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a 3D electromagnetic FDTD simulator written in Python

Project description

Python 3D FDTD Simulator

A 3D electromagnetic FDTD simulator written in Python. The FDTD simulator has an optional PyTorch backend, enabling FDTD simulations on a GPU.

NOTE: This library is under construction. Only some minimal features are implemented and the API might change considerably.

Installation

The fdtd-library can be installed with pip:

pip install fdtd

The development version can be installed by cloning the repository

git clone http://github.com/flaport/fdtd

and linking it with pip

pip install -e fdtd

Dependencies

  • python 3.6+
  • numpy
  • matplotlib
  • tqdm
  • pytorch (optional)

Contributing

The library is still in a very early stage of development, but all improvements or additions (for example new objects, sources or detectors) are welcome. Please make a pull-request 😊.

Introduction to the library

Imports

The fdtd library is simply imported as follows:

import fdtd

Setting the backend

The fdtd library allows to choose a backend. The "numpy" backend is the default one, but there are also several additional PyTorch backends:

  • "numpy" (defaults to float64 arrays)
  • "torch" (defaults to float64 tensors)
  • "torch.float32"
  • "torch.float64"
  • "torch.cuda" (defaults to float64 tensors)
  • "torch.cuda.float32"
  • "torch.cuda.float64"

For example, this is how to choose the "torch" backend:

fdtd.set_backend("torch")

In general, the "numpy" backend is preferred for standard CPU calculations with "float64" precision. In general, "float64" precision is always preferred over "float32" for FDTD simulations, however, "float32" might give a significant performance boost.

The "cuda" backends are only available for computers with a GPU.

The FDTD-grid

The FDTD grid defines the simulation region.

# signature
fdtd.Grid(
    shape: Tuple[Number, Number, Number],
    grid_spacing: float = 155e-9,
    permittivity: float = 1.0,
    permeability: float = 1.0,
    courant_number: float = None,
)

A grid is defined by its shape, which is just a 3D tuple of Number-types (integers or floats). If the shape is given in floats, it denotes the width, height and length of the grid in meters. If the shape is given in integers, it denotes the width, height and length of the grid in terms of the grid_spacing. Internally, these numbers will be translated to three integers: grid.Nx, grid.Ny and grid.Nz.

A grid_spacing can be given. For stability reasons, it is recommended to choose a grid spacing that is at least 10 times smaller than the smallest wavelength in the grid. This means that for a grid containing a source with wavelength 1550nm and a material with refractive index of 3.1, the recommended minimum grid_spacing turns out to be 50pm

For the permittivity and permeability floats or arrays with the following shapes

  • (grid.Nx, grid.Ny, grid.Nz)
  • or (grid.Nx, grid.Ny, grid.Nz, 1)
  • or (grid.Nx, grid.Ny, grid.Nz, 3)

are expected. In the last case, the shape implies the possibility for different permittivity for each of the major axes (so-called uniaxial or biaxial materials). Internally, these variables will be converted (for performance reasons) to their inverses grid.inverse_permittivity array and a grid.inverse_permeability array of shape (grid.Nx, grid.Ny, grid.Nz, 3). It is possible to change those arrays after making the grid.

Finally, the courant_number of the grid determines the relation between the time_step of the simulation and the grid_spacing of the grid. If not given, it is chosen to be the maximum number allowed by the Courant-Friedrichs-Lewy Condition: 1 for 1D simulations, 1/√2 for 2D simulations and 1/√3 for 3D simulations (the dimensionality will be derived by the shape of the grid). For stability reasons, it is recommended not to change this value.

grid = fdtd.Grid(
    shape = (25e-6, 15e-6, 1), # 25um x 15um x 1 (grid_spacing) --> 2D FDTD
)

print(grid)
Grid(shape=(161,97,1), grid_spacing=1.55e-07, courant_number=0.70)

Adding an object to the grid

An other option to locally change the permittivity or permeability in the grid is to add an Object to the grid.

# signature
fdtd.Object(
    permittivity: Tensorlike,
    name: str = None
)

An object defines a part of the grid with modified update equations, allowing to introduce for example absorbing materials or biaxial materials for which mixing between the axes are present through Pockels coefficients or many more. In this case we'll make an object with a different permittivity than the grid it is in.

Just like for the grid, the Object expects a permittivity to be a floats or an array of the following possible shapes

  • (obj.Nx, obj.Ny, obj.Nz)
  • or (obj.Nx, obj.Ny, obj.Nz, 1)
  • or (obj.Nx, obj.Ny, obj.Nz, 3)

Note that the values obj.Nx, obj.Ny and obj.Nz are not given to the object constructor. They are in stead derived from its placing in the grid:

grid[11:32, 30:84, 0] = fdtd.Object(permittivity=1.7**2, name="object")

Several things happen here. First of all, the object is given the space [11:32, 30:84, 0] in the grid. Because it is given this space, the object's Nx, Ny and Nz are automatically set. Furthermore, by supplying a name to the object, this name will become available in the grid:

print(grid.object)
    Object(name='object')
        @ x=11:32, y=30:84, z=0:1

A second object can be added to the grid:

grid[13e-6:18e-6, 5e-6:8e-6, 0] = fdtd.Object(permittivity=1.5**2)

Here, a slice with floating point numbers was chosen. These floats will be replaced by integer Nx, Ny and Nz during the registration of the object. Since the object did not receive a name, the object won't be available as an attribute of the grid. However, it is still available via the grid.objects list:

print(grid.objects)
[Object(name='object'), Object(name=None)]

This list stores all objects (i.e. of type fdtd.Object) in the order that they were added to the grid.

Adding a source to the grid

Similarly as to adding an object to the grid, an fdtd.LineSource can also be added:

# signature
fdtd.LineSource(
    period: Number = 15, # timesteps or seconds
    power: float = 1.0,
    phase_shift: float = 0.0,
    name: str = None,
)

And also just like an fdtd.Object, an fdtd.Source size is defined by its placement on the grid:

grid[7.5e-6:8.0e-6, 11.8e-6:13.0e-6, 0] = fdtd.LineSource(
    period = 1550e-9 / (3e8), name="source"
)

However, it is important to note that in this case a LineSource is added to the grid, i.e. the source spans the diagonal of the cube defined by the slices. Internally, these slices will be converted into lists to ensure this behavior:

print(grid.source)
    LineSource(period=14, power=1.0, phase_shift=0.0, name='source')
        @ x=[48, ... , 51], y=[76, ... , 83], z=[0, ... , 0]

Note that one could also have supplied lists to index the grid in the first place. This feature could be useful to create a LineSource of arbitrary shape.

Adding a detector to the grid

# signature
fdtd.LineDetector(
    name=None
)

Adding a detector to the grid works the same as adding a source

grid[12e-6, :, 0] = fdtd.LineDetector(name="detector")
print(grid.detector)
    LineDetector(name='detector')
        @ x=[77, ... , 77], y=[0, ... , 96], z=[0, ... , 0]

Adding grid boundaries

# signature
fdtd.PML(
    a: float = 1e-8, # stability factor
    name: str = None
)

Although, having an object, source and detector to simulate is in principle enough to perform an FDTD simulation, One also needs to define a grid boundary to prevent the fields to be reflected. One of those boundaries that can be added to the grid is a Perfectly Matched Layer or PML. These are basically absorbing boundaries.

# x boundaries
grid[0:10, :, :] = fdtd.PML(name="pml_xlow")
grid[-10:, :, :] = fdtd.PML(name="pml_xhigh")

# y boundaries
grid[:, 0:10, :] = fdtd.PML(name="pml_ylow")
grid[:, -10:, :] = fdtd.PML(name="pml_yhigh")

Grid summary

A simple summary of the grid can be shown by printing out the grid:

print(grid)
Grid(shape=(161,97,1), grid_spacing=1.55e-07, courant_number=0.70)

sources:
    LineSource(period=14, power=1.0, phase_shift=0.0, name='source')
        @ x=[48, ... , 51], y=[76, ... , 83], z=[0, ... , 0]

detectors:
    LineDetector(name='detector')
        @ x=[77, ... , 77], y=[0, ... , 96], z=[0, ... , 0]

boundaries:
    PML(name='pml_xlow')
        @ x=0:10, y=:, z=:
    PML(name='pml_xhigh')
        @ x=-10:, y=:, z=:
    PML(name='pml_ylow')
        @ x=:, y=0:10, z=:
    PML(name='pml_yhigh')
        @ x=:, y=-10:, z=:

objects:
    Object(name='object')
        @ x=11:32, y=30:84, z=0:1
    Object(name=None)
        @ x=84:116, y=32:52, z=0:1

Running a simulation

Running a simulation is as simple as using the grid.run method.

grid.run(
    total_time: Number,
    progress_bar: bool = True
)

Just like for the lengths in the grid, the total_time of the simulation can be specified as an integer (number of time_steps) or as a float (in seconds).

grid.run(total_time=100)

Grid visualization

Let's visualize the grid. This can be done with the grid.visualize method:

# signature
grid.visualize(
    grid,
    x=None,
    y=None,
    z=None,
    cmap="Blues",
    pbcolor="C3",
    pmlcolor=(0, 0, 0, 0.1),
    objcolor=(1, 0, 0, 0.1),
    srccolor="C0",
    detcolor="C2",
    show=True,
)

This method will by default visualize all objects in the grid, as well as the power at the current time_step at a certain x, y OR z-plane. By setting show=False, one can disable the immediate visualization of the matplotlib image.

grid.visualize(z=0)

png

Background

An as quick as possible explanation of the FDTD discretization of the Maxwell equations.

Update Equations

An electromagnetic FDTD solver solves the time-dependent Maxwell Equations

    curl(H) = ε*ε0*dE/dt
    curl(E) = -µ*µ0*dH/dt

These two equations are called Ampere's Law and Faraday's Law respectively.

In these equations, ε and µ are the relative permittivity and permeability tensors respectively. ε0 and µ0 are the vacuum permittivity and permeability and their square root can be absorbed into E and H respectively, such that E := √ε0*E and H := √µ0*H.

Doing this, the Maxwell equations can be written as update equations:

    E  += c*dt*inv(ε)*curl(H)
    H  -= c*dt*inv(µ)*curl(E)

The electric and magnetic field can then be discretized on a grid with interlaced Yee-coordinates, which in 3D looks like this:

grid discretization in 3D

According to the Yee discretization algorithm, there are inherently two types of fields on the grid: E-type fields on integer grid locations and H-type fields on half-integer grid locations.

The beauty of these interlaced coordinates is that they enable a very natural way of writing the curl of the electric and magnetic fields: the curl of an H-type field will be an E-type field and vice versa.

This way, the curl of E can be written as

    curl(E)[m,n,p] = (dEz/dy - dEy/dz, dEx/dz - dEz/dx, dEy/dx - dEx/dy)[m,n,p]
                   =( ((Ez[m,n+1,p]-Ez[m,n,p])/dy - (Ey[m,n,p+1]-Ey[m,n,p])/dz),
                      ((Ex[m,n,p+1]-Ex[m,n,p])/dz - (Ez[m+1,n,p]-Ez[m,n,p])/dx),
                      ((Ey[m+1,n,p]-Ey[m,n,p])/dx - (Ex[m,n+1,p]-Ex[m,n,p])/dy) )
                   =(1/du)*( ((Ez[m,n+1,p]-Ez[m,n,p]) - (Ey[m,n,p+1]-Ey[m,n,p])), [assume dx=dy=dz=du]
                             ((Ex[m,n,p+1]-Ex[m,n,p]) - (Ez[m+1,n,p]-Ez[m,n,p])),
                             ((Ey[m+1,n,p]-Ey[m,n,p]) - (Ex[m,n+1,p]-Ex[m,n,p])) )

this can be written efficiently with array slices (note that the factor (1/du) was left out):

def curl_E(E):
    curl_E = np.zeros(E.shape)
    curl_E[:,:-1,:,0] += E[:,1:,:,2] - E[:,:-1,:,2]
    curl_E[:,:,:-1,0] -= E[:,:,1:,1] - E[:,:,:-1,1]

    curl_E[:,:,:-1,1] += E[:,:,1:,0] - E[:,:,:-1,0]
    curl_E[:-1,:,:,1] -= E[1:,:,:,2] - E[:-1,:,:,2]

    curl_E[:-1,:,:,2] += E[1:,:,:,1] - E[:-1,:,:,1]
    curl_E[:,:-1,:,2] -= E[:,1:,:,0] - E[:,:-1,:,0]
    return curl_E

The curl for H can be obtained in a similar way (note again that the factor (1/du) was left out):

def curl_H(H):
    curl_H = np.zeros(H.shape)

    curl_H[:,1:,:,0] += H[:,1:,:,2] - H[:,:-1,:,2]
    curl_H[:,:,1:,0] -= H[:,:,1:,1] - H[:,:,:-1,1]

    curl_H[:,:,1:,1] += H[:,:,1:,0] - H[:,:,:-1,0]
    curl_H[1:,:,:,1] -= H[1:,:,:,2] - H[:-1,:,:,2]

    curl_H[1:,:,:,2] += H[1:,:,:,1] - H[:-1,:,:,1]
    curl_H[:,1:,:,2] -= H[:,1:,:,0] - H[:,:-1,:,0]
    return curl_H

The update equations can now be rewritten as

    E  += (c*dt/du)*inv(ε)*curl_H
    H  -= (c*dt/du)*inv(µ)*curl_E

The number (c*dt/du) is a dimensionless parameter called the courant number sc. For stability reasons, the Courant number should always be smaller than 1/√D, with D the dimension of the simulation. This can be intuitively be understood as the condition that information should always travel slower than the speed of light through the grid. in the FDTD method described here, information can only travel to the neighbouring grid cells (through application of the curl). It would therefore take D time steps to travel over the diagonal of a D-dimensional cube (square in 2D, cube in 3D), the Courant condition follows then automatically from the fact that the length of this diagonal is 1/√D.

This yields the final update equations for the FDTD algorithm:

    E  += sc*inv(ε)*curl_H
    H  -= sc*inv(µ)*curl_E

This is also how it is implemented:

class Grid:
    # ... [initialization]

    def step(self):
        self.update_E()
        self.update_H()

    def update_E(self):
        self.E += self.courant_number * self.inverse_permittivity * curl_H(self.H)

    def update_H(self):
        self.H -= self.courant_number * self.inverse_permeability * curl_E(self.E)

Sources

Ampere's Law can be updated to incorporate a current density:

    curl(H) = J + ε*ε0*dE/dt

Making again the usual substitutions sc := c*dt/du, E := √ε0*E and H := √µ0*H, the update equations can be modified to include the current density:

    E += sc*inv(ε)*curl_H - dt*inv(ε)*J/ε0

Making one final substitution Es := -dt*inv(ε)*J/√ε0 allows us to write this in a very clean way:

    E += sc*inv(ε)*curl_H + Es

Where we defined Es as the electric field source term.

It is often useful to also define a magnetic field source term Hs, which would be derived from the magnetic current density if it were to exist. In the same way, Faraday's update equation can be rewritten as

    H  -= sc*inv(µ)*curl_E + Hs
class Source:
    # ... [initialization]
    def update_E(self):
        # electric source function here

    def update_H(self):
        # magnetic source function here

class Grid:
    # ... [initialization]
    def update_E(self):
        # ... [electric field update equation]
        for source in self.sources:
            source.update_E()

    def update_H(self):
        # ... [magnetic field update equation]
        for source in self.sources:
            source.update_H()

Lossy Medium

When a material has a electric conductivity σe, a conduction-current will ensure that the medium is lossy. Ampere's law with a conduction current becomes

    curl(H) = σe*E + ε*ε0*dE/dt

making the usual substitutions, this becomes:

    E(t+dt) - E(t) = sc*inv(ε)*curl_H(t+dt/2) - dt*inv(ε)*σe*E(t+dt/2)/ε0

This update equation depends on the electric field on a half-integer time step (a magnetic field timestep). We need to make a substitution to interpolate the electric field to this time step:

    (1 + 0.5*dt*inv(ε)*σ/ε0)*E(t+dt) = sc*inv(ε)*curl_H(t+dt/2) + (1 - 0.5*dt*inv(ε)*σe/ε0)*E(t)

Which, after substitution σ := inv(ε)*σe/ε0 yield the new update equations:

    f = 0.5*dt*σ
    E *= inv(1 + f) * (1 - f)
    E += inv(1 + f)*sc*inv(ε)*curl_H

If we want to keep track of the absorbed energy:

Note that the more complicated the permittivity tensor ε is, the more time consuming this algorithm will be. It is therefore sometimes the right decision to transfer the absorption to the magnetic domain by introducing a (nonphysical) magnetic conductivity, because the permeability tensor µ is usually just equal to one.

Which, after substitution σ := inv(µ)*σm/µ0, we get the magnetic field update equations:

    f = 0.5*dt*σ
    H *= inv(1 + f) * (1 - f)
    H += inv(1 + f)*sc*inv(µ)*curl_E

Energy Density and Poynting Vector

The electromagnetic energy density can be given by

    e = (1/2)*ε*ε0*E**2 + (1/2)*µ*µ0*H**2

making the above substitutions, this becomes in simulation units:

    e = (1/2)*ε*E**2 + (1/2)*µ*H**2

The Poynting vector is given by

    P = E×H

Which in simulation units becomes

    P = c*E×H

The energy introduced by a source Es can be derived from tracking the change in energy density

    de = ε*Es·E + (1/2)*ε*Es**2

This could also be derived from Poyntings energy conservation law:

    de/dt = -grad(S) - J·E

where the first term just describes the redistribution of energy in a volume and the second term describes the energy introduced by a current density.

Note: although it is unphysical, one could also have introduced a magnetic source. This source would have introduced the following energy:

    de = ε*Hs·H + (1/2)*µ*Hs**2

Since the µ-tensor is usually just equal to one, using a magnetic source term is often more efficient.

Similarly, one can also keep track of the absorbed energy due to an electric conductivity in the following way:

    f = 0.5*dt*σ
    Enoabs = E + sc*inv(ε)*curl_H
    E *= inv(1 + f) * (1 - f)
    E += inv(1 + f)*sc*inv(ε)*curl_H
    dE = Enoabs - E
    e_abs += ε*E*dE + 0.5*ε*dE**2

or if we want to keep track of the absorbed energy by magnetic a magnetic conductivity:

    f = 0.5*dt*inv(µ)*σ
    Hnoabs = E + sc*inv(µ)*curl_E
    H *= inv(1 + f) * (1 - f)
    H += inv(1 + f)*sc*inv(µ)*curl_E
    dH = Hnoabs - H
    e_abs += µ*H*dH + 0.5*µ*dH**2

The electric term and magnetic term in the energy density are usually of the same size. Therefore, the same amount of energy will be absorbed by introducing a magnetic conductivity σm as by introducing a electric conductivity σe if:

    inv(µ)*σm/µ0 = inv(ε)*σe/ε0

Boundary Conditions

Periodic Boundary Conditions

Assuming we want periodic boundary conditions along the X-direction, then we have to make sure that the fields at Xlow and Xhigh are the same. This has to be enforced after performing the update equations:

Note that the electric field E is dependent on curl_H, which means that the first indices of E will not be updated through the update equations. It's those indices that need to be set through the periodic boundary condition. Concretely: E[0] needs to be set to equal E[-1]. For the magnetic field, the inverse is true: H is dependent on curl_E, which means that its last indices will not be set. This has to be done by the boundary condition: H[-1] needs to be set equal to H[0]:

class PeriodicBoundaryX:
    # ... [initialization]
    def update_E(self):
        self.grid.E[0, :, :, :] = self.grid.E[-1, :, :, :]

    def update_H(self):
        self.grid.H[-1, :, :, :] = self.grid.H[0, :, :, :]

class Grid:
    # ... [initialization]
    def update_E(self):
        # ... [electric field update equation]
        # ... [electric field source update equations]
        for boundary in self.boundaries:
            boundary.update_E()

    def update_H(self):
        # ... [magnetic field update equation]
        # ... [magnetic field source update equations]
        for boundary in self.boundaries:
            boundary.update_H()

Perfectly Matched Layer

a Perfectly Matched Layer (PML) is the state of the art for introducing absorbing boundary conditions in an FDTD grid. A PML is an impedance-matched absorbing area in the grid. It turns out that for a impedance-matching condition to hold, the PML can only be absorbing in a single direction. This is what makes a PML in fact a nonphysical material.

Consider Ampere's law for the Ez component, where the usual substitutions E := √ε0*E, H := √µ0*H and σ := inv(ε)*σe/ε0 are already introduced:

    ε*dEz/dt + ε*σ*Ez = c*dHy/dx - c*dHx/dy

This becomes in the frequency domain:

    *ε*Ez + ε*σ*Ez = c*dHy/dx - c*dHx/dy

We can split this equation in a x-propagating wave and a y-propagating wave:

    *ε*Ezx + ε*σx*Ezx = *ε*(1 + σx/)*Ezx = c*dHy/dx
    *ε*Ezy + ε*σy*Ezy = *ε*(1 + σy/)*Ezy = -c*dHx/dy

We can define the S-operators as follows

    Su = 1 + σu/          with u in {x, y, z}

In general, we prefer to add a stability factor au and a scaling factor ku to Su:

    Su = ku + σu/(+au)    with u in {x, y, z}

Summing the two equations for Ez back together after dividing by the respective S-operator gives

    *ε*Ez = (c/Sx)*dHy/dx - (c/Sy)*dHx/dy

Converting this back to the time domain gives

    ε*dEz/dt = c*sx[*]dHy/dx - c*sx[*]dHx/dy

where sx denotes the inverse Fourier transform of (1/Sx) and [*] denotes a convolution. The expression for su can be proven [after some derivation] to look as follows:

    su = (1/ku)*δ(t) + Cu(t)    with u in {x, y, z}

where δ(t) denotes the Dirac delta function and C(t) an exponentially decaying function given by:

    Cu(t) = -(σu/ku**2)*exp(-(au+σu/ku)*t)     for all t > 0 and u in {x, y, z}

Plugging this in gives:

    dEz/dt = (c/kx)*inv(ε)*dHy/dx - (c/ky)*inv(ε)*dHx/dy + c*inv(ε)*Cx[*]dHy/dx - c*inv(ε)*Cx[*]dHx/dy
           = (c/kx)*inv(ε)*dHy/dx - (c/ky)*inv(ε)*dHx/dy + c*inv(ε)*Фez/du      with du=dx=dy=dz

This can be written as an update equation:

    Ez += (1/kx)*sc*inv(ε)*dHy - (1/ky)*sc*inv(ε)*dHx + sc*inv(ε)*Фez

Where we defined Фeu as

    Фeu = Ψeuv - Ψezw           with u, v, w in {x, y, z}

and Ψeuv as the convolution updating the component Eu by taking the derivative of Hw in the v direction:

    Ψeuv = dv*Cv[*]dHw/dv     with u, v, w in {x, y, z}

This can be rewritten [after some derivation] as an update equation in itself:

     Ψeuv = bv*Ψeuv + cv*dv*(dHw/dv)
          = bv*Ψeuv + cv*dHw            with u, v, w in {x, y, z}

Where the constants bu and cu are derived to be:

    bu = exp(-(au + σu/ku)*dt)              with u in {x, y, z}
    cu = σu*(bu - 1)/(σu*ku + au*ku**2)     with u in {x, y, z}

The final PML algorithm for the electric field now becomes:

  1. Update Фe=[Фex, Фey, Фez] by using the update equation for the Ψ-components.
  2. Update the electric fields the normal way
  3. Add Фe to the electric fields.

or as python code:

class PML(Boundary):
    # ... [initialization]
    def update_phi_E(self): # update convolution
        self.psi_Ex *= self.bE
        self.psi_Ey *= self.bE
        self.psi_Ez *= self.bE

        c = self.cE
        Hx = self.grid.H[self.locx]
        Hy = self.grid.H[self.locy]
        Hz = self.grid.H[self.locz]

        self.psi_Ex[:, 1:, :, 1] += (Hz[:, 1:, :] - Hz[:, :-1, :]) * c[:, 1:, :, 1]
        self.psi_Ex[:, :, 1:, 2] += (Hy[:, :, 1:] - Hy[:, :, :-1]) * c[:, :, 1:, 2]

        self.psi_Ey[:, :, 1:, 2] += (Hx[:, :, 1:] - Hx[:, :, :-1]) * c[:, :, 1:, 2]
        self.psi_Ey[1:, :, :, 0] += (Hz[1:, :, :] - Hz[:-1, :, :]) * c[1:, :, :, 0]

        self.psi_Ez[1:, :, :, 0] += (Hy[1:, :, :] - Hy[:-1, :, :]) * c[1:, :, :, 0]
        self.psi_Ez[:, 1:, :, 1] += (Hx[:, 1:, :] - Hx[:, :-1, :]) * c[:, 1:, :, 1]

        self.phi_E[..., 0] = self.psi_Ex[..., 1] - self.psi_Ex[..., 2]
        self.phi_E[..., 1] = self.psi_Ey[..., 2] - self.psi_Ey[..., 0]
        self.phi_E[..., 2] = self.psi_Ez[..., 0] - self.psi_Ez[..., 1]

    def update_E(self): # update PML located at self.loc
        self.grid.E[self.loc] += (
            self.grid.courant_number
            * self.grid.inverse_permittivity[self.loc]
            * self.phi_E
        )

class Grid:
    # ... [initialization]
    def update_E(self):
        for boundary in self.boundaries:
            boundary.update_phi_E()
        # ... [electric field update equation]
        # ... [electric field source update equations]
        for boundary in self.boundaries:
            boundary.update_E()

The same has to be applied for the magnetic field.

These update equations for the PML were based on Schneider, Chap. 11.

License

© Floris laporte - MIT License

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