Sinusoidal actuation#
Ionic polymer-metal composites
Curling of an ipmc beam under sinusoidal voltage
This is a two-dimensional plane-strain simulation
Degrees of freedom#
Displacement: u
Species concentration: c
Electrostatic potential: phi
Units#
Length: mm
Mass: tonne (1000 kg)
Time: s
Amount of substance: mol
Electric charge: mC
Temperature: K
Force: N
Energy: mJ
Stress: MPa
Species concentration: mol/mm^3
Molar volume: mm^3/mol
Species diffusivity: mm^2/s
Electric potential: V (=mJ/mC)
Permittivity: mF/mm
Permittivity of free space = 8.85e-12 mF/mm
Faraday’s constant: 96.485e6 mC/mol
Gas constant: 8.3145e3 mJ/(mol K)
Software:#
Dolfinx v0.8.0
In the collection “Example Codes for Coupled Theories in Solid Mechanics,”
By Eric M. Stewart, Shawn A. Chester, and Lallit Anand.
Import modules#
# Import FEnicSx/dolfinx
import dolfinx
# For numerical arrays
import numpy as np
# For MPI-based parallelization
from mpi4py import MPI
comm = MPI.COMM_WORLD
rank = comm.Get_rank()
# PETSc solvers
from petsc4py import PETSc
# specific functions from dolfinx modules
from dolfinx import fem, mesh, io, plot, log
from dolfinx.fem import (Constant, dirichletbc, Function, functionspace, Expression )
from dolfinx.fem.petsc import NonlinearProblem
from dolfinx.nls.petsc import NewtonSolver
from dolfinx.io import VTXWriter, XDMFFile
# specific functions from ufl modules
import ufl
from ufl import (TestFunctions, TrialFunction, Identity, grad, det, div, dev, inv, tr, sqrt, conditional ,\
lt, gt, dx, inner, derivative, dot, ln, split, exp, eq, cos, sin, acos, ge, le, outer, tanh,\
cosh, atan, atan2)
# basix finite elements (necessary for dolfinx v0.8.0)
import basix
from basix.ufl import element, mixed_element
# Matplotlib for plotting
import matplotlib.pyplot as plt
plt.close('all')
# For timing the code
from datetime import datetime
# Set level of detail for log messages (integer)
# Guide:
# CRITICAL = 50, // errors that may lead to data corruption
# ERROR = 40, // things that HAVE gone wrong
# WARNING = 30, // things that MAY go wrong later
# INFO = 20, // information of general interest (includes solver info)
# PROGRESS = 16, // what's happening (broadly)
# TRACE = 13, // what's happening (in detail)
# DBG = 10 // sundry
#
log.set_log_level(log.LogLevel.WARNING)
Define geometry#
# We will first create a unit square mesh and then map it
# to create a graded mesh in the y-direction, which is refined towards its top and bottom edges.
# Finally we will map it to represent a beam of length 20mm and height 0.3 mm.
# Number of elements along the length (20 is fast, 100 is reasonable)
lElem = 100
# 2N number of elements in y-direction
N = 20
domain = mesh.create_rectangle(MPI.COMM_WORLD, [[0,0], [1,1]],\
[lElem, 2*N], mesh.CellType.triangle, diagonal=mesh.DiagonalType.crossed)
# Map the coordinates of the uniform square mesh to the biased spacing
# Be careful here, even though it is 2-D simulation xthe quantity Orig stores 3-D coordinates,
# so XMap should also be appropriately dimensioned.
#
# First get the mesh coordinates x for mapping
xOrig = domain.geometry.x
# create and empty xMap to stored the results from the mapping
xMap = np.zeros((len(xOrig),3))
# Perform the mapping to create a biased mesh. This is needed to resolve the electric double layers
# The parameter r is bias successive mesh spacing ratio in the vertical direction; it should be less than 1
r = 1/1.09
a = (1-r)/(1-r**N)
# Re-scale the y-coordinate so that it ranges from (-1, 1)
xMap[:,1] = xOrig[:,1]*2 - 1
# Bias the y-coordinate
xMap[:,1] = np.sign(xMap[:,1])*a*(r**(np.abs(xMap[:,1])*N)-1)/(r-1)
# Scale the whole thing to the desired dimensions
#
scaleX = 20.e0 # length of beam
scaleY = 0.3e0 # height of beam
#
xMap[:,0] = xOrig[:,0]*scaleX
xMap[:,1] = xMap[: ,1]*scaleY/2
# Uppdate the mesh coordinates
domain.geometry.x[:] = xMap
# This says "spatial coordinates" but is really the referential coordinates,
# since the mesh does not convect in FEniCS
x = ufl.SpatialCoordinate(domain)
Identify boundaries of the domain
# Identify the boundaries of the rectangle mesh
#
def xBot(x):
return np.isclose(x[0], 0)
def xTop(x):
return np.isclose(x[0], scaleX)
def yBot(x):
return np.isclose(x[1], - scaleY/2)
def yTop(x):
return np.isclose(x[1], scaleY/2)
# Mark the sub-domains
boundaries = [(1, xBot),(2, xTop), (3, yBot), (4, yTop)]
# Build collections of facets on each subdomain and mark them appropriately.
facet_indices, facet_markers = [], [] # initalize empty collections of indices and markers.
fdim = domain.topology.dim - 1 # geometric dimension of the facet (mesh dimension - 1)
for (marker, locator) in boundaries:
facets = mesh.locate_entities(domain, fdim, locator) # an array of all the facets in a
# given subdomain ("locator")
facet_indices.append(facets) # add these facets to the collection.
facet_markers.append(np.full_like(facets, marker)) # mark them with the appropriate index.
# Format the facet indices and markers as required for use in dolfinx.
facet_indices = np.hstack(facet_indices).astype(np.int32)
facet_markers = np.hstack(facet_markers).astype(np.int32)
sorted_facets = np.argsort(facet_indices)
#
# Add these marked facets as "mesh tags" for later use in BCs.
facet_tags = mesh.meshtags(domain, fdim, facet_indices[sorted_facets], facet_markers[sorted_facets])
Print out the unique facet index numbers
top_imap = domain.topology.index_map(1) # index map of 1D entities in domain
values = np.zeros(top_imap.size_global) # an array of zeros of the same size as number of 2D entities
values[facet_tags.indices]=facet_tags.values # populating the array with facet tag index numbers
print(np.unique(facet_tags.values)) # printing the unique
# Mark the sub-domains
# boundaries = [(1, xBot),(2, xTop),(3, yBot),(4, yTop)]
[1 2 3 4]
Visualize reference configuration
import pyvista
pyvista.set_jupyter_backend('html')
from dolfinx.plot import vtk_mesh
pyvista.start_xvfb()
# initialize a plotter
plotter = pyvista.Plotter()
# Add the mesh.
topology, cell_types, geometry = plot.vtk_mesh(domain, domain.topology.dim)
grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)
plotter.add_mesh(grid, show_edges=True, opacity=0.25)
plotter.view_xy()
#labels = dict(xlabel='X', ylabel='Y',zlabel='Z')
labels = dict(xlabel='X', ylabel='Y')
plotter.add_axes(**labels)
plotter.screenshot("results/ipmc_beam_mesh.png")
from IPython.display import Image
Image(filename='results/ipmc_beam_mesh.png')
# # Use the following commands for a zoom-able view
# if not pyvista.OFF_SCREEN:
# plotter.show()
# else:
# plotter.screenshot("ipmc_beam_mesh.png")
Define boundary and volume integration measure#
# Define the boundary integration measure "ds" using the facet tags,
# also specify the number of surface quadrature points.
ds = ufl.Measure('ds', domain=domain, subdomain_data=facet_tags, metadata={'quadrature_degree':2})
# Define the volume integration measure "dx"
# also specify the number of volume quadrature points.
dx = ufl.Measure('dx', domain=domain, metadata={'quadrature_degree': 2})
# Create facet to cell connectivity required to determine boundary facets.
domain.topology.create_connectivity(domain.topology.dim, domain.topology.dim)
domain.topology.create_connectivity(domain.topology.dim, domain.topology.dim-1)
domain.topology.create_connectivity(domain.topology.dim-1, domain.topology.dim)
# # Define facet normal
n2D = ufl.FacetNormal(domain)
n = ufl.as_vector([n2D[0], n2D[1], 0.0]) # define n as a 3D vector for later use
Material parmeters#
# Mechanics
#
Young = Constant(domain,PETSc.ScalarType(100)) # Young's modulus, MPa
Nu = Constant(domain,PETSc.ScalarType(0.4)) # Poisson's ratio
Gshear = Constant(domain,PETSc.ScalarType(Young/(2*(1+Nu)))) # Shear modulus, Mpa
Kbulk = Constant(domain,PETSc.ScalarType(Young/(3*(1-2*Nu)))) # Bulk modulus, MPa
# Chemistry
Omega = Constant(domain,PETSc.ScalarType(1.0e5)) # solvated cation molecular volume, mm^3/mol
D = Constant(domain,PETSc.ScalarType(1.3e-5) ) # Diffusivity, mm^2/s
RT = Constant(domain,PETSc.ScalarType(8.3145e3*(273.0+25.0))) # Gas constant*Temp
c_R0 = Constant(domain,PETSc.ScalarType(1500*1e-9)) # Initial concentration
cMax = Constant(domain,PETSc.ScalarType(4750*1e-9)) # Saturation concentration, mol/mm^3
c0 = Constant(domain,PETSc.ScalarType(c_R0/cMax)) # Initial cnormalized concentration
# Electrostatics
vareps0 = Constant(domain,PETSc.ScalarType(8.85e-12)) # Vacuum permittivity, mF/mm
vareps_r = Constant(domain,PETSc.ScalarType(50)) # Relative permittivity of polymer
vareps_num = Constant(domain,PETSc.ScalarType( (120/50)*1.0e7 )) # Numerical permittivity factor
Farad = Constant(domain,PETSc.ScalarType( 96.485e6 )) # Faraday constant, uC/mol
Function spaces#
# Define function spaces
#
U2 = element("Lagrange", domain.basix_cell(), 2, shape=(2,)) # For displacement
P1 = element("Lagrange", domain.basix_cell(), 1) # For species concentration and electric potential
#
TH = mixed_element([U2, P1, P1]) # Taylor-Hood style mixed element
ME = functionspace(domain, TH) # Total space for all DOFs
# Define actual functions with the required DOFs
w = Function(ME)
u, c, phi = split(w) # displacement u, concentration c, voltage phi
# A copy of functions to store values in the previous step for time-stepping
w_old = Function(ME)
u_old, c_old, phi_old = split(w_old)
# Define test functions
u_test, c_test, phi_test = TestFunctions(ME)
# Define trial functions needed for automatic differentiation
dw = TrialFunction(ME)
Initial conditions#
The initial conditions for degrees of freedom \(\mathbf{u}\) and \(\phi\) are zero everywhere.
These are imposed automatically, since we have not specified any non-zero initial conditions.
We do, however, need to impose the uniform initial condition \(\bar{c} = \bar{c}_0\). This is done below.
# Assign initial value of c0, the normalized concentration, to the domain
#
w.sub(1).interpolate(lambda x: np.full((x.shape[1],), c0))
w_old.sub(1).interpolate(lambda x: np.full((x.shape[1],), c0))
Subroutines for kinematics and constitutive equations#
# Special gradient operators for plane strain
#
# Gradient of vector field u
def pe_grad_vector(u):
grad_u = grad(u)
pe_grad_u = ufl.as_tensor([ [grad_u[0,0], grad_u[0,1], 0.0],
[grad_u[1,0], grad_u[1,1], 0.0],
[ 0.0, 0.0, 0.0] ])
return pe_grad_u
# Gradient of scalar field y
# (just need an extra zero for dimensions to work out)
def pe_grad_scalar(y):
grad_y = grad(y)
pe_grad_y = ufl.as_vector([grad_y[0], grad_y[1], 0.0])
return pe_grad_y
# Plane strain deformation gradient
def F_pe_calc(u):
dim = len(u) # dimension of problem (2)
Id = Identity(dim) # 2D Identity tensor
F = Id + grad(u) # 2D Deformation gradient
F_pe = ufl.as_tensor([ [F[0,0], F[0,1], 0.0],
[F[1,0], F[1,1], 0.0],
[ 0.0, 0.0, 1.0] ]) # Full plane strain F
return F_pe
def Piola_mech_calc(F, c):
Id = Identity(3)
# Elastic volumetric strain
Js = 1 + Omega*cMax*(c-c0)
Je = J/Js
#
Fs = Js**(1/3)*Id
Fe = F*inv(Fs)
Be = Fe*Fe.T
#
Bebar = Je**(-2/3)*Be
#
Piola_mech = Js*(Gshear*(Bebar - (1/3)*tr(Bebar)*Id) + Kbulk*(Je-1)*Id)*inv(F.T)
#
return Piola_mech
def p_tilde_calc(F, c):
#
# Elastic volumetric strain
Js = 1 + Omega*cMax*(c-c0)
Je = J/Js
#
p_tilde = -Kbulk*(Je-1.0)
#
return p_tilde
def Piola_maxw_calc(F, phi):
e_R = - pe_grad_scalar(phi)
e_sp = inv(F.T)*e_R
T_max = vareps0*vareps_r*(outer(e_sp,e_sp) - (1/2)*(inner(e_sp,e_sp))*Identity(3))
Piola_max = J*T_max*inv(F.T)
return Piola_max
def flux_calc(u, c, phi, press):
#
F = F_pe_calc(u)
J = det(F) # Total volumetric jacobian
Ci = inv(F.T*F)
#
Jmat = - D * Ci *( pe_grad_scalar(cMax*c) \
+ (Omega*cMax*c*(1-c)/RT)*pe_grad_scalar(press) \
+ (Farad*cMax*c*(1-c)/RT)*pe_grad_scalar(phi) )
#
return Jmat
def dmat_calc(F, phi):
dmat = - vareps0*vareps_r*vareps_num*Ci*pe_grad_scalar(phi)
return dmat
Evaluate kinematics and constitutive relations#
# Kinematics
F = F_pe_calc(u)
J = det(F) # Total volumetric jacobian
Js = 1 + Omega*cMax*(c - c0)
Ci = inv(F.T*F) # Inverse of the right Cauchy-Green strain tensor
# Compute the stress, pressure for echem. pot.
Piola_mech = Piola_mech_calc(F, c)
Piola_max = Piola_maxw_calc(F, phi)
Piola = Piola_mech + Piola_max
#
# Compute the mechanical pressure which drives species flux
press = p_tilde_calc(F, c)
# referential flux of mobile species
jmat = flux_calc(u, c, phi, press)
# referential electric displacement
dmat = dmat_calc(F, phi)
Weak forms#
# The weak form for the equilibrium equation
# Test fxn: u
Res_0 = inner(1/Gshear*Piola,pe_grad_vector(u_test))*dx
# The weak form for the mass balance
# Test fxn: c_test
Res_1 = dot((c-c_old)/dk,c_test)*dx \
- (1/cMax)*dot(jmat,pe_grad_scalar(c_test))*dx
# The weak form for Gauss' Law
# Test fxn: phi
Res_2 = dot(dmat,pe_grad_scalar(phi_test))*dx \
+inner(Farad*cMax*(c-c0),phi_test)*dx
# Total weak form
Res = Res_0 + Res_1 + Res_2
# Automatic differentiation tangent:
a = derivative(Res, w, dw)
Set-up output files#
# results file name
results_name = "ipmc_sinusoidal_curl"
# Function space for projection of results
U1 = element("DG", domain.basix_cell(), 1, shape=(2,)) # For displacement
P0 = element("DG", domain.basix_cell(), 1) # For concentration and electric potential
T1 = element("DG", domain.basix_cell(), 1, shape=(3,3)) # For stress tensor
V1 = fem.functionspace(domain, P0) # Scalar function space
V2 = fem.functionspace(domain, U1) # Vector function space
V3 = fem.functionspace(domain, T1) # Tensor function space
# basic fields to write to output file
u_vis = Function(V2)
u_vis.name = "disp"
c_vis = Function(V1)
c_vis.name = "c"
phi_vis = Function(V1)
phi_vis.name = "phi"
P11 = Function(V1)
P11.name = "P11"
P11_expr = Expression(Piola[0,0],V1.element.interpolation_points())
#
P22 = Function(V1)
P22.name = "P22"
P22_expr = Expression(Piola[1,1],V1.element.interpolation_points())
#
P33 = Function(V1)
P33.name = "P33"
P33_expr = Expression(Piola[2,2],V1.element.interpolation_points())
# Mises stress
T = Piola*F.T/J
T0 = T - (1/3)*tr(T)*Identity(3)
Mises = sqrt((3/2)*inner(T0, T0))
Mises_vis = Function(V1,name="Mises")
Mises_expr = Expression(Mises,V1.element.interpolation_points())
# set up the output VTX files.
file_results = VTXWriter(
MPI.COMM_WORLD,
"results/" + results_name + ".bp",
[ # put the functions here you wish to write to output
u_vis, c_vis, phi_vis, P11, P22, P33,Mises_vis,
],
engine="BP4",
)
def writeResults(t):
# Output field interpolation
u_vis.interpolate(w.sub(0))
c_vis.interpolate(w.sub(1))
phi_vis.interpolate(w.sub(2))
P11.interpolate(P11_expr)
P22.interpolate(P22_expr)
P33.interpolate(P33_expr)
Mises_vis.interpolate(Mises_expr)
# Write output fields
file_results.write(t)
Infrastructure for pulling out time history data (dispalcement, force, etc.)#
# Identify point for reporting potential and dispalcement
pointForDisp = np.array([scaleX,scaleY/2,0.0])
bb_tree = dolfinx.geometry.bb_tree(domain,domain.topology.dim)
cell_candidates = dolfinx.geometry.compute_collisions_points(bb_tree, pointForDisp)
colliding_cells = dolfinx.geometry.compute_colliding_cells(domain, cell_candidates, pointForDisp).array
Analysis Step#
# Give the step a descriptive name
step = "Sinusoidal Curl"
Boundary conditions#
# Sub-domain markers
# boundaries = [(1, xBot),(2, xTop),(3, yBot),(4, yTop)]
# Constant for applied electric potential
phi_cons = Constant(domain,PETSc.ScalarType(phiRamp(0)))
# Find the specific DOFs which will be constrained.
xBot_u1_dofs = fem.locate_dofs_topological(ME.sub(0).sub(0), facet_tags.dim, facet_tags.find(1))
xBot_u2_dofs = fem.locate_dofs_topological(ME.sub(0).sub(1), facet_tags.dim, facet_tags.find(1))
#
yBot_phi_dofs= fem.locate_dofs_topological(ME.sub(2), facet_tags.dim, facet_tags.find(3))
yTop_phi_dofs= fem.locate_dofs_topological(ME.sub(2), facet_tags.dim, facet_tags.find(4))
# Building Dirichlet BCs
bcs_1 = dirichletbc(0.0, xBot_u1_dofs, ME.sub(0).sub(0)) # u1 fix xBot
bcs_2 = dirichletbc(0.0, xBot_u2_dofs, ME.sub(0).sub(1)) # u2 fix xBot
#
bcs_3 = dirichletbc(0.0, yBot_phi_dofs, ME.sub(2)) # phi =0 yBot
bcs_4 = dirichletbc(phi_cons, yTop_phi_dofs, ME.sub(2)) # phi= mphi_cons- yTop
bcs = [bcs_1, bcs_2, bcs_3, bcs_4]
Define the nonlinear variational problem#
# Set up nonlinear problem
problem = NonlinearProblem(Res, w, bcs, a)
# The global newton solver and params
solver = NewtonSolver(MPI.COMM_WORLD, problem)
solver.convergence_criterion = "incremental"
solver.rtol = 1e-8
solver.atol = 1e-8
solver.max_it = 50
solver.report = True
# The Krylov solver parameters.
ksp = solver.krylov_solver
opts = PETSc.Options()
option_prefix = ksp.getOptionsPrefix()
opts[f"{option_prefix}ksp_type"] = "preonly"
opts[f"{option_prefix}pc_type"] = "lu" # do not use 'gamg' pre-conditioner
opts[f"{option_prefix}pc_factor_mat_solver_type"] = "mumps"
opts[f"{option_prefix}ksp_max_it"] = 30
ksp.setFromOptions()
Initialize arrays for storing output history#
# Arrays for storing output history
totSteps = 100000
timeHist0 = np.zeros(shape=[totSteps])
timeHist1 = np.zeros(shape=[totSteps])
timeHist2 = np.zeros(shape=[totSteps])
# timeHist3 = np.zeros(shape=[totSteps])
# Initialize a counter for reporting data
ii = 0
# Write initial state to file
writeResults(t=0.0)
Start calculation loop#
# Print message for simulation start
print("------------------------------------")
print("Simulation Start")
print("------------------------------------")
# Store start time
startTime = datetime.now()
# Time-stepping solution procedure loop
while (round(t + dt, 9) <= Ttot):
# increment time
t += dt
# increment counter
ii += 1
# update time variables in time-dependent BCs
phi_cons.value = float(phiRamp(t))
# Solve the problem
try:
(iter, converged) = solver.solve(w)
except: # Break the loop if solver fails
print("Ended Early")
break
# Collect results from MPI ghost processes
w.x.scatter_forward()
# Write output to file
writeResults(t)
# Update DOFs for next step
w_old.x.array[:] = w.x.array
# Store displacement at a particular point at this time
#
timeHist0[ii] = t
#
timeHist1[ii] = w.sub(2).eval([scaleX,scaleY/2,0.0],colliding_cells[0])[0] # time history of voltage (phi)# time
#
timeHist2[ii] = w.sub(0).sub(1).eval([scaleX,scaleY/2,0.0],colliding_cells[0])[0] # time history of tip displacement
# Print progress of calculation
if ii%1 == 0:
now = datetime.now()
current_time = now.strftime("%H:%M:%S")
print("Step: {} | Increment: {}, Iterations: {}".\
format(step, ii, iter))
print(" Simulation Time: {} s of {} s".\
format(round(t,4), Ttot))
print()
# close the output file.
file_results.close()
# End analysis
print("-----------------------------------------")
print("End computation")
# Report elapsed real time for the analysis
endTime = datetime.now()
elapseTime = endTime - startTime
print("------------------------------------------")
print("Elapsed real time: {}".format(elapseTime))
print("------------------------------------------")
------------------------------------
Simulation Start
------------------------------------
Step: Sinusoidal Curl | Increment: 1, Iterations: 4
Simulation Time: 0.04 s of 60.0 s
Step: Sinusoidal Curl | Increment: 2, Iterations: 4
Simulation Time: 0.08 s of 60.0 s
Step: Sinusoidal Curl | Increment: 3, Iterations: 5
Simulation Time: 0.12 s of 60.0 s
Step: Sinusoidal Curl | Increment: 4, Iterations: 5
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Step: Sinusoidal Curl | Increment: 5, Iterations: 5
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Step: Sinusoidal Curl | Increment: 6, Iterations: 5
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Simulation Time: 60.0 s of 60.0 s
-----------------------------------------
End computation
------------------------------------------
Elapsed real time: 0:29:18.140891
------------------------------------------
Plot results#
# Set up font size, initialize colors array
font = {'size' : 14}
plt.rc('font', **font)
#
prop_cycle = plt.rcParams['axes.prop_cycle']
colors = prop_cycle.by_key()['color']
# Only plot as far as we have time history data
ind = np.argmax(timeHist0[:])
# Two-axis plotting
fig, (ax1, ax2) = plt.subplots(2,1, sharex='col')
#---------------------------------------------
ax1.plot(timeHist0[0:ind], timeHist1[0:ind], c=colors[3], linewidth=2.0)
ax1.grid(linestyle="--", linewidth=0.5, color='b')
ax1.set_ylabel('Electric potential (V)')
from matplotlib.ticker import AutoMinorLocator,FormatStrFormatter
ax1.xaxis.set_minor_locator(AutoMinorLocator())
ax1.yaxis.set_minor_locator(AutoMinorLocator())
ax1.set_ylim([-4,4])
#-----------------------------------------------
ax2.plot(timeHist0[0:ind], timeHist2[0:ind], c=colors[0], linewidth=2.0)
ax2.grid(linestyle="--", linewidth=0.5, color='b')
ax2.set_xlabel('Time (s)')
ax2.set_ylabel('Displacement (mm)')
from matplotlib.ticker import AutoMinorLocator,FormatStrFormatter
ax2.xaxis.set_minor_locator(AutoMinorLocator())
ax2.yaxis.set_minor_locator(AutoMinorLocator())
ax2.set_ylim([-1.5,2.5])
#-----------------------------------------------
#plt.show()
fig = plt.gcf()
fig.set_size_inches(7,5)
plt.tight_layout()
plt.savefig("results/ipmc_beam_sinusoidal.png", dpi=600)
#---------------------------------------------------------------------
# New figure
# Two plots on the same Axes with different left and right scales
fig, ax1 = plt.subplots()
color = 'tab:red'
ax1.set_xlabel('Time (s)')
ax1.set_ylabel('Electric potential (V)', color=color)
ax1.plot(timeHist0[0:ind],timeHist1[0:ind], color=color, linewidth=2)
ax1.tick_params(axis='y', labelcolor=color)
ax1.set_xlim([50,60])
ax1.set_ylim([-4,4])
ax2 = ax1.twinx() # instantiate a second Axes that shares the same x-axis
color = 'tab:blue'
ax2.set_ylabel('Displacement (mm)', color=color) # we already handled the x-label with ax1
ax2.plot(timeHist0[0:ind],timeHist2[0:ind], color=color, linewidth=2)
ax2.tick_params(axis='y', labelcolor=color)
ax2.set_ylim([-1.5,1.5])
fig = plt.gcf()
fig.set_size_inches(7,5)
plt.tight_layout() # otherwise the right y-label is slightly clipped#
plt.savefig("results/IPMC_beam_sine_actuation_steady.png", dpi=600)
