implement lumped capacitor and resistor

[jackieyao0114]

This PR implemented lumped circuit components (inductor L, resistor R, and capacitor C) solving.
The R, L and/or C are added on top of the physical materials & structures . The contribution of these lumped elements to current density J is added into Ampere's law:

curl(H) = J_R + J_L + J_C + sigma*E + epsilon*dE/dt

Lumped resistor is verified by a R-terminated microstrip transmission line (https://chemandy.com/calculators/microstrip-transmission-line-calculator.htm). The input file is

################################
####### GENERAL PARAMETERS ######
#################################
max_step = 200000

amr.n_cell = n_cellx n_celly n_cellz
amr.max_grid_size = max_grid_sizex max_grid_sizey max_grid_sizez
amr.blocking_factor = blocking_factor
amr.refine_grid_layout = 1  # if n_MPI > n_grids, the grids will be successively divided in half until n_MPI <= n_grids

geometry.dims = 3
geometry.prob_lo = prob_lox prob_loy prob_loz
geometry.prob_hi = prob_hix prob_hiy prob_hiz

amr.max_level = 0

# use pec instead of pml overlaying current source so you don't get a reflection
boundary.field_lo = pml pml pec
boundary.field_hi = pml pml pml

#################################
############ NUMERICS ###########
#################################
warpx.verbose = 1

warpx.cfl = 0.8

# ESSENTIAL! This extends PEC metal lines into the PML region to avoid artificial reflections.
warpx.Apply_E_excitation_in_pml_region=1

# vacuum or macroscopic
algo.em_solver_medium = macroscopic

# laxwendroff or backwardeuler
algo.macroscopic_sigma_method = laxwendroff

######################## IMPORTANT ######################
# the resistor and capcitor flag belongs to warpx class
warpx.use_lumped_resistor = 1
warpx.use_lumped_capacitor = 0
#########################################################

################ IMPORTANT ##############
# the inductor flag belongs to algo class
algo.use_lumped_inductor = 0
#########################################

#################################
############ STRUCTURE ###########
#################################

my_constants.h_si = 0.5e-3
my_constants.W_metal = 2.0e-3
my_constants.length = 100.0e-3

######################################################################
####### resistor and capacitor belongs to macroscopic class ##########
######################################################################
macroscopic.lumped_resistor_x_function(x,y,z) = "0."
macroscopic.lumped_resistor_y_function(x,y,z) = "0."
macroscopic.lumped_resistor_z_function(x,y,z) = "10.0 * (z >= 0) * (z < h_si) * (y > length-ddy) * (y < length+ddy) * (x < ddx) * (x > - ddx)"
# total inductance devided by number of cells in the z direction

macroscopic.epsilon_function(x,y,z) = "eps_0 + eps_0 * (eps_r_si - 1.) * (z <= h_si)"

macroscopic.mu_function(x,y,z) = "mu_0 + mu_0 * (mu_r_test - 1.) * (z > h_si ) * (z < h_si + dz) * (y < length + ddy)  * (x < W_metal/2 + ddx) * (x > -W_metal/2 - ddx)"

###############
# excitation
###############

warpx.E_excitation_on_grid_style = parse_E_excitation_grid_function

warpx.Ex_excitation_flag_function(x,y,z) = "flag_hs * (z > h_si - ddz) * (z < h_si + ddz) *
                                           (y < length + ddy)  * (x < W_metal/2 + ddx) * (x > -W_metal/2 - ddx)"

warpx.Ey_excitation_flag_function(x,y,z) = "flag_hs * (z > h_si - ddz) * (z < h_si + ddz) *
                                           (y < length + ddy)  * (x < W_metal/2 + ddx) * (x > -W_metal/2 - ddx)"
warpx.Ez_excitation_flag_function(x,y,z) = "flag_ss * (z >= 0) * (z < h_si) * (y > dy-ddy) * (y < dy+ddy) * (x < ddx) * (x > - ddx)"

# This is a source on a qubit control line
warpx.Ex_excitation_grid_function(x,y,z,t) = "0."
warpx.Ey_excitation_grid_function(x,y,z,t) = "0."
warpx.Ez_excitation_grid_function(x,y,z,t) = "exp(-(t-3*TP)**2/(2*TP**2))*sin(2*pi*freq*t)"


###############
# material properties
###############
my_constants.sigma_0 = 0.0
my_constants.sigma_si = 0.0
my_constants.sigma_metal = 1.e11

my_constants.eps_0 = 8.8541878128e-12
my_constants.eps_r_si = 3.0

my_constants.mu_0 = 1.25663706212e-06
my_constants.mu_r_si = 1.0
my_constants.mu_r_test = 0.999999

###############
# domain size
# n_cellx/y/z and Lx/y/z are needed to calculate dx/dy/dz
###############
my_constants.n_cellx = 100
my_constants.n_celly = 1200
my_constants.n_cellz = 40

my_constants.dx = 0.1e-3
my_constants.dy = 0.1e-3
my_constants.dz = 0.05e-3

# grid spacing
my_constants.dx = Lx / n_cellx
my_constants.dy = Ly / n_celly
my_constants.dz = Lz / n_cellz

my_constants.ddx = dx/1.e6
my_constants.ddy = dy/1.e6
my_constants.ddz = dz/1.e6

my_constants.max_grid_sizex = 50
my_constants.max_grid_sizey = 600
my_constants.max_grid_sizez = 40
my_constants.blocking_factor = 1

my_constants.prob_lox = -5.0e-3
my_constants.prob_loy = 0.0e-3
my_constants.prob_loz = 0.0

my_constants.prob_hix = 5.0e-3
my_constants.prob_hiy = 120.0e-3
my_constants.prob_hiz = 2.0e-3

my_constants.Lx = prob_hix - prob_lox
my_constants.Ly = prob_hiy - prob_loy
my_constants.Lz = prob_hiz - prob_loz

my_constants.flag_none = 0
my_constants.flag_hs = 1
my_constants.flag_ss = 2


###############
# derived quantities and fundamental constants 
###############

my_constants.pi = 3.14159265358979

my_constants.freq = 5.e9
my_constants.TP = 1./freq

###############
# diagnostics
###############

diagnostics.diags_names = plt
###############
# full plotfiles
plt.intervals = 20
plt.fields_to_plot = Ex Ey Ez Bx By Bz mu
plt.diag_type = Full
plt.file_min_digits = 7

The simulated input impedance matches analytical calculation: