\chapter*{\sc \textbf{Preface}}
\markboth{\sc \textbf{Preface}}{\sc \textbf{Preface}}
\addcontentsline{toc}{chapter}{\sc \textbf{Preface}}
\vspace{2cm}
One century has elapsed since the discovery of superconductivity
by Heike Kamerlingh Onnes, opening a new world of significant
applications in technologies ranging from electric power devices such as
motors and generators, large magnet systems such as those needed in
storage rings for particle accelerators, and electricity transmission in power
lines. As it is well-known, the technological usage of any superconducting
material is based upon its ability
to carry and maintain a current with no applied voltage whatsoever, i.e., with
an almost negligible loss of energy even in those cases when the superconductor
is subjected to strong enough applied magnetic fields. Although electrical
currents can flow with a negligible loss of energy maintaining the
superconductor in an appropriate temperature environment, superconductivity
can be destroyed by the effect of a sufficiently intense magnetic field or the
flow of a current density exceeding a critical value. Indeed, most of the
technological applications of the superconductors are directly linked to their
magnetic properties, and in particular in the way that they expell the magnetic
fields. This fact leads to the classification of superconductors in two
different kinds. On the one hand, \textit{Type-I} superconductors are mainly
characterized by a unique curve for the maximal applied magnetic field which a
superconductor is able to expell before the sudden transition to the normal
state occurs. On the other hand, \textit{Type-II} superconductors are
characterized by a new phase or ``mixed-state'' where the transition from the
superconducting state to the normal state allows the existence of bundles of
magnetic flux penetrating the sample (vortices), before reaching the sudden
transition to the normal state. This remarkable property allows to preserve the
superconducting state with the advantage of sustaining much higher magnetic
fields, and therefore carrying much higher current densities. However, this
ability is directly related to the pinning efficiency of a given material as the
motion of vortices produces a high dissipation of energy which in turn can lead
to the \textit{quench} of the superconducting state.  It is worth noting that
all the superconductors, from metal-alloys to cuprates, fullerenes, $MgB_{2}$,
iron-based systems that have been discovered  along the last 60 years, are
\textit{Type-II} superconductors, and consequently almost all the actual
superconducting technology is based on these kind of materials.
Thus, since the vortices must be pinned by the underlying crystallographic
structure and the presence of different kind of defects, the knowledge of the
electromagnetic properties and laws governing the pinning of vortices becomes a
crucial but not trivial issue for the understanding and developing of devices in
the framework of applied superconductivity.

In spite of significant theoretical and practical interest, from the
macroscopical point of view, the material laws and the electromagnetic
properties of type-II superconductors still deserve attention,
and currently no book exists that covers all the aspects about this
topic in full depth. This thesis attempts to contribute with some novel
numerical methods in applied superconductivity, including a comprehensive
discussion of the different mechanisms involved in the vortex dynamics.

The book has been structured in three parts with sequential chapters increasing
the level of complexity, both from the mathematical point of view and as
concerns to the underlying phenomena. On the one hand, a general critical
state theory for type-II superconductors with magnetic anisotropy, its
computation, implications, and consequently some applications for particular
problems, is what the first and second part try to convey.
On the other hand, some microscopical aspects of the superconductivity have been
also considered and the attained results have been compiled along the third part
of this thesis.

In detail, the first part of this book is devoted to the study of the
electromagnetic properties of type-II superconductors in the critical state
regime.
After an introductory Chapter 1, which reviews the classical statements of the
critical state theory and derived approaches, Chapter 2 focuses on the
variational theory for critical state problems and the establishment of a
general material law for 3D critical states with an associated magnetic
anisotropy and the underlying physics for the mechanism of flux depinning and
cutting. Then, a technical but important issue arises and is covered in Chapter
3: how to deal with large-scale nonlinear minimization problems such as those
presented in the general critical state theory for applied superconductivity,
but in a personal computer. A well defined structure for the minimization
functional, constraints, bounds, and preconditioners, based upon a set of
FORTRAN packages, solve this problem.

Hopefully, at the end of the first part, the reader will feel either attracted
or at least intrigued by the scope of our theory and methods. In
this sense, the second part of this book is devoted to sketch some of the main
results obtained along this line, i.e., we show some examples where we
have implemented our general critical state theory whose impact affects not
only the understanding of the physical properties of a superconducting system
but also at its potential applications.

In Chapter 4, the advantages of the variational method are emphasized focusing
on its numerical performance that allows to explain a wide number of physical
scenarios. In particular, we present a thorough analysis of the underlying
effects derived of the three dimensional magnetic anisotropy and different
material laws (\textit{or models}) which allow us to treat with the flux
depinning and cutting mechanisms.

Chapter 5 deals with the study of the longitudinal transport problem
(the current is applied parallel to some bias magnetic field) in type II
superconductors. In particular, for the slab geometry with three dimensional
components of the local electromagnetic quantities, the complex interaction
between shielding and transport is solved. On the one hand, based on a
simplified analytical method for 2D configurations, and on the other hand,
based on a wide set of numerical studies for general scenarios (3D), it is shown
that an outstanding inversion of the current flow in a surface layer, and the
remarkable enhancement of the current density by their compression towards the
center of the sample, are straightforwardly predicted when the physical
mechanisms of flux depinning and consumption (via line cutting) are recalled. In
addition, a number of striking collateral effects, such as local and global
paramagnetic behavior, are predicted.

Chapter 6 addresses to a comprehensive study of the electromagnetic response
of superconducting wires subjected to diverse configurations of transverse
magnetic field and/or longitudinal transport current.
In particular, we have performed a wide set of numerical experiments dealing
with the local and global effects underlying to the distribution of field and
current for a straight, infinite, type II superconducting wire, it immersed in
an oscillating magnetic field applied perpendicular to its surface
($\textbf{B}_{0}$), and the simultaneous action of an AC transport current
($I_{tr}$). Thus, in a first part we have introduced the theoretical framework
of this problem focusing on the numerical advantages of our variational method.
Likewise, we provide a thorough discussion about some of the main macroscopic
quantities which may be experimentally measured, such as the magnetization curve
and the hysteretic AC loss, as well as on the local behavior of the
electromagnetic quantities \textbf{E}, \textbf{B}, and \textbf{J}. Three
different regimes of excitation have been considered: (\textit{i}) Isolated
electromagnetic excitations, where only the action of $\textbf{B}_{0}$ or
$I_{tr}$ is considered, (\textit{ii}) Synchronous electromagnetic sources,
where the concomitant action of $\textbf{B}_{0}$ and $I_{tr}$ shows a
unique oscillating phase and frequency, and (\textit{iii}) Asynchronous
electromagnetic sources, where $\textbf{B}_{0}$ and $I_{tr}$ do not show the
same oscillating frequency and therefore are out-phase. The underlying
effects of considering premagnetized wires under the above mentioned regimes are
also considered. Thus, several striking effects as the strong localization of
the local density of power loss, a distinct low-pass filtering effect intrinsic
to the wire's magnetic response, exotic magnetization loops, increases and
decreases of the hysteretic AC loss by power supplies with double frequency
effects, and significant differences between the widely used approximate
formulae and the actual AC loss numerically calculated, have been detected
and explained.

The last part of this dissertation concerns our contribution to another aspect
of superconductivity. By means of a specific integral method applied to
spectroscopic data, we have been able to draw some conclusions on the influence
of the Electron-Phonon (E-Ph) coupling mechanism in cuprate superconductors.
More specifically, we have focused on the analysis of high-resolution angle
resolved photoemission spectroscopies (ARPES) in several families of cuprate
superconductors. Although relying on solid (and sophisticated) techniques in
the realm of quantum theory, we describe a phenomenological procedure that
allows to obtain relevant physical parameters concerning the E-Ph interaction.

Thus, in chapter~\ref{ch-7}, we introduce a novel theoretical model which allows
a quite general explanation of the so-called nodal \textit{kink effect} observed
in ARPES, for several doping levels in the cuprate
families $La_{2-x}Sr_{x}CuO_{4}$, $Bi_{2}Sr_{2}CaCu_{2}O_{8+x}$, and
$YBa_{2}Cu_{3}O_{6+x}$.

Finally, in an effort to clarify the influence of
the E-Ph coupling mechanism to the boson mechanism which causes the pair
formation in the superconducting state, chapter~\ref{ch-8} addresses the study
of the superconducting thermodynamical quantities, $T_{c}$, the ratio gap
$2\Delta_{0}/k_{B}T_{c}$, and the zero temperature gap $\Delta_{0}$, for a wide
set of natural and empirical equations.

In reading this book, we want to remark that each one of its parts have its
own introduction and concluding sections, and also the list of references to the
literature have been placed forward. In addition, a small glossary can be found
at the end of this book.

\break

Hopefully, this thesis may serve to bring a bigger community interested in
the world of superconductivity, either in the application of their macroscopical
properties or the understanding of their microscopical ground.

\vspace*{2cm}

\hspace{7cm} February 2012, Zaragoza - Spain.
\cleardoublepage

\pagenumbering{arabic}

\part{\label{Part_1}\textsc{\textbf{Electromagnetism of type II
superconductors}}}

\chapter*{}
\vspace*{-3cm}
\section*{\label{Intro-P1}\hspace*{0.55cm} \sc \textbf{Introduction}}
\markboth{\sc \textbf{Introduction}}{\hspace*{0.55cm} \sc \textbf{Introduction}}
\addcontentsline{toc}{chapter}{\hspace*{0.55cm} \sc \textbf{Introduction}}

The high interest concerning the investigation of the macroscopic
magnetic properties of type-II superconductors in the mixed state is markedly
associated with its relevance to technological and industrial applications
achieving elevated transport currents with no discernible energy dissipation.
It relates to a wide list of physical phenomena concerning the physics of
vortices, which may be basically analyzed in terms of
interactions between the flux lines themselves (lattice elasticity and line
cutting), and interactions with the underlying crystal structure averaged
by the so-called flux pinning mechanism.

In a mesoscopic description of real type-II superconductors, the distribution
of vortices may be simplified through a mean-field approach for a volume
containing a big enough number of vortices and making use of an appropriate
material law incorporating the intrinsic properties of the material. This
picture of  coarse-grained fields, i.e.: magnetic induction ${\bf B}\equiv
\langle{\bf b}\rangle$, electric current density ${\bf J\equiv
\langle{\bf j}\rangle}$ and electric field ${\bf E\equiv \langle{\bf
e}\rangle}$, allows to state the problem of the driving force due to the
currents circulating in the superconducting sample and their balance with the
limiting pinning force acting on the vortex lattice so as to
prevent destabilization and the consequent propagation of dissipative states.
Per unit volume, this reads ${\bf J}\times{\bf B}={\bf F}_p$ (or $J_{\perp}B =
F_p$). The underlying concept behind this balance condition is already a
classical subject well known as the critical state model by Charles P.
Bean~\cite{P1-Bean_1964}. In this simple, but brilliant model, the response of
the superconducting sample is provided by assuming that the electrical current
density vector $\textbf{J}$ (oriented perpendicular to the direction of
the local magnetic field vector ${\bf B}$) compensates with the pinning force,
and then, it is constrained by a threshold value $\textbf{J}_{c}$ which defines
a local critical state for the array of magnetic flux lines. Thus, in
view of Faraday's law, external field variations are opposed by the maximum
current density $J_{c}$ within the material, and after the changes occur,
$J_{c}$ persists in those regions which have been affected by an electric field.
Although, such a model allows to capture the main features of the magnetic
response of superconductors with pinning at low frequencies and temperatures,
through the minimal mathematical complication, the stronger limitation
of Bean's ansatz is that one can just apply it to vortex lattices composed by
parallel flux lines perpendicular to the current flow, and unless for highly
symmetric situations \textbf{J} does not necessarily satisfy the condition
$\textbf{J}=\textbf{J}_{\perp}$. In fact, a proper theory for the critical state
must allow the coexistence of nonparallel flux lines. Thus, rotations of
\textbf{B} can lead to entanglement and recombination of neighboring flux lines
which brings a component of the current density along the local magnetic field,
$\textbf{J}_{\parallel}$. This component generates distortions which also become
unstable when a threshold value $J_{c\parallel}$ is exceeded, giving way to the
so-called flux cutting phenomenon.

When the conditions $J_{\parallel}=J_{c\parallel}$ and $J_{\perp}=J_{c\perp}$
become active, the so-called double critical state appears~\cite{P1-Clem_DCSM}.
In simple words, this upgraded theory ({\em double critical state model} or
DCSM) generalises the one-dimensional concept introduced by
Bean~\cite{P1-Bean_1964} to anisotropic scenarios for the material law in terms
of the natural concepts $E_{\parallel}(J_{\parallel})$ and
$E_{\perp}(J_{\perp})$~\cite{P1-Ruiz_PRB_2009,P1-Ruiz_SUST_2010,
P1-Ruiz_SUST_2011}.
From the mathematical point of view, the critical state problem can be
understood as finding the equilibrium distribution for the circulating
current density $\textbf{J}(\textbf{r})$ defined by the conditions
$J_{\parallel}\leq J_{c\parallel}$ and $J_{\perp}\leq J_{c\perp}$, both
consistent with the Maxwell equations in quasistationary form, and under
continuity boundary conditions that incorporate the influence of the sources.
Being a quasi-stationary approach, the critical state is customarily stated
without an explicit role for the transient electric field. Thus, Faraday's law
is implicitly used through Lenz' law by selecting the actual value $\pm
J_{c}$ or $0$ that minimizes flux variations when solving Ampere's law
$\nabla\times \textbf{B}=\mu_{0}\textbf{J}$ along the process. Customarily, one
also considers situations
where the local components of the magnetic field $\textbf{H}(\textbf{r})$ along
the superconductor (SC) are much higher than the lower critical field
$H_{c1}$ and well below $H_{c2}$ to allow the use of the linear relation
$\textbf{B}=\mu_{0}\textbf{H}$.

Within this picture of the electromagnetic problem, in this first \textit{part}
of the book we introduce the important definitions and concepts of those topics
behind the critical state theory, extending its scope for three dimensional
cases with help of numerical methods in the
framework of the variational formalism for optimal control problems. We want to
emphasize that although it is not our intent to develop a comprehensive study of
these mathematical topics, we will show common mathematical techniques which are
found to be particularly useful in applied superconductivity.  The reader is
referred to the
references~\cite{P1-Badia_PRL_2001,P1-Badia_PRB_2002,P1-Jackson,P1-Mayergoyz,
P1-Arfken,P1-Pontryagin,P1-Leitmann,P1-Knowles} for a more thorough discussion
of this
material.

Chapter 1 is devoted to introduce the theoretical background that
justifies the critical state concept as a valid
constitutive law for superconducting materials. First, the critical state is
described by the classical differential formalism of the Maxwell equations, and
then, the prescribed magnetoquasistationary approximation is thoroughly
discussed.

In chapter 2, our proposed general critical state theory is developed in two
parts. Firstly, the critical state problem is posed in terms of an equivalent
optimal control problem with variational statements, i.e., the classical Maxwell
equations are translated to the variational formalism where a simpler set of
integral equations with boundary conditions is to be solved by a minimization
procedure. Is to be noticed, that despite of the fact that the reader can feel
more familiar to the differential formalism, the numerical solution of the
differential set of equations is much more cumbersome than minimizing an
integral functional. Secondly, the underlying vortex physics is posed in terms
of a quite general material law for type-II
superconductors with magnetic anisotropy, which characterizes
the conducting behavior in terms of the threshold values for the current
density and the physical mechanisms of flux depinning and cutting.

Finally, chapter 3 covers the basic facts related to the computational method
adopted for the solution of general critical state problems such
as those tackled in the following part of this book. Here, no attempt
is made to scrutinize through the FORTRAN packages for large scale nonlinear
optimization. Instead, the presentation of this chapter must be
understood as a schematic tool for dealing with a wide variety of problems in
applied superconductivity.

\chapter{\label{ch-1} \sc \textbf{General Statements Of The Critical State}}

\subsection*{\label{ch-1-1}
\hspace*{0.1cm} \textsl{1.1 The Critical State In The Maxwell Equations
Formalism}}
\vspace*{1cm}
\markboth{\hspace*{0.1cm}\textsl{1.1 The Critical State In The Maxwell
Equations Formalism}}
{\hspace*{0.1cm} \textsl{1.1 The Critical State In The Maxwell Equations
Formalism}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{1.1 The Critical State In The Maxwell Equations
Formalism}}

The fundamental concept on which the critical state theory relies is that, in
many cases, the experimental conditions allow to analyze the evolution of the
system in a magnetoquasisteady (MQS) regime of the time-dependent Maxwell
equations accompanied by material constitutive laws,  ${\bf H}({\bf
B})$, ${\bf D}({\bf E})$, and ${\bf J}({\bf E})$. Thus, Faraday's and Ampere's
laws represent a coupled system of time evolution field equations
\begin{eqnarray}\label{Eq.1.1}
\partial_{t}{\bf B}=-\nabla\times{\bf E}\quad , \quad
\partial_{t}{\bf D}=\nabla\times{\bf H}-{\bf J} \, ,
\end{eqnarray}
which together determine the distribution of supercurrents within the sample.
Here, the induced transient electric field is determined through an appropriate
material relation ${\bf J}({\bf E})$, and is used to update the profile of $\bf
J$.

\footnotetext[1]{If there were some average surface currents, then the actual
density of magnetic flux $\textbf{B}$ would differ from
$\mu_{0}\textbf{H}$.}

Notice that, as equilibrium magnetization is usually neglected
in the critical-state regime, one is enabled to use the
relation $\textbf{B}=\mu_{0}\textbf{H}$, so that there are no average surface
currents.\footnotemark[1] Furthermore, as the magnetic fields of interest are
some fraction of the critical transition magnetic field $H_{c2}$ that is much
greater than the penetration field $H_{c1}$, the distribution of vortices and
the corresponding supercurrents will be thermodynamically favoured to
go into the superconductor, such that a ramp in the magnetic field is induced
by the external excitation within the interval $[t,t+dt]$,  see
Fig.~\ref{Figure_1_1}(a). Thus, as a consequence of a very fast diffusion
(elevated flux flow resistivity), the electric field quickly adjusts to a
constant value along the excitation interval, and once the
magnetic field ramp stops, $E$ goes back to zero again.

The {\em readjusting}
vertical bands are considered a second order effect and allow for charge
separation and recombination, according to the specific ${\bf E}({\bf J})$ model
[see Fig.~\ref{Figure_1_1}(b)]. Therefore,  we are allowed to model the
flux as entering the superconductor at zero field cooling, where the electric
field arises when some {\em critical} condition for the volume current density
is reached ($J_c$ in the 1D representation). Then, corresponding to the MQS
limit, the electric field instantaneously increases to a certain value
determined by the rate of variation of the magnetic field and then goes back to
zero.

By taking divergence in both sides
of the Faraday's and Ampere's laws, and recalling integrability
(permutation of space and time derivatives) it leads to
the additional conditions
\begin{eqnarray}\label{Eq.1.2}
\partial_{t}(\nabla\cdot{\bf B})=0 \quad ,\quad
 \partial_{t} (\nabla\cdot{\bf D}) + \nabla\cdot{\bf J} = 0 \, .
\end{eqnarray}

Within this picture, the remaining
Maxwell equations can be interpreted as ``\textit{spatial
initial conditions}'' for Eq.~(\ref{Eq.1.2}) which are defining the existence of
conserved electric charges, i.e.,
\begin{eqnarray}\label{Eq.1.3}
 \nabla\cdot{\bf B}(t=0) = 0 \quad ,\quad \nabla\cdot {\bf D} (t=0) = \rho
(t=0)\, .
\end{eqnarray}

In this sense, the set of equations~(\ref{Eq.1.1}), upon substitution of $\bf
H$,  $\bf D$ and $\bf J$ through the constitutive laws, and with appropriate
initial conditions, uniquely determine the evolution profiles ${\bf
B}({\bf r},t)$ and ${\bf E}({\bf r},t)$.

\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{Figure_1_1.pdf}
\caption{\label{Figure_1_1} (a) Schematic representation the time
dependence of the electromagnetic fields within the MQS
regime. (b) Pictorial drawing of the critical state model in terms of a one
dimensional ${E}({J})$ law.}
\end{figure}

Notice that, for {\em slow} and {\em uniform} sweep rates of the external
excitations (magnetic field sources and/or transport current), the
transient variables $\bf E$, $\bf D$ and $\rho$ are small, and proportional to
 $\dot{\bf B}$, whereas $\ddot {\bf B}$, $\dot {\bf E}$ and
$\dot {\rho}$ are negligible. Thus, the main hypothesis within the MQS regime is
that the {\em displacement} current densities $\partial _t{\bf D}$ are much
smaller than $\bf J$ in the bulk and vanish in a first order treatment. This
causes a crucial change in the
mathematical structure of the Maxwell equations: Ampere's law is no
longer a time evolution equation, but becomes a purely spatial
condition. It reads as
\begin{eqnarray}\label{Eq.1.4}
\nabla\times {\bf B} \simeq \mu_{0}{\bf J} \, ,
\end{eqnarray}
with approximate integrability condition $\nabla\cdot{\bf J}\simeq 0$.

In the MQS limit, Faraday's law is the unique time evolution equation. Then, one
can find the evolution profile ${\bf B}({\bf r},t)$ from
\begin{eqnarray}\label{Eq.1.5}
\partial_{t}{\bf B}=-\nabla\times{\bf
E}=-\nabla\times\left[\rho(\textbf{J})~\mu_{0}\nabla\times{\bf
B}\right] \, .
\end{eqnarray}
Here, $\rho ({\bf J})$ plays the role of a nonlinear and possibly nonscalar
resistivity
that should properly incorporate the physics of the threshold and dissipation
mechanisms associated to the flux depinning and flux cutting mechanisms.

We want to mention that, although the \textit{B-formulation} in
Eq.~(\ref{Eq.1.5}) is definitely the most extended one, the possibilities of
\textit{E-formulations}~\cite{P1-Barret_2006},
\textit{J-formulations}~\cite{P1-Wolsky_2008}, or a vector potential oriented
theory (\textit{A-formulation})~\cite{P1-Campbell_2009}, in which the dependent
variables are the fields ${\bf E}$, ${\bf J}$, or $\textbf{A}$ respectively,
have also been exploited by several authors.

\vspace*{0.5cm}
\subsection*{\label{ch-1-2}
\hspace*{0.1cm} \textsl{1.2 The Critical State Regime And The MQS
Limit}}
\vspace*{0.5cm}
\markboth{\hspace*{0.1cm}\textsl{1.2 The Critical State Regime And The MQS
Limit}}
{\hspace*{0.1cm} \textsl{1.2 The Critical State Regime And The MQS
Limit}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{1.2 The Critical State Regime And The MQS
Limit}}

In spite of the seeming simplicity of the MQS approach
($\partial_{t}\textbf{D}\approxeq0$), we want to emphasize that the numerical
procedure to solve a critical state problem is closely linked to the
consequences of having assumed this limit. Below, two of the most relevant
consequences of the MQS limit are highlighted.

\begin{enumerate}
\item  Notice that, making use of the conductivity law
through its inverse function ${\bf E}({\bf J})$, the successive field
penetration profiles within the superconductor
may be obtained by the finite-difference expression of Faraday's law,
\begin{eqnarray}\label{Eq.1.6}
\frac{B_{l+1}-B_{l}}{\delta t}=-\nabla\times{\bf
E}~\left(\mu_{0}\textbf{J}_{l+1}\approx\nabla\times\textbf{B}_{l+1}\right) \, .
\end{eqnarray}
Here we have assumed an evolutionary discretization scheme, where
$\textbf{B}_{l}$ stands for the local magnetic field induction at the time
layer $l\delta t$, and the current density profiles are related to some
magnetic diffusion process that takes place when the local condition for
critical state $\textbf{J}(\textbf{r})\leq \textbf{J}_{c}(\textbf{r})$ is
violated. On the other hand, the constitutive law ${\bf D}({\bf E})$ which is
not used in
Eq.~(\ref{Eq.1.6}), plays no role in the evolution of the magnetic variables
$\textbf{B}_{l+1}$ and $\textbf{J}_{l+1}$, which means that the magnetic
``\textit{sector}'' is decoupled from the charge density profile because the
coupling term (charge recombination) has disappeared. In this sense, notice that
the local profile $\textbf{B}_{l+1}$ can be solved in terms of the
previous field distribution $\textbf{B}_{l}$ and the boundary conditions at time
layer $(l+1)\delta t$.
\item As the initial conditions must fulfill the Ampere's
law $\nabla\times\textbf{B}_{l}=\mu_{0}\textbf{J}_{l}$ as well as
$\nabla\cdot\textbf{B}_{l}=0$ and $\nabla\cdot\textbf{J}_{l}=0$, only the
inductive component of $\textbf{E}$ (given by $\nabla\times {\bf E}_{\rm ind} =
- \dot {\bf B}$, $\nabla\cdot {\bf E}_{\rm ind} = 0$) determines the evolution
of $\bf B$ (Faraday's law). At this point, the conducting law in its inverse
formulation ${\bf E}({\bf J})$ seems show certain ambiguity, as far as
two different material laws related by ${\bf E}_2 ({\bf J}) = {\bf E}_1 ({\bf
J}) + \nabla\Phi ({\bf J})$ determine the same magnetic and current density
profiles. Going into some more detail, whereas for the complete Maxwell
equations  statement, the potential component of the electric field
($\nabla\times{\bf E}_{\rm pot} = 0$,
$\epsilon _0 \nabla\cdot {\bf E}_{\rm pot} = \rho$), is coupled to $\bf B$ and
${\bf E}_{\rm ind}$ through the $\dot {\bf D}$
term (which contains both inductive and potential parts), within the MQS limit
it is irrelevant for the magnetic quantities. In
fact, one is enabled to include the presence of charge densities without
contradiction with the condition $\nabla\cdot {\bf J} \simeq 0$ by means of
inhomogeneity or nonlinearity in the $\textbf{E}(\textbf{J})$ relation. Then
one has that the condition $\nabla\cdot\textbf{J}=0$ does not imply
$\nabla\cdot\textbf{E}=0$. The charge density ${\rho}$ can be
understood as a parametrized charge of {\it static} character as far as $\dot
{\rho}$ is
neglected. As indicated above, once the magnetic variables are computed, one
has the freedom to modify the ``\textit{electrostatic sector}''  if necessary
by the rule ${\bf E}({\bf J})+\nabla\Phi$ while still maintaining the values
of $\bf B$ and $\bf J$. This invariance can be of practical interest as far as
the ``electrostatic'' behavior in the critical state regime is still under
discussion, it because of the inherent difficulties in
the direct measurement of transient charge
densities~\cite{P1-Ruiz_SUST_2011,P1-Joos_2006,P1-Clem_2011_PRB,Clem_2011_SUST}.
\end{enumerate}

\chapter{\label{ch-2} \sc \textbf{Variational Theory for Critical State
Problems}}

\subsection*{\label{ch-2-1}
\hspace*{0.1cm} \textsl{2.1 General Principles Of The Variational
Method}}
\vspace*{1cm}
\markboth{\hspace*{0.1cm}\textsl{2.1 General Principles Of The Variational
Method}}
{\hspace*{0.1cm} \textsl{2.1 General Principles Of The Variational
Method}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{2.1 General Principles Of The Variational
Method}}

As we have mentioned before, the numerical solution of the critical state
problem from the differential formalism of the Maxwell equations may be
cumbersome. One
possibility for making the resolution of this system affordable is to find an
equivalent variational statement of Eq.~(\ref{Eq.1.6}). Then, one can avoid the
integration of these set of differential equations by \textit{inversion} of a
set of Euler-Lagrange equations
\begin{eqnarray}\label{Eq.2.1}
\mu_{0}\textbf{J}_{l+1}-\nabla\times\textbf{B}_{l+1}=0\, ,
\end{eqnarray}
and
\begin{eqnarray}\label{Eq.2.2}
\mu_{0}\nabla\times\textbf{p}_{l}+\textbf{B}_{l+1}-\textbf{B}_{l}=0\, ,
\end{eqnarray}
for arbitrary variations of the Lagrange multiplier
(i.e., $\delta\textbf{p}_{l}$), and the time-discretized local magnetic
induction field (i.e., $\delta\textbf{B}_{l+1}$).\footnotemark[1] Eventually,
the Lagrange multiplier, $\textbf{p}_{l}$, will be basically identified with the
electric field of the problem.
\footnotetext[1]{Recall that the magnetic field \textbf{H} is defined as
a modification of the induction field \textbf{B} due to magnetic fields produced
by material media. However, as in the critical state regime the use of the
linear relation $\textbf{B}=\mu_{0}\textbf{H}$ is allowed, henceforth, we
will refer to \textit{magnetic field} where either or both fields apply.}

Going into more detail, let us consider a small path step $\delta t$, from some
initial profile of the magnetic field $\textbf{B}_{l}(\textbf{r})$ to a
final profile $\textbf{B}_{l+1}(\textbf{r})$, and the
corresponding $\textbf{J}_{l}(\textbf{r})$ and $\textbf{J}_{l+1}(\textbf{r})$.
Defining $\Delta\textbf{B}=\textbf{B}_{l+1}-\textbf{B}_{l}$, both
configurations can be considered to be connected by a steady process
performing a small linear step, such that
$\textbf{B}_{l+1}=\textbf{B}_{l}+s\Delta\textbf{B}$ with $s\in[0,1]\delta t$.
Recalling that the initial condition fulfills Ampere's law
$\nabla\times\textbf{B}_{l}=\mu_{0}\textbf{J}_{l}$, as well as
$\nabla\cdot\textbf{B}_{l}=0$ and $\nabla\cdot\textbf{J}_{l}=0$, the
time-averaged Lagrange density (over whole space) is
\begin{eqnarray}\label{Eq.2.3}
{\cal L}
=\frac{1}{2}|\Delta\textbf{B}|^{2}+\textbf{E}\cdot(\nabla\times\textbf{B}_ { l+1
}
-\textbf{J}_{l+1})\delta t \, .
\end{eqnarray}
Thus, the physically admissible Lagrangian multipliers in the critical state
regime must satisfy the condition
\begin{eqnarray}\label{Eq.2.4}
\textbf{p}=\textbf{E}_{cs}\delta t  \, ,
\end{eqnarray}
where the critical state electric field $(\textbf{E}_{cs})$ must be properly
defined by the imposed material law $\textbf{E}(\textbf{J})$.

However, concerning the  ``{\it unknown parameter}'' $\textbf{J}_{l+1}$, as far
as it is not allowed to take arbitrary values, we cannot impose
arbitrary variations as it is customary for the typical steady condition
of the Euler-Lagrange equations. Instead, an Optimal Control-like Maximum
principle equivalent to a maximal projection rule
$\textbf{\^{E}}\cdot\textbf{J}$ must be
used (see Refs.~\cite{P1-Badia_PRL_2001,P1-Badia_PRB_1998}). For a
more comprehensive review
of
the optimal control theory which can be understood as a generalization of the
variational calculus, the interested reader is directed to see, for instance,
Refs.~\cite{P1-Pontryagin,P1-Leitmann,P1-Knowles}.

In simple terms, the optimal control concept introduces a geometrical picture of
the material law for the boundary conditions of the vector $\textbf{J}$ that may
be of much help when discussing the idea of a general
critical state theory.  Summarizing, it is necessary to declare that
there must be a region $\Delta_{\textbf{r}}$ within the \textbf{J}
space (possibly oriented according to the local magnetic field $\textbf{\^{B}}$,
and/or also depending on $|\textbf{B}|$ and \textbf{r}) such that nondissipative
current flow occurs when the condition $\textbf{J}\in\Delta_{\textbf{r}}$ is
verified. Thus, the minimum of the Lagrangian
must be sought within the set of current density vectors fulfilling
${\bf J}\in\Delta_{\textbf{r}}$, i.e.: $\textbf{J}_{l+1}$ is determined by
the condition
\begin{eqnarray}\label{Eq.2.5}
{\rm Min}\{ {\cal L} \}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}\equiv{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\, .
\end{eqnarray}

Notice that an $\textbf{E}(\textbf{J})$ law is needed in addition to
Eq.~\ref{Eq.2.3}. Thus, together with the concept of a very high dissipation
when \textbf{J} is driven outside $\Delta_{\textbf{r}}$ by some nonvanishing
electric field, Eq.~\ref{Eq.2.5} suffices to determine the relation between
the directions of \textbf{J} and \textbf{E}. Notice that the maximal shielding
condition is equivalent to the maximum projection rule, it means that the
orthogonality condition of the electric field direction with the surface of
$\Delta_{\textbf{r}}$ is recalled, and the Lagrange multiplier can be
straightforwardly identified with the electric field of the problem, i.e.,
\begin{eqnarray}\label{Eq.2.6}
{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{r}}}\equiv{\rm
Max}\{\textbf{E}\cdot\textbf{J}\}|_{\textbf{J}\in\Delta_{\textbf{r}}} \, .
\end{eqnarray}

Notice also that Ampere's law is imposed [Eq.~(\ref{Eq.2.1})] through the
Lagrange multiplier, while the discretized version of Faraday's law
[Eq.~(\ref{Eq.2.2})] is derived as an Euler-Lagrange equation for the
variational problem, so that absolute consistency with the Maxwell equations is
obtained. In fact, maximal global (integral) shielding is achieved through a
maximal local shielding rule [Eq.~(\ref{Eq.2.6})] that reproduces the elementary
evolution of $\partial_{t}{\bf J}$ for a perfect conductor with restricted
currents. Thus, in practice, if one explicitly introduces Ampere's
law $(\nabla \textbf{B}_{l+1}=\mu_{0}\textbf{J}_{l+1})$, minimization is made in
terms of
\begin{eqnarray}\label{Eq.2.7}
{\cal F}[\textbf{J}_{l+1}]={\rm
Min}\left\{ -\int_{\Re^{3}}\textbf{E}\cdot\textbf{J}_{l+1} \right\} \, ,
\end{eqnarray}
and the minimum is sought over $\textbf{J}_{l+1}\in\Delta_{r}$ for a fixed
$\textbf{E}$. However, as we have mentioned before, special attention must be
payed to the feasible ambiguity of the function $\textbf{E}(\textbf{J})$ as it
can lead to fake values of the variables $\textbf{J}$.

Likewise, the straightforward equivalence between the convex functionals for
Eqs.~(\ref{Eq.2.5}) and (\ref{Eq.2.7}) allows to establish an equivalent
minimization principle in terms of the general definitions
\begin{eqnarray}\label{Eq.2.8}
\textbf{B}=\nabla\times\textbf{A} \, ,
\end{eqnarray}
and
\begin{eqnarray}\label{Eq.2.9}
\textbf{E}=-\partial_{t}\textbf{A}-\nabla\varPhi \, ,
\end{eqnarray}
by imposing the material law $\textbf{E}(\textbf{J})$ through the Lagrange
multiplier $\textbf{p}_{l+1}$. Thus, the minimization problem turns to find out
the invariant gauge conditions $\nabla\varPhi_{l+1}$ and
$\textbf{J}_{l+1}\in\Delta_{\textbf{r}}$ for a given function
$\textbf{A}[\textbf{J}]$, in such manner that
\begin{eqnarray}\label{Eq.2.10}
{\cal F}[\textbf{B}_{l+1},\nabla \varPhi]
={\rm Min}\int_{\Re^{3}}&&
\frac{1}{2}|\Delta\textbf{B}|^{2}-\Delta\textbf{A}\cdot(\nabla\times\textbf{B}_{
l+1}-\textbf{J}_{l+1})\\
&&-\nabla\varPhi(\nabla\times\textbf{B}_{l+1}-\textbf{J}_{l
+1})\delta t \, . \nonumber
\end{eqnarray}

We call the readers' attention to notice that the uncoupling of the
electromagnetic potentials can be accomplished
by exploiting the arbitrariness involved in the definition of $\textbf{A}$. In
fact, since \textbf{B} is defined through Eq.~(\ref{Eq.2.8}) in terms of
\textbf{A}, the vector potential is arbitrary to the extent that the gradient of
some scalar function can be added. Therefore, the ``\textit{magnetic sector}''
could be decoupled of the ``\textit{electric sector}'' if the physical
admissible states in the time-averaged Lagrange density $L$ are invariant gauge
of the Lagrange multipliers \textbf{p}. As a consequence, if the problem is such
that there are no intrinsic electromagnetic sources, $\varPhi\equiv0$ (for
type-II superconductors it means absence of transport current), a proper choice
of \textbf{A} should satisfy the Coulomb gauge $(\nabla\cdot\textbf{A}\equiv0)$.
In this sense, by using the Laplace equation, the second term in
Eq.~(\ref{Eq.2.10}) is reduced to
$\Delta\textbf{A}\cdot\delta_{t}^{2}\textbf{A}$ meanwhile the third term
have vanished by assuming $\varPhi\equiv0$. Then, as the MQS approximation
relies in assume that the electric field quickly adjusts to a constant value
along the interval $[t+\delta t]$, for enough small time steps $\delta t$ (see
Fig.~\ref{Figure_1_1}) the action of
$\Delta\textbf{A}\cdot\delta_{t}^{2}\textbf{A}$ may be neglected, and therefore
the solution of the critical state problem can be also achieved from the
functional for the magnetic sector:
\begin{eqnarray}\label{Eq.2.11}
{\cal F}[\textbf{B}(\cdot)]={\rm Min}\int_{\Re^{3}}
\frac{1}{2}|\Delta\textbf{B}|^{2} \, .
\end{eqnarray}
Recall that, the minimization principle is based on a discretization of the path
followed by the external sources, meaning that it is an approximation to the
continuous evolution whose accuracy increases as the step diminishes.

Moreover, we must emphasize that the derived functionals
[Eqs.~(\ref{Eq.2.7}) \& (\ref{Eq.2.11})] are in matter of fact fully
equivalents, as long as the minimization procedure accomplishes the boundary
conditions imposed by the prescribed sources and the material law
$\textbf{J}\in\Delta_{\textbf{r}}$. Thus, in those cases when an intrinsic
electromagnetic source must be considered, i.e., $\nabla\varPhi\neq0$, the
global set of variables must me constrained by the prescribed conditions. For
example, if the superconductor is carrying a transport
current $I_{tr}$ flowing through the surface $s$, one has to mandatorily
consider the external constraint
\begin{eqnarray}\label{Eq.2.12}
\int_{s}\textbf{J}\cdot\hat{\textbf{n}}ds=I_{tr} \,
\end{eqnarray}
and further update the distribution of current to satisfy the physical condition
$\textbf{E}\cdot\textbf{J}=0$ (at those points where the magnetic flux does not
vary), by means the use of a \textit{calibrated} potential
$\textbf{\~{A}}$. Thus, one of the advantage
of the formulation in Eq.~(\ref{Eq.2.11}) is that the number of variables can be
reduced avoiding to include the intrinsic variables associated to $\varPhi$,
accordingly to the statement
$\textbf{E}=-\partial_{t}\textbf{A}-\nabla\varPhi\equiv\partial_{t}
\textbf{\~{A}}$

Concluding, for 3D problems, it must be emphasized that the introduced
minimization principle can be applied for any shape of the
superconducting volume $\Omega$ as well as for any general restriction (material
law) for the current density $\textbf{J}_{l+1}\in\Delta_{\textbf{r}}$.
Different possibilities for the material law are described in the following
chapter. It is
also to be noticed that the searching of the minimum for the allowed set of
current densities must fulfill the intrinsic condition
$\nabla\cdot\textbf{J}=0$ to be consistent with charge conservation
in the quasi-steady regime. Further, from the numerical point of view, the
advantage
of the variational formulation in  Eqs.~(\ref{Eq.2.7})~\&~(\ref{Eq.2.11}) is
that
one can avoid the integration of the equivalent partial differential equations
and straightforwardly minimize the discretized integral by using a numerical
algorithm for constrained minimization (see Chapter~\ref{ch-3}). This fact
represents an
important advantage in the performance and power of the numerical methods
applied to the design of superconducting devices, where symmetry arguments can
allow further simplifications and correspondingly faster numerical convergence.

\vspace*{1cm}
\subsection*{\label{ch-2-2}
\hspace*{0.1cm} \textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}
\vspace*{1cm}
\markboth{\hspace*{0.1cm}\textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}
{\hspace*{0.1cm} \textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}
\addcontentsline{toc}{chapter}
{\hspace*{0.1cm} \textsl{2.2 The Material Law: SCs with
magnetic anisotropy}}

In this section, we will continue our discussion of the critical state theory
which still needs the explicit inclusion of a material law
$\textbf{J}(\textbf{E})$ that dictates the magnetic response of a
superconducting sample for a given external excitation. For simplicity, we start
with an overview of the  material law for 1D systems, that will be gradually
generalized until a 3D formulation is reached.

\vspace*{0.5cm}
\hspace*{-0.3cm}
\textbf{\textit{2.2.1 ~Onto The 1D Critical States}}
\markboth{\hspace*{0.1cm} \textsl{2.2.1 ~1D Critical States}}{\hspace*{0.1cm}
\textsl{2.2.1 ~Onto The 1D Critical States}}
\addcontentsline{toc}{chapter}{\hspace*{0.1cm} \textsl{2.2.1 ~Onto the 1D
Critical States}}\label{ch-2.2.1}
\vspace*{0.5cm}

For our purposes, it is sufficient to recall that the basic structure of the
critical state problem (Fig.~\ref{Figure_1_1}) relates to an experimental graph
within the $\{V , I \}$ plane that basically contains two regions defined by
the critical current value $I_{c}$ as follows:
\begin{enumerate}
\item $-I_c \leq I \leq I_c$ with perfect conducting behavior, i.e.:
 $V=0$ and $\partial _t I = 0$.
\item For $I \gtrsim I_c$, the curve is characterized by a high
${\partial_{I} V}$ slope (and antisymmetric for $I \lesssim - I_c$). Further
steps, with $I$ increasing above the critical value $I_c$, i.e., the eventual
transition to the normal state, may be neglected for slow sweep rates of the
external sources, which produce moderate electric fields.
\end{enumerate}

Within the local description of the electromagnetic quantities involved in the
superconducting response, different models have been used for the
corresponding $E \leftrightarrow J$ graph, the most popular being

\break
\begin{enumerate}
\item The {\em power law} model~\footnotemark[2]\\
$E=\alpha\;{\rm sgn}(J)\left({|J|}/{J_c}\right) ^n$, with $\alpha$ a constant
and $n$ high.
\item The {\em piecewise continuous linear} approximation~\footnotemark[2]\\
$E=0$ for $|J|\leq J_c$, and $E=\beta\;{\rm sgn}(J)(|J|-J_c)$ for $|J|> J_c$,
$\beta$ having a high value.
\item {\em Bean's model}~\footnotemark[3]\\
$J$ constant for $E=0$, and $J={\rm
sgn}(E)J_c$ for $E\neq 0$.
\footnotetext[2]{Is to be noted that these models (1 \& 2) present a small
dependence on the sweep rate, as far as different values of $E$ give way to a
slightly different $J$. The feasibility of the selected model can only be
justified by agreement of the free parameters with the experimental results.}
\footnotetext[3]{This is the simplest model, without sweep rate dependence
because only the sign of $E$ enters the theory (see Fig.~\ref{Figure_1_1}).}
\end{enumerate}

In some treatments, the first or second models are
implemented, in order to transfer a full ${\bf E}({\bf J})$ law to the Maxwell
equations. Further, notice that Bean's model may be obtained from
the other representations through the limiting cases $n\to\infty$ and
$\beta\to\infty$ respectively.

The well known experimental evidence of a practical sweep rate independence for
magnetic moment measurements (unless for high frequency alternating sources or
at elevated temperatures) allows the use of the clearest Bean's model because
the critical state problem is no longer a time-dependent problem, but a
path-dependent one, meaning that the trajectory of the external sources
${\bf H}_{0}$ uniquely determines the magnetic evolution of the
sample. This makes an important difference when one compares to more
standard treatments, as far as Faraday's law is not completely determined from
the path~\cite{P1-Ruiz_PRB_2009}. Strictly speaking, one has
\begin{eqnarray}\label{Eq.2.13}
\Delta{\bf B} = - \nabla\times [{\bf E}\Delta t] \,\, ,
\end{eqnarray}
with $\Delta t$ (and therefore $|{\bf E}|$) depending on the external sources.
i.e., the absence of an intrinsic time constant gives way to the
arbitrariness in the time scale of the problem.

Furthermore, in the actual 1D applications of Beans's model,
Faraday's law is not strictly solved and \textbf{E} is absent from the theory.
It is just the sign rule (the {\it
vectorial} part of the material law), that is used to integrate Ampere's law.
Notice that such sign rule corresponds to a maximal shielding response against
magnetic vector variations, and thus, determines the selection of $J= \pm J_c$.

Regarding the direction of \textbf{E}, in ``1D'' problems one has ${\bf
J}\parallel {\bf E}$ and both orthogonal to $\bf B$, such that the physical
threshold related to a maximum value of the force balancing the magnetostatic
term ${\bf J}\times{\bf B}$ gives place to the material law
\begin{eqnarray}\label{Eq.2.14}
J_{\perp}={\rm sgn}(E_{\perp})J_{c\perp} \qquad{\rm for}\qquad E_{\perp}\neq
0\,\, .
\end{eqnarray}
Here, $E_{\perp}$ stands for the component of ${\bf E}$ along the direction
${\bf B}\times ({\bf J}\times{\bf B})$, and the material law falls in a ``1D''
scalar condition which describes the physical mechanism of vortex depinning.

At this point, the constitutive relation for the critical state describes the
underlying physics for the coarse-grained fields in homogeneous type-II
superconductors. However, it is well known that the coarse-grained behavior
approach straightforwardly depends on the manufacturing process of the
superconducting sample as far as inclusion of impurities, magnetic defects,
or deformation of their cristal structure imposes the local coupling of $J_{c}$
with the intrinsic variation of the magnetic field B. Thus, for practical
purposes we emphasize that the theoretical framework developed in this book is
fully general, with caution of suggest to the experimentalist the need of an
apriori measurement of the dependence $\textbf{J}_{c}[\textbf{r},\textbf{B},T]$
at least in those cases where the condition $\textbf{J}\perp \textbf{B}$ can be
asserted. Henceforth, the
implementation for a particular superconductor can be carried out.

\vspace*{0.5cm}
\hspace*{-0.3cm}
\textbf{\textit{2.2.2 ~Towards The 3D Critical States}}
\markboth{\hspace*{0.1cm} \textsl{2.2.2 ~Towards The 3D Critical
States}}{\hspace*{0.1cm} \textsl{2.2.2 ~Towards The 3D Critical States}}
\addcontentsline{toc}{chapter}{\hspace*{0.1cm} \textsl{2.2.2 ~Towards The 3D
Critical States}}\label{ch-2.2.2}
\vspace*{0.5cm}

In the case of superconductors with anisotropy of the critical current, the
description of their magnetic behavior requires the development of approaches
more sophisticated than 1D-Bean's model. The main issue is that, in general, the
parallelism of ${\bf E}$ and ${\bf J}$ and their perpendicularity to ${\bf B}$
are no longer warranted. Then, the {\em sign rule} of Eq.~(\ref{Eq.2.14})
does not suffice for determining the solution of Eq.~(\ref{Eq.2.6}), and the
optimal control condition with $\textbf{J}\in\Delta_{\textbf{r}}$
({\em a vectorial rule}) must be invoked.

The simplest assumption that translates the critical state problem to 3D
situations was already issued by Bean
in Ref.~\cite{P1-Bean_1970}. It has been called the \textbf{\textit{isotropic
critical state model}} (ICSM) and generalizes 1D Bean's law to
\begin{equation}\label{Eq.2.15}
{\bf J} =J_{c}\,\hat {\bf E}\quad{\rm if} \quad E \neq 0 \, ,
\end{equation}
i.e., the region $\Delta_{\textbf{r}}$ becomes a sphere. Noticeably, in spite
of lacking a solid physical basis, thanks to its mathematical simplicity, this
qualitative model has been widely used by several authors for reproducing a
number of experiments with rotating and crossed magnetic
fields~\cite{P1-Badia_PRL_2001,P1-Badia_PRB_2002,P1-ICSM_app}. In any case,
one could argue that statistical averaging over a system of entangled flux lines
within a random pinning structure might be responsible for the isotropization of
$\Delta_{\textbf{r}}$.

On the other hand, the general statement of the critical state in
terms of well accepted physical basis was firstly introduced by John R.
Clem~\cite{P1-Clem_DCSM}, and it is currently known as the
\textbf{\textit{double
critical
state model}} (DCSM). In particular, this theory
assumes two different critical parameters, $J_{c\parallel}$ and $J_{c\perp}$
acting as the thresholds for the components of ${\bf J}$ parallel and
perpendicular to ${\bf B}$ respectively. Notice
that, $J_{c\perp}$ relates to the flux depinning threshold induced by the
Lorentz force on flux tubes ($\textbf{J}\times\textbf{B}$),
while the additional $J_{c\parallel}$ is imposed by a maximum gradient in the
angle between adjacent vortices
($\textbf{B}\cdot\nabla\gamma=J_{\parallel}$) before mutual cutting
and recombination occurs  [see Fig.~\ref{Figure_2_1}~(a)]. In brief, the
DCSM may be expressed by the statement
\begin{eqnarray}\label{Eq.2.16}
\left\{
\begin{array}{ll}
{\bf J}_{\parallel} & =J_{c\parallel}\;\hat {\bf u}
\\
{\bf J}_{\perp} & =J_{c\perp}\,\hat {\bf v}
\end{array}
\right. \, ,
\end{eqnarray}
being $\textbf{\^{u}}$ the unit vector for the direction
of \textbf{B}, and $\textbf{\^{v}}$ a unit vector in the perpendicular plane
to \textbf{B}.

Within the DCSM, the region $\Delta_{\textbf{r}}$ is a cylinder with its axis
parallel to $\bf B$, and a rectangular longitudinal section in the plane defined
by the unit vectors $\hat{\bf B},\hat{\bf J}_{\perp}$ [see
Fig.~ \ref{Figure_2_1}~(b)]. The edges of the region $\Delta_{\textbf{r}}$
introduce a criterion for classifying the CS configurations into:
\begin{enumerate}
\item T zones
or flux transport zones ($J_{\perp}=J_{c\perp}$;
$J_{\parallel}<J_{c\parallel}$) where the
flux depinning threshold has been reached (${\bf J}$ belongs to the
horizontal sides of the rectangle),
\item C-zones or flux cutting zones
($J_{\parallel}= J_{c \parallel}~;~J_{\perp}<J_{c \perp}$) where the
cutting threshold has been reached (${\bf J}$ belongs to the vertical sides
of the rectangle),
\item CT zones ($J_{\parallel} = J_{c \parallel}$ and
$J_{\perp} = J_{c \perp}$) where both $J_{\parallel}$
and $J_{\perp}$ have reached their critical values (corners of the
rectangle), and
\item O zones depicted the regions without energy dissipation (the current
density vector belongs to the interior of the rectangle).
\end{enumerate}

\begin{figure}[t]
\centering
\includegraphics[width=1.0\textwidth]{Figure_2_1.pdf}
\caption{\label{Figure_2_1} {\it Pane}~(a), Top: Schematic representation of the
local relative
orientations of {\bf B} and {\bf J}. Also sketched is the direction of
the magnetic field at some neighboring point, at an angle $\gamma$.
The vectors ${\bf B}$, ${\bf B}'$ and ${\bf J}$ do not
necessarily lie at the same plane. The current is decomposed into its parallel
and perpendicular components, i.e.: ${\bf J}={\bf
J}_{\parallel}+{\bf J}_{\perp}$.
Bottom: the {\em perfect conducting} region within the plane perpendicular to
$\textbf{B}$. An induced electric field is shown. Initially (${\bf
J}_{0}$), the high dissipation region is touched, but almost instantaneously
{\bf J} shifts along the boundary, reaching a point where the condition ${\bf
E}\perp\partial\Delta_{\textbf{r}}$ is fulfilled. Anisotropy within the plane is
allowed. {\it Pane}~(b): Geometric interpretation of the DCSM. {\bf J} is
constrained to the
boundary of a rectangular region. T, C and CT states are related to
the horizontal and vertical sides, and to the corners. Coupling between
the components $J_{c\parallel}$ and $J_{c\perp}$ is envisaged by the EDCSM
(dotted red ellipse). {\it Pane}~(c): Our generalization of the material law for
critical
state problems or SDCST. Several regions are shown from the degree of the
superelliptical functions ($n=1,2,3,4,6,10,20,40,\infty$) and
$\chi=J_{c\parallel}/J_{c\perp}=1$.}
\end{figure}

Notice that $J_{c\parallel}$ and $J_{c\perp}$ are determined from different
physical phenomena, and their values may be very different (in general
$J_{c\parallel}>J_{c\perp}$ or even $J_{c\parallel}\gg J_{c\perp}$).
Nevertheless, the coupling of parallel and perpendicular effects has been longer
recognized by the experiments~\cite{Clem_2011_SUST,P1-Boyer_80} and, for
instance,
may be included in the
theory by the condition $J_{c\parallel}=K B J_{c\perp}$ with $K$ a
material dependent constant.

Recalling that the mesoscopic parameters $\textbf{J}_{c}$
are related to averages over the flux line lattice, interacting activation
barriers for the mechanisms of flux depinning and cutting are expected and this
may give place to deformations of the boundary $\partial\Delta_{\textbf{r}}$
[see Fig.~ \ref{Figure_2_1}(b)]. Thus, validated in those cases where a good
agreement with the experiments is achieved, the theoretical scenario can be
enlarged by a number of alternative approaches that focus on different aspects
of the vast number of experimental activities in this field, e.g. one
can identify the so-called:

\begin{enumerate}
\item \textit{Isotropic critical state models
(ICSM)}~\cite{P1-Ruiz_PRB_2009,P1-Ruiz_SUST_2010,P1-ICSM_app}\\
\hspace*{3.4cm} $J^{2}=J_{\parallel}^2+J_{\perp}^2 \leq J_{c}^2$
 \item \textit{Elliptical double critical state
models (EDCSM)}~\cite{P1-Ruiz_PRB_2009,P1-EDCSM_app}\\
\hspace*{3.26cm}
$J_{\parallel}^2/J_{c\parallel}^2+J_{\perp}^2/J_{c\perp}^2\leq1$
 \item \textit{T critical state model
(TCSM)}~\cite{P1-Ruiz_PRB_2009,P1-Ruiz_SUST_2010,P1-Ruiz_SUST_2011,
P1-Brandt_2007,
P1-Ruiz_PRB_2011}\\
\hspace*{2.87cm} $J_{\parallel}$ unbounded $\forall$ $J_{\perp}\leq J_{c\perp}$
.
\end{enumerate}

Remarkably, the whole set of models have been recently unified by us in
Ref.~\cite{P1-Ruiz_SUST_2010} within a continuous two-parameter theory that
poses
the critical state problem in terms of geometrical concepts within the
$J_{\parallel}-J_{\perp}$ plane (see Fig.~ \ref{Figure_2_1}~(c)). To be
specific, in this framework, we have shown that by the application of our
variational statement~\cite{P1-Ruiz_PRB_2009}, one is able to specify almost any
critical state law by means of an integer index $n$, that accounts for the
smoothness of the $J_{\parallel}(J_{\perp})$ relation, and a certain {\em
bandwidth} characterizing the magnetic anisotropy ratio $\chi\equiv
J_{c\parallel}/J_{c\perp}$. This and the variational formalism introduced
above constitutes the so-called \textbf{\textit{Smooth Double
Critical State Theory}} (SDCST), which allows to elucidate the relation
between diverse physical processes and the actual material law.

Mathematically, the material law introduced in our general theory for the
critical state problem or SDCST is based upon the idea that either material or
extrinsic anisotropy can be easily incorporated by prescribing a region
$\Delta_{\textbf{r}}$ where the physically admissible states of $\textbf{J}$ are
hosted as limiting cases of a smooth expression defined
by the two-parameter family of superelliptic functions,
\begin{equation}\label{Eq.2.17}
\left(\frac{J_{\parallel}}{J_{c\parallel}}\right)^{2n}+\left(\frac{J_{\perp}}{J_
{c\perp}}\right)^{2n}\leq 1.
\end{equation}

We call the readers' attention to the fact that an index $n=1$ and a bandwidth
defined by $\chi\equiv J_{c \parallel}/J_{c \perp}=1$ correspond to the standard
ICSM~\cite{P1-ICSM_app}.
On the other hand, when one assumes enlarged bandwidth (i.e.: $\chi>1$), the
region $\Delta_{\textbf{r}}$ of the SDCST becomes the standard
EDCSM introduced by Romero-Salazar and
P\'erez-Rodr\'iguez~\cite{P1-EDCSM_app}.
When the bandwidth $\chi$ is extremely large, i.e., $J_{c\parallel}\gg J_{c
\perp}$, one recovers the so-called $T-$zones treated by Brandt and
Mikitik~\cite{P1-Brandt_2007}. Rectangular regions strictly corresponding to the
DCSM~\cite{P1-Clem_DCSM} are
obtained for the limit $n\rightarrow\infty$ and arbitrary $\chi$. Finally,
allowing $n$ to take values over the positive integers, a wide scenario
describing anisotropy effects is envisioned [Fig.~\ref{Figure_2_1}(c)]. Such
regions will be named after superelliptical and their properties can be
understood in terms of the rounding (or smoothing) of the corners for the DCSM.

\chapter{\label{ch-3} \sc \textbf{Computational Method}}

In chapter~2.1 we have mentioned that the minimization functionals
[Eq.~(\ref{Eq.2.7}) or Eq.~(\ref{Eq.2.11})] may be transformed so as to get a
practical vector potential formulation. In turn, the resulting formulation can
be expressed in terms of the so-called magnetic inductance matrices which
allows a clearest identification of the set of elements playing some role in
the minimization procedure. In this chapter, we shall discuss how to implement
the above
statements for general critical states in the
framework of the computational methods for large scale nonlinear optimization
problems.

Being more specific, in Eq.~(\ref{Eq.2.11}) the integrand $\frac{1}{2}
(\Delta{\bf B}) ^2$ can be rewritten as $\frac {1}{2} (\Delta{\bf B})\cdot
(\nabla \times \Delta{\bf A})$,
and manipulated to get $\frac {1}{2} (\Delta{\bf A})\cdot (\nabla \times
\Delta{\bf B})$ plus a divergence term, fixed by the external sources at a
distant surface. Now, the integral is restricted to the superconducting sample
volume $\Omega$, because $\nabla \times \Delta{\bf B} = \mu _0 \Delta{\bf J}$ is
only unknown within the superconductor. In addition, assuming that local
sources such as an injected transport current may be introduced as an external
constraint, and with the boundary condition that $\textbf{A}$ goes
to zero sufficiently fast as they approach infinity, the vector
potential can be expressed as:\footnotemark[1]
\footnotetext[1]{Recall that, \textbf{A} is determined by the Maxwell's
equations solution in the Lorenz gauge condition, i.e., the vector potential
must satisfy the condition $\partial_{\mu}A^{\mu}=0$ for any transformation
gauge $A^{\mu}\rightarrow A^{\mu}+\partial^{\mu}\psi$ with $\psi$ a scalar
function. Thus, as no local-sources are present into the minimization
principle of Eq.~\ref{Eq.2.11}, one is enabled to apriori assume $\psi=0$, and
thence simplify the vector potential in terms of the Coulomb gauge.}
\begin{equation}\label{Eq.3.1}
 \Delta{\bf A} = \Delta{\bf A}_{0}
+ \frac {\mu _0 }{4 \pi} \int _\Omega \frac {\Delta{\bf J}}{|{\bf r} - {\bf
r}'|}
d^3 {\bf r}' \, .
\end{equation}

This transforms ${\cal F}$ into a double integral over the body of the sample,
i.e.:
\begin{eqnarray}\label{Eq.3.2}
{\cal F}[\textbf{A}(\cdot),J\in\Delta_{\textbf{r}}]= &&\frac {8\pi}{\mu_0}\int
_\Omega \Delta{\bf A}_{0}\cdot {\bf J}_{l+1}({\bf r})d^3 {\bf r}
\nonumber\\&&
+\int\!\int _{\Omega \times \Omega}
\frac {{\bf J}_{l+1}({\bf r}')
\cdot [{\bf J}_{l+1}({\bf r}) - 2 {\bf J}_{l}({\bf r})]}{|{\bf r} - {\bf
r}'|}d^3 {\bf r}d^3 {\bf r}'\;
\end{eqnarray}

As a consequence, only the unknown current components within
the superconductor $(\textbf{J}_{l+1})$
appear in the computation so reducing the number of unknown variables. At this
point let me emphasize that Eq.~(\ref{Eq.3.2}) can be applied for any shape of
the superconducting volume $\Omega$ as well as for any physical constraint
(material law $\Delta_{\textbf{r}}$) for the local current density
$\textbf{J}_{l+1}$, and further for any condition defined by the external
sources ($\textbf{A}_{0}$). Above this, minimization must ensure
the charge conservation
condition by searching the minimum for the allowed set of current densities
fulfilling $\nabla\cdot\textbf{J}=0$.

On the other hand, also it may be noticed that the double
integral in Eq.~(\ref{Eq.3.2}) can be (\textit{eventually}) identified as the
Neumann formula once it has been transformed into filamentary closed circuits. A
noteworthy fact is that regarding the superconducting volume, the coefficients
of the intrinsic inductance matrices are straightforwardly independent of time
and consequently, they appear in the root problem before going to minimize the
functional. Indeed, the proper description of the inductance coefficients
directly depends on the geometry of the superconductor and the boundary
conditions defined by the dynamics of the external electromagnetic sources,
where any symmetry of the problem allows further simplifications and
correspondingly faster numerical convergence. To be specific, upon
discretization in current elements ($I_{i}=J_{i}s_{i}$), the minimization
functional for critical state problems bears
the algebraic structure
\begin{eqnarray}\label{Eq.3.3}
{\cal
F}[I_{l+1}]=\frac{1}{2}\sum_{i,j}I_{i,l+1}M_{ij}I_{j,l+1}-\sum_{i,j}I_{i,l}M_{
ij } I_ {j,l+1}+\sum_{i}I_{i,l+1}\Delta A_{0}(M_{0}) \, ,
\end{eqnarray}
with $\{I_{i,l+1}\}$ the set of unknown currents at specific
circuits for the problem of interest, $M_{ij}$ their intrinsic {\em inductance
coupling coefficients}, and $M_{0}$ the inductance matrices associated to the
external sources $A_{0}$.

Corresponding to the critical state rule ${\bf J}\in\Delta_{\textbf{r}}$, in
order to minimize Eq.~(\ref{Eq.3.3}) each value $J_i$ must
be constrained. Thus, as it was described in chapter~\ref{ch-2}.2.2, we have
found that a number of constraints related to physically meaningful critical
state models may be expressed in the algebraic form
\begin{eqnarray}\label{Eq.3.4}
{\rm
F}_{\alpha}\left(\sum_{i}I_{i}C_{ij}^{\alpha}I_{j}\right)\leq f_{0\alpha} \,
~\forall ~ j
\end{eqnarray}
with $f_{0}$ some constant representing the physical threshold, and
${\rm F}_{\alpha}(\cdot)$ an algebraic function based upon a coupling matrix
$C_{ij}^{\alpha}$ whose elements depend on the physical model.
For example, in the simpler cases (isotropic models), the constraints
correspond to assume the matricial elements $C_{ij}=\delta_{ij}$, and the
physical threshold $f_{0}={J}_{c}^{2}$.

For simplicity, most technical procedures related to the introduction of
intricate models and either depict the minimization functional in terms of the
inductance coefficients (including those for external sources) will be left as
matter of study of the following chapters (Part~\ref{Part_2}). In return, below
we present a thorough analysis of the computational tools handled for critical
state problems at large scale.

With the purpose of obtaining a minimal understanding about how a critical state
problem can be tackled from the numerical point of view, the computational
method is sketched in the flow charts of
figures~\ref{Figure_3_1}~\&~\ref{Figure_3_2}.

\begin{figure}[t]
\centering
\includegraphics[height=13cm,width=13cm]{Figure_3_1.pdf}
\caption{\label{Figure_3_1} Flow chart describing  the preparation and
management of the input elements for the objective function.}
\end{figure}

The first step is designing a grid which will allows to describe the
superconducting volume $\Omega$ as a set of elements
$\delta\Omega_{i}$, each of them characterized by a well defined current
density flowing along the coordinates $\textbf{r}_{i}$. Then, the matrices for
the intrinsic inductance coefficients between the elements
$\textbf{J}(\textbf{r}_{i})$ and $\textbf{J}(\textbf{r}_{j})$ for all the set
of possible couples $(\textbf{r}_{i},\textbf{r}_{j})\in\Omega$ must be
calculated and stored on disk. As the mesh of points
$(\textbf{r}_{i},\textbf{r}_{j})$ can be considerably large, we suggest take
advantage on the matricial formalism provided by Matlab$^{\textregistered}$
and
their own language for storage data. Once the spatial elements playing some role
into the functional have been properly defined, the temporal \textit{sector}
must be introduced by means enough small path steps of the external
electromagnetic sources, i.e., the experimental conditions must be connected by
the finite difference expressions such a
$\Delta\textbf{B}_{0}=\textbf{B}_{l+1}-\textbf{B}_{l}$, where the
associated distribution of currents $\textbf{I}_{l+1}$ plays the role of
unknown. To be specific, in those cases where the superconductor is subjected
to an external magnetic field $\textbf{B}_{0}$, additional inductance matrices
($M_{0}$)
must be introduced according to the definition
\begin{equation}\label{Eq.3.5}
 \textbf{A}_{0}
(\textbf{B}_{0},\textbf{r}_{j})=\textbf{B}_{0}\times\textbf{r}_{ j } \, .
\end{equation}
On the other side, the vector potential $\textbf{A}_{0}$ not only allows to
define the contribution at the local potential $\textbf{A}$ produced by an
external magnetic field ($\textbf{A}_{{\rm B}}$), rather it also allows
consider the coupling with another materials such as ferromagnets.

Before going within the minimization procedure, it has to be noticed that those
cases
considering a transport current along the superconducting sample must
be understood as a problem where the minimization variables are required to
satisfy a set of auxiliary constraints [see Eq.~(\ref{Eq.2.12})] under the
global critical state condition $I_{tr}\leq I_{c}$. Also, as the
values for the elements $\textbf{I}_{l}(\textbf{r})$ are
assumed to be known in advance, the linear elements into the argument of the
functional (objective function) can be calculated before minimizing.

At this point, it is probably worthwhile to argue on what we mean by the
computational method for minimization of an objective function. Firstly, this
notion is clearly computer dependent, as the size of large scale problems can
require a substantial amount of memory and store. Moreover, what is large in a
personal computer can be significantly different from what is large on a super
computer. The  first machine just to have a smaller memory and storage than the
second one, and therefore has more difficulty handling problems involving a
large amount of data. Secondly, the size of the objective function strongly
depends on the structure and the mathematical formulation of the problem
and exploiting it is often crucial if one wants to obtain an answer
efficiently. The complexity of this structure is often a central key in
assessing the size of a problem\footnotemark[2]. For example, for linear
objective functions
(not our case) it is possible to solve pretty large size problems (say four
million variables). However, the objective function for problems in applied
superconductivity is in general highly nonlinear and, for instance, the
quadratic terms suggest to reduce the number of variables in a root square
factor (say two thousand variables). One advisable possibility for reducing the
number of elements in the objective function is subdividing the problem into
loosely connected subsystems, i.e., all the internal operations which do not
depend of the minimization variables must
be preallocated to a well structured data (Figure.~\ref{Figure_3_1}).
Lastly, an efficient algorithm for nonlinear optimization problems
must be either invoked or built. Fortunately, nowadays there is a significant
amount of available software with standard optimization tools which allow a
faster foray in this matter~\cite{P1-Matlab,P1-Lancelot}.
\footnotetext[2]{Customarily memory access violations (segmentation faults)
appear on nonlinear large systems without a well designed structure.}

We must call reader's attention on the fact that efficient algorithms for
small-scale problems (in the sense that, assuming infinite precision,
quasi-Newton methods for unconstrained optimization are invariant under linear
transformations) do not necessarily translate into efficient algorithms
for large scale problems. Perhaps the main reason is that, in order to be able
to handle large problems with a high accuracy, the structure of the objective
function and the minimization algorithms have both of them to be enough simple
and tractable to avoid a wasting of time in the scaling of variables for the
inner iteration subproblem (the minimization itself) and the finding of an
optimum value for each one of the variables with respect to the remaining
variables sought. In this context, one  of the most powerful algorithms for
large and nonlinear constrained optimization problems, known as
LANCELOT, has been developed by the professors Andrew Conn (IBM corporation,
USA), Nick Gould (Rutherford Appleton Laboratory, UK), Philippe Toint
(Facult\'es Universitaries Notre-dame de la Paix, Belgium), and Dominique Orban
(Ecole Polytechnique de Montreal, Canada)~\cite{P1-Lancelot}. The wide number of
optimization techniques provided by this package and their flexibility in
handle and storage of large amounts of variables, make this program a
clever choice for tackle highly complicated systems as those described by
Eq.~(\ref{Eq.3.2}).

\begin{figure}[t]
\centering
\includegraphics[height=13cm,width=13cm]{Figure_3_2.pdf}
\caption{\label{Figure_3_2} Flow chart describing the main structure of the
computational method implemented along this book.}
\end{figure}

A thorough study of the minimization techniques and the computational language
allocated in this package is far away of the purpose of this thesis. However,
the structure of a general problem can be understood via the flow chart in
figure~\ref{Figure_3_2}. In brief, the
minimization functional is translated into a suite of FORTRAN
procedures for minimizing an objective function, where the minimization
variables are required to satisfy a set of auxiliary constraints and possibly
internal bounds. Here, the major advantage of LANCELOT is the use of a Standard
Input Format (SIF) as a unified method for communicating numerical data and
FORTRAN subprograms with any optimization algorithm. Thus, when an optimization
problem (minimizing or maximizing a sought of variables) is specified in the SIF
decoder, one is required to write one or more files in ordered sections which
accomplish the role of introduce the set of preconditioners for the objective
function.

Once the set of input data has been structured accordingly to the number of
variables and further on the temporal dependence of the experimental conditions
(see Fig.~\ref{Figure_3_1}), one is enabled to predefine a set of input cards
allowing the knowledgeable user to specify a priori known limits on the possible
values of the objective function, as well as on the specific optimization
variables, accuracy parameters, and scaling factors (see Fig.~\ref{Figure_3_2}).
Then, the minimization functional or so-called objective function is
subdivided in a set of groups, whose purpose is twofold: On the one hand, the
linear and nonlinear (\textit{quadratic}) elements for the minimization
procedure are identified in a fore. Likewise, the specification of analytical
first derivatives is optional, but recommended whenever possible. The SIF
decoder allows also include the Hessian matrix of the objective function, if the
second-order partial derivatives of the whole set of minimization variables are
known,\footnotemark[3] otherwise the derivatives of the nonlinear element
functions can be approximated by some finite difference method.

Actually, the full Hessian matrix can be difficult to compute in practice; in
such situations, quasi-Newton algorithms\footnotemark[4] can be
straightforwardly called by LANCELOT where at least ten
different minimization algorithms have been already implemented and
coded according to the standard input format provided by the SIF
decoder~\cite{P1-Lancelot}. Notwithstanding, the solution obtained by LANCELOT
may be compromised if finite difference approximations are used, it has been our
experience that, once
understood, the programming language of SIF is in fact quite efficient for
problem specifications, in such manner that for objective functions
correctly written, and constraint functions well defined, any method can
efficiently reach to the solution sought. In this sense, additional groups may
be announced to make up the objective function by including an ``starting
point'' for the envisaged solution (if it is more or less known, or by defect
it is equals to zero) or, for introducing additional constraints (external
functions conditioning the system) as it is the case when the superconductor
implies a flow of transport current.

\footnotetext[3]{Given the real-valued function $f(x_{1},x_{2},...,x_{n}$, if
all second derivatives of $f$ exists, then the Hessian matrix of $f$ is the
matrix $H(f)_{ij}(x)=\partial_{i}\partial_{j} f(x)$. Hessian matrices are used
in large-scale optimization problems within Newton-type methods because they are
the coefficient of the quadratic term of the local Taylor expansion
$f(x+\Delta x)\approx f(x)+\breve{J}(x)\Delta x + \frac{1}{2}\Delta x^{~
\textbf{T}} H(x)\Delta x$, where $\breve{J}$ is the Jacobian matrix, which is a
vector (the gradient for scalar-valued functions)}

\footnotetext[4]{Also known as variable metric methods, are algorithms for
finding local maxima and minima of a function, with the aim of find out the
stationary point of their local Taylor expansion where the gradient is 0. Thus,
these methods are so called quasi-Newton methods, because the Hessian matrix
does not need to be a priori computed, as the Hessian is straightforwardly
updated by analyzing successive gradient vectors instead.}

It may happen that a specific problem uses variables or general constraints
whose numerical values are widely different in magnitude, causing
significant difficulties in the numerical convergence. However, LANCELOT also
gives the chance of incorporate a list of scaling factors which are applied to
the general constraints and variables separately before the optimization
commences, allowing a clearest handling of the group elements in highly
nonlinear problems as those herein considered. Thus, assuring a good
convergence, whether single or double precision,\footnotemark[5] is
implying  in turn to modify the experimental time stepping and either, the
accuracy
parameters such as, the number of iterations allowed, the constraint and
gradient accuracy, the penalty parameter and the trust region for the
optimizing.

\footnotetext[5]{For large scale programming in FORTRAN based languages, one
must care about exceeding the largest positive (or negative)
floating-point number defined by the FORTRAN distribution. By default, we have
defined an architecture of double precision in 64 bits conforming to the IEEE
standard 754 for the latest versions of Intel FORTRAN Compiler (see
Ref.~\cite{P1-IEEE}).}

Finally, in applied superconductivity, a set of complementary programs have to
be developed in an effort to provide a comprehensive understanding of the
temporal evolution of the electromagnetic quantities. For example, if the
set of minimization variables corresponds to the local profiles of current
density $\textbf{J}_{i}(\textbf{r}_{i})$, additional codes must to be used for
calculating $\textbf{A}_{i}(\textbf{r})$, $\textbf{B}_{i}(\textbf{r})$,
and $\textbf{E}_{i}(\textbf{r})$ in the whole $\Re^{3}$-space. Thus,
although integrated quantities such as the magnetic moment
$\textbf{M}(\textbf{J}_{i},\textbf{r}_{i})$ may be revealing a smooth trend
despite the use of a poor numerical accuracy, it is of utter importance testing
the numerical convergence by calculating the local profiles for the
electromagnetic quantities concerning to derived quantities. In this sense, the
following part of this book is devoted to the reliable solution of some
interesting problems in applied superconductivity, where the critical state
statement  falls into a large scale optimization problem.

 \chapter*{}
 \vspace*{-3cm}
 \section*{\Huge{Conclusions I}}
 \markboth{\sc \textbf{Conclusions I}}{\sc \textbf{Conclusions I}}
 \addcontentsline{toc}{chapter}{\sc \textbf{Conclusions I}}
 \vspace*{2cm}

In summary, in this part we have shown that the critical state theory for the
magnetic response of type-II superconductors may be built in a quite general
framework and in turn it may be solved by several means. As our interest is to
deal with highly nonlinear problems at large-scale, we
have emphasized in the performance of variational methods and computational
techniques for solving problems on personal computers.

We remark that the basic concepts underlying our generalization of the critical
state theory can be identified as follows:

\begin{enumerate}
\item The critical state theory bears a Magneto Quasi Steady (MQS) approximation
for the Maxwell equations in which $\ddot{\textbf{B}}$, $\dot{\bf E}$ and
$\dot{\rho}$
are second order quantities and consequently, the displacement current
densities $\dot{\textbf{D}}$ are much smaller than $\textbf{J}$ in the bulk and
vanish
in a first order treatment. This means that the {\em magnetic flux dynamics}
can be entirely described by the finite-difference expression of Faraday's law
\begin{eqnarray}\label{Eq.3.6}
\Delta\textbf{B}=-\nabla\times(\textbf{E}\delta t) \, ,
\end{eqnarray}
where the physically admissible states must accomplish the MQS Ampere's law,
i.e., $\nabla\times\textbf{B}=\mu_{0}\textbf{J}$. Here, the inductive
part of ${\bf E}$ may be introduced through Faraday's law, whereas the role of
electrostatic quantities is irrelevant for the magnetic sector. In other
words, {\bf E} may be modified by a gradient function (${\bf E}\to{\bf
E}+\nabla\phi$) with no effect on the magnetic response.

\item In type-II superconductors, the law that characterizes the
\textit{conducting behavior} of the material may be written in terms of
thresholds values for the current density constrained to a geometrical region
$(\textbf{J}\in\Delta_{\textbf{r}})$ which suffices to determine the relation
between the directions of \textbf{E} and \textbf{J}. Thus, \textbf{E} is no
longer an unknown variable but rather plays the role of a parameter to be
adjusted in a direct algebraic minimization, i.e.,
\begin{eqnarray}\label{Eq.3.7}
{\rm Min}\{L\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}\equiv{\rm
Max}\{\textbf{J}\cdot\textbf{p}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\equiv{\rm
Max}\{\textbf{E}\cdot\textbf{J}\}|_{\textbf{J}\in\Delta_{\textbf{\textbf{r}}}}
\, .
\end{eqnarray}
In physical terms, the material ``\textit{reacts}'' with a maximal shielding
rule when electric fields are induced, and a perfect conducting behavior
characterizes the magnetostatic equilibrium when external variations cease. In
fact, is to be noticed that the above representation can be
understood as the macroscopic counterpart of the underlying vortex physics.
Thus, recalling that, in type II superconductors an incomplete isotropy for the
limitations of the current density relative to the orientation of the local
magnetic field arises from the different physical conditions of current flow
either along or across the Abrikosov vortices, one may talk about
magnetically induced anisotropy where the physical barriers of flux depinning
and cutting are customarily depicted by the
condition $\textbf{J}\leq\textbf{J}_{c}\in\Delta_{\textbf{r}}$.
The evolution
from one magnetostatic configuration to another
occurs through the local violation of this condition, i.e.: ${\bf
J}\notin\Delta_{\textbf{r}}$ ($\textbf{J}>\textbf{J}_{c}$). However, owing
to the high dissipation, an almost instantaneous response may be assumed,
represented by a {\em maximum shielding} rule in the form ${\rm Max}\{{\bf
J}\cdot\hat{\bf E}\}\left.\right|_{{\bf J}\in\Delta_{\textbf{r}}}$.

\item With the aim of offering a meaningful reduction of the number of
variables,
we have shown that the problem can be simplified by solving a minimization
functional with a underlying structure based upon inductance matrices [see
Eq.~(\ref{Eq.3.3})]. In
particular, the mutual inductance representation with ${\bf J}({\bf
r})$ as the unknown, offers two important advantages:

(i) intricate boundary
conditions and infinite domains are avoided, and

(ii) the transparency of the
numerical statement and its performance (stability) are outlined.

Then, the quantities of interest (flux penetration profiles and magnetic moment)
are obtained by integration.

\item Most popular models for critical state problems have been
generalized in our so-called \textit{smooth double critical state theory}
(SDCST) for
anisotropic material laws~\cite{P1-Ruiz_SUST_2010}. This theory relies on our
variational framework for general critical state
problems~\cite{P1-Ruiz_PRB_2009} that allows us to
incorporate the above-mentioned physical structure in the form of mathematical
restrictions for the circulating current density. Two fundamental
material-dependent quantities play key roles in this theory
$(J_{c\parallel},J_{c\perp})$ related to the flux cutting and flux depinning
thresholds. Notoriously, the boundary condition for the material law
$\textbf{J}\in\Delta_{\textbf{r}}$ and the mutual interaction between the
critical thresholds have been described in a quite general picture, based upon
the relation between the coupling parameters
$\chi\equiv J_{c\parallel}/J_{c\perp}$
 and the smoothing index $n$ of the \textit{superelliptical} condition
$(J_{\parallel}/J_{c\parallel})^{2n}+(J_{\perp}/J_{c\perp})^{2n}\leq1$.

Hence, our SDCST cover a wide range of laws:

(i) the isotropic model ($\chi^{2}=1$, $n=1\Rightarrow \Delta_{\textbf{r}}$ is a
circle),

(ii) the elliptical model ($\chi^{2}>1$, $n=1\Rightarrow \Delta_{\textbf{r}}$ is
an ellipse),

(iii) the rectangular model ($\chi^{2}\geq 1$, $n\to\infty\Rightarrow
\Delta_{\textbf{r}}$ is a rectangle),

(iv) the infinite band model ($\chi^{2}\rightarrow\infty$), and

(v) else others with smooth magnetic anisotropy ($\chi^{2}\geq 1$, $n\in{\rm N}
\geqslant 1 \Rightarrow \Delta_{\textbf{r}}$ is a rectangle with smoothed
corners).

\end{enumerate}

Finally, let me emphasize that the scope of our
theory is rather beyond the actual examples treated in the following part of
this thesis. On the one side, we have shown that the critical state concept
allows arbitrariness in the presence of electrostatic charge and potential, and
one could simply upgrade the models by the rule ${\bf E}\to{\bf E}+\nabla\phi$
if necessary. For instance, a scalar function  $\phi$ may be introduced if the
direction of ${\bf E}$ has to be modified respect to the maximum shielding rule
in the MQS limit. On the other side, the extension of the theory to arbitrary
sample geometries is intrinsically allowed by the mutual inductance
representation. Thus, this first part has laid necessary groundwork for
attacking general critical state problems in 3D geometry.

\renewcommand\bibname{References I}