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---

# Electromagnetic scattering from a wire with scattering boundary conditions

Copyright (C) 2022 Michele Castriotta, Igor Baratta, Jørgen S. Dokken

This demo is implemented in two files: one for the mesh generation
with gmsh, and one for the variational forms and the solver. It
illustrates how to:

- Use complex quantities in FEniCSx
- Setup and solve Maxwell's equations
- Implement Scattering Boundary Conditions

## Equations, problem definition and implementation

First of all, let's import the modules that will be used:

```python
import sys
from typing import Tuple

from analytical_efficiencies_wire import calculate_analytical_efficiencies
from mesh_wire import generate_mesh_wire

import ufl
from basix.ufl import element
from dolfinx import default_scalar_type, fem, io, plot
from dolfinx.fem.petsc import LinearProblem

from mpi4py import MPI

try:
    import gmsh
except ModuleNotFoundError:
    print("This demo requires gmsh to be installed")
    sys.exit(0)
import numpy as np

try:
    import pyvista
    have_pyvista = True
except ModuleNotFoundError:
    print("pyvista and pyvistaqt are required to visualise the solution")
    have_pyvista = False
```

Since we want to solve time-harmonic Maxwell's equation, we need to
solve a complex-valued PDE, and therefore need to use PETSc compiled
with complex numbers.

```python
if not np.issubdtype(default_scalar_type, np.complexfloating):
    print("Demo should only be executed with DOLFINx complex mode")
    exit(0)
```

Now, let's consider an infinite metallic wire immersed in a background
medium (e.g. vacuum or water). Let's now consider the plane cutting
the wire perpendicularly to its axis at a generic point. Such plane
$\Omega=\Omega_{m} \cup\Omega_{b}$ is formed by the cross-section of
the wire $\Omega_m$ and the background medium $\Omega_{b}$ surrounding
the wire. Let's consider just the portion of this plane delimited by
an external circular boundary $\partial \Omega$. We want to calculate
the electric field $\mathbf{E}_s$ scattered by the wire when a
background wave $\mathbf{E}_b$ impinges on it. We will consider a
background plane wave at $\lambda_0$ wavelength, that can be written
analytically as:

$$
\mathbf{E}_b = \exp(\mathbf{k}\cdot\mathbf{r})\hat{\mathbf{u}}_p
$$

with $\mathbf{k} = \frac{2\pi}{\lambda_0}n_b\hat{\mathbf{u}}_k$ being
the wavevector of the plane wave, pointing along the propagation
direction, with $\hat{\mathbf{u}}_p$ being the polarization direction,
and with $\mathbf{r}$ being a point in $\Omega$. We will only consider
$\hat{\mathbf{u}}_k$ and $\hat{\mathbf{u}}_p$ with components
belonging to the $\Omega$ domain and perpendicular to each other, i.e.
$\hat{\mathbf{u}}_k \perp \hat{\mathbf{u}}_p$ (transversality
condition of plane waves). Using a Cartesian coordinate system for
$\Omega$, and by defining $k_x = n_bk_0\cos\theta$ and $k_y =
n_bk_0\sin\theta$, with $\theta$ being the angle defined by the
propagation direction $\hat{\mathbf{u}}_k$ and the horizontal axis
$\hat{\mathbf{u}}_x$, we have:

$$
\mathbf{E}_b = -\sin\theta e^{j (k_xx+k_yy)}\hat{\mathbf{u}}_x
+ \cos\theta e^{j (k_xx+k_yy)}\hat{\mathbf{u}}_y
$$

The following class implements this functions. The inputs to the
function are the angle $\theta$, the background refractive index $n_b$
and the vacuum wavevector $k_0$.

```python

class BackgroundElectricField:

    def __init__(self, theta: float, n_bkg: float, k0: complex):
        self.theta = theta  # incident angle
        self.k0 = k0  # vacuum wavevector
        self.n_bkg = n_bkg  # background refractive index

    def eval(self, x: np.typing.NDArray[np.float64]) -> Tuple[np.typing.NDArray[np.complex128],
                                                              np.typing.NDArray[np.complex128]]:
        kx = self.n_bkg * self.k0 * np.cos(self.theta)
        ky = self.n_bkg * self.k0 * np.sin(self.theta)
        phi = kx * x[0] + ky * x[1]
        ax, ay = np.sin(self.theta), np.cos(self.theta)
        return (-ax * np.exp(1j * phi), ay * np.exp(1j * phi))
```

The Maxwell's equation for scattering problems takes the following
form:

$$
-\nabla \times \nabla \times \mathbf{E}_s+\varepsilon_{r} k_{0}^{2}
\mathbf{E}_s
+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_{b}\right)
\mathbf{E}_{\mathrm{b}}=0 \textrm{ in } \Omega,
$$

where $k_0 = 2\pi/\lambda_0$ is the vacuum wavevector of the
background field, $\varepsilon_b$ is the background relative
permittivity and $\varepsilon_r$ is the relative permittivity as a
function of space, i.e.:

$$
\varepsilon_r = \begin{cases}
\varepsilon_m & \textrm{on }\Omega_m \\
\varepsilon_b & \textrm{on }\Omega_b
\end{cases}
$$

with $\varepsilon_m$ being the relative permittivity of the metallic
wire. As reference values, we will consider $\lambda_0 =
400\textrm{nm}$ (violet light), $\varepsilon_b = 1.33^2$ (relative
permittivity of water), and $\varepsilon_m = -1.0782 +
5.8089\textrm{j}$ (relative permittivity of gold at $400\textrm{nm}$).

To form a well-determined system, we add boundary conditions on
$\partial \Omega$. It is common to use scattering boundary conditions
(ref), which make the boundary transparent for $\mathbf{E}_s$,
allowing us to restrict the computational boundary to a finite
$\Omega$ domain. The first-order boundary conditions in the 2D case
take the following form:

$$\mathbf{n} \times
\nabla \times \mathbf{E}_s+\left(j k_{0}n_b + \frac{1}{2r}
\right) \mathbf{n} \times \mathbf{E}_s
\times \mathbf{n}=0\quad \textrm{ on } \partial \Omega,
$$


with $n_b = \sqrt{\varepsilon_b}$ being the background refractive
index, $\mathbf{n}$ being the normal vector to $\partial \Omega$, and
$r = \sqrt{(x-x_s)^2 + (y-y_s)^2}$ being the distance of the $(x, y)$
point on $\partial\Omega$ from the wire centered in $(x_s, y_s)$. We
consider a wired centered at the origin, i.e. $r =\sqrt{x^2 + y^2}$.

The radial distance function $r(x)$ and $\nabla \times$ operator for a
2D vector (in UFL syntax) is defined below.

```python
def radial_distance(x: ufl.SpatialCoordinate):
    """Returns the radial distance from the origin"""
    return ufl.sqrt(x[0]**2 + x[1]**2)


def curl_2d(f: fem.Function):
    """Returns the curl of two 2D vectors as a 3D vector"""
    return ufl.as_vector((0, 0, f[1].dx(0) - f[0].dx(1)))
```

Next we define some mesh specific parameters. Please notice that the
length units are normalized with respect to $1\mu m$.

```python
pi = np.pi
epsilon_0 = 8.8541878128 * 10**-12
mu_0 = 4 * pi * 10**-7

# Radius of the wire and of the boundary of the domain
radius_wire = 0.050
radius_dom = 1

# The smaller the mesh_factor, the finer is the mesh
mesh_factor = 1.2

# Mesh size inside the wire
in_wire_size = mesh_factor * 7.0e-3

# Mesh size at the boundary of the wire
on_wire_size = mesh_factor * 3.0e-3

# Mesh size in the background
bkg_size = mesh_factor * 60.0e-3

# Mesh size at the boundary
boundary_size = mesh_factor * 30.0e-3

# Tags for the subdomains
au_tag = 1          # gold wire
bkg_tag = 2         # background
boundary_tag = 3    # boundary
```

We generate the mesh using GMSH and convert it to a
`dolfinx.mesh.Mesh`.

```python
model = None
gmsh.initialize(sys.argv)
if MPI.COMM_WORLD.rank == 0:
    model = generate_mesh_wire(radius_wire, radius_dom, in_wire_size, on_wire_size, bkg_size,
                               boundary_size, au_tag, bkg_tag, boundary_tag)

model = MPI.COMM_WORLD.bcast(model, root=0)
domain, cell_tags, facet_tags = io.gmshio.model_to_mesh(model, MPI.COMM_WORLD, 0, gdim=2)

gmsh.finalize()
MPI.COMM_WORLD.barrier()
```

The mesh is visualized with [PyVista](https://docs.pyvista.org/)

```python
if have_pyvista:
    topology, cell_types, geometry = plot.vtk_mesh(domain, 2)
    grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)
    plotter = pyvista.Plotter()
    num_local_cells = domain.topology.index_map(domain.topology.dim).size_local
    grid.cell_data["Marker"] = cell_tags.values[cell_tags.indices < num_local_cells]
    grid.set_active_scalars("Marker")
    plotter.add_mesh(grid, show_edges=True)
    plotter.view_xy()
    if not pyvista.OFF_SCREEN:
        plotter.show()
    else:
        pyvista.start_xvfb()
        figure = plotter.screenshot("wire_mesh.png", window_size=[8000, 8000])
```

Now we define some other problem specific parameters:

```python
wl0 = 0.4  # Wavelength of the background field
n_bkg = 1.33  # Background refractive index
eps_bkg = n_bkg**2  # Background relative permittivity
k0 = 2 * np.pi / wl0  # Wavevector of the background field
theta = np.pi / 4  # Angle of incidence of the background field
```

We use a function space consisting of degree 3 [Nedelec (first
kind)](https://defelement.com/elements/nedelec1.html) elements to
represent the electric field

```python
degree = 3
curl_el = element("N1curl", domain.basix_cell(), degree)
V = fem.functionspace(domain, curl_el)
```

Next, we can interpolate $\mathbf{E}_b$ into the function space $V$:

```python
f = BackgroundElectricField(theta, n_bkg, k0)
Eb = fem.Function(V)
Eb.interpolate(f.eval)

x = ufl.SpatialCoordinate(domain)
r = radial_distance(x)

# Create test and trial functions
Es = ufl.TrialFunction(V)
v = ufl.TestFunction(V)

# Definition of 3d fields for cross and curl operations
Es_3d = ufl.as_vector((Es[0], Es[1], 0))
v_3d = ufl.as_vector((v[0], v[1], 0))

# Measures for subdomains
dx = ufl.Measure("dx", domain, subdomain_data=cell_tags)
ds = ufl.Measure("ds", domain, subdomain_data=facet_tags)
dDom = dx((au_tag, bkg_tag))
dsbc = ds(boundary_tag)

# Normal to the boundary
n = ufl.FacetNormal(domain)
n_3d = ufl.as_vector((n[0], n[1], 0))
```

We turn our focus to the permittivity $\varepsilon$. First, we define
the relative permittivity $\varepsilon_m$ of the gold wire at $400nm$.
This data can be found in [*Olmon et al.
2012*](https://doi.org/10.1103/PhysRevB.86.235147) or at
[refractiveindex.info](
https://refractiveindex.info/?shelf=main&book=Au&page=Olmon-sc)):

```python
eps_au = -1.0782 + 1j * 5.8089
```

We define a permittivity function $\varepsilon$ that takes the value
of the gold permittivity $\varepsilon_m$ for cells inside the wire,
while it takes the value of the background permittivity otherwise:

```python
D = fem.functionspace(domain, ("DG", 0))
eps = fem.Function(D)
au_cells = cell_tags.find(au_tag)
bkg_cells = cell_tags.find(bkg_tag)
eps.x.array[au_cells] = np.full_like(au_cells, eps_au, dtype=eps.x.array.dtype)
eps.x.array[bkg_cells] = np.full_like(bkg_cells, eps_bkg, dtype=eps.x.array.dtype)
eps.x.scatter_forward()
```

Next we derive the weak formulation of the Maxwell's equation plus
with scattering boundary conditions. First, we take the inner products
of the equations with a complex test function $\mathbf{v}$, and
integrate the terms over the corresponding domains:

$$
& \int_{\Omega}-\nabla \times( \nabla \times \mathbf{E}_s) \cdot
\bar{\mathbf{v}}+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot
\bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right)
\mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d}x \\ +& \int_{\partial \Omega}
(\mathbf{n} \times \nabla \times \mathbf{E}_s) \cdot \bar{\mathbf{v}}
+\left(j n_bk_{0}+\frac{1}{2r}\right) (\mathbf{n} \times \mathbf{E}_s
\times \mathbf{n}) \cdot \bar{\mathbf{v}}~\mathrm{d}s=0
$$

By using $(\nabla \times \mathbf{A}) \cdot \mathbf{B}=\mathbf{A}
\cdot(\nabla \times \mathbf{B})+\nabla \cdot(\mathbf{A} \times
\mathbf{B}),$ we can change the first term into:

$$
& \int_{\Omega}-\nabla \cdot(\nabla\times\mathbf{E}_s \times
\bar{\mathbf{v}})-\nabla \times \mathbf{E}_s \cdot \nabla
\times\bar{\mathbf{v}}+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s
\cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right)
\mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{dx} \\ +&\int_{\partial \Omega}
(\mathbf{n} \times \nabla \times \mathbf{E}_s) \cdot \bar{\mathbf{v}}
+\left(j n_bk_{0}+\frac{1}{2r}\right) (\mathbf{n} \times \mathbf{E}_s
\times \mathbf{n}) \cdot \bar{\mathbf{v}}~\mathrm{d}s=0,
$$

using the divergence theorem
$\int_\Omega\nabla\cdot\mathbf{F}~\mathrm{d}x = \int_{\partial\Omega}
\mathbf{F}\cdot\mathbf{n}~\mathrm{d}s$, we can write:

$$
& \int_{\Omega}-(\nabla \times \mathbf{E}_s) \cdot (\nabla \times
\bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot
\bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right)
\mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d}x \\ +&\int_{\partial \Omega}
-(\nabla\times\mathbf{E}_s \times \bar{\mathbf{v}})\cdot\mathbf{n}
+ (\mathbf{n} \times \nabla \times \mathbf{E}_s) \cdot \bar{\mathbf{v}}
+\left(j n_bk_{0}+\frac{1}{2r}\right) (\mathbf{n} \times \mathbf{E}_s
\times \mathbf{n}) \cdot \bar{\mathbf{v}}~\mathrm{d}s=0.
$$

Cancelling $-(\nabla\times\mathbf{E}_s \times \bar{\mathbf{V}})
\cdot\mathbf{n}$  and $\mathbf{n} \times \nabla \times \mathbf{E}_s
\cdot \bar{\mathbf{V}}$ using the triple product rule $\mathbf{A}
\cdot(\mathbf{B} \times \mathbf{C})=\mathbf{B} \cdot(\mathbf{C} \times
\mathbf{A})=\mathbf{C} \cdot(\mathbf{A} \times \mathbf{B})$, we get:

$$
& \int_{\Omega}-(\nabla \times \mathbf{E}_s) \cdot (\nabla \times
\bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot
\bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right)
\mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d}x \\ +&\int_{\partial \Omega}
\left(j n_bk_{0}+\frac{1}{2r}\right)( \mathbf{n} \times \mathbf{E}_s \times
\mathbf{n}) \cdot \bar{\mathbf{v}} ~\mathrm{d} s = 0.
$$

We use the [UFL](https://github.com/FEniCS/ufl/) to implement the
residual

```python
# Weak form
F = - ufl.inner(ufl.curl(Es), ufl.curl(v)) * dDom \
    + eps * (k0**2) * ufl.inner(Es, v) * dDom \
    + (k0**2) * (eps - eps_bkg) * ufl.inner(Eb, v) * dDom \
    + (1j * k0 * n_bkg + 1 / (2 * r)) \
    * ufl.inner(ufl.cross(Es_3d, n_3d), ufl.cross(v_3d, n_3d)) * dsbc
```

We split the residual into a sesquilinear (lhs) and linear (rhs) form
and solve the problem. We store the scattered field $\mathbf{E}_s$ as
`Esh`:

```python
a, L = ufl.lhs(F), ufl.rhs(F)
problem = LinearProblem(a, L, bcs=[], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
Esh = problem.solve()
```

We save the solution as an [ADIOS2
bp](https://adios2.readthedocs.io/en/latest/ecosystem/visualization.html)
folder. In order to do so, we need to interpolate our solution
discretized with Nedelec elements into a suitable discontinuous
Lagrange space.

```python
gdim = domain.geometry.dim
V_dg = fem.functionspace(domain, ("Discontinuous Lagrange", degree, (gdim,)))
Esh_dg = fem.Function(V_dg)
Esh_dg.interpolate(Esh)

if hasattr(io,"VTXWriter"):
    with io.VTXWriter(domain.comm, "Esh.bp", Esh_dg) as vtx:
        vtx.write(0.0)
else:
    print("Cannot write Esh.bp: VTXWriter (adios2) is not available")
```

We visualize the solution using PyVista. For more information about
saving and visualizing vector fields discretized with Nedelec
elements, check [this](
https://docs.fenicsproject.org/dolfinx/main/python/demos/demo_interpolation-io.html)
DOLFINx demo.

```python
if have_pyvista:
    V_cells, V_types, V_x = plot.vtk_mesh(V_dg)
    V_grid = pyvista.UnstructuredGrid(V_cells, V_types, V_x)
    Esh_values = np.zeros((V_x.shape[0], 3), dtype=np.float64)
    Esh_values[:, : domain.topology.dim] = Esh_dg.x.array.reshape(V_x.shape[0], domain.topology.dim).real

    V_grid.point_data["u"] = Esh_values

    plotter = pyvista.Plotter()
    plotter.add_text("magnitude", font_size=12, color="black")
    plotter.add_mesh(V_grid.copy(), show_edges=True)
    plotter.view_xy()
    plotter.link_views()
    if not pyvista.OFF_SCREEN:
        plotter.show()
    else:
        pyvista.start_xvfb()
        plotter.screenshot("Esh.png", window_size=[800, 800])
```

Next we can calculate the total electric field
$\mathbf{E}=\mathbf{E}_s+\mathbf{E}_b$ and save it.

```python
E = fem.Function(V)
E.x.array[:] = Eb.x.array[:] + Esh.x.array[:]
E_dg = fem.Function(V_dg)
E_dg.interpolate(E)
if hasattr(io,"VTXWriter"):
    with io.VTXWriter(domain.comm, "E.bp", E_dg) as vtx:
        vtx.write(0.0)
else:
    print("Cannot write E.bp: VTXWriter (adios2) is not available")
```

We validate our numerical solution by computing the absorption,
scattering and extinction efficiencies, which are quantities that
define how much light is absorbed and scattered by the wire. First of
all, we calculate the analytical efficiencies with the
`calculate_analytical_efficiencies` function defined in a separate
file:

```python
# Calculation of analytical efficiencies
q_abs_analyt, q_sca_analyt, q_ext_analyt = calculate_analytical_efficiencies(
    eps_au, n_bkg, wl0, radius_wire)
```

Now we can calculate the numerical efficiencies. The formula for the
absorption, scattering and extinction are:

$$
& Q_{abs} = \operatorname{Re}\left(\int_{\Omega_{m}} \frac{1}{2}
  \frac{\operatorname{Im}(\varepsilon_m)k_0}{Z_0n_b}
  \mathbf{E}\cdot\hat{\mathbf{E}}dx\right) \\
& Q_{sca} = \operatorname{Re}\left(\int_{\partial\Omega} \frac{1}{2}
  \left(\mathbf{E}_s\times\bar{\mathbf{H}}_s\right)
  \cdot\mathbf{n}ds\right)\\ \\
& Q_{ext} = Q_{abs} + Q_{sca},
$$

with $Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}}$ being the vacuum
impedance, and $\mathbf{H}_s =
-j\frac{1}{Z_0k_0n_b}\nabla\times\mathbf{E}_s$ being the scattered
magnetic field. We can then normalize these values over the intensity
of the electromagnetic field $I_0$ and the geometrical cross section
of the wire, $\sigma_{gcs} = 2r_w$:

$$
& q_{abs} = \frac{Q_{abs}}{I_0\sigma_{gcs}} \\
& q_{sca} = \frac{Q_{sca}}{I_0\sigma_{gcs}} \\
& q_{ext} = q_{abs} + q_{sca}.
$$

We can calculate these values in the following way:

```python
# Vacuum impedance
Z0 = np.sqrt(mu_0 / epsilon_0)

# Magnetic field H
Hsh_3d = -1j * curl_2d(Esh) / (Z0 * k0 * n_bkg)

Esh_3d = ufl.as_vector((Esh[0], Esh[1], 0))
E_3d = ufl.as_vector((E[0], E[1], 0))

# Intensity of the electromagnetic fields I0 = 0.5*E0**2/Z0 E0 =
# np.sqrt(ax**2 + ay**2) = 1, see background_electric_field
I0 = 0.5 / Z0

# Geometrical cross section of the wire
gcs = 2 * radius_wire

# Quantities for the calculation of efficiencies
P = 0.5 * ufl.inner(ufl.cross(Esh_3d, ufl.conj(Hsh_3d)), n_3d)
Q = 0.5 * np.imag(eps_au) * k0 * (ufl.inner(E_3d, E_3d)) / Z0 / n_bkg

# Define integration domain for the wire
dAu = dx(au_tag)

# Normalized absorption efficiency
q_abs_fenics_proc = (fem.assemble_scalar(fem.form(Q * dAu)) / gcs / I0).real
q_abs_fenics = domain.comm.allreduce(q_abs_fenics_proc, op=MPI.SUM)

# Normalized scattering efficiency
q_sca_fenics_proc = (fem.assemble_scalar(fem.form(P * dsbc)) / gcs / I0).real
q_sca_fenics = domain.comm.allreduce(q_sca_fenics_proc, op=MPI.SUM)

# Extinction efficiency
q_ext_fenics = q_abs_fenics + q_sca_fenics

# Error calculation
err_abs = np.abs(q_abs_analyt - q_abs_fenics) / q_abs_analyt
err_sca = np.abs(q_sca_analyt - q_sca_fenics) / q_sca_analyt
err_ext = np.abs(q_ext_analyt - q_ext_fenics) / q_ext_analyt

# Check if errors are smaller than 1%
assert err_abs < 0.01
assert err_sca < 0.01
assert err_ext < 0.01

if domain.comm.rank == 0:
    print()
    print(f"The analytical absorption efficiency is {q_abs_analyt}")
    print(f"The numerical absorption efficiency is {q_abs_fenics}")
    print(f"The error is {err_abs*100}%")
    print()
    print(f"The analytical scattering efficiency is {q_sca_analyt}")
    print(f"The numerical scattering efficiency is {q_sca_fenics}")
    print(f"The error is {err_sca*100}%")
    print()
    print(f"The analytical extinction efficiency is {q_ext_analyt}")
    print(f"The numerical extinction efficiency is {q_ext_fenics}")
    print(f"The error is {err_ext*100}%")
```
