgraham 1702
387 gross 1841 \section{Isotropic Kelvin Material \label{IKM}}
388 gross 1859 As proposed by Kelvin~\ref{KELVN} material strain $D\hackscore{ij}=v\hackscore{i,j}+v\hackscore{j,i}$ can be decomposed into
389     an elastic part $D\hackscore{ij}^{el}$ and visco-plastic part $D\hackscore{ij}^{vp}$:
390 gross 1841 \begin{equation}\label{IKM-EQU-2}
391 gross 1859 D\hackscore{ij}=D\hackscore{ij}^{el}+D\hackscore{ij}^{vp}
392 gross 1841 \end{equation}
393 gross 1859 with the elastic strain given as
394 gross 1841 \begin{equation}\label{IKM-EQU-3}
395 gross 1878 D\hackscore{ij}'^{el}=\frac{1}{2 \mu} \dot{\sigma}\hackscore{ij}'
396 gross 1841 \end{equation}
397 gross 1859 where $\sigma'\hackscore{ij}$ is the deviatoric stress (Notice that $\sigma'\hackscore{ii}=0$).
398     If the material is composed by materials $q$ the visco-plastic strain can be decomposed as
399 gross 1841 \begin{equation}\label{IKM-EQU-4}
400 gross 1859 D\hackscore{ij}'^{vp}=\sum\hackscore{q} D\hackscore{ij}'^{q}
401 gross 1841 \end{equation}
402 gross 1859 where $D\hackscore{ij}^{q}$ is the strain in material $q$ given as
403 gross 1841 \begin{equation}\label{IKM-EQU-5}
404 gross 1859 D\hackscore{ij}'^{q}=\frac{1}{2 \eta^{q}} \sigma'\hackscore{ij}
405 gross 1841 \end{equation}
406 gross 1859 where $\eta^{q}$ is the viscosity of material $q$. We assume the following
407     betwee the the strain in material $q$
408     \begin{equation}\label{IKM-EQU-5b}
409     \eta^{q}=\eta^{q}\hackscore{N} \left(\frac{\tau}{\tau\hackscore{t}^q}\right)^{\frac{1}{n^{q}}-1}
410     \mbox{ with } \tau=\sqrt{\frac{1}{2}\sigma'\hackscore{ij} \sigma'\hackscore{ij}}
411     \end{equation}
412     for a given power law coefficients $n^{q}$ and transition stresses $\tau\hackscore{t}^q$, see~\ref{POERLAW}.
413     Notice that $n^{q}=1$ gives a constant viscosity.
414 gross 1841 After inserting equation~\ref{IKM-EQU-5} into equation \ref{IKM-EQU-4} one gets:
415 gross 1859 \begin{equation}\label{IKM-EQU-6}
416 gross 2100 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}} \;.
417 gross 1841 \end{equation}
418 gross 1859 With
419     \begin{equation}\label{IKM-EQU-8}
420     \dot{\gamma}=\sqrt{2 D\hackscore{ij} D\hackscore{ij}}
421     \end{equation}
422     one gets
423     \begin{equation}\label{IKM-EQU-8b}
424     \tau = \eta^{vp} \dot{\gamma}^{vp} \;.
425     \end{equation}
426     With the Drucker-Prager cohesion factor $\tau\hackscore{Y}$, Drucker-Prager friction $\beta$ and total pressure $p$ we want to achieve
427     \begin{equation}\label{IKM-EQU-8c}
428     \tau \le \tau\hackscore{Y} + \beta \; p
429     \end{equation}
430     which leads to the condition
431     \begin{equation}\label{IKM-EQU-8d}
432     \eta^{vp} \le \frac{\tau\hackscore{Y} + \beta \; p}{ \dot{\gamma}^{vp}} \; .
433     \end{equation}
434     Therefore we modify the definition of $\eta^{vp}$ to the form
435     \begin{equation}\label{IKM-EQU-6b}
436     \frac{1}{\eta^{vp}}=\max(\sum\hackscore{q} \frac{1}{\eta^{q}}, \frac{\dot{\gamma}^{vp}} {\tau\hackscore{Y} + \beta \; p})
437     \end{equation}
438     The deviatoric stress needs to fullfill the equilibrion equation
439 gross 1841 \begin{equation}\label{IKM-EQU-1}
440 gross 1859 -\sigma'\hackscore{ij,j}+p\hackscore{,i}=F\hackscore{i}
441 gross 1841 \end{equation}
442 gross 1859 where $F\hackscore{j}$ is a given external fource. We assume an incompressible media:
443 gross 1841 \begin{equation}\label{IKM-EQU-2}
444 gross 1859 -v\hackscore{i,i}=0
445 gross 1841 \end{equation}
446 gross 1878 Natural boundary conditions are taken in the form
447     \begin{equation}\label{IKM-EQU-Boundary}
448     \sigma'\hackscore{ij}n\hackscore{j}-n\hackscore{i}p=f
449     \end{equation}
450     which can be overwritten by a constraint
451     \begin{equation}\label{IKM-EQU-Boundary0}
452     v\hackscore{i}(x)=0
453     \end{equation}
454     where the index $i$ may depend on the location $x$ on the bondary.
455    
456     \subsection{Solution Method \label{IKM-SOLVE}}
457     By using a first order finite difference approximation wit step size $dt>0$~\ref{IKM-EQU-3} get the form
458     \begin{equation}\label{IKM-EQU-3b}
459     D\hackscore{ij}'^{el}=\frac{1}{2 \mu dt } \left( \sigma\hackscore{ij}' - \sigma\hackscore{ij}^{'-} \right)
460     \end{equation}
461     where $\sigma\hackscore{ij}^{'-}$ is the deviatoric stress at the precious time step.
462     Now we can combine equations~\ref{IKM-EQU-2}, \ref{IKM-EQU-3b} and~\ref{IKM-EQU-6b} to get
463     \begin{equation}\label{IKM-EQU-10}
464 gross 2100 \sigma\hackscore{ij}' = 2 \eta\hackscore{eff} \left( D\hackscore{ij}' +
465     \frac{1}{ 2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right) \mbox{ with }
466 gross 1878 \frac{1}{\eta\hackscore{eff}}=\frac{1}{\mu \; dt}+\frac{1}{\eta^{vp}}
467     \end{equation}
468 gross 2100 Notice that $\eta\hackscore{eff}$ is a function of diatoric stress $\sigma\hackscore{ij}'$.
469 gross 1859 After inserting~\ref{IKM-EQU-10} into~\ref{IKM-EQU-1} we get
470     \begin{equation}\label{IKM-EQU-1ib}
471 gross 2100 -\left(\eta\hackscore{eff} (v\hackscore{i,j}+ v\hackscore{i,j})
472     \right)\hackscore{,j}+p\hackscore{,i}=F\hackscore{i}+
473 gross 1878 \frac{\eta\hackscore{eff}}{\mu dt } \sigma\hackscore{ij,j}^{'-}
474 gross 1859 \end{equation}
475 gross 1878 Together with the incomressibilty condition~\ref{IKM-EQU-2} we need to solve a problem with a form almost identical
476 jfenwick 1966 to the Stokes problem discussed in section~\ref{STOKES SOLVE} but with the difference that $\eta\hackscore{eff}$ is depending on the solution. Analog to the iteration scheme~\ref{SADDLEPOINT iteration 2} we can run
477 gross 1878 \begin{equation}
478     \begin{array}{rcl}
479 gross 2100 -\left(\eta\hackscore{eff}(dv\hackscore{i,j}+ dv\hackscore{i,j}
480     )\right)\hackscore{,j} & = & F\hackscore{i}+ \sigma\hackscore{ij,j}'-p\hackscore{,i} \\
481     \frac{1}{\eta\hackscore{eff}} dp & = & - v\hackscore{i,i}^+
482 gross 1878 \end{array}
483     \label{IKM iteration 2}
484     \end{equation}
485     where $v^+=v+dv$. As this problem is non-linear the Jacobi-free Newton-GMRES method is used with the norm
486     \begin{equation}
487 gross 2100 \|(v, p)\|^2= \int\hackscore{\Omega} v\hackscore{i,j}^2 + \frac{1}{\bar{\eta}^2\hackscore{eff}} p^2 \; dx
488 gross 1878 \label{IKM iteration 3}
489     \end{equation}
490 gross 2100 where $\bar{\eta}\hackscore{eff}$ is the caracteristic viscosity, for instance:
491     \begin{equation}
492     \frac{1}{\bar{\eta}\hackscore{eff}} = \frac{1}{\tau^{-}}+\sum\hackscore{q} \frac{1}{\eta^{q}\hackscore{N}}
493     \label{IKM iteration 4}
494     \end{equation}
495     In oder to perform step~\ref{IKM iteration 2} we need to calculate the $\eta\hackscore{eff}$ as well as $\sigma\hackscore{ij}'$ while via $\tau$ the first is a function of the latter. The priority is the
496     calculation of $\eta\hackscore{eff}$ with the Newton-Raphson scheme. This value can then be used to calculate
497     $\sigma\hackscore{ij}'$ via~\ref{IKM-EQU-10}. We need to solve
498     \begin{equation}
499     \tau = \eta\hackscore{eff} \cdot \epsilon \mbox{ with }
500     \epsilon = \sqrt{ 2 \left( D\hackscore{ij}' +
501     \frac{1}{ 2 \mu \; dt} \sigma\hackscore{ij}^{'-}\right)^2}
502     \label{IKM iteration 5}
503     \end{equation}
504     The Newton scheme takes the form
505     \begin{equation}
506     \tau\hackscore{n+1} = \min(\tau\hackscore{n} - \frac{\tau\hackscore{n} - \eta\hackscore{eff} \cdot \epsilon}{1 - \eta\hackscore{eff}' \cdot \epsilon}, \tau\hackscore{Y} + \beta \; p)
507     = \min(\frac{\eta\hackscore{eff} - \tau\hackscore{n} \eta\hackscore{eff}'}
508     {1 - \eta\hackscore{eff}' \cdot \epsilon}, \frac{\tau\hackscore{Y} + \beta \; p}{\epsilon}) \epsilon
509     \label{IKM iteration 6}
510     \end{equation}
511     where $\eta\hackscore{eff}'$ denotes the derivative of $\eta\hackscore{eff}$ with respect of $\tau$. The second term in $\min$ is droped of $\tau\hackscore{Y} + \beta \; p<0$ or $\epsilon=0$. In fact we have
512     \begin{equation}
513     \eta\hackscore{eff}' = - \eta\hackscore{eff}^2 \left(\frac{1}{\eta\hackscore{eff}}\right)'
514     \mbox{ with }
515     \left(\frac{1}{\eta\hackscore{eff}}\right)' = \sum\hackscore{q} \left(\frac{1}{\eta^{q}} \right)'
516     \label{IKM iteration 7}
517     \end{equation}
518     \begin{equation}\label{IKM-EQU-5XX}
519     \left(\frac{1}{\eta^{q}} \right)'
520     = \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}}}}
521     = \frac{1-\frac{1}{n^{q}}}{ \tau \eta^{q}}
522     \end{equation}
523     Notice that allways $\eta\hackscore{eff}'\le 0$ which makes the denomionator in~\ref{IKM iteration 6}
524     positive.
525 gross 1841
526 gross 1878
527 gross 2100

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