pages={15-32}
}
+%-------------------------------------------------------------------------------
+% Step 97
+%-------------------------------------------------------------------------------
+
+@book{griffiths2017b,
+ place = {Cambridge},
+ edition = {4},
+ title = {Introduction to electrodynamics},
+ author = {Griffiths, David J.},
+ year = {2017},
+ publisher = {Cambridge University Press},
+ isbn = {978-1108420419},
+ doi = {10.1017/9781108333511},
+ url = {https://www.cambridge.org/highereducation/books/introduction-to-electrodynamics/3AB220820DBB628E5A43D52C4B011ED4#overview}
+}
+
+@book{uzunbajakau2024b,
+ title = {Electromagnetics with finite elements: static problems},
+ author = {Uzunbajakau, Siarhei},
+ year = {2024},
+ publisher = {CEM Books},
+ isbn = {978-9090380933},
+ url = {https://www.cembooks.nl}
+}
+
+@article{manges1995a,
+ title = {A generalized tree-cotree gauge for magnetic field computation},
+ author = {Manges, John B. and Cendes, Zoltan J.},
+ journal = {IEEE Transactions on Magnetics},
+ volume = {31},
+ number = {3},
+ pages = {1342-1347},
+ year = {1995},
+ doi = {10.1109/20.376275},
+ url = {https://ieeexplore.ieee.org/document/376275}
+}
+
+@book{hypre1998b,
+ title = {Documentation for hypre},
+ publisher = {Lawrence Livermore National Security, LLC and other HYPRE Project Developers},
+ url = {https://hypre.readthedocs.io/en/latest/solvers-ams.html}
+}
+
+@inproceedings{gratkowski2010p,
+ title = {Implementation of Asymptotic Boundary Conditions in Commercial FEM Software},
+ author = {Gratkowski, Stanislaw and Ziolkowski, Marek and Brauer, Hartmut},
+ booktitle = {14th IGTE Symposium},
+ year = {2010},
+ url = {https://www.researchgate.net/profile/Marek_Ziolkowski/publication/279994830_Implementation_of_Asymptotic_Boundary_Conditions_in_Commercial_FEM_Software/links/55a22c5508aea54aa81457ae.pdf}
+}
+
+@article{meeker2013a,
+ title = {Improvised Open Boundary Conditions for Magnetic Finite Elements},
+ author = {Meeker, David C.},
+ journal = {IEEE Transactions on Magnetics},
+ volume = {49},
+ number = {10},
+ pages = {5243-5247},
+ year = {2013},
+ doi = {10.1109/TMAG.2013.2260348},
+ url = {https://ieeexplore.ieee.org/document/6509478}
+ }
%-------------------------------------------------------------------------------
% References used elsewhere
* <td> Using nonlocal dofs to solve a simple optimization problem with the Poisson equation
* <br/> Keywords: nonlocal dofs, optimization, Poisson
* </td></tr>
+ *
+ * <tr valign="top">
+ * <td>step-97</td>
+ * <td> Application of the Nedelec and Raviart-Thomas finite elements to a magnetic
+ * problem formulated in terms of the magnetic vector potential and current vector potential.
+ * Solving the curl-curl equation.
+ * <br/> Keywords: FE_Nedelec, FE_RaviartThomas, WorkStream::run()
+ * </td></tr>
* </table>
*
*
* <td>step-46</td>
* </tr>
*
+ * <tr valign="top">
+ * <td> The curl-curl equation
+ * </td>
+ * <td>
+ * step-81,
+ * step-97
+ * </td>
+ * </tr>
+ *
* </table>
*
*
* <td>step-46</td>
* </tr>
*
+ * <tr valign="top">
+ * <td> The magnetic vector potential and current vector potential
+ * </td>
+ * <td>step-97</td>
+ * </tr>
+ *
* </table>
*
*
PATTERN "topography.txt.gz" # step-53
PATTERN "input/initial_mesh_3d.vtk" # step-54
PATTERN "input/DTMB-5415_bulbous_bow.iges" # step-54
+ PATTERN "sphere_r6.geo" # step-97
+ PATTERN "sphere_r6.msh" # step-97
+ PATTERN "sphere_r7.geo" # step-97
+ PATTERN "sphere_r7.msh" # step-97
+ PATTERN "sphere_r8.geo" # step-97
+ PATTERN "sphere_r8.msh" # step-97
+ PATTERN "sphere_r9.geo" # step-97
+ PATTERN "sphere_r9.msh" # step-97
)
message(STATUS "Setting up examples - Done")
--- /dev/null
+##
+# CMake script for the step-97 tutorial program:
+##
+
+# Set the name of the project and target:
+set(TARGET "step-97")
+
+# Declare all source files the target consists of. Here, this is only
+# the one step-X.cc file, but as you expand your project you may wish
+# to add other source files as well. If your project becomes much larger,
+# you may want to either replace the following statement by something like
+# file(GLOB_RECURSE TARGET_SRC "source/*.cc")
+# file(GLOB_RECURSE TARGET_INC "include/*.h")
+# set(TARGET_SRC ${TARGET_SRC} ${TARGET_INC})
+# or switch altogether to the large project CMakeLists.txt file discussed
+# in the "CMake in user projects" page accessible from the "User info"
+# page of the documentation.
+set(TARGET_SRC
+ ${TARGET}.cc
+)
+
+# Define the output that should be cleaned:
+set(CLEAN_UP_FILES *.vtu *.tex)
+
+# Usually, you will not need to modify anything beyond this point...
+
+cmake_minimum_required(VERSION 3.13.4)
+
+find_package(deal.II 9.7.0
+ HINTS ${deal.II_DIR} ${DEAL_II_DIR} ../ ../../ $ENV{DEAL_II_DIR}
+)
+if(NOT ${deal.II_FOUND})
+ message(FATAL_ERROR "\n"
+ "*** Could not locate a (sufficiently recent) version of deal.II. ***\n\n"
+ "You may want to either pass a flag -DDEAL_II_DIR=/path/to/deal.II to cmake\n"
+ "or set an environment variable \"DEAL_II_DIR\" that contains this path."
+ )
+endif()
+
+deal_ii_initialize_cached_variables()
+project(${TARGET})
+deal_ii_invoke_autopilot()
--- /dev/null
+<i>
+This program was contributed by Siarhei Uzunbajakau, <a href="https://www.cembooks.nl"> CEM Books</a>, 2025.
+</i>
+
+<h1>Introduction</h1>
+
+<h3>Problem definition</h3>
+
+This tutorial illustrates application of the FE_Nedelec and FE_RaviartThomas
+finite elements to problems in electromagnetics. To this end, we will compute
+the magnetic field induced by a coil with a magnetic core and compare the
+result to an exact closed-form analytical expression of the magnetic field. We
+will repeat the comparison on a set of progressively refined meshes and observe
+that the $L^2$ error norm decreases with the size of mesh cells. We will
+also vary the degree of the finite elements and observe that the higher-degree
+finite elements imply faster convergence of the $L^2$ error norms. Experimenting
+with boundary conditions is suggested as a possibility for extension.
+
+The first figure below illustrates a cross section of the coil. The second
+figure illustrates the corresponding problem domain, coordinate system, and the
+relevant notations. The coil consists of two spherical shells. The inner shell,
+depicted in green, is the magnetic core. The walls of the core are
+filled with a soft magnetic material of permeability $\mu_1$. The magnetic
+material is assumed to be homogeneous, linear, lossless, and isotropic. The
+permeability of the rest of the space (the ball-shaped space inside the core and
+the space outside the core) equals to that of the free space, $\mu_0$. The
+permeability of the entire space can be expressed as
+\f{equation}{
+\mu = \left\{
+\begin{aligned}
+& \mu_0 &\text{if } \text{ } & r < a_1\\
+& \mu_1 &\text{if } \text{ } & a_1 \le r \le b_1\\
+& \mu_0 &\text{if } \text{ } & r > b_1.
+\end{aligned}
+\right.
+\f}
+The outer shell, depicted in blue, contains the current-carrying windings of
+the coil. The windings are modeled by a prescribed free-current density. The
+prescribed free-current density fills all the space available in the wall of
+the shell. In the figure below, however, only three cross sections of the
+free-current density are shown. The free-current density is the highest on the
+equator and equals zero at the two poles. It can be described as
+\f[
+\vec{J}_f = \left\{
+\begin{aligned}
+& 0 &&\text{if }&& r < a_2\\
+& K_0 (-y\hat{i} + x\hat{j})&&\text{if }&& a_2 \le r \le b_2\\
+& 0 &&\text{if }&& r > b_2\\
+\end{aligned}
+\right.
+\f]
+in the Cartesian coordinate system.
+
+@htmlonly
+<p align="center">
+ <img src="images/steps/developer/step-97-coil.svg" alt="The coil" height="425">
+</p>
+@endhtmlonly
+
+There is no consensus in the literature on what to call a magnetic field.
+On this page we adopt the nomenclature from @cite griffiths2017b. Consequently,
+we call a magnetic field the vector quantity $\vec{B}$ such that the Ampere's
+law in free space in static approximation reads
+\f[
+\begin{equation}
+\vec{\nabla} \times\vec{B} = \mu_0\vec{J},
+\end{equation}
+\f]
+where $\vec{J}$ is the total current density, free plus bound.
+
+<h3>Problem domain and coordinate systems</h3>
+
+We adopt the Cartesian coordinate system, the coordinates being $x$, $y$, and
+$z$, as shown in the figure below. We denote the corresponding unitary basis
+vectors as $\hat{i}$, $\hat{j}$, and $\hat{k}$. The problem domain consists
+of a ball centered at the origin and four spherical shells around it.
+The spheres $\Gamma_{I1}$ and $\Gamma_{I2}$ are interfaces between the magnetic
+material and free space. Strictly speaking, the magnetic field induced by the
+coil extends to infinity. Therefore, the problem we would like to solve
+is unbounded. As the mesh needs to have finite dimensions, we truncate the
+space with the sphere $\Gamma_{R1}$. This sphere will represent infinity.
+We denote the spherical coordinates as $r$, $\theta$, and $\phi$, with $\theta$
+and $\phi$ being polar and azimuthal angles, respectively. The figure below
+illustrates the basis vectors of the spherical coordinate system, $\hat{r}$,
+$\hat{\theta}$, and $\hat{\phi}$.
+
+@htmlonly
+<p align="center">
+ <img src="images/steps/developer/step-97-domain.svg" alt="The domain" height="496">
+</p>
+@endhtmlonly
+
+<h3>Exact solution</h3>
+
+The problem of a permeable spherical shell exposed to a uniform magnetic field
+is ubiquitous in the literature on electromagnetics. The problem of the magnetic
+field generated by a spherical coil can be found in the literature as well.
+Remarkably, the magnetic field inside the coil is uniform. Due to this fact one
+can combine the solution to these two problems to get the exact closed-form
+analytical expressions for the magnetic field generated by the coil described
+above. The exact solution presented below is a combination of the solutions to
+the problems of permeable shell and spherical coil. The derivation of the
+equations presented below can be found in @cite uzunbajakau2024b.
+
+We express the magnetic field induced by the coil as a sum of the magnetic
+field induced by the free-current density in free space, $\vec{B}_J$,
+and the magnetic field induced by the magnetization of the magnetic core,
+$\vec{B}_{\mu}$,
+\f[
+\vec{B} = \vec{B}_J +\vec{B}_{\mu}.
+\f]
+
+It is convenient to express the term $\vec{B}_J$ in a spherical coordinate
+system,
+
+\f[
+\vec{B}_J = \left\{
+\begin{aligned}
+& \frac{1}{2} (b_2^2-a_2^2) \vec{F}_1(\theta)
+ &&\text{if }&& r \le a_2\\
+& \frac{1}{2} (b_2^2-r^2) \vec{F}_1(\theta) +
+\frac{1}{5} (r^5-a_2^5) \vec{F}_2(r, \theta)
+&& \text{if } && a_2 \le r \le b_2\\
+& \frac{1}{5} (b_2^5-a_2^5) \vec{F}_2(r, \theta) &&\text{if }&& r \ge b_2,\\
+\end{aligned}
+\right.
+\f]
+where
+\f[
+\begin{aligned}
+&\vec{F}_1(\theta) = \dfrac{2}{3}\mu_0
+K_0\big(\cos(\theta)\hat{r}-\sin(\theta)\hat{\theta}\big), \\
+&\vec{F}_2(r, \theta) = \dfrac{2}{3}\mu_0 K_0
+\frac{1}{r^3}\big(\cos(\theta)\hat{r}+\frac{1}{2}\sin(\theta)\hat{\theta}\big).
+\end{aligned}
+\f]
+
+We, however, need to program in the Cartesian coordinate system. The expressions
+above can be converted into the Cartesian coordinate system by invoking the
+following identities:
+\f[
+r = \sqrt{x^2+y^2+z^2},
+\f]
+\f[
+\cos(\theta) = \frac{z}{r},
+\f]
+\f[
+\sin(\theta) = \frac{\sqrt{x^2+y^2}}{r},
+\f]
+\f[
+\cos(\phi) = \frac{x}{\sqrt{x^2+y^2}},
+\f]
+\f[
+\sin(\phi) = \frac{y}{\sqrt{x^2+y^2}},
+\f]
+\f[
+\hat{r} = \frac{1}{r} (x\hat{i}+y\hat{j}+z\hat{k}),
+\f]
+and
+\f[
+\hat{\theta}= \cos(\theta)\cos{\phi}\hat{i} + \cos(\theta)\sin{\phi}\hat{j} - \sin(\theta) \hat{k}.
+\f]
+
+The magnetic field induced by the magnetic core can be expressed in Cartesian
+coordinates as
+\f[
+\vec{B}_{\mu} = - \mu\vec{\nabla} \Psi - \mu_0 H_0 \hat{k},
+\f]
+where
+\f[
+H_0 = \frac{1}{3} K_0 (b_2^2-a_2^2),
+\f]
+\f{equation}{
+\mu = \left\{
+\begin{aligned}
+& \mu_0 &\text{if } \text{ } & r < a_1\\
+& \mu_1 &\text{if } \text{ } & a_1 \le r \le b_1\\
+& \mu_0 &\text{if } \text{ } & r > b_1,
+\end{aligned}
+\right.
+\f}
+and
+\f{equation}{
+\vec{\nabla} \Psi = \left\{
+\begin{aligned}
+&\delta_1 \hat{k} &\text{if } \text{ } & r \le a_1\\
+&-3\gamma_1\frac{xz}{r^5}\hat{i} -3\gamma_1\frac{yz}{r^5}\hat{j} +
+\bigg(\beta_1 + \gamma_1\frac{1}{r^3} - 3\gamma_1\frac{z^2}{r^5} \bigg)\hat{k}
+&\text{if } \text{ } & a_1 \le r \le b_1\\
+&-3\alpha_1\frac{xz}{r^5}\hat{i} -3\alpha_1\frac{yz}{r^5}\hat{j} +
+\bigg(-H_0 + \alpha_1\frac{1}{r^3} - 3\alpha_1\frac{z^2}{r^5} \bigg)\hat{k}
+&\text{if } \text{ } & r \ge b_1.
+\end{aligned}
+\right.
+\f}
+The following constants were used in the last equation:
+\f{equation}{
+\begin{aligned}
+&\Omega = \frac{(\mu_r - 1)}{(\mu_r + 2)} \frac{a_1^3}{b_1^3}, \\
+&\gamma_1 = \frac{-3b_1^3 H_0 \Omega}{(2\mu_r+1)-2 (\mu_r-1) \Omega}, \\
+&\beta_1 = \frac{(2\mu_r+1) \gamma_1}{(\mu_r-1)a_1^3}, \\
+&\alpha_1 = \frac{-b_1^3 H_0+2\mu_r\gamma_1-\mu_r b_1^3\beta_1}{2}, \\
+&\delta_1 = \frac{\mu_r a_1^3\beta_1-2\mu_r\gamma_1}{a_1^3}.
+\end{aligned}
+\f}
+
+The relative permeability in the last five equations is computed as
+\f[
+\mu_r = \frac{\mu_1}{\mu_0}.
+\f]
+
+<h3>Boundary value problems</h3>
+
+<h4>Magnetic vector potential - A</h4>
+
+Ampere's law in static approximation can be written in terms of the
+auxiliary vector field $\vec{H}$ as
+\f[
+\vec{\nabla}\times\vec{H} = \vec{J}_f,
+\f]
+where $\vec{J}_f$ is the free current and $\vec{H}$ is given by
+\f{equation}{
+\begin{array}{ll}
+\vec{H} = \dfrac{1}{\mu}\vec{B} & \text{(constitutive relation)}.
+\end{array}
+\f}
+We introduce the magnetic vector potential, $\vec{A}$, as
+\f{equation}{
+\begin{array}{ll}
+\vec{B} = \vec{\nabla}\times\vec{A} &
+\text{(definition of magnetic vector potential)}.
+\end{array}
+\f}
+By combining the last three equations we arrive at
+\f[
+\vec{\nabla}\times\bigg(\dfrac{1}{\mu} \vec{\nabla}\times\vec{A}\bigg) =
+\vec{J}_f,
+\f]
+which is the curl-curl partial differential equation we would like to solve.
+There are, however, two subtleties we need to address before solving any
+curl-curl equations: the gauge and compatibility condition.
+
+<h5>Gauge</h5>
+
+The Helmholtz decomposition theorem suggests that any vector field studied in
+electromagnetics, say $\vec{A}$, can be represented as a sum of a conservative
+and a solenoidal vector field,
+\f[
+\vec{A} =
+\underbrace{- \vec{\nabla} V}_{\text{conservative}} +
+\underbrace{\vec{\nabla} \times \vec{W}}_{\text{solenoidal}} =
+\vec{C} + \vec{S}.
+\f]
+The curl of a conservative vector field always equals zero (conservative fields
+are curl free),
+\f[
+\vec{\nabla} \times \vec{C} =
+-\vec{\nabla} \times \big( \vec{\nabla} V \big) = 0.
+\f]
+Consequently, any vector field derived by adding a conservative vector field to
+a solution to the curl-curl equation is also a solution,
+\f[
+\vec{\nabla}\times\bigg(\dfrac{1}{\mu} \vec{\nabla}\times\Big(\vec{A}
+- \vec{\nabla} V'\Big)\bigg) =
+\vec{\nabla}\times\bigg(\dfrac{1}{\mu} \vec{\nabla}\times\vec{A}\bigg).
+\f]
+We can restrict this freedom by specifying the divergence of the solution.
+Indeed, the divergence of a solenoidal vector field always equals zero (solenoidal
+fields are divergence free),
+\f[
+\vec{\nabla} \cdot \vec{S} =
+\vec{\nabla} \cdot \big(\vec{\nabla} \times \vec{W}\big) = 0.
+\f]
+Then the divergence of a solution can be expressed as
+\f[
+\vec{\nabla} \cdot \vec{A} =
+\vec{\nabla} \cdot \big(\vec{C} + \vec{S}\big) =
+\vec{\nabla} \cdot \vec{C}.
+\f]
+By specifying the divergence of the solution, we specify the divergence of the
+conservative part of the solution, and, thus, we specify the conservative part
+of the solution itself. In short: the magnetic vector potential consists of
+conservative and solenoidal components; the curl-curl equation fixes the
+solenoidal component, the gauge fixes the conservative component. For example,
+the Coulomb gauge,
+\f[
+\vec{\nabla} \cdot \vec{A} = 0,
+\f]
+specifies that there is no conservative component in $\vec{A}$. That is, the
+Coulomb gauge selects a purely solenoidal solution. We are interested in the
+magnetic field, $\vec{B}$. In the realm of classical electromagnetics, the
+magnetic vector potential, $\vec{A}$, has an instrumental value - we just use
+it to compute the magnetic field, $\vec{B}$. The magnetic vector potential is
+a useful mathematical tool but it has no physical meaning as it cannot be
+measured. On the contrary, the magnetic field, $\vec{B}$, can be measured and,
+as a consequence, has a distinctive physical meaning. From a physical
+perspective, there are (infinitely) many magnetic vector potentials that lead
+to the same physically measurable magnetic field and for a physicist, it doesn't
+matter which of these potentials we choose. But from a mathematical perspective,
+we need to pick one to make the problem well-posed. A gauge is a way to pick
+one. Because the magnetic field is computed as
+\f[
+\vec{B} = \vec{\nabla}\times\vec{A} =
+\vec{\nabla}\times\big(\vec{C} + \vec{S} \big) =
+\vec{\nabla}\times\vec{S},
+\f]
+we are interested in the solenoidal part of the solution. Which conservative
+component of the solution is brought by the gauge is not important.
+
+A good question is: how to enforce the gauge? The literature suggests that the
+best approach is not to gauge the magnetic vector potential explicitly. Instead,
+one should invoke an advanced linear solver (tree-cotree splitting, domain
+decomposition, geometric multigrid, algebraic multigrid, etc.). For example,
+in @cite manges1995a it is suggested that the tree-cotree splitting can be used
+us a gauge. The hypre AMS utilizes a more modern approach @cite hypre1998b. It
+can solve positive semidefinite linear systems yielded by the ungauged curl-curl
+equation. In general, if a solver is able to choose one solution out of many
+ungauged solutions, we can claim that such solution is implicitly gauged by
+the linear solver. The linear solver will not communicate to us which gauge has
+been applied, i.e., the conservative part of the solution will be fixed, but we
+will not be able to tell what it is, exactly. We will call such gauge an
+implicit gauge. An implicit gauge is good enough for our purpose.
+
+We will use the CG solver which is the next best choice to the advanced linear
+solvers mentioned above. The CG solver is designed to handle symmetric positive
+definite matrices. This means that all eigenvalues of the system matrix are
+expected to be positive. On the other hand, the curl-curl equation above
+being augmented with proper boundary conditions will always yield a symmetric
+positive semidefinite matrix. The last means that most of the eigenvalues of
+the system matrix will be positive but some of them will be zero. We elevate
+the zero eigenvalues above zero just a bit by adding a very small gauging
+term, $\eta^2\vec{A}$, to the partial differential equation,
+\f[
+\vec{\nabla}\times\bigg(\dfrac{1}{\mu} \vec{\nabla}\times\vec{A}\bigg) +
+\eta^2\vec{A} = \vec{J}_f,
+\f]
+so all eigenvalues become positive. Formally, the system matrix with
+elevated eigenvalues is symmetric positive definite. We can feed it to
+the CG solver. The partial differential equation is different now. We can
+expect a different solution. Moreover, $\eta^2 \vec{A}$ belongs to the
+$H(\text{curl})$ function space, while the rest of the terms of the partial
+differential equation belong to the $H(\text{div})$ function space, see the
+Bossavit's diagram below. They exhibit different behavior on interfaces
+between dissimilar materials. Strictly speaking, we destroy the compatibility
+between the two sides of the curl-curl equation by adding the gauging term.
+However, by choosing the parameter $\eta^2$ to be small enough we can make
+these horrible crimes insignificant. In practice, we need to set $\eta^2$
+to zero. If the CG solver chokes, we will increase $\eta^2$ just a bit. Setting
+$\eta^2=\dfrac{10^{-6}}{\mu_0}$ can save the day. The $\eta^2$-trick is an
+implicit gauge: it helps the CG solver to select one solution out of many,
+but it is impossible to predict what the conservative part of the solution
+will look like. The last is not important for the problem we would like to
+solve because, as mentioned above, the vector potential $\vec{A}$ has no
+physical reality and all choices of a gauge (including the implicit gauge used here)
+lead to the same physical fields.
+
+<h5>Compatibility condition</h5>
+
+To solve the curl-curl equation numerically, we need to feed to the solver the
+right-hand side of the equation, $\vec{J}_f$. Whatever we do, we will end up
+feeding to the solver a discretized version of $\vec{J}_f$ which will be a
+combination of conservative and solenoidal vector fields even if the problem
+we would like to solve specifies a closed-form analytical expression of a purely
+solenoidal $\vec{J}_f$. The conservative component will be added implicitly by
+discretization. On the other hand, the left-hand side of the curl-curl equation
+is purely solenoidal as the curl of any vector field is always solenoidal
+(assume $\eta^2 = 0$ for a moment). Therefore, the two sides of the equation
+will be somewhat incompatible: pure solenoidal vector field on the left-hand
+side and a combination of a conservative and solenoidal vector fields on the
+right-hand side. As a consequence, we cannot expect that we can get the norm
+of the residual to zero. In other words, the linear solver cannot converge
+to a solution that has zero residual. In practice, this means that CG will
+either not converge at all, or terminate only after an unreasonably large
+number of iterations. To make the two sides of the equation compatible, we derive
+the free-current density from a current vector potential, $\vec{T}$, as
+\f[
+\vec{J}_f = \vec{\nabla}\times\vec{T}.
+\f]
+Consequently, the curl-curl equation becomes
+\f[
+\vec{\nabla}\times\bigg(\dfrac{1}{\mu} \vec{\nabla}\times\vec{A}\bigg) +
+\eta^2\vec{A} = \vec{\nabla}\times\vec{T}.
+\f]
+This equation contains purely solenoidal vector fields on both sides if $\eta^2=0$.
+Now, however, we need to convert $\vec{J}_f$ into $\vec{T}$. There are a number
+of ways of doing so. One of them is solving another curl-curl equation as
+discussed below. For now we assume that $\vec{T}$ is known.
+
+<h5>Boundary and interface conditions</h5>
+
+Even if the magnetic vector potential is gauged and the compatibility issue
+is solved, one still needs to set up the boundary conditions to pick up one
+solution out of many gauged vector potentials that satisfy the curl-curl
+equation. Moreover, the behavior of the magnetic field on interfaces between
+dissimilar materials is defined by the Maxwell's equations. Consequently, the
+behavior of the magnetic vector potential on interfaces is defined by the
+Maxwell's equations as well. For this reason, we need to add the interface
+conditions next to the boundary conditions to ensure the correct behavior of
+the current vector potential on interfaces. The curl-curl partial differential
+equation augmented with the boundary and interface conditions constitutes the
+boundary value problem. We compose the following boundary value problem for the
+problem discussed above:
+
+\f{equation}{
+\begin{array}{lrcll}
+\text{ } & \vec{\nabla}\times\bigg(\dfrac{1}{\mu} \vec{\nabla}\times\vec{A}\bigg)
++ \eta^2 \vec{A} = \vec{\nabla}\times\vec{T} & \text{in} & \Omega & \text{(i)},\\
+\text{(n)}& \dfrac{1}{\mu}\hat{n}\times\bigg(\vec{\nabla}\times\vec{A}\bigg) +
+\gamma \hat{n}\times\bigg(\hat{n}\times\vec{A}\bigg)
+=0 &\text{on} & \Gamma_{R1} & \text{(ii)}, \\
+\text{(e)}&\hat{n}\times\vec{A}_{+} =
+\hat{n}\times\vec{A}_{-}&\text{on}&\Gamma_{I1}\cup\Gamma_{I2}&\text{(iii)},\\
+\text{(n)}&\dfrac{1}{\mu}_{+}\hat{n}\times \bigg( \vec{\nabla} \times \vec{A}_{+} \bigg) -
+\dfrac{1}{\mu}_{-}\hat{n}\times \bigg( \vec{\nabla} \times \vec{A}_{-} \bigg) = 0
+& \text{on} & \Gamma_{I1}\cup\Gamma_{I2} & \text{(iv)},
+\end{array}
+\f}
+where
+\f[
+\gamma = \frac{1}{\mu r}.
+\f]
+In this boundary value problem the vector $\hat{n}$ is the unit vector normal
+to the surface on which the boundary or interface condition is defined. By
+convention, $\hat{n}$ always points outside a closed surface. The subscript
+"+" refers to the space immediately next to the interface in the direction
+of vector $\hat{n}$. The subscript "-" refers to the space immediately next
+to the interface in the direction opposite to $\hat{n}$.
+
+Equation (i) in the boundary value problem has been discussed above and needs
+no further comments.
+
+The surface $\Gamma_{R1}$ represents infinity. As most problems in magnetics,
+the problem we would like to solve is unbounded. That is to say, the magnetic
+field as well as the magnetic vector potential extend to infinity. A
+straightforward modeling of an unbounded domain by means of finite elements is
+impossible as such modeling will require an infinite amount of cells. To
+overcome this unfortunate predicament, we artificially truncate the unbounded
+problem domain with the surface $\Gamma_{R1}$. This truncation will introduce a
+simulation error. We, however, will try to minimize this error by making the
+interior of the artificial surface $\Gamma_{R1}$ as spacious as possible.
+The magnetic field induced by the coil vanishes at infinity, so does the
+magnetic vector potential. For this reason, the homogeneous Dirichlet,
+\f{equation}{
+\begin{array}{ll}
+\hat{n} \times \vec{A} = 0 & \text{(Dirichlet)},
+\end{array}
+\f}
+and homogeneous Neumann,
+\f{equation}{
+\begin{array}{ll}
+\dfrac{1}{\mu}\hat{n}\times\bigg(\vec{\nabla}\times\vec{A}\bigg) = 0 &
+\text{(Neumann)},
+\end{array}
+\f}
+boundary conditions are used on $\Gamma_{R1}$ most often. For this approach to be
+efficient, one needs to choose the radius of the sphere $\Gamma_{R1}$ large enough so that
+the tangential component of the magnetic vector potential (Dirichlet) or the
+tangential component of the magnetic field (Neumann) is negligibly small on
+$\Gamma_{R1}$. There is, however, a more efficient approach.
+
+The first term of the multipole expansion of magnetic vector potential is
+the magnetic vector potential of a magnetic dipole moment unless the source
+of the magnetic field is explicitly configured as a higher-order multipole.
+For this reason, the magnetic vector potential induced by the coil looks more
+like that of a magnetic dipole at a sufficient distance. If a magnetic dipole
+is placed at the center of a sphere, the magnetic vector potential induced by
+the dipole on the sphere satisfies the following identity:
+
+\f[
+\hat{n} \times \bigg( \vec{\nabla}\times \vec{A} \bigg) = - \frac{1}{r}
+\hat{n} \times \bigg( \hat{n} \times \vec{A} \bigg).
+\f]
+
+Note that this is a purely geometric statement. It does not depend on how we
+choose our coordinate system. Consequently, the identity given by the last
+equation is true for a magnetic dipole of any orientation as long as it is
+situated at the center of the sphere. By multiplying the last equation by
+$\dfrac{1}{\mu}$ and rearranging the terms we arrive into
+
+\f{equation}{
+\begin{array}{ll}
+\dfrac{1}{\mu}\hat{n}\times\bigg(\vec{\nabla}\times\vec{A}\bigg) +
+\dfrac{1}{\mu r}\hat{n}\times\bigg(\hat{n}\times\vec{A}\bigg)
+=0 & \text{(Robin)},
+\end{array}
+\f}
+which is, essentially, the Robin boundary condition (ii) in the boundary value
+problem above. The Robin boundary condition in the current context is also called
+the first-order asymptotic boundary condition, first-order ABC for short.
+The Robin boundary condition is expected to be superior to both Dirichlet and
+Neumann boundary conditions as it allows to reach the same level of simulation
+error with a sphere $\Gamma_{R1}$ of a smaller radius.
+
+All three boundary conditions, Dirichlet, Neumann, and Robin, guarantee the
+uniqueness of the curl of the solution, $\vec{\nabla}\times\vec{A}$. The
+uniqueness of the solution itself, $\vec{A}$, can only be guaranteed if
+one of the boundary conditions is applied in combination with a gauge, see the
+discussion above. The program uses the Robin boundary condition by default,
+but can easily be switched to Dirichlet or Robin boundary conditions.
+
+The behavior of the magnetic field on the interfaces between dissimilar
+materials can be described by
+\f{equation}{
+\begin{array}{ll}
+\hat{n}\cdot\vec{B}_+ = \hat{n}\cdot\vec{B}_- ,\\
+\hat{n}\times\vec{H}_+ - \hat{n}\times\vec{H}_- = \vec{K}_f,
+\end{array}
+\f}
+where $\vec{K}_f$ is a surface free-current density that can be present on an
+interface. No free currents flow on the surface of the magnetic core of the coil
+described above. Therefore, we set $\vec{K}_f = 0$. Then by substituting the
+constitutive relation for the magnetic field and the definition of the magnetic
+vector potential, see above, into the last two equations we get
+\f{equation}{
+\begin{array}{ll}
+\hat{n}\cdot\bigg(\vec{\nabla}\times\vec{A}_+\bigg) =
+\hat{n}\cdot\bigg(\vec{\nabla}\times\vec{A}_-\bigg) &\text{(condition A)},\\
+\hat{n}\times\bigg(\dfrac{1}{\mu_+}\vec{\nabla}\times\vec{A}_+\bigg) -
+\hat{n}\times\bigg(\dfrac{1}{\mu_-}\vec{\nabla}\times\vec{A}_-\bigg) = 0.
+\end{array}
+\f}
+Next, we simply replace the first condition with another condition
+and rearrange the terms of the second condition:
+\f{equation}{
+\begin{array}{ll}
+\hat{n}\times\vec{A}_+ = \hat{n}\times\vec{A}_- & \text{(condition B)}, \\
+\dfrac{1}{\mu_+}\hat{n}\times\bigg(\vec{\nabla}\times\vec{A}_+\bigg) -
+\dfrac{1}{\mu_-}\hat{n}\times\bigg(\vec{\nabla}\times\vec{A}_-\bigg) = 0,
+\end{array}
+\f}
+The last two equations are identical to equations (iii) and (iv) in the boundary
+value problem above. These are the interface conditions that must be observed on
+the inner and outer surfaces of the magnetic core, $\Gamma_{I1}\cup\Gamma_{I2}$.
+
+We have replaced condition A with condition B. This replacement deserves an
+explanation. The are two reasons for this replacement. The first reason is the
+following. A magnetic flux through an infinitesimally small closed loop must be
+infinitesimally small even if an interface between dissimilar magnetic materials
+passes through the loop. We can guarantee this only if we require that
+condition B holds. Condition A will not do. The second reason is the
+uniqueness of the curl of the solution. If we include condition A into the
+boundary value problem above, the uniqueness of the curl of the solution,
+$\vec{\nabla}\times\vec{A}$, cannot be guaranteed. It is possible
+to guarantee it if condition B is included instead. A good question is:
+does this replacement disturb the behavior of the magnetic field on the
+interfaces? Not at all. Consider the following. The normal component
+of the curl, $\hat{n}\cdot\big(\vec{\nabla}\times\vec{A}\big)$, is defined by
+the derivatives of the tangential components of the vector potential. If two
+vector potentials, $\vec{A}_-$ and $\vec{A}_+$, have the same tangential
+components on the interface, i.e., condition B, the derivatives of these
+tangential components along the interface will be the same as well. Therefore,
+condition B implicitly implies condition A and, thus, the replacement is
+justified. We can conclude that condition B is more restrictive than
+condition A, so the behavior of the magnetic vector potential on the
+interfaces is observed. Note, that the interface conditions (iii) and (iv)
+in the boundary value problem above have absolutely no influence on the
+uniqueness of the solution regardless which boundary condition, Dirichlet,
+Neumann, or Robin, is used.
+
+It remains to comment on the labels (e) and (n) in the boundary value problem
+above. They label (e)ssential and (n)atural boundary and interface conditions.
+How to sort conditions in these two categories and why it is important to do
+so is discussed below in the section on variational formulations.
+
+<h4>Current vector potential - T</h4>
+
+To be able to solve the boundary value problem above, we need to convert the
+free current density, $\vec{J}_f$, into current vector potential, $\vec{T}$. We
+define the current vector potential as
+\f[
+\vec{J}_f = \vec{\nabla}\times\vec{T}.
+\f]
+
+By taking the curl of the last equation and rearranging the terms we arrive into
+
+\f[
+\vec{\nabla}\times\bigg(\vec{\nabla}\times\vec{T}\bigg)
+= \vec{\nabla}\times\vec{J}_f.
+\f]
+We will compute the current vector potential by solving this equation. As soon
+as there are curls of vector fields on both sides of the equation, the
+compatibility condition is observed. We add to this equation a gauging term,
+\f[
+\vec{\nabla}\times\bigg(\vec{\nabla}\times\vec{T}\bigg) + \eta^2 \vec{T}
+= \vec{\nabla}\times\vec{J}_f.
+\f]
+The discussion on the gauging term in the preceding section is valid in the
+current context as well.
+
+The current vector potential is computed in free space, i.e., $\mu = \mu_0$
+everywhere. For this reason, there is no need for interface conditions as
+there are no interfaces in free space.
+
+We still need to apply a boundary condition to the surface $\Gamma_{R1}$.
+It is beneficial to apply the homogeneous Dirichlet boundary condition as it
+allows to nullify the integral $I_{b3-2}$ in the recipe for calculating the
+magnetic vector potential, see below. Consequently, we can neglect this
+integral when programming and this makes the code more efficient.
+
+We compose a boundary value problem by combining the curl-curl equation for
+the current vector potential with the Dirichlet boundary condition:
+
+\f{equation}{
+\begin{array}{lrcll}
+\text{ } & \vec{\nabla}\times\bigg(\vec{\nabla}\times\vec{T}\bigg)
++ \eta^2 \vec{T} = \vec{\nabla}\times\vec{J}_f & \text{in} & \Omega &
+\text{(i)},\\
+\text{(e)}& \hat{n}\times\vec{T} = 0 &\text{on} & \Gamma_{R1} & \text{(ii)}.
+\end{array}
+\f}
+
+<h3>Variational formulations</h3>
+
+<h4>Magnetic vector potential</h4>
+
+The problem of solving the boundary value problem for the magnetic vector
+potential can be replaced by the problem of minimizing the following
+functional:
+\f{equation}{
+\begin{aligned}
+&F(\vec{A}) =
+\iiint_{\Omega}\frac{1}{\mu}\bigg|\vec{\nabla}\times\vec{A}\bigg|^2 dV +
+\iint_{\Gamma_{R1}} \gamma \bigg|\hat{n}\times\vec{A}\bigg|^2 dS +
+\eta^2 \iiint_{\Omega}\mid\vec{A}\mid^2 dV
+-2\iiint_{\Omega} \vec{T} \cdot \bigg( \vec{\nabla} \times\vec{A} \bigg) dV
++2\iint_{\Gamma_{R1}} \vec{T} \cdot \bigg(\hat{n}\times\vec{A}\bigg) dS.
+\end{aligned}
+\f}
+Any magnetic vector potential that minimizes this functional will satisfy
+the curl-curl equation (i), the boundary condition (ii) and the interface
+condition (iv). On the other hand, this functional is invariant to the
+interface condition (iii). This can be verified as follows: The
+first variation of the functional at the minimum vanishes,
+\f[
+\delta F(\vec{A}) = 0.
+\f]
+We can express the first variation and deduce that the identities given by
+equations (i), (ii), and (iv) in the boundary value problem must be satisfied
+for the last equation to be true. The interface condition (iii), however,
+is not needed for the last equation to be satisfied. The boundary and
+interface conditions that are needed to drive the first variation of the
+functional to zero are called natural and are labeled by (n) in the boundary
+value problem. If a condition is not natural, it is essential. We label
+essential conditions by (e).
+
+The natural conditions are enforced by minimization of the functional. The
+essential boundary and interface conditions must be enforced by other means.
+The interface condition (iii) in the boundary value problem above is enforced
+by choosing the finite elements wisely. The FE_Nedelec finite elements ensure
+continuity of the tangential component of the vector field they model on the
+faces of the cells. Therefore, if we construct the mesh such that all interfaces
+are made of cell faces, i.e., no interface runs through a cell, and choose the
+FE_Nedelec finite elements to model the magnetic vector potential, the
+interface condition (iii) will be enforced automatically. The Dirichlet boundary
+condition is essential. It is enforced by constraining the degrees of freedom
+in the system of linear equations. This can be done by invoking the deal.II
+function VectorTools::project_boundary_values_curl_conforming_l2(). The other
+two boundary conditions, Neumann and Robin, are natural.
+
+Note that the homogeneous Neumann boundary condition is embedded into the
+first integral of the functional. This integral is present in all functionals
+related to the curl-curl equation, even in the most minimalistic ones.
+Therefore, application of no boundary condition is quite impossible: the
+homogeneous Neumann boundary condition,
+\f[
+\frac{1}{\mu}\hat{n}\times\bigg(\vec{\nabla}\times\vec{A}\bigg) = 0,
+\f]
+will be applied implicitly by default as a result of the functional
+minimization.
+
+We can switch between the three boundary conditions as the following.
+- The functional above as it is corresponds to a boundary value problem with
+ Robin boundary condition, i.e., the boundary value problem given above.
+- To switch to Neumann boundary condition, we need to discard the second
+ integral from the functional. Such functional will correspond to the boundary
+ value problem above with equation (ii) replaced by the Neumann boundary
+ condition.
+- To switch to Dirichlet boundary condition, we use the functional with
+ discarded second integral and use
+ VectorTools::project_boundary_values_curl_conforming_l2() to constrain the
+ system of linear equations. Such configuration will correspond to the
+ boundary value problem above with equation (ii) replaced by the Dirichlet
+ boundary condition.
+
+In general, we need to sort all boundary and interface conditions into two
+categories, natural and essential. The reason for doing so is simple: we need to
+know which conditions are not taken care of by the functional (the essential
+conditions), so we can take care of them by other means (choice of finite
+elements, restricting dofs). Interrogating the first variation of the functional
+as discussed above is a common method of sorting the conditions. A more detailed
+discussion on boundary conditions can be found in step-22.
+(@dealiiVideoLectureSeeAlso{21.5,21.55,21.6,21.65})
+
+Next, we need to convert the functional above into a numerical recipe that can
+be programmed into a computer. To do so, we note that the final result, i.e., the
+magnetic vector potential computed by the finite element method, is represented
+as a sum,
+\f[
+\vec{A}(\vec{r}) = \sum_{j=0}^{m-1} c_j \vec{N}_j(\vec{r}),
+\f]
+where $c_j$ are the degrees of freedom and $\vec{N}_j$ are the vector-valued
+shape functions of the FE_Nedelec finite elements. We convert the functional into a
+multivariate function by substituting the last equation into the functional,
+\f[
+f(c_0, c_1, ..., c_{m-1}) = F\bigg(\sum_{j=0}^{m-1} c_j \vec{N}_j\bigg).
+\f]
+Then we compose the system of linear equations by computing the partial
+derivatives of the multivariate function:
+\f{equation}{
+\begin{array}{lr}
+\dfrac{\partial}{\partial c_0} f(c_0, c_1, ..., c_{m-1})& = 0, \\
+\dfrac{\partial}{\partial c_1} f(c_0, c_1, ..., c_{m-1})& = 0, \\
+... & \\
+\dfrac{\partial}{\partial c_{m-1}} f(c_0, c_1, ..., c_{m-1})& = 0.
+\end{array}
+\f}
+
+This system of linear equations can be written in matrix form as
+\f[
+\boldsymbol{A} \boldsymbol{c} = \boldsymbol{b},
+\f]
+where $\boldsymbol{c}$ is a column vector filled with the degrees of freedom,
+\f[
+\boldsymbol{c} = [c_0, c_1, ... c_{m-1}]^T.
+\f]
+The system matrix and the right-hand side column vector are computed as
+\f{equation}{
+\begin{aligned}
+& A_{ij} =
+\underbrace{\iiint_{\Omega}\frac{1}{\mu}
+\bigg(\vec{\nabla}\times\vec{N}_i\bigg) \cdot
+\bigg(\vec{\nabla}\times\vec{N}_j\bigg) dV}_{I_{a1}}
++
+\underbrace{\iint_{\Gamma_{R1}} \gamma
+\bigg(\hat{n}\times \vec{N}_i \bigg) \cdot
+\bigg(\hat{n}\times \vec{N}_j \bigg) dS}_{I_{a2}}
++
+\underbrace{\eta^2\iiint_{\Omega} \vec{N}_i \cdot \vec{N}_j dV}_{I_{a3}}
+\end{aligned}
+\f}
+and
+\f{equation}{
+b_i =
+\underbrace{\iiint_{\Omega}\vec{T}\cdot\bigg(\vec{\nabla}\times\vec{N}_i\bigg) dV
+}_{I_{b3-1}} -
+\underbrace{\iint_{\Gamma_{R1}}\vec{T}\cdot\bigg(\hat{n}\times\vec{N}_i\bigg)dS
+}_{I_{b3-2} = 0}.
+\f}
+The last two equations are implemented by the computer code. The integral $I_{b3-2}$
+equals zero as the tangential component of the current vector potential,
+$\hat{n} \times \vec{T}$, is forced to zero on the boundary $\Gamma_{R1}$ by
+the Dirichlet boundary condition. We will neglect this integral in the computer
+code.
+
+<h4>Current vector potential</h4>
+
+The problem of solving the boundary value problem
+for the current vector potential can be replaced by the problem of minimizing
+the following functional:
+\f{equation}{
+\begin{aligned}
+&F(\vec{T}) =
+\iiint_{\Omega}\bigg|\vec{\nabla}\times\vec{T}\bigg|^2 dV
++ \eta^2 \iiint_{\Omega}\mid\vec{T}\mid^2 dV
+-2\iiint_{\Omega} \vec{J}_f \cdot \bigg( \vec{\nabla} \times\vec{T} \bigg) dV
++2\iint_{\Gamma_{R1}} \vec{J}_f \cdot \bigg(\hat{n}\times\vec{T}\bigg) dS,
+\end{aligned}
+\f}
+and constraining the system of linear equations with a help of
+VectorTools::project_boundary_values_curl_conforming_l2().
+
+This functional is minimized by solving a system of linear equations with the
+following system matrix and right-hand side column vector:
+\f{equation}{
+\begin{aligned}
+& A_{ij} =
+\underbrace{\iiint_{\Omega}
+\bigg(\vec{\nabla}\times\vec{N}_i\bigg) \cdot
+\bigg(\vec{\nabla}\times\vec{N}_j\bigg) dV}_{I_{a1}}
++
+\underbrace{\eta^2\iiint_{\Omega} \vec{N}_i \cdot \vec{N}_j dV}_{I_{a3}},
+\end{aligned}
+\f}
+
+\f{equation}{
+b_i =
+\underbrace{\iiint_{\Omega}\vec{J}_f\cdot\bigg(\vec{\nabla}\times\vec{N}_i\bigg) dV
+}_{I_{b3-1}} -
+\underbrace{\iint_{\Gamma_{R1}}\vec{J}_f\cdot\bigg(\hat{n}\times\vec{N}_i\bigg)dS
+}_{I_{b3-2}=0}.
+\f}
+Here again, the integral $I_{b3-2}$ equals zero as the free-current density,
+$\vec{J}_f$, equals zero at the boundary $\Gamma_{R1}$ by definition of the
+problem. We will neglect this integral in the computer code.
+
+<h4>Converting potentials into fields</h4>
+
+As discussed above, we are interested in the magnetic field induced by the coil.
+For this reason, we convert the numerically computed vector potential into magnetic
+field as
+\f[
+\vec{B} = \vec{\nabla} \times \vec{A}.
+\f]
+The problem of computing this equation can be replaced by the problem of
+minimizing the following functional:
+\f[
+F(\vec{B}) = \iiint_{\Omega} \big| \vec{B} \big|^2 dV -
+2 \iiint_{\Omega} \bigg( \vec{\nabla} \times \vec{A} \bigg) \cdot \vec{B} dV.
+\f]
+This functional is minimized by solving a system of linear equations with the
+system matrix
+\f[
+A_{ij} = \underbrace{\iiint_{\Omega} \vec{N}_i \cdot \vec{N}_j dV}_{I_a}
+\f]
+and right-hand side column vector
+\f[
+b_i = \underbrace{\iiint_{\Omega} \bigg( \vec{\nabla} \times \vec{A} \bigg)
+\cdot \vec{N}_i dS}_{I_b}.
+\f]
+This time, however, $\vec{N}_i$ are the shape functions of the FE_RaviartThomas
+finite elements. The vector field $\vec{A}$ in the last equation is the numerically
+computed magnetic vector potential, i.e., a linear combination of the shape
+functions of the FE_Nedelec finite elements.
+
+It is informative to verify the quality of the computed current vector potential,
+$\vec{T}$. We can do this by converting it back into the free-current
+density, $\vec{J}_f$, and comparing the result to the closed-form analytical
+expression given by the definition of the problem, see above. The current vector
+potential relates to the free-current density as
+\f[
+\vec{J}_f = \vec{\nabla} \times \vec{T}.
+\f]
+This equation is, essentially, the same as the equation for converting
+$\vec{A}$ into $\vec{B}$, see the first equation of this section. Consequently,
+we can simply adapt the equations for the functional, system matrix, and the
+right-hand side given above by making the following substitutions:
+\f{equation}{
+\begin{array}{lcr}
+\vec{A} \rightarrow \vec{T} &\text{ and }& \vec{B} \rightarrow \vec{J}_f.
+\end{array}
+\f}
+
+The two conversions, $\vec{A}$ into $\vec{B}$ and $\vec{T}$ into $\vec{J}_f$,
+can be done by the same piece of code. We will call it a projector from
+$H(\text{curl})$ to $H(\text{div})$.
+
+<h3>Selecting finite elements</h3>
+
+The preceding sections leave an impression that we are going to model $\vec{A}$
+and $\vec{T}$ by the FE_Nedelec finite elements, while $\vec{B}$ and $\vec{J}_f$
+by the FE_RaviartThomas finite elements. It makes sense to discuss how this
+kind of choice can be made.
+
+The simplest method of assigning a particular type of finite elements to a
+physical quantity studied in electromagnetics is a contemplation of the Bossavit's
+diagram. It is presented below. The Bossavit's diagram describes the relations
+between physical quantities as they are given by Maxwell's equations and
+constitutive relations. Most equations derived from Maxwell's equations
+can be captured by this diagram.
+
+@htmlonly
+<p align="center">
+ <img src="images/steps/developer/step-97-diagram.svg" alt="The Bossavit's diagram" height="283">
+</p>
+@endhtmlonly
+
+Let us consider assigning the correct type of finite elements to the magnetic
+vector potential, $\vec{A}$. First, we contemplate the Bossavit's diagram and
+observe that $\vec{A}$ belongs to the $H(\text{curl})$ function space. Second,
+we look at the table below and conclude that the physical quantities that belong
+to the $H(\text{curl})$ function space are modeled by the FE_Nedelec finite
+elements. Therefore, we need to model the magnetic vector potential, $\vec{A}$,
+by the FE_Nedelec finite elements. The same assigning procedure can be applied
+to the rest of the physical quantities, $\vec{B}$, $\vec{T}$, and $\vec{J}_f$.
+
+|Funct. space |Finite elements |
+|-----------------|-------------------|
+|$H(\text{grad})$ | FE_Q |
+|$H(\text{curl})$ | FE_Nedelec |
+|$H(\text{div})$ | FE_RaviartThomas |
+|$L^2$ | FE_DGQ |
+
+Alternatively, one of the types of the finite elements listed in the table can
+be assigned to a physical quantity by considering the behavior of the physical
+quantity on the interfaces between dissimilar materials. Let us consider the
+magnetic field $\vec{B}$ as an example. The normal component of the magnetic
+field is continuous on interfaces. On the contrary, the tangential component is
+discontinuous, see the first figure on this page. This type of behavior can be
+modeled by the FE_RaviartThomas finite elements as they guarantee the continuity
+of the normal component of the vector field on the faces of mesh cells.
+
+<h3>Mesh</h3>
+
+The mesh is, essentially, a discretization of the problem domain shown above.
+The mesh is constructed such that all spherical surfaces of the problem domain
+are delineated by cell faces. That is, no spherical surface runs through a mesh
+cell. The figure below illustrates the mesh. The main elements of the mesh are
+listed below.
+
+- The cube with a side of $2d_1$ in the middle of the mesh. The purpose of
+ this cube is instrumental. This is the only way to construct a mesh that has
+ spherical interfaces and contains no tetrahedral cells as the cells around the
+ origin must be hexahedral.
+- The spherical shell nr.1. This shell represents the magnetic core. The
+ permeability of this region is $\mu_1$. The permeability outside this region
+ is $\mu_0$.
+- The spherical shell nr.2. This region contains the free current. There is no
+ free current outside this region.
+- Sphere nr.1. The $L^2$ error norms are computed inside the region delineated
+ by this surface.
+- Sphere nr.2. The boundary of the problem domain. Represents infinity. The
+ boundary condition is applied to this surface.
+
+The program uses four meshes created with a help of gmsh. Each mesh has a
+different degree of refinement. The degree of refinement in each mesh is
+defined by a number of mesh nodes on the transfinite lines, $r$. The figure
+below illustrates a mesh with $r=5$. The program uses four meshes with the
+amount of nodes on the transfinite lines in the interval $r = [6, 9]$.
+
+Note that this tutorial uses only globally refined meshes. There are no
+non-conforming cells, hanging nodes, and hanging node constrains in this
+tutorial. We will use constrains only for the purpose of enforcing the
+Dirichlet boundary condition.
+
+@htmlonly
+<p align="center">
+ <img src="images/steps/developer/step-97-mesh.svg" alt="The mesh" height="531">
+</p>
+@endhtmlonly
+
+<h3>Overview of the program</h3>
+
+The program runs in a loop. In each iteration of the loop a different parameter
+$r$ (the mesh refinement parameter, i.e., the amount of nodes on transfinite
+lines) is assumed. Each iteration consists of four stages:
+
+- The current vector potential, $\vec{T}$, is computed given the closed-form
+ analytical expression for the free-current density, $\vec{J}_f$, see above.
+ No error norms are computed. The mesh is loaded at this stage. This mesh is
+ simply reused at all other stages.
+- The current vector potential computed at the preceding stage, $\vec{T}$, is
+ converted numerically back into free-current density, $\vec{J}_f$, for
+ verification. The $L^2$ error norm is computed by comparing the numerical
+ result with the closed-form analytical expression. The $L^2$ error norm is
+ saved in a convergence table `table_Jf`.
+- The magnetic vector potential, $\vec{A}$, is computed given the current vector
+ potential computed at the first stage, $\vec{T}$. No error norms are computed
+ at this stage.
+- The magnetic vector potential computed at the preceding stage, $\vec{A}$, is
+ converted into magnetic field, $\vec{B}$. The $L^2$ norm is computed by
+ comparing the numerical result with the closed-form analytical solution. The
+ $L^2$ error norm is saved in a convergence table `table_B`.
+
+The first stage is implemented by the code contained in the name space
+`SolverT`. The third stage is implemented by the code contained in the name
+space `SolverA`. The second and the fourth stages are implemented by the code
+contained in the name space `ProjectorHcurlToHdiv`. The name space
+`ExactSolutions` contains exact closed-form analytical expressions for
+$\vec{B}$ and $\vec{J}_f$.
+
+No error norms are computed at the first and the third stages. Computing error
+norms at these stages makes no sense as we apply implicit gauges and the
+conservative portion of the solution is unknown. We can, however, judge the
+quality of the simulations at these two stages indirectly by evaluating the
+quality of the simulations at the second and the fourth stages.
--- /dev/null
+techniques
--- /dev/null
+<h1>Results</h1>
+
+The program generates the following output in the command line interface by
+default.
+
+@code
+Solving for (p = 0): T Jf A B T Jf A B T Jf A B T Jf A B
+@endcode
+
+The program assumes the finite elements of the lowermost degree, $p = 0$.
+To change the degree of the finite elements, say $p = 2$, one needs to change
+the setting `Settings::fe_degree = 2` and rebuild the program.
+
+The program also dumps a number of files in the current directory. In the default
+configuration these files are:
+- vtu files. They contain the computed vector fields. Recall that the spherical
+ manifold is attached to many cell faces. Consequently, these cell faces are
+ curved. They look more like patches of a sphere. Furthermore, the shape
+ functions are mapped from the reference cell to the real mesh cells by the
+ second-order mapping to accommodate the cells with curved faces. For these
+ reasons, one needs to use a visualization software that can deal with curved
+ faces and the higher-order mapping. A fresh version of ParaView is recommended.
+ Visit will not do. The <a href="https://github.com/dealii/dealii/wiki/Notes-on-visualizing-high-order-output">
+ Notes on visualizing high order output</a> provide more information on this topic.
+- tex files. These files contain the convergence tables.
+
+The following provides examples of the convergence tables simulated with the
+default settings for three different degrees of the finite elements,
+$p = 0, 1, 2$.
+
+| p | r | cells | dofs |$\|e\|_{L^2}$|$\alpha_{L^2}$|
+|-- |---|-------|---------|----------|------|
+| 0 | 6 | 4625 | 13950 | 1.66e-01 | - |
+| 0 | 7 | 7992 | 24084 | 1.38e-01 | 0.99 |
+| 0 | 8 | 12691 | 38220 | 1.19e-01 | 0.99 |
+| 0 | 9 | 18944 | 57024 | 1.04e-01 | 0.99 |
+| 1 | 6 | 4625 | 111300 | 8.12e-04 | - |
+| 1 | 7 | 7992 | 192240 | 4.97e-04 | 2.69 |
+| 1 | 8 | 12691 | 305172 | 3.32e-04 | 2.61 |
+| 1 | 9 | 18944 | 455424 | 2.37e-04 | 2.54 |
+| 2 | 6 | 4625 | 375300 | 6.78e-04 | - |
+| 2 | 7 | 7992 | 648324 | 3.94e-04 | 2.97 |
+| 2 | 8 | 12691 | 1029294 | 2.49e-04 | 2.98 |
+| 2 | 9 | 18944 | 1536192 | 1.67e-04 | 2.99 |
+
+**Table 1. Convergence table. Free-current density, $\vec{J}_f$.
+
+| p | r | cells | dofs |$\|e\|_{L^2}$|$\alpha_{L^2}$|
+|-- |---|-------|---------|----------|------|
+| 0 | 6 | 4625 | 13950 | 8.84e-08 | - |
+| 0 | 7 | 7992 | 24084 | 7.36e-08 | 1.00 |
+| 0 | 8 | 12691 | 38220 | 6.30e-08 | 1.01 |
+| 0 | 9 | 18944 | 57024 | 5.51e-08 | 1.00 |
+| 1 | 6 | 4625 | 111300 | 4.41e-09 | - |
+| 1 | 7 | 7992 | 192240 | 3.11e-09 | 1.91 |
+| 1 | 8 | 12691 | 305172 | 2.23e-09 | 2.18 |
+| 1 | 9 | 18944 | 455424 | 1.71e-09 | 1.96 |
+| 2 | 6 | 4625 | 375300 | 1.84e-10 | - |
+| 2 | 7 | 7992 | 648324 | 1.03e-10 | 3.21 |
+| 2 | 8 | 12691 | 1029294 | 6.08e-11 | 3.40 |
+| 2 | 9 | 18944 | 1536192 | 4.04e-11 | 3.07 |
+
+**Table 2. Convergence table. Magnetic field, $\vec{B}$.
+
+The following notations were used in the headers of the tables:
+
+- p - the degree of the finite elements.
+
+- r - the mesh refinement parameter, i.e., the number of nodes on the transfinite
+lines.
+
+- cells - the total amount of active cells.
+
+- dofs - the amount of degrees of freedom.
+
+-$\|e\|_{L^2}$ - the $L^2$ error norm.
+
+-$\alpha_{L^2}$ - the order of convergence of the $L^2$ error norm.
+
+If `Settings::log_cg_convergence = true`, the program saves the convergence data
+of the CG solver into csv files.
+
+The vector representations of the calculated vector fields, $\vec{J}_f$ and
+$\vec{B}$, are illustrated above by the first figure on this page. The figures
+below illustrate slices of the magnitudes of these fields. The figures below
+were simulated with $p = 2$ and $r = 9$. Visual inspection of the vector
+potentials is not very informative as their conservative portions are unknown.
+
+@htmlonly
+<p align="center">
+ <img src="images/steps/developer/step-97-Jf.svg" alt="The result - free-current
+ density" height="531">
+</p>
+@endhtmlonly
+
+@htmlonly
+<p align="center">
+ <img src="images/steps/developer/step-97-B.svg" alt="The result - magnetic field
+ density" height="531">
+</p>
+@endhtmlonly
+
+<h3>Possibilities for extensions</h3>
+
+Repeat the simulations for the three types of the boundary conditions,
+Dirichlet, Neumann, and Robin. The Robin boundary condition is supposed to be
+superior to the other two. Look at the simulated data to see that this is indeed
+the case. You can save the projected exact solution next to the simulated
+solutions into the vtu files, just set `Settings::project_exact_solution = true`.
+ParaView has "Plot Over Line" filter. You can use this filter to visualize the
+difference between the exact solution and a solution simulated with a particular
+boundary condition. You can also draw conclusions by observing the convergence
+tables. Keep in mind the $\eta^2$ parameter. Increase it if the CG solver chokes
+while you are experimenting. Note that the benefits offered by the Robin
+boundary condition are observed the best when higher-order finite elements are
+used, i.e., $p = 1$ and $p = 2$.
+
+The Robin boundary condition as described above is also called the first-order
+asymptotic boundary condition (ABC). There exist ABCs of higher orders
+@cite gratkowski2010p. Implement and test the second-order ABC to see if it
+performs any better. There exist improvised asymptotic boundary conditions, IABCs,
+@cite meeker2013a. Try to implement the first order IABC.
--- /dev/null
+The curl-curl equation. FE_Nedelec, FE_RaviartThomas. Magnetic vector potential. Current vector potential.
--- /dev/null
+/* ------------------------------------------------------------------------
+ *
+ * SPDX-License-Identifier: LGPL-2.1-or-later
+ * Copyright (C) 2024 by the deal.II authors
+ *
+ * This file is part of the deal.II library.
+ *
+ * Part of the source code is dual licensed under Apache-2.0 WITH
+ * LLVM-exception OR LGPL-2.1-or-later. Detailed license information
+ * governing the source code and code contributions can be found in
+ * LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
+ *
+ * ------------------------------------------------------------------------*/
+
+r = 6;
+
+d1 = 0.1;
+a1 = 0.3;
+b1 = 0.6;
+a2 = 0.9;
+b2 = 1.2;
+d2 = 2.0;
+d3 = 3.0;
+
+a13 = a1/Sqrt(3);
+b13 = b1/Sqrt(3);
+
+a23 = a2/Sqrt(3);
+b23 = b2/Sqrt(3);
+
+d23 = d2/Sqrt(3);
+d33 = d3/Sqrt(3);
+
+Point(0) = { 0, 0, 0};
+
+Point(1) = { d1, d1, d1};
+Point(2) = { d1,-d1, d1};
+
+Point(3) = { a13, a13, a13};
+Point(4) = { a13,-a13, a13};
+
+Point(5) = { b13, b13, b13};
+Point(6) = { b13,-b13, b13};
+
+Point(7) = { a23, a23, a23};
+Point(8) = { a23,-a23, a23};
+
+Point(9) = { b23, b23, b23};
+Point(10) = { b23,-b23, b23};
+
+Point(11) = { d23, d23, d23};
+Point(12) = { d23,-d23, d23};
+
+Point(13) = { d33, d33, d33};
+Point(14) = { d33,-d33, d33};
+
+Line(1) = {1, 3};
+Line(2) = {3, 5};
+Line(3) = {5, 7};
+Line(4) = {7, 9};
+Line(5) = {9, 11};
+
+Line(7) = {2, 4};
+Line(8) = {4, 6};
+Line(9) = {6, 8};
+Line(10) = {8, 10};
+Line(11) = {10, 12};
+
+Line(13) = {1, 2};
+
+Circle(14) = {3, 0, 4};
+Circle(15) = {5, 0, 6};
+Circle(16) = {7, 0, 8};
+Circle(17) = {9, 0, 10};
+
+Circle(18) = {11, 0, 12};
+
+Line(19) = {11,13};
+Line(20) = {12,14};
+
+Circle(21) = {13, 0, 14};
+
+Line Loop(1) = {7, -14, -1, 13};
+Plane Surface(1) = {1};
+
+Line Loop(2) = {8, -15, -2, 14};
+Plane Surface(2) = {2};
+
+Line Loop(3) = {9, -16, -3, 15};
+Plane Surface(3) = {3};
+
+Line Loop(4) = {10, -17, -4, 16};
+Plane Surface(4) = {4};
+
+Line Loop(5) = {11, -18, -5, 17};
+Plane Surface(5) = {5};
+
+Line Loop(6) = {20, -21, -19, 18};
+Plane Surface(6) = {6};
+
+Q1[] = Symmetry {0, 1, 1, 0} {Duplicata {Surface{:};}};
+Q2[] = Symmetry {0, -1, 1, 0} {Duplicata {Surface{:};}};
+
+ll = newll; Line Loop(ll) = {13, -26, 85, -55};
+Plane Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {14, 24, -83, 53};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {15, 29, -88, 58};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {16, 34, -93, 63};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {17, 39, -98, 68};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {18, 44, -103, 73};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {21, 49, -108, 78};
+Surface(ll) = {ll};
+
+Surface Loop(1) = {1, 22, 81, 51, 109, 110};
+Volume(1) = {1};
+
+Surface Loop(2) = {2, 27, 86, 56, 110, 111};
+Volume(2) = {2};
+
+Surface Loop(3) = {3, 32, 91, 61, 111, 112};
+Volume(3) = {3};
+
+Surface Loop(4) = {4, 37, 96, 66, 112, 113};
+Volume(4) = {4};
+
+Surface Loop(5) = {5, 42, 101, 71, 113, 114};
+Volume(5) = {5};
+
+Surface Loop(6) = {6, 47, 106, 76, 114, 115};
+Volume(6) = {6};
+
+V1[] = Symmetry {1, 1, 0, 0} {Duplicata {Volume{:};}};
+V2[] = Symmetry {-1, 1, 0, 0} {Duplicata {Volume{:};}};
+V3[] = Symmetry {1, 0, 1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+V4[] = Symmetry {1, 0, -1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+
+Surface Loop(7) = {109, 319, 505, 137, 687, 869};
+Volume(7) = {7};
+
+Physical Volume(1) = {Volume{:}};
+
+Physical Surface(2) = {115, 479, 665, 297, 847, 1029};
+
+Recombine Surface "*";
+Recombine Volume "*";
+
+Transfinite Volume "*";
+Transfinite Surface "*";
+Transfinite Line "*" = r;
--- /dev/null
+$MeshFormat
+2.2 0 8
+$EndMeshFormat
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+$EndElements
--- /dev/null
+/* ------------------------------------------------------------------------
+ *
+ * SPDX-License-Identifier: LGPL-2.1-or-later
+ * Copyright (C) 2024 by the deal.II authors
+ *
+ * This file is part of the deal.II library.
+ *
+ * Part of the source code is dual licensed under Apache-2.0 WITH
+ * LLVM-exception OR LGPL-2.1-or-later. Detailed license information
+ * governing the source code and code contributions can be found in
+ * LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
+ *
+ * ------------------------------------------------------------------------*/
+
+r = 7;
+
+d1 = 0.1;
+a1 = 0.3;
+b1 = 0.6;
+a2 = 0.9;
+b2 = 1.2;
+d2 = 2.0;
+d3 = 3.0;
+
+a13 = a1/Sqrt(3);
+b13 = b1/Sqrt(3);
+
+a23 = a2/Sqrt(3);
+b23 = b2/Sqrt(3);
+
+d23 = d2/Sqrt(3);
+d33 = d3/Sqrt(3);
+
+Point(0) = { 0, 0, 0};
+
+Point(1) = { d1, d1, d1};
+Point(2) = { d1,-d1, d1};
+
+Point(3) = { a13, a13, a13};
+Point(4) = { a13,-a13, a13};
+
+Point(5) = { b13, b13, b13};
+Point(6) = { b13,-b13, b13};
+
+Point(7) = { a23, a23, a23};
+Point(8) = { a23,-a23, a23};
+
+Point(9) = { b23, b23, b23};
+Point(10) = { b23,-b23, b23};
+
+Point(11) = { d23, d23, d23};
+Point(12) = { d23,-d23, d23};
+
+Point(13) = { d33, d33, d33};
+Point(14) = { d33,-d33, d33};
+
+Line(1) = {1, 3};
+Line(2) = {3, 5};
+Line(3) = {5, 7};
+Line(4) = {7, 9};
+Line(5) = {9, 11};
+
+Line(7) = {2, 4};
+Line(8) = {4, 6};
+Line(9) = {6, 8};
+Line(10) = {8, 10};
+Line(11) = {10, 12};
+
+Line(13) = {1, 2};
+
+Circle(14) = {3, 0, 4};
+Circle(15) = {5, 0, 6};
+Circle(16) = {7, 0, 8};
+Circle(17) = {9, 0, 10};
+
+Circle(18) = {11, 0, 12};
+
+Line(19) = {11,13};
+Line(20) = {12,14};
+
+Circle(21) = {13, 0, 14};
+
+Line Loop(1) = {7, -14, -1, 13};
+Plane Surface(1) = {1};
+
+Line Loop(2) = {8, -15, -2, 14};
+Plane Surface(2) = {2};
+
+Line Loop(3) = {9, -16, -3, 15};
+Plane Surface(3) = {3};
+
+Line Loop(4) = {10, -17, -4, 16};
+Plane Surface(4) = {4};
+
+Line Loop(5) = {11, -18, -5, 17};
+Plane Surface(5) = {5};
+
+Line Loop(6) = {20, -21, -19, 18};
+Plane Surface(6) = {6};
+
+Q1[] = Symmetry {0, 1, 1, 0} {Duplicata {Surface{:};}};
+Q2[] = Symmetry {0, -1, 1, 0} {Duplicata {Surface{:};}};
+
+ll = newll; Line Loop(ll) = {13, -26, 85, -55};
+Plane Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {14, 24, -83, 53};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {15, 29, -88, 58};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {16, 34, -93, 63};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {17, 39, -98, 68};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {18, 44, -103, 73};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {21, 49, -108, 78};
+Surface(ll) = {ll};
+
+Surface Loop(1) = {1, 22, 81, 51, 109, 110};
+Volume(1) = {1};
+
+Surface Loop(2) = {2, 27, 86, 56, 110, 111};
+Volume(2) = {2};
+
+Surface Loop(3) = {3, 32, 91, 61, 111, 112};
+Volume(3) = {3};
+
+Surface Loop(4) = {4, 37, 96, 66, 112, 113};
+Volume(4) = {4};
+
+Surface Loop(5) = {5, 42, 101, 71, 113, 114};
+Volume(5) = {5};
+
+Surface Loop(6) = {6, 47, 106, 76, 114, 115};
+Volume(6) = {6};
+
+V1[] = Symmetry {1, 1, 0, 0} {Duplicata {Volume{:};}};
+V2[] = Symmetry {-1, 1, 0, 0} {Duplicata {Volume{:};}};
+V3[] = Symmetry {1, 0, 1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+V4[] = Symmetry {1, 0, -1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+
+Surface Loop(7) = {109, 319, 505, 137, 687, 869};
+Volume(7) = {7};
+
+Physical Volume(1) = {Volume{:}};
+
+Physical Surface(2) = {115, 479, 665, 297, 847, 1029};
+
+Recombine Surface "*";
+Recombine Volume "*";
+
+Transfinite Volume "*";
+Transfinite Surface "*";
+Transfinite Line "*" = r;
--- /dev/null
+$MeshFormat
+2.2 0 8
+$EndMeshFormat
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+$EndElements
--- /dev/null
+/* ------------------------------------------------------------------------
+ *
+ * SPDX-License-Identifier: LGPL-2.1-or-later
+ * Copyright (C) 2024 by the deal.II authors
+ *
+ * This file is part of the deal.II library.
+ *
+ * Part of the source code is dual licensed under Apache-2.0 WITH
+ * LLVM-exception OR LGPL-2.1-or-later. Detailed license information
+ * governing the source code and code contributions can be found in
+ * LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
+ *
+ * ------------------------------------------------------------------------*/
+
+r = 8;
+
+d1 = 0.1;
+a1 = 0.3;
+b1 = 0.6;
+a2 = 0.9;
+b2 = 1.2;
+d2 = 2.0;
+d3 = 3.0;
+
+a13 = a1/Sqrt(3);
+b13 = b1/Sqrt(3);
+
+a23 = a2/Sqrt(3);
+b23 = b2/Sqrt(3);
+
+d23 = d2/Sqrt(3);
+d33 = d3/Sqrt(3);
+
+Point(0) = { 0, 0, 0};
+
+Point(1) = { d1, d1, d1};
+Point(2) = { d1,-d1, d1};
+
+Point(3) = { a13, a13, a13};
+Point(4) = { a13,-a13, a13};
+
+Point(5) = { b13, b13, b13};
+Point(6) = { b13,-b13, b13};
+
+Point(7) = { a23, a23, a23};
+Point(8) = { a23,-a23, a23};
+
+Point(9) = { b23, b23, b23};
+Point(10) = { b23,-b23, b23};
+
+Point(11) = { d23, d23, d23};
+Point(12) = { d23,-d23, d23};
+
+Point(13) = { d33, d33, d33};
+Point(14) = { d33,-d33, d33};
+
+Line(1) = {1, 3};
+Line(2) = {3, 5};
+Line(3) = {5, 7};
+Line(4) = {7, 9};
+Line(5) = {9, 11};
+
+Line(7) = {2, 4};
+Line(8) = {4, 6};
+Line(9) = {6, 8};
+Line(10) = {8, 10};
+Line(11) = {10, 12};
+
+Line(13) = {1, 2};
+
+Circle(14) = {3, 0, 4};
+Circle(15) = {5, 0, 6};
+Circle(16) = {7, 0, 8};
+Circle(17) = {9, 0, 10};
+
+Circle(18) = {11, 0, 12};
+
+Line(19) = {11,13};
+Line(20) = {12,14};
+
+Circle(21) = {13, 0, 14};
+
+Line Loop(1) = {7, -14, -1, 13};
+Plane Surface(1) = {1};
+
+Line Loop(2) = {8, -15, -2, 14};
+Plane Surface(2) = {2};
+
+Line Loop(3) = {9, -16, -3, 15};
+Plane Surface(3) = {3};
+
+Line Loop(4) = {10, -17, -4, 16};
+Plane Surface(4) = {4};
+
+Line Loop(5) = {11, -18, -5, 17};
+Plane Surface(5) = {5};
+
+Line Loop(6) = {20, -21, -19, 18};
+Plane Surface(6) = {6};
+
+Q1[] = Symmetry {0, 1, 1, 0} {Duplicata {Surface{:};}};
+Q2[] = Symmetry {0, -1, 1, 0} {Duplicata {Surface{:};}};
+
+ll = newll; Line Loop(ll) = {13, -26, 85, -55};
+Plane Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {14, 24, -83, 53};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {15, 29, -88, 58};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {16, 34, -93, 63};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {17, 39, -98, 68};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {18, 44, -103, 73};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {21, 49, -108, 78};
+Surface(ll) = {ll};
+
+Surface Loop(1) = {1, 22, 81, 51, 109, 110};
+Volume(1) = {1};
+
+Surface Loop(2) = {2, 27, 86, 56, 110, 111};
+Volume(2) = {2};
+
+Surface Loop(3) = {3, 32, 91, 61, 111, 112};
+Volume(3) = {3};
+
+Surface Loop(4) = {4, 37, 96, 66, 112, 113};
+Volume(4) = {4};
+
+Surface Loop(5) = {5, 42, 101, 71, 113, 114};
+Volume(5) = {5};
+
+Surface Loop(6) = {6, 47, 106, 76, 114, 115};
+Volume(6) = {6};
+
+V1[] = Symmetry {1, 1, 0, 0} {Duplicata {Volume{:};}};
+V2[] = Symmetry {-1, 1, 0, 0} {Duplicata {Volume{:};}};
+V3[] = Symmetry {1, 0, 1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+V4[] = Symmetry {1, 0, -1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+
+Surface Loop(7) = {109, 319, 505, 137, 687, 869};
+Volume(7) = {7};
+
+Physical Volume(1) = {Volume{:}};
+
+Physical Surface(2) = {115, 479, 665, 297, 847, 1029};
+
+Recombine Surface "*";
+Recombine Volume "*";
+
+Transfinite Volume "*";
+Transfinite Surface "*";
+Transfinite Line "*" = r;
--- /dev/null
+$MeshFormat
+2.2 0 8
+$EndMeshFormat
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+12982 5 2 1 1003 12941 4919 744 3654 12942 4920 743 3653
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+12984 5 2 1 1003 12943 4921 742 3652 12944 4922 741 3651
+12985 5 2 1 1003 12944 4922 741 3651 4371 837 55 740
+$EndElements
--- /dev/null
+/* ------------------------------------------------------------------------
+ *
+ * SPDX-License-Identifier: LGPL-2.1-or-later
+ * Copyright (C) 2024 by the deal.II authors
+ *
+ * This file is part of the deal.II library.
+ *
+ * Part of the source code is dual licensed under Apache-2.0 WITH
+ * LLVM-exception OR LGPL-2.1-or-later. Detailed license information
+ * governing the source code and code contributions can be found in
+ * LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
+ *
+ * ------------------------------------------------------------------------*/
+
+r = 9;
+
+d1 = 0.1;
+a1 = 0.3;
+b1 = 0.6;
+a2 = 0.9;
+b2 = 1.2;
+d2 = 2.0;
+d3 = 3.0;
+
+a13 = a1/Sqrt(3);
+b13 = b1/Sqrt(3);
+
+a23 = a2/Sqrt(3);
+b23 = b2/Sqrt(3);
+
+d23 = d2/Sqrt(3);
+d33 = d3/Sqrt(3);
+
+Point(0) = { 0, 0, 0};
+
+Point(1) = { d1, d1, d1};
+Point(2) = { d1,-d1, d1};
+
+Point(3) = { a13, a13, a13};
+Point(4) = { a13,-a13, a13};
+
+Point(5) = { b13, b13, b13};
+Point(6) = { b13,-b13, b13};
+
+Point(7) = { a23, a23, a23};
+Point(8) = { a23,-a23, a23};
+
+Point(9) = { b23, b23, b23};
+Point(10) = { b23,-b23, b23};
+
+Point(11) = { d23, d23, d23};
+Point(12) = { d23,-d23, d23};
+
+Point(13) = { d33, d33, d33};
+Point(14) = { d33,-d33, d33};
+
+Line(1) = {1, 3};
+Line(2) = {3, 5};
+Line(3) = {5, 7};
+Line(4) = {7, 9};
+Line(5) = {9, 11};
+
+Line(7) = {2, 4};
+Line(8) = {4, 6};
+Line(9) = {6, 8};
+Line(10) = {8, 10};
+Line(11) = {10, 12};
+
+Line(13) = {1, 2};
+
+Circle(14) = {3, 0, 4};
+Circle(15) = {5, 0, 6};
+Circle(16) = {7, 0, 8};
+Circle(17) = {9, 0, 10};
+
+Circle(18) = {11, 0, 12};
+
+Line(19) = {11,13};
+Line(20) = {12,14};
+
+Circle(21) = {13, 0, 14};
+
+Line Loop(1) = {7, -14, -1, 13};
+Plane Surface(1) = {1};
+
+Line Loop(2) = {8, -15, -2, 14};
+Plane Surface(2) = {2};
+
+Line Loop(3) = {9, -16, -3, 15};
+Plane Surface(3) = {3};
+
+Line Loop(4) = {10, -17, -4, 16};
+Plane Surface(4) = {4};
+
+Line Loop(5) = {11, -18, -5, 17};
+Plane Surface(5) = {5};
+
+Line Loop(6) = {20, -21, -19, 18};
+Plane Surface(6) = {6};
+
+Q1[] = Symmetry {0, 1, 1, 0} {Duplicata {Surface{:};}};
+Q2[] = Symmetry {0, -1, 1, 0} {Duplicata {Surface{:};}};
+
+ll = newll; Line Loop(ll) = {13, -26, 85, -55};
+Plane Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {14, 24, -83, 53};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {15, 29, -88, 58};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {16, 34, -93, 63};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {17, 39, -98, 68};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {18, 44, -103, 73};
+Surface(ll) = {ll};
+
+ll = newll; Line Loop(ll) = {21, 49, -108, 78};
+Surface(ll) = {ll};
+
+Surface Loop(1) = {1, 22, 81, 51, 109, 110};
+Volume(1) = {1};
+
+Surface Loop(2) = {2, 27, 86, 56, 110, 111};
+Volume(2) = {2};
+
+Surface Loop(3) = {3, 32, 91, 61, 111, 112};
+Volume(3) = {3};
+
+Surface Loop(4) = {4, 37, 96, 66, 112, 113};
+Volume(4) = {4};
+
+Surface Loop(5) = {5, 42, 101, 71, 113, 114};
+Volume(5) = {5};
+
+Surface Loop(6) = {6, 47, 106, 76, 114, 115};
+Volume(6) = {6};
+
+V1[] = Symmetry {1, 1, 0, 0} {Duplicata {Volume{:};}};
+V2[] = Symmetry {-1, 1, 0, 0} {Duplicata {Volume{:};}};
+V3[] = Symmetry {1, 0, 1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+V4[] = Symmetry {1, 0, -1, 0} {Duplicata {Volume{1, 2, 3, 4, 5, 6};}};
+
+Surface Loop(7) = {109, 319, 505, 137, 687, 869};
+Volume(7) = {7};
+
+Physical Volume(1) = {Volume{:}};
+
+Physical Surface(2) = {115, 479, 665, 297, 847, 1029};
+
+Recombine Surface "*";
+Recombine Volume "*";
+
+Transfinite Volume "*";
+Transfinite Surface "*";
+Transfinite Line "*" = r;
--- /dev/null
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+$EndElements
--- /dev/null
+/* ------------------------------------------------------------------------
+ *
+ * SPDX-License-Identifier: LGPL-2.1-or-later
+ * Copyright (C) 2024 by the deal.II authors
+ *
+ * This file is part of the deal.II library.
+ *
+ * Part of the source code is dual licensed under Apache-2.0 WITH
+ * LLVM-exception OR LGPL-2.1-or-later. Detailed license information
+ * governing the source code and code contributions can be found in
+ * LICENSE.md and CONTRIBUTING.md at the top level directory of deal.II.
+ *
+ * ------------------------------------------------------------------------
+ *
+ * This program was contributed by Siarhei Uzunbajakau, www.cembooks.nl, 2025.
+ */
+
+#include <deal.II/base/tensor.h>
+#include <deal.II/base/function.h>
+#include <deal.II/base/convergence_table.h>
+#include <deal.II/base/timer.h>
+#include <deal.II/base/types.h>
+#include <deal.II/base/work_stream.h>
+#include <deal.II/base/multithread_info.h>
+
+#include <deal.II/grid/tria.h>
+#include <deal.II/grid/grid_in.h>
+#include <deal.II/grid/grid_out.h>
+#include <deal.II/grid/manifold_lib.h>
+
+#include <deal.II/dofs/dof_handler.h>
+#include <deal.II/dofs/dof_tools.h>
+
+#include <deal.II/fe/fe_nedelec.h>
+#include <deal.II/fe/fe_raviart_thomas.h>
+#include <deal.II/fe/mapping_q.h>
+#include <deal.II/fe/fe_values.h>
+
+#include <deal.II/lac/vector.h>
+#include <deal.II/lac/sparse_matrix.h>
+#include <deal.II/lac/affine_constraints.h>
+#include <deal.II/lac/sparsity_pattern.h>
+#include <deal.II/lac/dynamic_sparsity_pattern.h>
+#include <deal.II/lac/solver_control.h>
+#include <deal.II/lac/vector_memory.h>
+#include <deal.II/lac/solver_cg.h>
+#include <deal.II/lac/precondition.h>
+
+#include <deal.II/numerics/vector_tools.h>
+#include <deal.II/numerics/data_component_interpretation.h>
+#include <deal.II/numerics/data_out.h>
+
+#include <string>
+#include <ostream>
+
+using namespace dealii;
+
+// This enumeration is used to switch between boundary conditions when solving
+// for the magnetic vector potential, $\vec{A}$. The boundary condition is fixed
+// (Dirichlet) when solving for the current vector potential, $\vec{T}$.
+enum BoundaryConditionType
+{
+ Dirichlet,
+ Neumann,
+ Robin
+};
+
+// @sect3{Settings}
+
+// The following is the control panel of the program. The scaling of the
+// program can be changed by setting `mu_0 = 1.0`. Then the computed magnetic
+// field, $\vec{B}$, will have to be multiplied by a factor of
+// $\mu_0 = 1.25664 \cdot 10^{-6}$. The free-current density, $\vec{J}_f$, does
+// not depend on scaling. The setting `boundary_condition_type_A` can be used
+// to switch between Dirichlet, Neumann, and Robin (ABC) boundary conditions.
+// This has an effect only on the solver that solves for the magnetic vector
+// potential, $\vec{A}$. The solver that solves for the current vector
+// potential,
+// $\vec{T}$, applies the Dirichlet boundary condition. This cannot be changed.
+// Recall, that forcing to zero the tangential component of $\vec{T}$ on the
+// boundary allows us to skip the boundary integral $I_{b3-2}$. The boundary ID
+// is set in the geo files that describe the mesh geometry. This is done by
+// specifying the physical surface, i.e.,
+// @code
+// Physical Surface(2) = {115, 479, 665, 297, 847, 1029};
+// @endcode
+// The boundary ID in the geo file must match the boundary ID setting below.
+// If `project_exact_solution = true`, the program projects the exact
+// solutions for $\vec{J}_f$ and $\vec{B}$ onto $H(\text{div})$ function space
+// and saves the results into corresponding vtu files next to the numerical
+// solutions. If the projected exact solution and the numerical solution look
+// alike, it is a good sign. In case of problems with the convergence of the
+// CG solver the parameter $\eta^2$ must be increased just a bit.
+namespace Settings
+{
+ const double permeability_fs = 1.2566370614359172954e-6;
+ const double mu_0 = permeability_fs; // Permeability of free space.
+
+ const double mu_r = 4; // Relative permeability of the magnetic material.
+ const double mu_1 = mu_0 * mu_r; // Permeability of the magnetic material.
+ const double d1 = 0.1; // Half-side of the cube in the center of the mesh.
+ const double a1 = 0.3; // Inner radius of the magnetic core.
+ const double b1 = 0.6; // Outer radius of the magnetic core.
+ const double a2 = 0.9; // Inner radius of the free-current region.
+ const double b2 = 1.2; // Outer radius of the free-current region.
+ const double d2 = 2.0; // Error norms are computed if r < d2.
+ const double K0 = 1.0; // Magnitude of the free-current density.
+ const double H0 = // Magnitude of H-field inside the coil, core removed.
+ (1.0 / 3.0) * K0 * (std::pow(b2, 2) - std::pow(a2, 2));
+ const types::material_id material_id_free_space =
+ 1; // Material ID of the free space.
+ const types::material_id material_id_core =
+ 2; // Material ID of the magnetic core.
+ const types::material_id material_id_free_current =
+ 3; // Material ID of the free-current region.
+ const types::boundary_id boundary_id_infinity =
+ 2; // Boundary ID. Set in the geo file.
+
+ const unsigned int mapping_degree = 2; // Mapping degree used in all solvers.
+ const unsigned int fe_degree = 0; // Degree of the finite elements.
+ const BoundaryConditionType boundary_condition_type_A = Robin;
+
+ const double eta_squared_T = 0.0; // eta^2 when solving for T$.
+ const double eta_squared_A = 0.0; // eta^2 when solving for A$.
+
+ const unsigned int n_threads_max = 0; // If >0 limits the number of threads.
+ const double eps = 1e-12; // Two doubles are equal if their difference < eps.
+ const bool log_cg_convergence = false; // Save the CG convergence data.
+ const bool print_time_tables = false; // Print the time tables on the screen.
+ const bool project_exact_solution = false; // Save the exact solution.
+} // namespace Settings
+
+// Next comes the weight function used to limit the region in which the $L^2$
+// error norms are computed. The error norms are computed if $r < d2$.
+class Weight : public Function<3>
+{
+public:
+ virtual double value(const Point<3> &p,
+ const unsigned int) const override final
+ {
+ if (p.norm() > Settings::d2)
+ return 0.0;
+
+ return 1.0;
+ }
+};
+
+// This class describes a convergence table. The convergence tables are saved on
+// disk in TeX format.
+class MainOutputTable : public ConvergenceTable
+{
+public:
+ MainOutputTable() = delete;
+
+ MainOutputTable(const unsigned int dimensions)
+ : ConvergenceTable()
+ , dimensions(dimensions)
+ {}
+
+ void set_new_order(const std::vector<std::string> &new_order_in)
+ {
+ new_order = new_order_in;
+ }
+
+ void append_new_order(const std::string &new_column)
+ {
+ new_order.push_back(new_column);
+ }
+
+ void format()
+ {
+ set_precision("L2", 2);
+
+ set_scientific("L2", true);
+
+ evaluate_convergence_rates("L2",
+ "ncells",
+ ConvergenceTable::reduction_rate_log2,
+ dimensions);
+
+ set_tex_caption("p", "p");
+ set_tex_caption("r", "r");
+ set_tex_caption("ncells", "nr. cells");
+ set_tex_caption("ndofs", "nr. dofs");
+ set_tex_caption("L2", "L2 norm");
+
+ set_column_order(new_order);
+ }
+
+ void save(const std::string &fname)
+ {
+ format();
+
+ std::ofstream ofs(fname + ".tex");
+ write_tex(ofs);
+ }
+
+private:
+ const unsigned int dimensions;
+ std::vector<std::string> new_order = {"p", "r", "ncells", "ndofs", "L2"};
+};
+
+// @sect3{Equations}
+
+// This name space contains all closed-form analytical expressions mentioned in
+// the introduction to this tutorial.
+namespace ExactSolutions
+{
+
+ // This function describes the free-current density, $\vec{J}_f$, inside the
+ // current region.
+ inline Tensor<1, 3> volume_free_current_density(const Point<3> &p,
+ const double K0)
+ {
+ return Tensor<1, 3>({-K0 * p[1], K0 * p[0], 0.0});
+ }
+
+ // This function computes the magnetic field induced by the free current in
+ // absence of the magnetic core, see the expression for $\vec{B}_J$ in the
+ // introduction.
+ inline Tensor<1, 3> magnetic_field_coil(const Point<3> &p,
+ const double K0,
+ const double mu_0,
+ const double a2,
+ const double b2)
+ {
+ const double r = p.norm();
+
+ const double cos_theta = p[2] / r;
+ const double sin_theta =
+ std::sqrt(std::pow(p[0], 2) + std::pow(p[1], 2)) / r;
+
+ const double cos_phi =
+ p[0] / std::sqrt(std::pow(p[0], 2) + std::pow(p[1], 2));
+ const double sin_phi =
+ p[1] / std::sqrt(std::pow(p[0], 2) + std::pow(p[1], 2));
+
+ const Tensor<1, 3> r_hat({p[0] / r, p[1] / r, p[2] / r});
+ const Tensor<1, 3> theta_hat(
+ {cos_theta * cos_phi, cos_theta * sin_phi, -sin_theta});
+
+ const Tensor<1, 3> F1 =
+ (2.0 / 3.0) * mu_0 * K0 * (cos_theta * r_hat - sin_theta * theta_hat);
+ const Tensor<1, 3> F2 = (2.0 / 3.0) * mu_0 * K0 *
+ (cos_theta * r_hat + 0.5 * sin_theta * theta_hat) /
+ std::pow(r, 3);
+
+ if (r <= a2)
+ {
+ return 0.5 * (std::pow(b2, 2) - std::pow(a2, 2)) * F1;
+ }
+ else if (r >= b2)
+ {
+ return 0.2 * (std::pow(b2, 5) - std::pow(a2, 5)) * F2;
+ }
+ else
+ {
+ return 0.5 * (std::pow(b2, 2) - std::pow(r, 2)) * F1 +
+ 0.2 * (std::pow(r, 5) - std::pow(a2, 5)) * F2;
+ }
+
+ return Tensor<1, 3>({0.404, 0.404, 0.404});
+ }
+
+ // This function computes the magnetic field induced by the magnetic core, see
+ // the expression for $\vec{B}_{\mu}$ in the introduction.
+ inline Tensor<1, 3> magnetic_field_core(const Point<3> &p,
+ const double H0,
+ const double mur,
+ const double mu0,
+ const double a1,
+ const double b1)
+ {
+ const double a3 = std::pow(a1, 3);
+ const double b3 = std::pow(b1, 3);
+
+ const double OMEGA = ((mur - 1.0) / (mur + 2.0)) * (a3 / b3);
+ const double gamma_1 = (-3.0 * b3 * H0 * OMEGA) /
+ ((2.0 * mur + 1.0) - 2.0 * (mur - 1.0) * OMEGA);
+ const double beta_1 = ((2.0 * mur + 1.0) * gamma_1) / ((mur - 1.0) * a3);
+ const double alpha_1 =
+ (-b3 * H0 + 2.0 * mur * gamma_1 - mur * b3 * beta_1) / 2.0;
+ const double delta_1 = (mur * a3 * beta_1 - 2.0 * mur * gamma_1) / a3;
+
+ const double r = p.norm();
+ const double r3 = std::pow(r, 3);
+ const double r5 = std::pow(r, 5);
+
+ const double zz = -3.0 * p[2] * p[2] / r5;
+ const double xz = -3.0 * p[0] * p[2] / r5;
+ const double yz = -3.0 * p[1] * p[2] / r5;
+
+ if (r <= a1)
+ {
+ return -mu0 * (Tensor<1, 3>({0.0, 0.0, delta_1}) +
+ Tensor<1, 3>({0.0, 0.0, H0}));
+ }
+ else if (r >= b1)
+ {
+ return -mu0 * (Tensor<1, 3>({alpha_1 * xz,
+ alpha_1 * yz,
+ -H0 + alpha_1 / r3 + alpha_1 * zz}) +
+ Tensor<1, 3>({0.0, 0.0, H0}));
+ }
+ else
+ {
+ return -mu0 *
+ (mur * Tensor<1, 3>({gamma_1 * xz,
+ gamma_1 * yz,
+ beta_1 + gamma_1 / r3 + gamma_1 * zz}) +
+ Tensor<1, 3>({0.0, 0.0, H0}));
+ }
+
+ return Tensor<1, 3>({0.404, 0.404, 0.404});
+ }
+
+ // This class implements the closed-form analytical expression for the
+ // free-current density, $\vec{J}_f$, in the entire domain.
+ class FreeCurrentDensity : public Function<3>
+ {
+ public:
+ FreeCurrentDensity()
+ : Function<3>(3)
+ {}
+
+ virtual void
+ vector_value_list(const std::vector<Point<3>> &p,
+ std::vector<Vector<double>> &values) const override final
+ {
+ Assert(values.size() == p.size(),
+ ExcDimensionMismatch(values.size(), p.size()));
+
+ double r;
+
+ for (unsigned int i = 0; i < values.size(); i++)
+ {
+ r = p[i].norm();
+
+ if ((r >= Settings::a2) && (r <= Settings::b2))
+ {
+ const Tensor<1, 3> Jf =
+ volume_free_current_density(p[i], Settings::K0);
+
+ values[i][0] = Jf[0];
+ values[i][1] = Jf[1];
+ values[i][2] = Jf[2];
+ }
+ else
+ {
+ values[i][0] = 0.0;
+ values[i][1] = 0.0;
+ values[i][2] = 0.0;
+ }
+ }
+ }
+ };
+
+ // This class implements the closed-form analytical expression for the
+ // magnetic field induced by the coil, $\vec{B} = \vec{B}_J + \vec{B}_{\mu}$,
+ // see the introduction.
+ class MagneticField : public Function<3>
+ {
+ public:
+ MagneticField()
+ : Function<3>(3)
+ {}
+
+ virtual void
+ vector_value_list(const std::vector<Point<3>> &p,
+ std::vector<Vector<double>> &values) const override final
+ {
+ Assert(values.size() == p.size(),
+ ExcDimensionMismatch(values.size(), p.size()));
+
+ Tensor<1, 3> B;
+
+ for (unsigned int i = 0; i < values.size(); i++)
+ {
+ using namespace Settings;
+ B = magnetic_field_coil(p[i], K0, mu_0, a2, b2) +
+ magnetic_field_core(p[i], H0, mu_r, mu_0, a1, b1);
+
+ values[i][0] = B[0];
+ values[i][1] = B[1];
+ values[i][2] = B[2];
+ }
+ }
+ };
+
+} // namespace ExactSolutions
+
+// @sect3{Solver - T}
+
+// This name space contains all the code related to the computation of the
+// current vector potential, $\vec{T}$.
+namespace SolverT
+{
+ // This class describes the free-current density, $\vec{J}_f$, on the
+ // right-hand side of the curl-curl equation, see equation (i) in the
+ // boundary value problem for $\vec{T}$. The free-current density is given as
+ // a closed-form analytical expression by the definition of the problem.
+ class FreeCurrentDensity
+ {
+ public:
+ void value_list(const std::vector<Point<3>> &p, // Quadrature points.
+ const types::material_id mid, // Material ID of the cell.
+ std::vector<Tensor<1, 3>> &values) const
+ {
+ Assert(p.size() == values.size(),
+ ExcDimensionMismatch(p.size(), values.size()));
+
+ if ((mid == Settings::material_id_free_space) ||
+ (mid == Settings::material_id_core))
+ for (unsigned int i = 0; i < values.size(); i++)
+ values[i] = Tensor<1, 3>({0.0, 0.0, 0.0});
+
+ if (mid == Settings::material_id_free_current)
+ for (unsigned int i = 0; i < values.size(); i++)
+ values[i] =
+ ExactSolutions::volume_free_current_density(p[i], Settings::K0);
+ }
+ };
+
+ // This class implements the solver that minimizes the functional
+ // $F(\vec{T})$, see the introduction. The mesh is loaded in this class.
+ // All other solvers use a reference to this mesh.
+ class Solver
+ {
+ public:
+ Solver() = delete;
+ Solver(const unsigned int p, // Degree of the Nedelec finite elements.
+ const unsigned int r, // The mesh refinement parameter.
+ const unsigned int mapping_degree,
+ const double eta_squared = 0.0,
+ const std::string &fname = "data");
+
+ void make_mesh(); // Loads the mesh. Assigns mat. IDs. Attaches manifold.
+ void setup(); // Initializes dofs, vectors, matrices.
+ void assemble(); // Assembles the system of linear equations.
+ void solve(); // Solves the system of linear equations.
+ void save() const; // Saves computed T into a vtu file.
+ void clear()
+ { // Clears the memory for the next solver.
+ system_matrix.clear();
+ system_rhs.reinit(0);
+ }
+ void run(); /* Executes the last six functions in the proper order
+ and measures the execution time for each function. */
+
+ // These three get-functions are used to channel the mesh and the solution
+ // to the next solver.
+ const Triangulation<3> &get_tria() const
+ {
+ return triangulation;
+ }
+ const DoFHandler<3> &get_dof_handler() const
+ {
+ return dof_handler;
+ }
+ const Vector<double> &get_solution() const
+ {
+ return solution;
+ }
+
+ private:
+ // The following data members are typical for all deal.II simulations:
+ // triangulation, finite elements, dof handlers, etc. The constraints
+ // are used to enforce the Dirichlet boundary conditions. The names of the
+ // data members are self-explanatory.
+
+ Triangulation<3> triangulation;
+
+ const FE_Nedelec<3> fe;
+ DoFHandler<3> dof_handler;
+ Vector<double> solution;
+
+ SparseMatrix<double> system_matrix;
+ Vector<double> system_rhs;
+
+ AffineConstraints<double> constraints;
+ SparsityPattern sparsity_pattern;
+
+ SphericalManifold<3> sphere;
+
+ const unsigned int refinement_parameter;
+ const unsigned int mapping_degree;
+ const double eta_squared;
+ const std::string fname;
+ TimerOutput timer;
+
+ // The program utilizes the WorkStream technology. The Step-9 tutorial
+ // does a much better job of explaining the workings of WorkStream.
+ // Reading the "WorkStream paper", see the glossary, is recommended.
+ // The following structures and functions are related to WorkStream.
+ struct AssemblyScratchData
+ {
+ AssemblyScratchData(const FiniteElement<3> &fe,
+ const double eta_squared,
+ const unsigned int mapping_degree);
+
+ AssemblyScratchData(const AssemblyScratchData &scratch_data);
+
+ const FreeCurrentDensity Jf;
+
+ MappingQ<3> mapping;
+ FEValues<3> fe_values;
+
+ const unsigned int dofs_per_cell;
+ const unsigned int n_q_points;
+
+ std::vector<Tensor<1, 3>> Jf_list;
+
+ const double eta_squared;
+ const FEValuesExtractors::Vector ve;
+ };
+
+ struct AssemblyCopyData
+ {
+ FullMatrix<double> cell_matrix;
+ Vector<double> cell_rhs;
+ std::vector<types::global_dof_index> local_dof_indices;
+ };
+
+ void system_matrix_local(
+ const typename DoFHandler<3>::active_cell_iterator &cell,
+ AssemblyScratchData &scratch_data,
+ AssemblyCopyData ©_data);
+
+ void copy_local_to_global(const AssemblyCopyData ©_data);
+ };
+
+ Solver::Solver(const unsigned int p,
+ const unsigned int r,
+ const unsigned int mapping_degree,
+ const double eta_squared,
+ const std::string &fname)
+ : fe(p)
+ , refinement_parameter(r)
+ , mapping_degree(mapping_degree)
+ , eta_squared(eta_squared)
+ , fname(fname)
+ , timer(std::cout,
+ (Settings::print_time_tables) ? TimerOutput::summary :
+ TimerOutput::never,
+ TimerOutput::cpu_and_wall_times_grouped)
+ {}
+
+ // The following function loads the mesh, assigns material IDs to all cells,
+ // and attaches the spherical manifold to the mesh. The material IDs are
+ // assigned on the basis of the distance from the center of a cell to the
+ // origin. The spherical manifold is attached to a face if all vertices of
+ // the face are at the same distance from the origin provided the cell is
+ // outside the cube in the center of the mesh, see mesh description in the
+ // introduction.
+ void Solver::make_mesh()
+ {
+ GridIn<3> gridin;
+
+ gridin.attach_triangulation(triangulation);
+ std::ifstream ifs("sphere_r" + std::to_string(refinement_parameter) +
+ ".msh");
+ gridin.read_msh(ifs);
+
+ triangulation.reset_all_manifolds();
+
+ for (auto cell : triangulation.active_cell_iterators())
+ {
+ cell->set_material_id(
+ Settings::material_id_free_space); // The cell is in free space.
+
+ if ((cell->center().norm() > Settings::a1) &&
+ (cell->center().norm() < Settings::b1))
+ cell->set_material_id(
+ Settings::material_id_core); // The cell is inside the core.
+
+ if ((cell->center().norm() > Settings::a2) &&
+ (cell->center().norm() < Settings::b2))
+ cell->set_material_id(
+ Settings::material_id_free_current); /* The cell is inside the Jf
+ region. */
+
+ for (unsigned int f = 0; f < cell->n_faces(); f++)
+ {
+ double dif_norm = 0.0;
+ for (unsigned int v = 1; v < cell->face(f)->n_vertices(); v++)
+ dif_norm += std::abs(cell->face(f)->vertex(0).norm() -
+ cell->face(f)->vertex(v).norm());
+
+ if ((dif_norm < Settings::eps) &&
+ (cell->center().norm() > Settings::d1))
+ cell->face(f)->set_all_manifold_ids(1);
+ }
+ }
+
+ triangulation.set_manifold(1, sphere);
+ }
+
+ // This function initializes the dofs, applies the Dirichlet boundary
+ // condition, and initializes the vectors and matrices.
+ void Solver::setup()
+ {
+ dof_handler.reinit(triangulation);
+ dof_handler.distribute_dofs(fe);
+
+ // The following segment of the code applies the homogeneous Dirichlet
+ // boundary condition. As discussed in the introduction, the Dirichlet
+ // boundary condition is an essential condition and must be enforced
+ // by constraining the system matrix. This segment of the code does the
+ // constraining.
+ constraints.clear();
+
+ DoFTools::make_hanging_node_constraints(dof_handler, constraints);
+
+ VectorTools::project_boundary_values_curl_conforming_l2(
+ dof_handler,
+ 0, // The first vector component.
+ Functions::ZeroFunction<3>(3),
+ Settings::boundary_id_infinity,
+ constraints,
+ MappingQ<3>(mapping_degree));
+
+ constraints.close();
+
+ // The rest of the function arranges the dofs in a sparsity pattern and
+ // initializes the system matrices and vectors.
+ DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
+ DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints, false);
+ sparsity_pattern.copy_from(dsp);
+
+ system_matrix.reinit(sparsity_pattern);
+ solution.reinit(dof_handler.n_dofs());
+ system_rhs.reinit(dof_handler.n_dofs());
+ }
+
+ // Formally, the following function assembles the system of linear equations.
+ // In reality, however, it just spells all the magic words to get the
+ // WorkStream going. The interesting part, i.e., the actual assembling of the
+ // system matrix and the right-hand side, happens below in the function
+ // Solver::system_matrix_local().
+ void Solver::assemble()
+ {
+ WorkStream::run(dof_handler.begin_active(),
+ dof_handler.end(),
+ *this,
+ &Solver::system_matrix_local,
+ &Solver::copy_local_to_global,
+ AssemblyScratchData(fe, eta_squared, mapping_degree),
+ AssemblyCopyData());
+ }
+
+ // The following two constructors initialize scratch data from the input
+ // parameters and from another object of the same type, i.e., a copy
+ // constructor.
+ Solver::AssemblyScratchData::AssemblyScratchData(
+ const FiniteElement<3> &fe,
+ const double eta_squared,
+ const unsigned int mapping_degree)
+ : Jf()
+ , mapping(mapping_degree)
+ , fe_values(mapping,
+ fe,
+ QGauss<3>(fe.degree + 2),
+ update_gradients | update_values | update_quadrature_points |
+ update_JxW_values)
+ , dofs_per_cell(fe_values.dofs_per_cell)
+ , n_q_points(fe_values.get_quadrature().size())
+ , Jf_list(n_q_points, Tensor<1, 3>())
+ , eta_squared(eta_squared)
+ , ve(0)
+ {}
+
+ Solver::AssemblyScratchData::AssemblyScratchData(
+ const AssemblyScratchData &scratch_data)
+ : Jf()
+ , mapping(scratch_data.mapping.get_degree())
+ , fe_values(mapping,
+ scratch_data.fe_values.get_fe(),
+ scratch_data.fe_values.get_quadrature(),
+ update_gradients | update_values | update_quadrature_points |
+ update_JxW_values)
+ , dofs_per_cell(fe_values.dofs_per_cell)
+ , n_q_points(fe_values.get_quadrature().size())
+ , Jf_list(n_q_points, Tensor<1, 3>())
+ , eta_squared(scratch_data.eta_squared)
+ , ve(0)
+ {}
+
+ // This function assembles a fraction of the system matrix and the system
+ // right-hand side related to a single cell. These fractions are
+ // `copy_data.cell_matrix` and `copy_data.cell_rhs`. They are copied into
+ // the system matrix, $A_{ij}$, and the right-hand side, $b_i$, by the
+ // function `Solver::copy_local_to_global()`.
+ void Solver::system_matrix_local(
+ const typename DoFHandler<3>::active_cell_iterator &cell,
+ AssemblyScratchData &scratch_data,
+ AssemblyCopyData ©_data)
+ {
+ // First, we reinitialize the matrices and vectors related to the current
+ // cell and compute the FE values.
+ copy_data.cell_matrix.reinit(scratch_data.dofs_per_cell,
+ scratch_data.dofs_per_cell);
+
+ copy_data.cell_rhs.reinit(scratch_data.dofs_per_cell);
+
+ copy_data.local_dof_indices.resize(scratch_data.dofs_per_cell);
+
+ scratch_data.fe_values.reinit(cell);
+
+ // Second, we compute the free-current density, $\vec{J}_f$, at the
+ // quadrature points.
+ scratch_data.Jf.value_list(scratch_data.fe_values.get_quadrature_points(),
+ cell->material_id(),
+ scratch_data.Jf_list);
+
+ // Third, we compute the components of the cell matrix and cell right-hand
+ // side. The labels of the integrals are the same as in the introduction to
+ // this tutorial.
+ for (unsigned int q_index = 0; q_index < scratch_data.n_q_points; ++q_index)
+ {
+ for (unsigned int i = 0; i < scratch_data.dofs_per_cell; ++i)
+ {
+ for (unsigned int j = 0; j < scratch_data.dofs_per_cell; ++j)
+ {
+ copy_data.cell_matrix(i, j) += // Integral I_a1+I_a3.
+ (scratch_data.fe_values[scratch_data.ve].curl(
+ i, q_index) * // curl N_i
+ scratch_data.fe_values[scratch_data.ve].curl(
+ j, q_index) // curl N_j
+ + scratch_data.eta_squared * // eta^2
+ scratch_data.fe_values[scratch_data.ve].value(
+ i, q_index) * // N_i
+ scratch_data.fe_values[scratch_data.ve].value(
+ j, q_index) // N_j
+ ) *
+ scratch_data.fe_values.JxW(q_index); // dV
+ }
+ copy_data.cell_rhs(i) += // Integral I_b3-1.
+ (scratch_data.Jf_list[q_index] *
+ scratch_data.fe_values[scratch_data.ve].curl(i, q_index)) *
+ scratch_data.fe_values.JxW(q_index); // J_f.(curl N_i)dV.
+ }
+ }
+
+ // Finally, we query the dof indices on the current cell and store them
+ // in the copy data structure, so we know to which locations of the system
+ // matrix and right-hand side the components of the cell matrix and
+ // cell right-hand side must be copied.
+ cell->get_dof_indices(copy_data.local_dof_indices);
+ }
+
+ // This function copies the components of a cell matrix and a cell right-hand
+ // side into the system matrix, $A_{i,j}$, and the system right-hand side,
+ // $b_i$.
+ void Solver::copy_local_to_global(const AssemblyCopyData ©_data)
+ {
+ constraints.distribute_local_to_global(copy_data.cell_matrix,
+ copy_data.cell_rhs,
+ copy_data.local_dof_indices,
+ system_matrix,
+ system_rhs);
+ }
+
+ // This function solves the system of linear equations.
+ // If `Settings::log_cg_convergence == true`, the convergence data is saved
+ // into a file. In theory, a CG solver can solve an $m \times m$ system of
+ // linear equations in at most $m$ steps. In practice, it can take more steps
+ // to converge. The convergence of the algorithm depends on the spectral
+ // properties of the system matrix. The best case is if the eigenvalues form a
+ // compact cluster away from zero. In our case, however, the eigenvalues are
+ // spread in between zero and the maximal eigenvalue. Consequently, we expect
+ // a poor convergence and increase the maximal number of iteration steps by a
+ // factor of 10, i.e., `10*system_rhs.size()`. The stopping condition is \f[
+ // |\boldsymbol{b} - \boldsymbol{A}\boldsymbol{c}|
+ // < 10^{-6} |\boldsymbol{b}|.
+ // \f]
+ // As soon as we use constraints, we must not forget to distribute them.
+ void Solver::solve()
+ {
+ SolverControl control(10 * system_rhs.size(),
+ 1.0e-6 * system_rhs.l2_norm(),
+ false,
+ false);
+
+ if (Settings::log_cg_convergence)
+ control.enable_history_data();
+
+ GrowingVectorMemory<Vector<double>> memory;
+ SolverCG<Vector<double>> cg(control, memory);
+
+ PreconditionSSOR<SparseMatrix<double>> preconditioner;
+ preconditioner.initialize(system_matrix, 1.2);
+
+ cg.solve(system_matrix, solution, system_rhs, preconditioner);
+
+ constraints.distribute(solution);
+
+ if (Settings::log_cg_convergence)
+ {
+ const std::vector<double> history_data = control.get_history_data();
+
+ std::ofstream ofs(fname + "_cg_convergence.csv");
+
+ for (unsigned int i = 1; i < history_data.size(); i++)
+ ofs << i << ", " << history_data[i] << "\n";
+ }
+ }
+
+ // This function saves the computed current vector potential into a vtu file.
+ void Solver::save() const
+ {
+ const std::vector<std::string> solution_names(3, "VectorField");
+ const std::vector<DataComponentInterpretation::DataComponentInterpretation>
+ interpretation(3,
+ DataComponentInterpretation::component_is_part_of_vector);
+
+ DataOut<3> data_out;
+
+ data_out.add_data_vector(dof_handler,
+ solution,
+ solution_names,
+ interpretation);
+
+ DataOutBase::VtkFlags flags;
+ flags.write_higher_order_cells = true;
+ data_out.set_flags(flags);
+
+ const MappingQ<3> mapping(mapping_degree);
+
+ data_out.build_patches(mapping,
+ fe.degree + 2,
+ DataOut<3>::CurvedCellRegion::curved_inner_cells);
+
+ std::ofstream ofs(fname + ".vtu");
+ data_out.write_vtu(ofs);
+ }
+
+ void Solver::run()
+ {
+ {
+ TimerOutput::Scope timer_section(timer, "Make mesh");
+ make_mesh();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Setup");
+ setup();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Assemble");
+ assemble();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Solve");
+ solve();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Save");
+ save();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Clear");
+ clear();
+ }
+ }
+} // namespace SolverT
+
+// @sect3{Solver - A}
+
+// This name space contains all the code related to the computation of the
+// magnetic vector potential, $\vec{A}$. The main difference between this
+// solver and the solver for the current vector potential, $\vec{T}$, is in how
+// the information on the source is fed to respective solvers. The solver for
+// $\vec{T}$ is fed data sampled from the analytical closed-form expression for
+// $\vec{J}_f$. The solver for $\vec{A}$ is fed a field function, i.e., a
+// numerically computed current vector potential, $\vec{T}$.
+namespace SolverA
+{
+ // This class describes the permeability in the entire problem domain. The
+ // permeability is given by the definition of the problem, see the
+ // introduction.
+ class Permeability
+ {
+ public:
+ void value_list(const types::material_id mid,
+ std::vector<double> &values) const
+ {
+ if ((mid == Settings::material_id_free_space) ||
+ (mid == Settings::material_id_free_current))
+ std::fill(values.begin(), values.end(), Settings::mu_0);
+
+ if (mid == Settings::material_id_core)
+ std::fill(values.begin(), values.end(), Settings::mu_1);
+ }
+ };
+
+ // This class describes the parameter $\gamma$ in the Robin boundary
+ // condition. As soon as it is evaluated on the boundary, the permeability
+ // equals to that of free space. Therefore, we evaluate the parameter gamma as
+ // \f[
+ // \gamma = \dfrac{1}{\mu_0 r}.
+ // \f]
+ class Gamma
+ {
+ public:
+ void value_list(const std::vector<Point<3>> &r,
+ std::vector<double> &values) const
+ {
+ Assert(r.size() == values.size(),
+ ExcDimensionMismatch(r.size(), values.size()));
+
+ for (unsigned int i = 0; i < values.size(); i++)
+ values[i] = 1.0 / (Settings::mu_0 * r[i].norm());
+ }
+ };
+
+ // This class implements the solver that minimizes the functional
+ // $F(\vec{A})$. The numerically computed current vector potential, $\vec{T}$,
+ // is fed to this solver by means of the input parameters `dof_handler_T` and
+ // `solution_T`. Moreover, this solver reuses the mesh on which $\vec{T}$ has
+ // been computed. The reference to the mesh is passed via the input parameter
+ // `triangulation_T`.
+ class Solver
+ {
+ public:
+ Solver() = delete;
+ Solver(const unsigned int p, // Degree of the Nedelec finite elements.
+ const unsigned int mapping_degree,
+ const Triangulation<3> &triangulation_T,
+ const DoFHandler<3> &dof_handler_T,
+ const Vector<double> &solution_T,
+ const double eta_squared = 0.0,
+ const std::string &fname = "data");
+
+ void setup(); // Initializes dofs, vectors, matrices.
+ void assemble(); // Assembles the system of linear equations.
+ void solve(); // Solves the system of linear equations.
+ void save() const; // Saves computed T into a vtu file.
+ void clear() // Clears the memory for the next solver.
+ {
+ system_matrix.clear();
+ system_rhs.reinit(0);
+ }
+ void run(); /* Executes the last five functions in the proper order
+ and measures the execution time for each function. */
+
+ const DoFHandler<3> &get_dof_handler() const
+ {
+ return dof_handler;
+ }
+ const Vector<double> &get_solution() const
+ {
+ return solution;
+ }
+
+ private:
+ const Triangulation<3> &triangulation_T;
+ const DoFHandler<3> &dof_handler_T;
+ const Vector<double> &solution_T;
+
+ // The following data members are typical for all deal.II simulations:
+ // triangulation, finite elements, dof handlers, etc. The constraints
+ // are used to enforce the Dirichlet boundary conditions. The names of the
+ // data members are self-explanatory.
+ const FE_Nedelec<3> fe;
+ DoFHandler<3> dof_handler;
+ Vector<double> solution;
+
+ SparseMatrix<double> system_matrix;
+ Vector<double> system_rhs;
+
+ AffineConstraints<double> constraints;
+ SparsityPattern sparsity_pattern;
+
+ const unsigned int mapping_degree;
+ const double eta_squared;
+ const std::string fname;
+ TimerOutput timer;
+
+ // This time we have two dof handlers, `dof_handler_T` for $\vec{T}$ and
+ // `dof_handler` for $\vec{A}$. The WorkStream needs to walk through
+ // the two dof handlers synchronously. For this purpose we will pair two
+ // active cell iterators (one from `dof_handler_T`, another from
+ // `dof_handler`). For that we need the `IteratorPair` type.
+ using IteratorTuple =
+ std::tuple<typename DoFHandler<3>::active_cell_iterator,
+ typename DoFHandler<3>::active_cell_iterator>;
+
+ using IteratorPair = SynchronousIterators<IteratorTuple>;
+
+ // The program utilizes the WorkStream technology. The Step-9 tutorial
+ // does a much better job of explaining the workings of WorkStream.
+ // Reading the "WorkStream paper", see the glossary, is recommended.
+ // The following structures and functions are related to WorkStream.
+ struct AssemblyScratchData
+ {
+ AssemblyScratchData(const FiniteElement<3> &fe,
+ const DoFHandler<3> &dof_hand_T,
+ const Vector<double> &dofs_T,
+ const unsigned int mapping_degree,
+ const double eta_squared,
+ const BoundaryConditionType boundary_condition_type);
+
+ AssemblyScratchData(const AssemblyScratchData &scratch_data);
+
+ const Permeability permeability;
+ const Gamma gamma;
+
+ MappingQ<3> mapping;
+ FEValues<3> fe_values;
+ FEFaceValues<3> fe_face_values;
+
+ FEValues<3> fe_values_T;
+
+ const unsigned int dofs_per_cell;
+ const unsigned int n_q_points;
+ const unsigned int n_q_points_face;
+
+ std::vector<double> permeability_list;
+ std::vector<double> gamma_list;
+ std::vector<Tensor<1, 3>> T_values;
+
+ const FEValuesExtractors::Vector ve;
+
+ const DoFHandler<3> &dof_hand_T;
+ const Vector<double> &dofs_T;
+
+ const double eta_squared;
+ const BoundaryConditionType boundary_condition_type;
+ };
+
+ struct AssemblyCopyData
+ {
+ FullMatrix<double> cell_matrix;
+ Vector<double> cell_rhs;
+ std::vector<types::global_dof_index> local_dof_indices;
+ };
+
+ void system_matrix_local(const IteratorPair &IP,
+ AssemblyScratchData &scratch_data,
+ AssemblyCopyData ©_data);
+
+ void copy_local_to_global(const AssemblyCopyData ©_data);
+ };
+
+ Solver::Solver(const unsigned int p,
+ const unsigned int mapping_degree,
+ const Triangulation<3> &triangulation_T,
+ const DoFHandler<3> &dof_handler_T,
+ const Vector<double> &solution_T,
+ const double eta_squared,
+ const std::string &fname)
+ : triangulation_T(triangulation_T)
+ , dof_handler_T(dof_handler_T)
+ , solution_T(solution_T)
+ , fe(p)
+ , mapping_degree(mapping_degree)
+ , eta_squared(eta_squared)
+ , fname(fname)
+ , timer(std::cout,
+ (Settings::print_time_tables) ? TimerOutput::summary :
+ TimerOutput::never,
+ TimerOutput::cpu_and_wall_times_grouped)
+ {}
+
+ // This function initializes the dofs, applies the Dirichlet boundary
+ // condition, and initializes the vectors and matrices.
+ void Solver::setup()
+ {
+ dof_handler.reinit(triangulation_T);
+ dof_handler.distribute_dofs(fe);
+
+ // The following segment of the code applies the homogeneous Dirichlet
+ // boundary condition. As discussed in the introduction, the Dirichlet
+ // boundary condition is an essential condition and must be enforced
+ // by constraining the system matrix. This segment of code does the
+ // constraining.
+ constraints.clear();
+
+ DoFTools::make_hanging_node_constraints(dof_handler, constraints);
+
+ if (Settings::boundary_condition_type_A == Dirichlet)
+ VectorTools::project_boundary_values_curl_conforming_l2(
+ dof_handler,
+ 0, // The first vector component.
+ Functions::ZeroFunction<3>(3),
+ Settings::boundary_id_infinity,
+ constraints,
+ MappingQ<3>(mapping_degree));
+
+ constraints.close();
+
+ // The rest of the function arranges the dofs in a sparsity pattern and
+ // initializes the system matrix and the system vectors.
+ DynamicSparsityPattern dsp(dof_handler.n_dofs(), dof_handler.n_dofs());
+ DoFTools::make_sparsity_pattern(dof_handler, dsp, constraints, false);
+
+ sparsity_pattern.copy_from(dsp);
+ system_matrix.reinit(sparsity_pattern);
+ solution.reinit(dof_handler.n_dofs());
+ system_rhs.reinit(dof_handler.n_dofs());
+ }
+
+ // Formally, this function assembles the system of linear equations. In
+ // reality, however, it just spells all the magic words to get the WorkStream
+ // going. The interesting part, i.e., the actual assembling of the system
+ // matrix and the right-hand side happens below in the
+ // Solver::system_matrix_local function. Note that this time the first two
+ // input parameters to `WorkStream::run` are pairs of iterators, not
+ // iterators themselves as per usual. Note also the order in which we package
+ // the iterators: first the iterator of `dof_handler`, then the iterator of
+ // the `dof_handler_T`. We will extract them in the same order.
+ void Solver::assemble()
+ {
+ WorkStream::run(IteratorPair(IteratorTuple(dof_handler.begin_active(),
+ dof_handler_T.begin_active())),
+ IteratorPair(
+ IteratorTuple(dof_handler.end(), dof_handler_T.end())),
+ *this,
+ &Solver::system_matrix_local,
+ &Solver::copy_local_to_global,
+ AssemblyScratchData(fe,
+ dof_handler_T,
+ solution_T,
+ mapping_degree,
+ eta_squared,
+ Settings::boundary_condition_type_A),
+ AssemblyCopyData());
+ }
+
+ // The following two constructors initialize scratch data from the input
+ // parameters and from another object of the same type, i.e., a copy
+ // constructor.
+ Solver::AssemblyScratchData::AssemblyScratchData(
+ const FiniteElement<3> &fe,
+ const DoFHandler<3> &dof_hand_T,
+ const Vector<double> &dofs_T,
+ const unsigned int mapping_degree,
+ const double eta_squared,
+ const BoundaryConditionType boundary_condition_type)
+ : permeability()
+ , gamma()
+ , mapping(mapping_degree)
+ , fe_values(mapping,
+ fe,
+ QGauss<3>(fe.degree + 2),
+ update_gradients | update_values | update_JxW_values)
+ , fe_face_values(mapping,
+ fe,
+ QGauss<2>(fe.degree + 2),
+ update_values | update_normal_vectors |
+ update_quadrature_points | update_JxW_values)
+ , fe_values_T(mapping,
+ dof_hand_T.get_fe(),
+ QGauss<3>(fe.degree + 2),
+ update_values | update_gradients)
+ , dofs_per_cell(fe_values.dofs_per_cell)
+ , n_q_points(fe_values.get_quadrature().size())
+ , n_q_points_face(fe_face_values.get_quadrature().size())
+ , permeability_list(n_q_points)
+ , gamma_list(n_q_points_face)
+ , T_values(n_q_points)
+ , ve(0)
+ , dof_hand_T(dof_hand_T)
+ , dofs_T(dofs_T)
+ , eta_squared(eta_squared)
+ , boundary_condition_type(boundary_condition_type)
+ {}
+
+ Solver::AssemblyScratchData::AssemblyScratchData(
+ const AssemblyScratchData &scratch_data)
+ : permeability()
+ , gamma()
+ , mapping(scratch_data.mapping.get_degree())
+ , fe_values(mapping,
+ scratch_data.fe_values.get_fe(),
+ scratch_data.fe_values.get_quadrature(),
+ update_gradients | update_values | update_JxW_values)
+ , fe_face_values(mapping,
+ scratch_data.fe_face_values.get_fe(),
+ scratch_data.fe_face_values.get_quadrature(),
+ update_values | update_normal_vectors |
+ update_quadrature_points | update_JxW_values)
+ , fe_values_T(mapping,
+ scratch_data.fe_values_T.get_fe(),
+ scratch_data.fe_values_T.get_quadrature(),
+ update_values | update_gradients)
+ , dofs_per_cell(fe_values.dofs_per_cell)
+ , n_q_points(fe_values.get_quadrature().size())
+ , n_q_points_face(fe_face_values.get_quadrature().size())
+ , permeability_list(n_q_points)
+ , gamma_list(n_q_points_face)
+ , T_values(n_q_points)
+ , ve(0)
+ , dof_hand_T(scratch_data.dof_hand_T)
+ , dofs_T(scratch_data.dofs_T)
+ , eta_squared(scratch_data.eta_squared)
+ , boundary_condition_type(scratch_data.boundary_condition_type)
+ {}
+
+ // This function assembles a fraction of the system matrix and the system
+ // right-hand side related to a single cell. These fractions are
+ // `copy_data.cell_matrix` and `copy_data.cell_rhs`. They are copied into
+ // to the system matrix, $A_{ij}$, and the right-hand side, $b_i$, by the
+ // function `Solver::copy_local_to_global()`.
+ void Solver::system_matrix_local(const IteratorPair &IP,
+ AssemblyScratchData &scratch_data,
+ AssemblyCopyData ©_data)
+ {
+ // First we reinitialize the matrices and vectors related to the current
+ // cell.
+ copy_data.cell_matrix.reinit(scratch_data.dofs_per_cell,
+ scratch_data.dofs_per_cell);
+
+ copy_data.cell_rhs.reinit(scratch_data.dofs_per_cell);
+
+ copy_data.local_dof_indices.resize(scratch_data.dofs_per_cell);
+
+ // Second, we extract the cells from the pair. We extract them in the
+ // correct order, see above.
+ auto cell = std::get<0>(*IP);
+ auto cell_T = std::get<1>(*IP);
+
+ // Third, we compute the ordered FE values, the permeability, and the values
+ // of the current vector potential, $\vec{T}$, on the cell.
+ scratch_data.fe_values.reinit(cell);
+ scratch_data.fe_values_T.reinit(cell_T);
+
+ scratch_data.permeability.value_list(cell->material_id(),
+ scratch_data.permeability_list);
+
+ scratch_data.fe_values_T[scratch_data.ve].get_function_values(
+ scratch_data.dofs_T, scratch_data.T_values);
+
+ // Fourth, we compute the components of the cell matrix and cell right-hand
+ // side. The labels of the integrals are the same as in the introduction to
+ // this tutorial.
+ for (unsigned int q_index = 0; q_index < scratch_data.n_q_points; ++q_index)
+ {
+ for (unsigned int i = 0; i < scratch_data.dofs_per_cell; ++i)
+ {
+ for (unsigned int j = 0; j < scratch_data.dofs_per_cell; ++j)
+ {
+ copy_data.cell_matrix(i, j) += // Integral I_a1+I_a3.
+ (1.0 / scratch_data.permeability_list[q_index]) * // 1 / mu
+ (scratch_data.fe_values[scratch_data.ve].curl(
+ i, q_index) * // curl N_i
+ scratch_data.fe_values[scratch_data.ve].curl(
+ j, q_index) // curl N_j
+ + scratch_data.eta_squared * // eta^2
+ scratch_data.fe_values[scratch_data.ve].value(
+ i, q_index) * // N_i
+ scratch_data.fe_values[scratch_data.ve].value(
+ j, q_index) // N_j
+ ) *
+ scratch_data.fe_values.JxW(q_index); // dV
+ }
+ copy_data.cell_rhs(i) += // Integral I_b3-1.
+ (scratch_data.T_values[q_index] *
+ scratch_data.fe_values[scratch_data.ve].curl(i, q_index)) *
+ scratch_data.fe_values.JxW(q_index); // T.(curl N_i)dV
+ }
+ }
+
+ // If the Robin boundary condition (first-order ABC) is ordered,
+ // we compute an extra integral over the boundary.
+ if (scratch_data.boundary_condition_type == BoundaryConditionType::Robin)
+ {
+ for (unsigned int f = 0; f < cell->n_faces(); ++f)
+ {
+ if (cell->face(f)->at_boundary())
+ {
+ scratch_data.fe_face_values.reinit(cell, f);
+
+ for (unsigned int q_index_face = 0;
+ q_index_face < scratch_data.n_q_points_face;
+ ++q_index_face)
+ {
+ for (unsigned int i = 0; i < scratch_data.dofs_per_cell;
+ ++i)
+ {
+ scratch_data.gamma.value_list(
+ scratch_data.fe_face_values.get_quadrature_points(),
+ scratch_data.gamma_list);
+
+ for (unsigned int j = 0; j < scratch_data.dofs_per_cell;
+ ++j)
+ {
+ copy_data.cell_matrix(i, j) += // Integral I_a2.
+ scratch_data.gamma_list[q_index_face] * // gamma
+ (cross_product_3d(
+ scratch_data.fe_face_values.normal_vector(
+ q_index_face),
+ scratch_data.fe_face_values[scratch_data.ve]
+ .value(i, q_index_face)) *
+ cross_product_3d(
+ scratch_data.fe_face_values.normal_vector(
+ q_index_face),
+ scratch_data.fe_face_values[scratch_data.ve]
+ .value(j,
+ q_index_face))) // (n x N_i).(n x N_j)
+ * scratch_data.fe_face_values.JxW(
+ q_index_face); // dS
+ } // for j
+ } // for i
+ } // for q_index_face
+ } // if (cell->face(f)->at_boundary())
+ } // for f
+ } /* if (scratch_data.boundary_condition_type ==
+ BoundaryConditionType::Robin) */
+
+ // Finally, we query the dof indices on the current cell and store them
+ // in the copy data structure, so we know to which locations of the system
+ // matrix and right-hand side the components of the cell matrix and
+ // cell right-hand side must be copied.
+ cell->get_dof_indices(copy_data.local_dof_indices);
+ }
+
+ // This function copies the components of a cell matrix and a cell right-hand
+ // side into the system matrix, $A_{i,j}$, and the system right-hand side,
+ // $b_i$.
+ void Solver::copy_local_to_global(const AssemblyCopyData ©_data)
+ {
+ constraints.distribute_local_to_global(copy_data.cell_matrix,
+ copy_data.cell_rhs,
+ copy_data.local_dof_indices,
+ system_matrix,
+ system_rhs);
+ }
+
+ // This function solves the system of linear equations.
+ // If `Settings::log_cg_convergence == true`, the convergence data is saved
+ // into a file. In theory, a CG solver can solve an $m \times m$ system of
+ // linear equations in at most $m$ steps. In practice, it can take more steps
+ // to converge. The convergence of the algorithm depends on the spectral
+ // properties of the system matrix. The best case is if the eigenvalues form a
+ // compact cluster away from zero. In our case, however, the eigenvalues are
+ // spread in between zero and the maximal eigenvalue. Consequently, we expect
+ // a poor convergence and increase the maximal number of iteration steps by a
+ // factor of 10, i.e., `10*system_rhs.size()`. The stopping condition is \f[
+ // |\boldsymbol{b} - \boldsymbol{A}\boldsymbol{c}|
+ // < 10^{-6} |\boldsymbol{b}|.
+ // \f]
+ // As soon as we use constraints, we must not forget to distribute them.
+ void Solver::solve()
+ {
+ SolverControl control(10 * system_rhs.size(),
+ 1.0e-6 * system_rhs.l2_norm(),
+ false,
+ false);
+
+ if (Settings::log_cg_convergence)
+ control.enable_history_data();
+
+ GrowingVectorMemory<Vector<double>> memory;
+ SolverCG<Vector<double>> cg(control, memory);
+
+ PreconditionSSOR<SparseMatrix<double>> preconditioner;
+ preconditioner.initialize(system_matrix, 1.2);
+
+ cg.solve(system_matrix, solution, system_rhs, preconditioner);
+
+ constraints.distribute(solution);
+
+ if (Settings::log_cg_convergence)
+ {
+ const std::vector<double> history_data = control.get_history_data();
+
+ std::ofstream ofs(fname + "_cg_convergence.csv");
+
+ for (unsigned int i = 1; i < history_data.size(); i++)
+ ofs << i << ", " << history_data[i] << "\n";
+ }
+ }
+
+ // This function saves the computed magnetic vector potential into a vtu file.
+ void Solver::save() const
+ {
+ std::vector<std::string> solution_names(3, "VectorField");
+ std::vector<DataComponentInterpretation::DataComponentInterpretation>
+ interpretation(3,
+ DataComponentInterpretation::component_is_part_of_vector);
+
+ DataOut<3> data_out;
+
+ data_out.add_data_vector(dof_handler,
+ solution,
+ solution_names,
+ interpretation);
+
+ DataOutBase::VtkFlags flags;
+ flags.write_higher_order_cells = true;
+ data_out.set_flags(flags);
+
+ const MappingQ<3> mapping(mapping_degree);
+
+ data_out.build_patches(mapping,
+ fe.degree + 2,
+ DataOut<3>::CurvedCellRegion::curved_inner_cells);
+
+ std::ofstream ofs(fname + ".vtu");
+ data_out.write_vtu(ofs);
+ }
+
+ void Solver::run()
+ {
+ {
+ TimerOutput::Scope timer_section(timer, "Setup");
+ setup();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Assemble");
+ assemble();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Solve");
+ solve();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Save");
+ save();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Clear");
+ clear();
+ }
+ }
+} // namespace SolverA
+
+
+// @sect3{Projector from H(curl) to H(div)}
+// This name space contains all the code related to the conversion of the
+// magnetic vector potential, $\vec{A}$, into magnetic field, $\vec{B}$.
+// The magnetic vector potential is modeled by the FE_Nedelec finite elements,
+// while the magnetic field is modeled by the FE_RaviartThomas finite elements.
+// This code is also used for converting the current vector potential,
+// $\vec{T}$ into the free-current density, $\vec{J}_f$.
+namespace ProjectorHcurlToHdiv
+{
+
+ // This class implements the solver that minimizes the functional $F(\vec{B})$
+ // or $F(\vec{J}_f)$, see the introduction. The input vector field,
+ // $\vec{A}$ or $\vec{T}$, is fed to the solver by means of the input
+ // parameters `dof_handler_Hcurl` and `solution_Hcurl`. Moreover, this solver
+ // reuses the mesh on which the input vector field has been computed. The
+ // reference to the mesh is passed via the input parameter
+ // `triangulation_Hcurl`. There are no constraints this time around as we are
+ // not going to apply the Dirichlet boundary condition.
+ class Solver
+ {
+ public:
+ Solver() = delete;
+ Solver(const unsigned int p, // Degree of the Raviart-Thomas finite elements
+ const unsigned int mapping_degree,
+ const Triangulation<3> &triangulation_Hcurl,
+ const DoFHandler<3> &dof_handler_Hcurl,
+ const Vector<double> &solution_Hcurl,
+ const std::string &fname = "data",
+ const Function<3> *exact_solution = nullptr);
+
+ double get_L2_norm()
+ {
+ return L2_norm;
+ };
+
+ unsigned int get_n_cells() const
+ {
+ return triangulation_Hcurl.n_active_cells();
+ }
+
+ types::global_dof_index get_n_dofs() const
+ {
+ return dof_handler_Hdiv.n_dofs();
+ }
+
+ void setup(); // Initializes dofs, vectors, matrices.
+ void assemble(); // Assembles the system of linear equations.
+ void solve(); // Solves the system of linear equations.
+ void save() const; // Saves computed T into a vtu file.
+ void compute_error_norms(); // Computes L^2 error norm.
+ void project_exact_solution_fcn(); // Projects exact solution.
+ void clear()
+ {
+ system_matrix.clear();
+ system_rhs.reinit(0);
+ }
+ void run(); /* Executes the last seven functions in the proper order
+ and measures the execution time for each function. */
+
+ private:
+ const Triangulation<3> &triangulation_Hcurl;
+ const DoFHandler<3> &dof_handler_Hcurl;
+ const Vector<double> &solution_Hcurl;
+
+ // The following data members are typical for all deal.II simulations:
+ // triangulation, finite elements, dof handlers, etc. The constraints
+ // are used to enforce the Dirichlet boundary conditions. The names of the
+ // data members are self-explanatory.
+ const FE_RaviartThomas<3> fe_Hdiv;
+ DoFHandler<3> dof_handler_Hdiv;
+
+ SparsityPattern sparsity_pattern;
+ SparseMatrix<double> system_matrix;
+
+ Vector<double> solution_Hdiv;
+ Vector<double> system_rhs;
+
+ Vector<double> projected_exact_solution;
+
+ AffineConstraints<double> constraints;
+
+ const Function<3> *exact_solution;
+
+ const unsigned int mapping_degree;
+
+ Vector<double> L2_per_cell;
+ double L2_norm;
+
+ const std::string fname;
+ TimerOutput timer;
+
+ // This time we have two dof handlers, `dof_handler_Hcurl` for the input
+ // vector field and `dof_handler_Hdiv` for the output vector field. The
+ // WorkStream needs to walk through the two dof handlers synchronously.
+ // For this purpose we will pair two active cells iterators (one from
+ // `dof_handler_Hcurl`, another from `dof_handler_Hdiv`) to be walked
+ // through synchronously. For that we need the `IteratorPair` type.
+ using IteratorTuple =
+ std::tuple<typename DoFHandler<3>::active_cell_iterator,
+ typename DoFHandler<3>::active_cell_iterator>;
+
+ using IteratorPair = SynchronousIterators<IteratorTuple>;
+
+ // The program utilizes the WorkStream technology. The Step-9 tutorial
+ // does a much better job of explaining the workings of WarkStream.
+ // Reading the "WorkStream paper", see the glossary, is recommended.
+ // The following structures and functions are related to WorkStream.
+ struct AssemblyScratchData
+ {
+ AssemblyScratchData(const FiniteElement<3> &fe,
+ const DoFHandler<3> &dof_handr_Hcurl,
+ const Vector<double> &dofs_Hcurl,
+ const unsigned int mapping_degree);
+
+ AssemblyScratchData(const AssemblyScratchData &scratch_data);
+
+ MappingQ<3> mapping;
+ FEValues<3> fe_values_Hdiv;
+ FEValues<3> fe_values_Hcurl;
+
+ const unsigned int dofs_per_cell;
+ const unsigned int n_q_points;
+
+ std::vector<Tensor<1, 3>> curl_vec_in_Hcurl;
+
+ const FEValuesExtractors::Vector ve;
+
+ const DoFHandler<3> &dof_hand_Hcurl;
+ const Vector<double> &dofs_Hcurl;
+ };
+
+ struct AssemblyCopyData
+ {
+ FullMatrix<double> cell_matrix;
+ Vector<double> cell_rhs;
+ std::vector<types::global_dof_index> local_dof_indices;
+ };
+
+ void system_matrix_local(const IteratorPair &IP,
+ AssemblyScratchData &scratch_data,
+ AssemblyCopyData ©_data);
+
+ void copy_local_to_global(const AssemblyCopyData ©_data);
+ };
+
+ Solver::Solver(const unsigned int p,
+ const unsigned int mapping_degree,
+ const Triangulation<3> &triangulation_Hcurl,
+ const DoFHandler<3> &dof_handler_Hcurl,
+ const Vector<double> &solution_Hcurl,
+ const std::string &fname,
+ const Function<3> *exact_solution)
+ : triangulation_Hcurl(triangulation_Hcurl)
+ , dof_handler_Hcurl(dof_handler_Hcurl)
+ , solution_Hcurl(solution_Hcurl)
+ , fe_Hdiv(p)
+ , exact_solution(exact_solution)
+ , mapping_degree(mapping_degree)
+ , fname(fname)
+ , timer(std::cout,
+ (Settings::print_time_tables) ? TimerOutput::summary :
+ TimerOutput::never,
+ TimerOutput::cpu_and_wall_times_grouped)
+ {
+ Assert(exact_solution != nullptr,
+ ExcMessage("The exact solution is missing."));
+ }
+
+ // This function initializes the dofs, vectors and matrices. This time there
+ // are no constraints as we do not apply Dirichlet boundary condition.
+ void Solver::setup()
+ {
+ constraints.close();
+
+ dof_handler_Hdiv.reinit(triangulation_Hcurl);
+ dof_handler_Hdiv.distribute_dofs(fe_Hdiv);
+
+ DynamicSparsityPattern dsp(dof_handler_Hdiv.n_dofs(),
+ dof_handler_Hdiv.n_dofs());
+ DoFTools::make_sparsity_pattern(dof_handler_Hdiv, dsp, constraints, false);
+
+ sparsity_pattern.copy_from(dsp);
+ system_matrix.reinit(sparsity_pattern);
+ solution_Hdiv.reinit(dof_handler_Hdiv.n_dofs());
+ system_rhs.reinit(dof_handler_Hdiv.n_dofs());
+
+ if (Settings::project_exact_solution && exact_solution)
+ projected_exact_solution.reinit(dof_handler_Hdiv.n_dofs());
+
+ if (exact_solution)
+ L2_per_cell.reinit(triangulation_Hcurl.n_active_cells());
+ }
+
+ // Formally, this function assembles the system of linear equations. In
+ // reality, however, it just spells all the magic words to get the WorkStream
+ // going. The interesting part, i.e., the actual assembling of the system
+ // matrix and the right-hand side happens below in the
+ // Solver::system_matrix_local function.
+ void Solver::assemble()
+ {
+ WorkStream::run(IteratorPair(
+ IteratorTuple(dof_handler_Hdiv.begin_active(),
+ dof_handler_Hcurl.begin_active())),
+ IteratorPair(IteratorTuple(dof_handler_Hdiv.end(),
+ dof_handler_Hcurl.end())),
+ *this,
+ &Solver::system_matrix_local,
+ &Solver::copy_local_to_global,
+ AssemblyScratchData(fe_Hdiv,
+ dof_handler_Hcurl,
+ solution_Hcurl,
+ mapping_degree),
+ AssemblyCopyData());
+ }
+
+ // The following two constructors initialize scratch data from the input
+ // parameters and from another object of the same type, i.e., a copy
+ // constructor.
+ Solver::AssemblyScratchData::AssemblyScratchData(
+ const FiniteElement<3> &fe,
+ const DoFHandler<3> &dof_hand_Hcurl,
+ const Vector<double> &dofs_Hcurl,
+ const unsigned int mapping_degree)
+ : mapping(mapping_degree)
+ , fe_values_Hdiv(mapping,
+ fe,
+ QGauss<3>(fe.degree + 2),
+ update_values | update_JxW_values)
+ , fe_values_Hcurl(mapping,
+ dof_hand_Hcurl.get_fe(),
+ QGauss<3>(fe.degree + 2),
+ update_gradients)
+ , dofs_per_cell(fe_values_Hdiv.dofs_per_cell)
+ , n_q_points(fe_values_Hdiv.get_quadrature().size())
+ , curl_vec_in_Hcurl(n_q_points)
+ , ve(0)
+ , dof_hand_Hcurl(dof_hand_Hcurl)
+ , dofs_Hcurl(dofs_Hcurl)
+ {}
+
+ Solver::AssemblyScratchData::AssemblyScratchData(
+ const AssemblyScratchData &scratch_data)
+ : mapping(scratch_data.mapping.get_degree())
+ , fe_values_Hdiv(mapping,
+ scratch_data.fe_values_Hdiv.get_fe(),
+ scratch_data.fe_values_Hdiv.get_quadrature(),
+ update_values | update_JxW_values)
+ , fe_values_Hcurl(mapping,
+ scratch_data.fe_values_Hcurl.get_fe(),
+ scratch_data.fe_values_Hcurl.get_quadrature(),
+ update_gradients)
+ , dofs_per_cell(fe_values_Hdiv.dofs_per_cell)
+ , n_q_points(fe_values_Hdiv.get_quadrature().size())
+ , curl_vec_in_Hcurl(scratch_data.n_q_points)
+ , ve(0)
+ , dof_hand_Hcurl(scratch_data.dof_hand_Hcurl)
+ , dofs_Hcurl(scratch_data.dofs_Hcurl)
+ {}
+
+ // This function assembles a fraction of the system matrix and the system
+ // right-hand side related to a single cell. These fractions are
+ // `copy_data.cell_matrix` and `copy_data.cell_rhs`. They are copied into
+ // to the system matrix, $A_{ij}$, and the right-hand side, $b_i$, by the
+ // function `Solver::copy_local_to_global()`.
+ void Solver::system_matrix_local(const IteratorPair &IP,
+ AssemblyScratchData &scratch_data,
+ AssemblyCopyData ©_data)
+ {
+ // First we reinitialize the matrices and vectors related to the current
+ // cell, update the FE values, and compute the curl of the input vector
+ // field.
+ copy_data.cell_matrix.reinit(scratch_data.dofs_per_cell,
+ scratch_data.dofs_per_cell);
+
+ copy_data.cell_rhs.reinit(scratch_data.dofs_per_cell);
+
+ copy_data.local_dof_indices.resize(scratch_data.dofs_per_cell);
+
+ scratch_data.fe_values_Hdiv.reinit(std::get<0>(*IP));
+ scratch_data.fe_values_Hcurl.reinit(std::get<1>(*IP));
+
+ // The variable `curl_vec_in_Hcurl` denotes the curl of the input vector
+ // field, $\vec{\nabla} \times \vec{T}$ or $\vec{\nabla} \times \vec{A}$,
+ // depending on the context.
+ scratch_data.fe_values_Hcurl[scratch_data.ve].get_function_curls(
+ scratch_data.dofs_Hcurl, scratch_data.curl_vec_in_Hcurl);
+
+ // Second, we compute the components of the cell matrix and cell right-hand
+ // side. The labels of the integrals are the same as in the introduction to
+ // this tutorial.
+ for (unsigned int q_index = 0; q_index < scratch_data.n_q_points; ++q_index)
+ {
+ for (unsigned int i = 0; i < scratch_data.dofs_per_cell; ++i)
+ {
+ for (unsigned int j = 0; j < scratch_data.dofs_per_cell; ++j)
+ {
+ copy_data.cell_matrix(i, j) += // Integral I_a
+ scratch_data.fe_values_Hdiv[scratch_data.ve].value(i,
+ q_index) *
+ scratch_data.fe_values_Hdiv[scratch_data.ve].value(j,
+ q_index) *
+ scratch_data.fe_values_Hdiv.JxW(q_index);
+ }
+
+ copy_data.cell_rhs(i) += // Integral I_b
+ scratch_data.curl_vec_in_Hcurl[q_index] *
+ scratch_data.fe_values_Hdiv[scratch_data.ve].value(i, q_index) *
+ scratch_data.fe_values_Hdiv.JxW(q_index);
+ }
+ }
+
+ // Finally, we query the dof indices on the current cell and store them
+ // in the copy data structure, so we know to which locations of the system
+ // matrix and right-hand side the components of the cell matrix and
+ // cell right-hand side must be copied.
+ std::get<0>(*IP)->get_dof_indices(copy_data.local_dof_indices);
+ }
+
+ // This function copies the components of a cell matrix and a cell right-hand
+ // side into the system matrix, $A_{i,j}$, and the system right-hand side,
+ // $b_i$.
+ void Solver::copy_local_to_global(const AssemblyCopyData ©_data)
+ {
+ constraints.distribute_local_to_global(copy_data.cell_matrix,
+ copy_data.cell_rhs,
+ copy_data.local_dof_indices,
+ system_matrix,
+ system_rhs);
+ }
+
+ // The following two functions compute the error norms and project the exact
+ // solution.
+ void Solver::compute_error_norms()
+ {
+ const Weight weight;
+ const Function<3, double> *mask = &weight;
+
+ VectorTools::integrate_difference(MappingQ<3>(mapping_degree),
+ dof_handler_Hdiv,
+ solution_Hdiv,
+ *exact_solution,
+ L2_per_cell,
+ QGauss<3>(fe_Hdiv.degree + 4),
+ VectorTools::L2_norm,
+ mask);
+
+ L2_norm = VectorTools::compute_global_error(triangulation_Hcurl,
+ L2_per_cell,
+ VectorTools::L2_norm);
+ }
+
+ void Solver::project_exact_solution_fcn()
+ {
+ AffineConstraints<double> constraints_empty;
+ constraints_empty.close();
+
+ VectorTools::project(MappingQ<3>(mapping_degree),
+ dof_handler_Hdiv,
+ constraints_empty,
+ QGauss<3>(fe_Hdiv.degree + 2),
+ *exact_solution,
+ projected_exact_solution);
+ }
+
+ // This function solves the system of linear equations. This time we are
+ // dealing with a mass matrix. It has good spectral properties. Consequently,
+ // we do not use the factor of 10 as in preceding two solvers.
+ // The stopping condition is
+ // \f[
+ // |\boldsymbol{b} - \boldsymbol{A}\boldsymbol{c}|
+ // < 10^{-6} |\boldsymbol{b}|.
+ // \f]
+ void Solver::solve()
+ {
+ SolverControl control(system_rhs.size(),
+ 1.0e-6 * system_rhs.l2_norm(),
+ false,
+ false);
+
+ if (Settings::log_cg_convergence)
+ control.enable_history_data();
+
+ GrowingVectorMemory<Vector<double>> memory;
+ SolverCG<Vector<double>> cg(control, memory);
+
+ PreconditionSSOR<SparseMatrix<double>> preconditioner;
+ preconditioner.initialize(system_matrix, 1.2);
+
+ cg.solve(system_matrix, solution_Hdiv, system_rhs, preconditioner);
+
+ if (Settings::log_cg_convergence)
+ {
+ const std::vector<double> history_data = control.get_history_data();
+
+ std::ofstream ofs(fname + "_cg_convergence.csv");
+
+ for (unsigned int i = 1; i < history_data.size(); i++)
+ ofs << i << ", " << history_data[i] << "\n";
+ }
+ }
+
+ // This function saves the computed fields into a vtu file. This time we also
+ // save the projected exact solution and the $L^2$ error norm. The exact
+ // solution is only saved if the `Settings::project_exact_solution = true`
+ void Solver::save() const
+ {
+ std::vector<std::string> solution_names(3, "VectorField");
+ std::vector<DataComponentInterpretation::DataComponentInterpretation>
+ interpretation(3,
+ DataComponentInterpretation::component_is_part_of_vector);
+
+ DataOut<3> data_out;
+
+ data_out.add_data_vector(dof_handler_Hdiv,
+ solution_Hdiv,
+ solution_names,
+ interpretation);
+
+ if (Settings::project_exact_solution)
+ {
+ std::vector<std::string> solution_names_ex(3, "VectorFieldExact");
+
+ data_out.add_data_vector(dof_handler_Hdiv,
+ projected_exact_solution,
+ solution_names_ex,
+ interpretation);
+ }
+
+ if (exact_solution)
+ {
+ data_out.add_data_vector(L2_per_cell, "L2norm");
+ }
+
+ DataOutBase::VtkFlags flags;
+ flags.write_higher_order_cells = true;
+ data_out.set_flags(flags);
+
+ const MappingQ<3> mapping(mapping_degree);
+
+ data_out.build_patches(mapping,
+ fe_Hdiv.degree + 2,
+ DataOut<3>::CurvedCellRegion::curved_inner_cells);
+
+ std::ofstream ofs(fname + ".vtu");
+ data_out.write_vtu(ofs);
+ }
+
+ void Solver::run()
+ {
+ {
+ TimerOutput::Scope timer_section(timer, "Setup");
+ setup();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Assemble");
+ assemble();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Solve");
+ solve();
+ }
+
+ if (exact_solution)
+ {
+ {
+ TimerOutput::Scope timer_section(timer, "Compute error norms");
+ compute_error_norms();
+ }
+
+ if (Settings::project_exact_solution)
+ {
+ {
+ TimerOutput::Scope timer_section(timer, "Project exact solution");
+ project_exact_solution_fcn();
+ }
+ }
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Save");
+ save();
+ }
+ {
+ TimerOutput::Scope timer_section(timer, "Clear");
+ clear();
+ }
+ }
+} // namespace ProjectorHcurlToHdiv
+
+
+// @sect3{The main loop}
+
+// This class contains the main loop of the program.
+class MagneticProblem
+{
+public:
+ void run()
+ {
+ if (Settings::n_threads_max)
+ MultithreadInfo::set_thread_limit(Settings::n_threads_max);
+
+ MainOutputTable table_Jf(3);
+ MainOutputTable table_B(3);
+
+ table_Jf.clear();
+ table_B.clear();
+
+ std::cout << "Solving for (p = " << Settings::fe_degree
+ << "): " << std::flush;
+
+ for (unsigned int r = 6; r < 10; r++) // Mesh refinement parameter.
+ {
+ table_Jf.add_value("r", r);
+ table_Jf.add_value("p", Settings::fe_degree);
+
+ table_B.add_value("r", r);
+ table_B.add_value("p", Settings::fe_degree);
+
+ // Stage 1. Computing $\vec{T}$.
+
+ std::cout << "T " << std::flush;
+
+ SolverT::Solver T(Settings::fe_degree,
+ r,
+ Settings::mapping_degree,
+ Settings::eta_squared_T,
+ "T_p" + std::to_string(Settings::fe_degree) + "_r" +
+ std::to_string(r));
+
+ T.run();
+
+ // Stage 2. Computing $\vec{J}_f$.
+
+ std::cout << "Jf " << std::flush;
+
+ ExactSolutions::FreeCurrentDensity Jf_exact;
+
+ ProjectorHcurlToHdiv::Solver Jf(Settings::fe_degree,
+ Settings::mapping_degree,
+ T.get_tria(),
+ T.get_dof_handler(),
+ T.get_solution(),
+ "Jf_p" +
+ std::to_string(Settings::fe_degree) +
+ "_r" + std::to_string(r),
+ &Jf_exact);
+
+ Jf.run();
+
+ table_Jf.add_value("ndofs", Jf.get_n_dofs());
+ table_Jf.add_value("ncells", Jf.get_n_cells());
+ table_Jf.add_value("L2", Jf.get_L2_norm());
+
+ // Stage 3. Computing $\vec{A}$.
+
+ std::cout << "A " << std::flush;
+
+ SolverA::Solver A(Settings::fe_degree,
+ Settings::mapping_degree,
+ T.get_tria(),
+ T.get_dof_handler(),
+ T.get_solution(),
+ Settings::eta_squared_A,
+ "A_p" + std::to_string(Settings::fe_degree) + "_r" +
+ std::to_string(r));
+
+ A.run();
+
+ // Stage 4. Computing $\vec{B}$.
+
+ std::cout << "B " << std::flush;
+
+ ExactSolutions::MagneticField B_exact;
+
+ ProjectorHcurlToHdiv::Solver B(Settings::fe_degree,
+ Settings::mapping_degree,
+ T.get_tria(),
+ A.get_dof_handler(),
+ A.get_solution(),
+ "B_p" +
+ std::to_string(Settings::fe_degree) +
+ "_r" + std::to_string(r),
+ &B_exact);
+ B.run();
+
+ table_B.add_value("ndofs", B.get_n_dofs());
+ table_B.add_value("ncells", B.get_n_cells());
+ table_B.add_value("L2", B.get_L2_norm());
+ // End stage 4.
+
+ table_Jf.save("table_Jf_p" + std::to_string(Settings::fe_degree));
+ table_B.save("table_B_p" + std::to_string(Settings::fe_degree));
+ }
+ std::cout << std::endl;
+ }
+};
+
+int main()
+{
+ try
+ {
+ MagneticProblem problem;
+ problem.run();
+ }
+ catch (std::exception &exc)
+ {
+ std::cerr << std::endl
+ << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+ std::cerr << "Exception on processing: " << std::endl
+ << exc.what() << std::endl
+ << "Aborting!" << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+
+ return 1;
+ }
+ catch (...)
+ {
+ std::cerr << std::endl
+ << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+ std::cerr << "Unknown exception!" << std::endl
+ << "Aborting!" << std::endl
+ << "----------------------------------------------------"
+ << std::endl;
+ return 1;
+ }
+
+ return 0;
+}
--- /dev/null
+560,561c560,561
+< std::ifstream ifs("sphere_r" + std::to_string(refinement_parameter) +
+< ".msh");
+---
+> std::ifstream ifs("../../../source/step-97/sphere_r" +
+> std::to_string(refinement_parameter) + ".msh");
+1938c1938
+< for (unsigned int r = 6; r < 10; r++) // Mesh refinement parameter.
+---
+> for (unsigned int r = 6; r < 7; r++) // Mesh refinement parameter.
--- /dev/null
+Solving for (p = 0): T Jf A B