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Convergence#
Convergence of the various FMM formulations.
import time
import matplotlib.pyplot as plt
import nannos as nn
bk = nn.backend
We will study the convergence on a benchmark case from [Li1997]. First we define the main function that performs the simulation.
wavelength = 1
sq_size = 1.25 * wavelength
eps_diel = 2.25
def checkerboard_cellA(nh, formulation):
d = 2 * sq_size
Nx = 2**9
Ny = 2**9
lattice = nn.Lattice(([d, 0], [0, d]), discretization=(Nx, Ny))
pw = nn.PlaneWave(wavelength=wavelength, angles=(0, 0, 0))
epsgrid = lattice.ones() * eps_diel
sq1 = lattice.square((0.25 * d, 0.25 * d), sq_size)
sq2 = lattice.square((0.75 * d, 0.75 * d), sq_size)
epsgrid[sq1] = 1
epsgrid[sq2] = 1
sup = lattice.Layer("Superstrate", epsilon=eps_diel)
sub = lattice.Layer("Substrate", epsilon=1)
st = lattice.Layer("Structured", wavelength)
st.epsilon = epsgrid
sim = nn.Simulation([sup, st, sub], pw, nh, formulation=formulation)
# this actually corresponds to order (0,-1) for the other unit cell in [Li1997]
order = (-1, -1)
R, T = sim.diffraction_efficiencies(orders=True)
t = sim.get_order(T, order)
return t, sim
def checkerboard_cellB(nh, formulation):
d = sq_size * 2**0.5
Nx = 2**9
Ny = 2**9
lattice = nn.Lattice(([d, 0], [0, d]), discretization=(Nx, Ny))
pw = nn.PlaneWave(wavelength=wavelength, angles=(0, 45, 0))
epsgrid = lattice.ones() * eps_diel
sq = lattice.square((0.5 * d, 0.5 * d), sq_size, rotate=45)
epsgrid[sq] = 1
sup = lattice.Layer("Superstrate", epsilon=eps_diel)
sub = lattice.Layer("Substrate", epsilon=1)
st = lattice.Layer("Structured", wavelength)
st.epsilon = epsgrid
# st.plot()
sim = nn.Simulation([sup, st, sub], pw, nh, formulation=formulation)
order = (0, -1)
R, T = sim.diffraction_efficiencies(orders=True)
t = sim.get_order(T, order)
return t, sim
Perform the simulation for different formulations and number of retained harmonics:
def plot_cell(sim):
axin = plt.gca().inset_axes([0.77, 0.0, 0.25, 0.25])
lay = sim.get_layer_by_name("Structured")
lay.plot(ax=axin)
axin.set_axis_off()
NH = [100, 200, 300, 400, 600]
formulations = ["original", "tangent", "pol", "jones"]
for icell, cell_fun in enumerate([checkerboard_cellA, checkerboard_cellB]):
celltype = "A" if icell == 0 else "B"
print("============================")
print(f"cell type {celltype}")
print("============================")
nhs = {f: [] for f in formulations}
ts = {f: [] for f in formulations}
times = {f: [] for f in formulations}
for nh in NH:
print("number of harmonics = ", nh)
for formulation in formulations:
t0 = -time.time()
t, sim = cell_fun(nh, formulation=formulation)
t0 += time.time()
print("formulation = ", formulation)
print(f"number of harmonics: asked={nh}, actual={sim.nh}")
print(f"elapsed time = {t0}s")
print("T(0,-1) = ", t)
print("-----------------")
nhs[formulation].append(sim.nh)
ts[formulation].append(t)
times[formulation].append(t0)
#########################################################################
# Plot the results:
markers = {"original": "^", "tangent": "o", "jones": "s", "pol": "^"}
colors = {
"original": "#d4b533",
"tangent": "#d46333",
"jones": "#3395d4",
"pol": "#54aa71",
}
plt.figure()
for formulation in formulations:
plt.plot(
nhs[formulation],
ts[formulation],
"-",
color=colors[formulation],
marker=markers[formulation],
label=formulation,
)
plt.pause(0.1)
plt.legend(loc=5, ncols=2)
plt.xlabel("number of Fourier harmonics $n_h$")
plt.ylabel("$T_{0,-1}$")
# plt.ylim(0.1255, 0.129)
plt.title(f"cell {celltype}")
plot_cell(sim)
plt.tight_layout()
plt.figure()
for formulation in formulations:
plt.plot(
nhs[formulation],
times[formulation],
"-",
color=colors[formulation],
marker=markers[formulation],
label=formulation,
)
plt.pause(0.1)
plt.yscale("log")
plt.legend(ncols=2)
plt.xlabel("number of Fourier harmonics $n_h$")
plt.ylabel("CPU time (s)")
plt.title(f"cell {celltype}")
plot_cell(sim)
plt.tight_layout()
============================
cell type A
============================
number of harmonics = 100
formulation = original
number of harmonics: asked=100, actual=97
elapsed time = 0.16234850883483887s
T(0,-1) = 0.12583443545828732
-----------------
formulation = tangent
number of harmonics: asked=100, actual=97
elapsed time = 0.38002657890319824s
T(0,-1) = 0.1280522866532793
-----------------
formulation = pol
number of harmonics: asked=100, actual=97
elapsed time = 0.29866957664489746s
T(0,-1) = 0.128052286653268
-----------------
formulation = jones
number of harmonics: asked=100, actual=97
elapsed time = 0.32566332817077637s
T(0,-1) = 0.12829938643405658
-----------------
number of harmonics = 200
formulation = original
number of harmonics: asked=200, actual=197
elapsed time = 1.294370412826538s
T(0,-1) = 0.1267736966566059
-----------------
formulation = tangent
number of harmonics: asked=200, actual=197
elapsed time = 1.489173173904419s
T(0,-1) = 0.1284389495515869
-----------------
formulation = pol
number of harmonics: asked=200, actual=197
elapsed time = 1.4316017627716064s
T(0,-1) = 0.12843894955158688
-----------------
formulation = jones
number of harmonics: asked=200, actual=197
elapsed time = 1.475632905960083s
T(0,-1) = 0.12874832484252968
-----------------
number of harmonics = 300
formulation = original
number of harmonics: asked=300, actual=293
elapsed time = 3.243748188018799s
T(0,-1) = 0.12700879783995656
-----------------
formulation = tangent
number of harmonics: asked=300, actual=293
elapsed time = 3.5353994369506836s
T(0,-1) = 0.12852262525193675
-----------------
formulation = pol
number of harmonics: asked=300, actual=293
elapsed time = 3.5288069248199463s
T(0,-1) = 0.12852262525193467
-----------------
formulation = jones
number of harmonics: asked=300, actual=293
elapsed time = 3.630852222442627s
T(0,-1) = 0.12868178902297692
-----------------
number of harmonics = 400
formulation = original
number of harmonics: asked=400, actual=385
elapsed time = 8.174145698547363s
T(0,-1) = 0.12725054007633035
-----------------
formulation = tangent
number of harmonics: asked=400, actual=385
elapsed time = 8.266616106033325s
T(0,-1) = 0.12854946017234484
-----------------
formulation = pol
number of harmonics: asked=400, actual=385
elapsed time = 8.598104000091553s
T(0,-1) = 0.12854946017234717
-----------------
formulation = jones
number of harmonics: asked=400, actual=385
elapsed time = 8.470607995986938s
T(0,-1) = 0.12867166821042042
-----------------
number of harmonics = 600
formulation = original
number of harmonics: asked=600, actual=593
elapsed time = 30.996415853500366s
T(0,-1) = 0.1274294323989178
-----------------
formulation = tangent
number of harmonics: asked=600, actual=593
elapsed time = 30.444187879562378s
T(0,-1) = 0.12852617563412577
-----------------
formulation = pol
number of harmonics: asked=600, actual=593
elapsed time = 32.32937836647034s
T(0,-1) = 0.12852617563411897
-----------------
formulation = jones
number of harmonics: asked=600, actual=593
elapsed time = 32.36756253242493s
T(0,-1) = 0.12865899294967859
-----------------
============================
cell type B
============================
number of harmonics = 100
formulation = original
number of harmonics: asked=100, actual=97
elapsed time = 0.2957303524017334s
T(0,-1) = 0.12638413481714725
-----------------
formulation = tangent
number of harmonics: asked=100, actual=97
elapsed time = 0.4401991367340088s
T(0,-1) = 0.12821645033384166
-----------------
formulation = pol
number of harmonics: asked=100, actual=97
elapsed time = 0.46117520332336426s
T(0,-1) = 0.12821645033383877
-----------------
formulation = jones
number of harmonics: asked=100, actual=97
elapsed time = 0.4781177043914795s
T(0,-1) = 0.12812319496479274
-----------------
number of harmonics = 200
formulation = original
number of harmonics: asked=200, actual=197
elapsed time = 1.1963269710540771s
T(0,-1) = 0.12700548284814542
-----------------
formulation = tangent
number of harmonics: asked=200, actual=197
elapsed time = 1.6700668334960938s
T(0,-1) = 0.12823255616464457
-----------------
formulation = pol
number of harmonics: asked=200, actual=197
elapsed time = 1.4159190654754639s
T(0,-1) = 0.12823255616465107
-----------------
formulation = jones
number of harmonics: asked=200, actual=197
elapsed time = 1.4853930473327637s
T(0,-1) = 0.12808777176019862
-----------------
number of harmonics = 300
formulation = original
number of harmonics: asked=300, actual=293
elapsed time = 3.9687180519104004s
T(0,-1) = 0.12718554135686563
-----------------
formulation = tangent
number of harmonics: asked=300, actual=293
elapsed time = 4.29416036605835s
T(0,-1) = 0.12825634085494536
-----------------
formulation = pol
number of harmonics: asked=300, actual=293
elapsed time = 4.139264822006226s
T(0,-1) = 0.12825634085494877
-----------------
formulation = jones
number of harmonics: asked=300, actual=293
elapsed time = 4.333812952041626s
T(0,-1) = 0.12809359760124372
-----------------
number of harmonics = 400
formulation = original
number of harmonics: asked=400, actual=385
elapsed time = 8.171222925186157s
T(0,-1) = 0.12731775874049386
-----------------
formulation = tangent
number of harmonics: asked=400, actual=385
elapsed time = 8.840601205825806s
T(0,-1) = 0.12825655476841305
-----------------
formulation = pol
number of harmonics: asked=400, actual=385
elapsed time = 8.926902770996094s
T(0,-1) = 0.1282565547684097
-----------------
formulation = jones
number of harmonics: asked=400, actual=385
elapsed time = 9.400936603546143s
T(0,-1) = 0.12808885995391947
-----------------
number of harmonics = 600
formulation = original
number of harmonics: asked=600, actual=593
elapsed time = 31.018673419952393s
T(0,-1) = 0.1275044325581457
-----------------
formulation = tangent
number of harmonics: asked=600, actual=593
elapsed time = 32.60936188697815s
T(0,-1) = 0.1282448075086175
-----------------
formulation = pol
number of harmonics: asked=600, actual=593
elapsed time = 35.93264150619507s
T(0,-1) = 0.12824480750861617
-----------------
formulation = jones
number of harmonics: asked=600, actual=593
elapsed time = 36.11417269706726s
T(0,-1) = 0.1280828249565236
-----------------
Total running time of the script: (6 minutes 23.015 seconds)
Estimated memory usage: 4762 MB
