# Electromagnetic scattering from a wire with perfectly matched layer condition

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

This demo is implemented in three files: one for the mesh generation
with Gmsh, one for the calculation of analytical efficiencies, 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 (rectangular) perfectly matched layers

## Equations, problem definition and implementation

First, we import the required modules

```python
import sys
from functools import partial
from typing import Union

from mpi4py import MPI

from efficiencies_pml_demo import calculate_analytical_efficiencies
from mesh_wire_pml import generate_mesh_wire

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

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 require
that the demo is executed with DOLFINx (PETSc) complex mode.

```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. We limit the background medium with a squared perfectly
matched layer (or shortly PML), which will act as an absorber for
outgoing scattered waves.

The goal of this demo is 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, which 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 function `background_field` below implements this analytical
formula:

```python
def background_field(theta: float, n_b: float, k0: complex, x: np.typing.NDArray[np.float64]):
    kx = n_b * k0 * np.cos(theta)
    ky = n_b * k0 * np.sin(theta)
    phi = kx * x[0] + ky * x[1]
    return (-np.sin(theta) * np.exp(1j * phi), np.cos(theta) * np.exp(1j * phi))
```

For convenience, we define the $\nabla\times$ operator for a 2D vector

```python
def curl_2d(a: fem.Function):
    return ufl.as_vector((0, 0, a[1].dx(0) - a[0].dx(1)))
```

Let's now see how we can implement PMLs for our problem. PMLs are
artificial layers surrounding the real domain that gradually absorb
waves impinging them. Mathematically, we can use a complex coordinate
transformation of this kind to obtain this absorption:

$$
x^\prime= x\left\{1+j\frac{\alpha}{k_0}\left[\frac{|x|-l_{dom}/2}
{(l_{pml}/2 - l_{dom}/2)^2}\right] \right\}
$$

with $l_{dom}$ and $l_{pml}$ being the lengths of the domain without
and with PML, respectively, and with $\alpha$ being a parameter that
tunes the absorption within the PML (the bigger the $\alpha$, the
faster the absorption). In DOLFINx, we can define this coordinate
transformation in the following way:

```python
def pml_coordinates(x: ufl.indexed.Indexed, alpha: float, k0: complex, l_dom: float, l_pml: float):
    return x + 1j * alpha / k0 * x * (ufl.algebra.Abs(x) - l_dom / 2) / (l_pml / 2 - l_dom / 2) ** 2
```

We use the following domain specific parameters.

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

# Radius of the wire and of the boundary of the domain
radius_wire = 0.05
l_dom = 0.8
radius_scatt = 0.8 * l_dom / 2
l_pml = 1

# The smaller the mesh_factor, the finer is the mesh
mesh_factor = 1

# Mesh size inside the wire
in_wire_size = mesh_factor * 6e-3

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

# Mesh size in the background
scatt_size = mesh_factor * 15.0e-3

# Mesh size at the boundary
pml_size = mesh_factor * 15.0e-3

# Tags for the subdomains
au_tag = 1
bkg_tag = 2
scatt_tag = 3
pml_tag = 4
```

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_scatt,
        l_dom,
        l_pml,
        in_wire_size,
        on_wire_size,
        scatt_size,
        pml_size,
        au_tag,
        bkg_tag,
        scatt_tag,
        pml_tag,
    )
model = MPI.COMM_WORLD.bcast(model, root=0)
msh, cell_tags, facet_tags = gmshio.model_to_mesh(model, MPI.COMM_WORLD, 0, gdim=2)

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

We visualize the mesh and subdomains with
[PyVista](https://docs.pyvista.org/)

```python
if have_pyvista:
    topology, cell_types, geometry = plot.vtk_mesh(msh, 2)
    grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)
    plotter = pyvista.Plotter()
    num_local_cells = msh.topology.index_map(msh.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(interactive=True)
    else:
        pyvista.start_xvfb()
        figure = plotter.screenshot("wire_mesh_pml.png", window_size=[800, 800])
```

We observe five different subdomains: one for the gold wire
(`au_tag`), one for the background medium (`bkg_tag`), one for the PML
corners (`pml_tag`), one for the PML rectangles along $x$ (`pml_tag +
1`), and one for the PML rectangles along $y$ (`pml_tag + 2`). These
different PML regions have different coordinate transformation, as
specified here below:

$$
\begin{align}
\text{PML}_\text{corners} \rightarrow \mathbf{r}^\prime &= (x^\prime, y^\prime) \\
\text{PML}_\text{rectangles along x} \rightarrow
                                      \mathbf{r}^\prime &= (x^\prime, y) \\
\text{PML}_\text{rectangles along y} \rightarrow
                                      \mathbf{r}^\prime &= (x, y^\prime).
\end{align}
$$

Now we define some other problem specific parameters:

```python
wl0 = 0.4  # Wavelength of the background field
n_bkg = 1  # Background refractive index
eps_bkg = n_bkg**2  # Background relative permittivity
k0 = 2 * np.pi / wl0  # Wavevector of the background field
theta = 0  # Angle of incidence of the background field
```

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

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

Next, we interpolate $\mathbf{E}_b$ into the function space $V$,
define our trial and test function, and the integration domains:

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

# Definition of Trial and Test functions
Es = ufl.TrialFunction(V)
v = ufl.TestFunction(V)

# Definition of 3d fields
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", msh, subdomain_data=cell_tags)
dDom = dx((au_tag, bkg_tag))
dPml_xy = dx(pml_tag)
dPml_x = dx(pml_tag + 1)
dPml_y = dx(pml_tag + 2)
```

Let's now define the relative permittivity $\varepsilon_m$ of the gold
wire at $400nm$ (data taken from [*Olmon et al.
2012*](https://doi.org/10.1103/PhysRevB.86.235147) , and for a quick
reference have a look at [refractiveindex.info](
https://refractiveindex.info/?shelf=main&book=Au&page=Olmon-sc)):

```python
# Definition of relative permittivity for Au @400nm
eps_au = -1.0782 + 1j * 5.8089
```

We can now define a space function for the permittivity $\varepsilon$
that takes the value $\varepsilon_m$ for cells inside the wire, while
it takes the value of the background permittivity $\varepsilon_b$ in
the background region:

```python
D = fem.functionspace(msh, ("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()
```

Now we need to define our weak form in DOLFINx. Let's write the PML
weak form first. As a first step, we can define our new complex
coordinates as:

```python
x = ufl.SpatialCoordinate(msh)
alpha = 1

# PML corners
xy_pml = ufl.as_vector(
    (pml_coordinates(x[0], alpha, k0, l_dom, l_pml), pml_coordinates(x[1], alpha, k0, l_dom, l_pml))
)

# PML rectangles along x
x_pml = ufl.as_vector((pml_coordinates(x[0], alpha, k0, l_dom, l_pml), x[1]))

# PML rectangles along y
y_pml = ufl.as_vector((x[0], pml_coordinates(x[1], alpha, k0, l_dom, l_pml)))
```

We can then express this coordinate systems as a material
transformation within the PML region. In other words, the PML region
can be interpreted as a material having, in general, anisotropic,
inhomogeneous and complex permittivity
$\boldsymbol{\varepsilon}_{pml}$ and permeability
$\boldsymbol{\mu}_{pml}$. To do this, we need to calculate the
Jacobian of the coordinate transformation:

$$
\mathbf{J}=\mathbf{A}^{-1}= \nabla\boldsymbol{x}^
\prime(\boldsymbol{x}) =
\left[\begin{array}{ccc}
\frac{\partial x^{\prime}}{\partial x} &
\frac{\partial y^{\prime}}{\partial x} &
\frac{\partial z^{\prime}}{\partial x} \\
\frac{\partial x^{\prime}}{\partial y} &
\frac{\partial y^{\prime}}{\partial y} &
\frac{\partial z^{\prime}}{\partial y} \\
\frac{\partial x^{\prime}}{\partial z} &
\frac{\partial y^{\prime}}{\partial z} &
\frac{\partial z^{\prime}}{\partial z}
\end{array}\right]=\left[\begin{array}{ccc}
\frac{\partial x^{\prime}}{\partial x} & 0 & 0 \\
0 & \frac{\partial y^{\prime}}{\partial y} & 0 \\
0 & 0 & \frac{\partial z^{\prime}}{\partial z}
\end{array}\right]=\left[\begin{array}{ccc}
J_{11} & 0 & 0 \\
0 & J_{22} & 0 \\
0 & 0 & 1
\end{array}\right]
$$

Then, our $\boldsymbol{\varepsilon}_{pml}$ and
$\boldsymbol{\mu}_{pml}$ can be calculated with the following formula,
from [Ward & Pendry, 1996](
https://www.tandfonline.com/doi/abs/10.1080/09500349608232782):

$$
\begin{align}
{\boldsymbol{\varepsilon}_{pml}} &=
A^{-1} \mathbf{A} {\boldsymbol{\varepsilon}_b}\mathbf{A}^{T},\\
{\boldsymbol{\mu}_{pml}} &=
A^{-1} \mathbf{A} {\boldsymbol{\mu}_b}\mathbf{A}^{T},
\end{align}
$$

with $A^{-1}=\operatorname{det}(\mathbf{J})$.

We use `ufl.grad` to calculate the Jacobian of our coordinate
transformation for the different PML regions, and then we can
implement this Jacobian for calculating
$\boldsymbol{\varepsilon}_{pml}$ and $\boldsymbol{\mu}_{pml}$. The
here below function named `create_eps_mu()` serves this purpose:

```python


def create_eps_mu(
    pml: ufl.tensors.ListTensor,
    eps_bkg: Union[float, ufl.tensors.ListTensor],
    mu_bkg: Union[float, ufl.tensors.ListTensor],
) -> tuple[ufl.tensors.ComponentTensor, ufl.tensors.ComponentTensor]:
    J = ufl.grad(pml)

    # Transform the 2x2 Jacobian into a 3x3 matrix.
    J = ufl.as_matrix(((J[0, 0], 0, 0), (0, J[1, 1], 0), (0, 0, 1)))

    A = ufl.inv(J)
    eps_pml = ufl.det(J) * A * eps_bkg * ufl.transpose(A)
    mu_pml = ufl.det(J) * A * mu_bkg * ufl.transpose(A)
    return eps_pml, mu_pml


eps_x, mu_x = create_eps_mu(x_pml, eps_bkg, 1)
eps_y, mu_y = create_eps_mu(y_pml, eps_bkg, 1)
eps_xy, mu_xy = create_eps_mu(xy_pml, eps_bkg, 1)
```

The final weak form in the PML region is:

$$
\int_{\Omega_{pml}}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}
\right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2}
\left[\boldsymbol{\varepsilon}_{pml} \mathbf{E} \right]\cdot
\bar{\mathbf{v}}~ d x=0,
$$


while in the rest of the domain is:

$$
\int_{\Omega_m\cup\Omega_b}-(\nabla \times \mathbf{E}_s)
\cdot (\nabla \times \bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2}
\mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}
-\varepsilon_b\right)\mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d}x.
= 0.
$$

Let's solve this equation in DOLFINx:

```python
# Definition of the weak form
F = (
    -ufl.inner(curl_2d(Es), curl_2d(v)) * dDom
    + eps * (k0**2) * ufl.inner(Es, v) * dDom
    + (k0**2) * (eps - eps_bkg) * ufl.inner(Eb, v) * dDom
    - ufl.inner(ufl.inv(mu_x) * curl_2d(Es), curl_2d(v)) * dPml_x
    - ufl.inner(ufl.inv(mu_y) * curl_2d(Es), curl_2d(v)) * dPml_y
    - ufl.inner(ufl.inv(mu_xy) * curl_2d(Es), curl_2d(v)) * dPml_xy
    + (k0**2) * ufl.inner(eps_x * Es_3d, v_3d) * dPml_x
    + (k0**2) * ufl.inner(eps_y * Es_3d, v_3d) * dPml_y
    + (k0**2) * ufl.inner(eps_xy * Es_3d, v_3d) * dPml_xy
)

a, L = ufl.lhs(F), ufl.rhs(F)

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

Let's now save the solution in a `bp`-file. In order to do so, we need
to interpolate our solution discretized with Nedelec elements into a
compatible discontinuous Lagrange space.

```python
gdim = msh.geometry.dim
V_dg = fem.functionspace(msh, ("DG", degree, (gdim,)))
Esh_dg = fem.Function(V_dg)
Esh_dg.interpolate(Esh)

with VTXWriter(msh.comm, "Esh.bp", Esh_dg) as vtx:
    vtx.write(0.0)
```

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[:, : msh.topology.dim] = Esh_dg.x.array.reshape(V_x.shape[0], msh.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)

with VTXWriter(msh.comm, "E.bp", E_dg) as vtx:
    vtx.write(0.0)
```

## Post-processing

To validate the formulation we calculate 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
q_abs_analyt, q_sca_analyt, q_ext_analyt = calculate_analytical_efficiencies(
    eps_au, n_bkg, wl0, radius_wire
)
```

We calculate the numerical efficiencies in the same way as done in
`demo_scattering_boundary_conditions.py`, with the only difference
that now the scattering efficiency needs to be calculated over an
inner facet, and therefore it requires a slightly different approach:

```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

n = ufl.FacetNormal(msh)
n_3d = ufl.as_vector((n[0], n[1], 0))

# Create a marker for the integration boundary for the scattering
# efficiency
marker = fem.Function(D)
scatt_facets = facet_tags.find(scatt_tag)
incident_cells = mesh.compute_incident_entities(
    msh.topology, scatt_facets, msh.topology.dim - 1, msh.topology.dim
)

midpoints = mesh.compute_midpoints(msh, msh.topology.dim, incident_cells)
inner_cells = incident_cells[(midpoints[:, 0] ** 2 + midpoints[:, 1] ** 2) < (radius_scatt) ** 2]

marker.x.array[inner_cells] = 1

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

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

# Define integration facet for the scattering efficiency
dS = ufl.Measure("dS", msh, subdomain_data=facet_tags)

# Normalized absorption efficiency
q_abs_fenics_proc = (fem.assemble_scalar(fem.form(Q * dAu)) / (gcs * I0)).real
# Sum results from all MPI processes
q_abs_fenics = msh.comm.allreduce(q_abs_fenics_proc, op=MPI.SUM)

# Normalized scattering efficiency
q_sca_fenics_proc = (
    fem.assemble_scalar(fem.form((P("+") + P("-")) * dS(scatt_tag))) / (gcs * I0)
).real

# Sum results from all MPI processes
q_sca_fenics = msh.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

if msh.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}%")
```

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