Stretch-induced heating#
Gough-Joule effect: Under adiabatic conditions a rubber increases in temperature when stretched
This is an axisymmetric simulation
Units#
Length: mm
Mass: kg
Time: s
Mass density: kg/mm^3
Force: milliN
Stress: kPa
Energy: microJ
Temperature: K
Boltzmann Constant: 1.38E-17 microJ/K
Number of polymer chains per unit vol: #/mm^3
Thermal expansion coefficient: #/K
Specific heat: microJ/(mm^3 K)
Thermal conductivity: microW/(mm 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 ,\
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#
# Create mesh
R0 = 10.0 # radius of bar
H0 = 10.0 # height of bar
domain = mesh.create_rectangle(MPI.COMM_WORLD, [[0.0,0.0], [R0,H0]],\
[20,20], mesh.CellType.triangle, diagonal=mesh.DiagonalType.crossed)
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], R0)
def yBot(x):
return np.isclose(x[1], 0)
def yTop(x):
return np.isclose(x[1], H0)
# Mark the sub-domains
boundaries = [(1, xBot),(2,xTop),(3,yBot),(4,yTop)]
# Identify the bottom left corner
def Ground(x):
return np.isclose(x[0], 0) and np.isclose(x[1], 0)
# 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])
Visualize reference configuration and boundary facets
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)
plotter.view_xy()
#labels = dict(zlabel='Z', xlabel='X', ylabel='Y')
labels = dict(xlabel='X', ylabel='Y')
plotter.add_axes(**labels)
plotter.screenshot("results/bar_mesh.png")
from IPython.display import Image
Image(filename='results/bar_mesh.png')
# # Use the following commands for a zoom-able view
# if not pyvista.OFF_SCREEN:
# plotter.show()
# else:
# plotter.screenshot("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 parameters#
k_B = Constant(domain,PETSc.ScalarType(1.38E-17)) # Boltzmann's constant
theta0 = Constant(domain,PETSc.ScalarType(298)) # Initial temperature
Gshear_0 = Constant(domain,PETSc.ScalarType(280)) # Ground sate shear modulus
N_R = Constant(domain,PETSc.ScalarType(Gshear_0/(k_B*theta0)))# Number polymer chains per unit ref. volume
lambdaL = Constant(domain,PETSc.ScalarType(5.12)) # Locking stretch
Kbulk = Constant(domain,PETSc.ScalarType(1000*Gshear_0)) # Bulk modulus
alpha = Constant(domain,PETSc.ScalarType(180.0E-6)) # Coefficient of thermal expansion
c_v = Constant(domain,PETSc.ScalarType(1839)) # Specific heat
k_therm = Constant(domain,PETSc.ScalarType(0.16E3)) # Thermal conductivity
Function spaces#
# Define function space, both vectorial and scalar
#
U2 = element("Lagrange", domain.basix_cell(), 2, shape=(2,)) # For displacement
P1 = element("Lagrange", domain.basix_cell(), 1) # For pressure and temperature
#
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, p, theta = split(w) # displacement u, pressure p, temperature theta
# A copy of functions to store values in the previous step for time-stepping
w_old = Function(ME)
u_old, p_old, theta_old = split(w_old)
# Define test functions
u_test, p_test, theta_test = TestFunctions(ME)
# Define trial functions needed for automatic differentiation
dw = TrialFunction(ME)
Initial conditions#
The initial conditions for \(\mathbf{u}\) and \(p\) 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 for temperature \(\vartheta = \vartheta_0\). This is done below.
# Assign initial temperature theta0 to the body
w.sub(2).interpolate(lambda x: np.full((x.shape[1],), theta0))
w_old.sub(2).interpolate(lambda x: np.full((x.shape[1],), theta0))
Subroutines for kinematics and constitutive equations#
# Special gradient operators for axisymmetric functions
#
# Gradient of vector field u
def axi_grad_vector(u):
grad_u = grad(u)
axi_grad_33_exp = conditional(eq(x[0], 0), 0.0, u[0]/x[0])
axi_grad_u = ufl.as_tensor([[grad_u[0,0], grad_u[0,1], 0],
[grad_u[1,0], grad_u[1,1], 0],
[0, 0, axi_grad_33_exp]])
return axi_grad_u
# Gradient of scalar field y
# (just need an extra zero for dimensions to work out)
def axi_grad_scalar(y):
grad_y = grad(y)
axi_grad_y = ufl.as_vector([grad_y[0], grad_y[1], 0.])
return axi_grad_y
# Axisymmetric deformation gradient
def F_axi_calc(u):
dim = len(u) # dimension of problem (2)
Id = Identity(dim) # 2D Identity tensor
F = Id + grad(u) # 2D Deformation gradient
F33_exp = 1.0 + u[0]/x[0] # axisymmetric F33, R/R0
F33 = conditional(eq(x[0], 0.0), 1.0, F33_exp) # avoid divide by zero at r=0
F_axi = ufl.as_tensor([[F[0,0], F[0,1], 0],
[F[1,0], F[1,1], 0],
[0, 0, F33]]) # Full axisymmetric F
return F_axi
def lambdaBar_calc(u):
F = F_axi_calc(u)
C = F.T*F
Cdis = J**(-2/3)*C
I1 = tr(Cdis)
lambdaBar = sqrt(I1/3.0)
return lambdaBar
def zeta_calc(u):
lambdaBar = lambdaBar_calc(u)
# Use Pade approximation of Langevin inverse
z = lambdaBar/lambdaL
z = conditional(gt(z,0.95), 0.95, z) # Keep simulation from blowing up
beta = z*(3.0 - z**2.0)/(1.0 - z**2.0)
zeta = (lambdaL/(3*lambdaBar))*beta
return zeta
#----------------------------------------------
# Subroutine for calculating the Piola stress
#----------------------------------------------
def Piola_calc(u, p, theta):
F = F_axi_calc(u)
J = det(F)
C = F.T*F
#
zeta = zeta_calc(u)
Gshear = N_R * k_B * theta * zeta
#
Piola = J**(-2/3) * Gshear * (F - (1/3)*tr(C)*inv(F.T) ) - J * p * inv(F.T)
return Piola
#--------------------------------------------------------------
# Subroutine for calculating the stress temperature tensor
#--------------------------------------------------------------
def M_calc(u):
Id = Identity(3)
F = F_axi_calc(u)
#
C = F.T*F
Cinv = inv(C)
J = det(F)
zeta = zeta_calc(u)
#
fac1 = N_R * k_B * zeta
fac2 = (3*Kbulk*alpha)/J
#
M = J**(-2/3) * fac1 * (Id - (1/3)*tr(C)*Cinv) - J * fac2 * Cinv
return M
#--------------------------------------------------------------
# Subroutine for calculating the Heat flux
#--------------------------------------------------------------
def Heat_flux_calc(u, theta):
F = F_axi_calc(u)
J = det(F)
#
Cinv = inv(F.T*F)
#
Tcond = J * k_therm * Cinv # Thermal conductivity tensor
#
Qmat = - Tcond * axi_grad_scalar(theta)
return Qmat
#--------------------------------------------------------------------------------
# Subroutine for calculating the principal Cauchy stress for visualization only
#--------------------------------------------------------------------------------
def tensor_eigs(T):
# invariants of T
I1 = tr(T)
I2 = (1/2)*(tr(T)**2 - tr(T*T))
I3 = det(T)
# Intermediate quantities b, c, d
b = -I1
c = I2
d = -I3
# intermediate quantities E, F, G
E = (3*c - b*b)/3
F = (2*(b**3) - 9*b*c + 27*d)/27
G = (F**2)/4 + (E**3)/27
# Intermediate quantities H, I, J, K, L
H = sqrt(-(E**3)/27)
I = H**(1/3)
J = acos(-F/(2*H))
K = cos(J/3)
L = sqrt(3)*sin(J/3)
# Finally, the (not necessarily ordered) eigenvalues
t1 = 2*I*K - b/3
t2 = -I*(K+L) - b/3
t3 = -I*(K-L) - b/3
# Order the eigenvalues using conditionals
#
T1_temp = conditional(lt(t1, t3), t3, t1 ) # returns the larger of t1 and t3.
T1 = conditional(lt(T1_temp, t2), t2, T1_temp ) # returns the larger of T1_temp and t2.
#
T3_temp = conditional(gt(t3, t1), t1, t3 ) # returns the smaller of t1 and t3.
T3 = conditional(gt(T3_temp, t2), t2, T1_temp ) # returns the smaller of T3_temp and t2.
#
# use the trace to report the middle eigenvalue.
T2 = I1 - T1 - T3
return T1, T2, T3
Evaluate kinematics and constitutive relations#
# Kinematics
F = F_axi_calc(u)
J = det(F)
#
lambdaBar = lambdaBar_calc(u)
#
F_old = F_axi_calc(u_old)
J_old = det(F_old)
#
C = F.T*F
C_old = F_old.T*F_old
# Piola stress
Piola = Piola_calc(u, p, theta)
# Calculate the stress-temperature tensor
M = M_calc(u)
# Calculate the heat flux
Qmat = Heat_flux_calc(u,theta)
Weak forms#
# Residuals:
# Res_0: Balance of forces (test fxn: u)
# Res_1: Coupling pressure (test fxn: p)
# Res_2: Balance of energy (test fxn: theta)
# Time step field, constant within body
dk = Constant(domain, PETSc.ScalarType(dt))
# The weak form for the equilibrium equation
Res_0 = inner(Piola, axi_grad_vector(u_test) )*x[0]*dx
# The weak form for the pressure
fac_p = ( ln(J) - 3*alpha*(theta-theta0) )/J
#
Res_1 = dot( (p/Kbulk + fac_p), p_test)*x[0]*dx
# The weak form for heat equation
Res_2 = dot( c_v*(theta - theta_old), theta_test)*x[0]*dx \
- (1/2)*theta * inner(M, (C - C_old)) * theta_test*x[0]*dx \
- dk*dot(Qmat , axi_grad_scalar(theta_test) )*x[0]*dx
# Total weak form
Res = Res_0 + Res_1 + Res_2
# Automatic differentiation tangent:
a = derivative(Res, w, dw)
Set-up output files#
# Set up projection problem for fixing visualization of fields in axisymmetric formulation
def setup_projection(u, V):
trial = ufl.TrialFunction(V)
test = ufl.TestFunction(V)
a = ufl.inner(trial, test)*x[0]*dx
L = ufl.inner(u, test)*x[0]*dx
projection_problem = dolfinx.fem.petsc.LinearProblem(a, L, [], \
petsc_options={"ksp_type": "cg", "ksp_rtol": 1e-16, "ksp_atol": 1e-16, "ksp_max_it": 1000})
return projection_problem
# results file name
results_name = "Axi_adiabatic_stretch"
# 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 pressure and temperature
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) # linear tensor function space
# fields to write to output file
u_vis = Function(V2)
u_vis.name = "disp"
p_vis = Function(V1)
p_vis.name = "p"
theta_vis = Function(V1)
theta_vis.name = "theta"
J_vis = Function(V1)
J_vis.name = "J"
J_expr = Expression(J,V1.element.interpolation_points())
lambdaBar_vis = Function(V1)
lambdaBar_vis.name = "lambdaBar"
lambdaBar_expr = Expression(lambdaBar,V1.element.interpolation_points())
# Project the Piola stress tensor for visualization.
Piola_projection = setup_projection(Piola, V3)
Piola_temp = Piola_projection.solve()
T = Piola_temp*F.T/J
T0 = T - (1/3)*tr(T)*Identity(3)
Mises = sqrt((3/2)*inner(T0, T0))
Mises_projection = setup_projection(Mises, V1)
Mises_vis = Mises_projection.solve()
Mises_vis.name = "Mises"
P11 = Function(V1)
P11.name = "P11"
P11_expr = Expression(Piola_temp[0,0],V1.element.interpolation_points())
P22 = Function(V1)
P22.name = "P22"
P22_expr = Expression(Piola_temp[1,1],V1.element.interpolation_points())
P33 = Function(V1)
P33.name = "P33"
P33_expr = Expression(Piola_temp[2,2],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, p_vis, theta_vis, J_vis, P11, P22, P33,
lambdaBar_vis,Mises_vis,
],
engine="BP4",
)
def writeResults(t):
# re-project stress-related quantities for visualization
# This is necessary here to remove visual artifacts which arise due to the axisymmetric formulation as r -> 0.
Piola_temp = Piola_projection.solve()
Mises_vis = Mises_projection.solve()
# Output field interpolation
u_vis.interpolate(w.sub(0))
p_vis.interpolate(w.sub(1))
theta_vis.interpolate(w.sub(2))
J_vis.interpolate(J_expr)
P11.interpolate(P11_expr)
P22.interpolate(P22_expr)
P33.interpolate(P33_expr)
lambdaBar_vis.interpolate(lambdaBar_expr)
# (Piola stress and Mises stress do not need to be re-interpolated after projection)
# Write output fields
file_results.write(t)
Infrastructure for pulling out time history data (force, displacement, etc.)#
# Identify point for reporting displacement and temperature at a given point
pointForDisp = np.array([0.0,H0,0.0])
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
# boundaries = [(1, xBot),(2,xTop),(3,yBot),(4,yTop)]
# Compute the reaction force using the Piola stress field
RxnForce = fem.form(2*np.pi*P22*x[0]*ds(4))
Analysis Step#
# Give the step a descriptive name
step = "Stretch"
Boundary conditions#
# Constant for applied displacement
disp_cons = Constant(domain,PETSc.ScalarType(dispRamp(0)))
# Recall the sub-domains names and numbers
# boundaries = [(1, xBot),(2,xTop),(3,yBot),(4,yTop)]
# 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))
yBot_u2_dofs = fem.locate_dofs_topological(ME.sub(0).sub(1), facet_tags.dim, facet_tags.find(3))
yTop_u2_dofs = fem.locate_dofs_topological(ME.sub(0).sub(1), 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, yBot_u2_dofs, ME.sub(0).sub(1)) # u2 fix - yBot
bcs_3 = dirichletbc(disp_cons, yTop_u2_dofs, ME.sub(0).sub(1)) # disp_cons - yTop
bcs = [bcs_1, bcs_2, bcs_3]
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" # "preonly" works equally well
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])
#
timeHist3[0] = theta0 # Initialize the temperature
# 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
disp_cons.value = dispRamp(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 and stress at a particular point at this time
#
timeHist0[ii] = t # time
#
timeHist1[ii] = w.sub(0).sub(1).eval([0.0,H0,0.0],colliding_cells[0])[0] # displacement
#
timeHist2[ii] = domain.comm.gather(fem.assemble_scalar(RxnForce))[0] # reaction force
#
timeHist3[ii] = w.sub(2).eval([0.0,H0,0.0],colliding_cells[0])[0] # temperature
# 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: Stretch | Increment: 1, Iterations: 4
Simulation Time: 1.0 s of 100 s
Step: Stretch | Increment: 2, Iterations: 4
Simulation Time: 2.0 s of 100 s
Step: Stretch | Increment: 3, Iterations: 4
Simulation Time: 3.0 s of 100 s
Step: Stretch | Increment: 4, Iterations: 4
Simulation Time: 4.0 s of 100 s
Step: Stretch | Increment: 5, Iterations: 4
Simulation Time: 5.0 s of 100 s
Step: Stretch | Increment: 6, Iterations: 4
Simulation Time: 6.0 s of 100 s
Step: Stretch | Increment: 7, Iterations: 4
Simulation Time: 7.0 s of 100 s
Step: Stretch | Increment: 8, Iterations: 4
Simulation Time: 8.0 s of 100 s
Step: Stretch | Increment: 9, Iterations: 4
Simulation Time: 9.0 s of 100 s
Step: Stretch | Increment: 10, Iterations: 4
Simulation Time: 10.0 s of 100 s
Step: Stretch | Increment: 11, Iterations: 4
Simulation Time: 11.0 s of 100 s
Step: Stretch | Increment: 12, Iterations: 4
Simulation Time: 12.0 s of 100 s
Step: Stretch | Increment: 13, Iterations: 4
Simulation Time: 13.0 s of 100 s
Step: Stretch | Increment: 14, Iterations: 4
Simulation Time: 14.0 s of 100 s
Step: Stretch | Increment: 15, Iterations: 4
Simulation Time: 15.0 s of 100 s
Step: Stretch | Increment: 16, Iterations: 4
Simulation Time: 16.0 s of 100 s
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Simulation Time: 17.0 s of 100 s
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Simulation Time: 18.0 s of 100 s
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Simulation Time: 19.0 s of 100 s
Step: Stretch | Increment: 20, Iterations: 4
Simulation Time: 20.0 s of 100 s
Step: Stretch | Increment: 21, Iterations: 4
Simulation Time: 21.0 s of 100 s
Step: Stretch | Increment: 22, Iterations: 4
Simulation Time: 22.0 s of 100 s
Step: Stretch | Increment: 23, Iterations: 4
Simulation Time: 23.0 s of 100 s
Step: Stretch | Increment: 24, Iterations: 4
Simulation Time: 24.0 s of 100 s
Step: Stretch | Increment: 25, Iterations: 4
Simulation Time: 25.0 s of 100 s
Step: Stretch | Increment: 26, Iterations: 4
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-----------------------------------------
End computation
------------------------------------------
Elapsed real time: 0:00:09.593202
------------------------------------------
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[:])
# Create figure for stress-stretch curve.
#
fig = plt.figure()
ax=fig.gca()
#---------------------------------------------------------------------------------------------
plt.plot((timeHist1[0:ind]/H0 +1), timeHist2[0:ind]/(np.pi*R0*R0)/1000, c='b', linewidth=2.0)
#---------------------------------------------------------------------------------------------
plt.xlim(1,8)
plt.ylim(0,7)
plt.grid(linestyle="--", linewidth=0.5, color='b')
ax.set_xlabel("$\lambda$",size=14)
ax.set_ylabel("Stress (MPa)",size=14)
#ax.set_title("Stress versus stretch curve", size=14, weight='normal')
#
from matplotlib.ticker import AutoMinorLocator,FormatStrFormatter
ax.xaxis.set_minor_locator(AutoMinorLocator())
ax.yaxis.set_minor_locator(AutoMinorLocator())
#plt.legend()
import matplotlib.ticker as ticker
ax.xaxis.set_major_formatter(ticker.FormatStrFormatter('%0.2f'))
# Save figure
fig = plt.gcf()
fig.set_size_inches(7,5)
plt.tight_layout()
plt.savefig("results/axi_adiabatic_stress_stretch.png", dpi=600)
# Create figure for temperature change versus stretch curve.
#
fig = plt.figure()
ax=fig.gca()
#---------------------------------------------------------------------------------------------
plt.plot((timeHist1[0:ind]/H0 +1), timeHist3[0:ind]- timeHist3[0], c='b', linewidth=2.0)
#---------------------------------------------------------------------------------------------
plt.xlim(1,8)
plt.ylim(0,8)
plt.grid(linestyle="--", linewidth=0.5, color='b')
# ax.set_ylabel("Surface temperature change (K)",size=14)
ax.set_xlabel("$\lambda$",size=14)
ax.set_ylabel(r'$ \Delta \theta = \theta - \theta_0$ (K)',size=14)
#ax.set_title(r' $\Delta\theta$ versus $\lambda$ curve', size=14, weight='normal')
#
from matplotlib.ticker import AutoMinorLocator,FormatStrFormatter
ax.xaxis.set_minor_locator(AutoMinorLocator())
ax.yaxis.set_minor_locator(AutoMinorLocator())
#plt.legend()
import matplotlib.ticker as ticker
ax.xaxis.set_major_formatter(ticker.FormatStrFormatter('%0.2f'))
# Save figure
fig = plt.gcf()
fig.set_size_inches(7,5)
plt.tight_layout()
plt.savefig("results/axi_adiabatic_temp_stretch.png", dpi=600)
