Swelling of a cylindrical packer

Swelling of a cylindrical packer#

Swelling-deswelling-reswelling of an axisymmetric packer
This is a two-dimensional axisymmetric simulation

Degrees of freedom#

Displacement: u
pressure: p
chemical potential: mu
concentration: c
temperature: theta

Units#

Length: mm
Mass: tonne (1000kg)
Time: s
Mass density: tonne/mm^3
Force: N
Stress: MPa
Energy: mJ
Temperature: K
Amount of substance: mol
Species concentration: mol/mm^3
Chemical potential: mJ/mol
Molar volume: mm^3/mol
Species diffusivity: mm^2/s
Thermal expansion coefficient: #/K
Specific heat: mJ/(mm^3 K)
Thermal conductivity: mW/(mm K)
Boltzmann Constant: 1.38E-20 mJ/K
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.

https://solidmechanicscoupledtheories.github.io/

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 
r_in   = 3 # inner packer radius
r_out  = 9 # outer packer radius
r_wall = 11 # outer rigid wall radius
L0     = 6 # axial length of packer

domain = mesh.create_rectangle(MPI.COMM_WORLD, [[r_in,0.0], [r_out,L0]],\
                         [20,20], mesh.CellType.triangle, diagonal=mesh.DiagonalType.crossed)

# Note: Qudrilaterals do not seem to work in axi-symmetric simulations
# domain = mesh.create_rectangle(MPI.COMM_WORLD, [[0.0,0.0], [L0,L0]],\
#                          [15,15], mesh.CellType.quadrilateral)
                         
x = ufl.SpatialCoordinate(domain)

Identify the boundaries of the domain

# Identify the boundaries of the  rectangle mesh
#
def xBot(x):
    return np.isclose(x[0],r_in)
def xTop(x):
    return np.isclose(x[0], r_out)
def yBot(x):
    return np.isclose(x[1], 0)
def yTop(x):
    return np.isclose(x[1], L0)
    
# 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])

# A single point at the center of the left edge for "grounding" the displacement
def ground(x):
    return np.logical_and(np.isclose(x[0], r_in), np.isclose(x[1], L0/2))

x = ufl.SpatialCoordinate(domain)

Visualize the 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/packer_mesh.png")

from IPython.display import Image
Image(filename='results/packer_mesh.png') 

# # Use the following  commands for a  zoom-able  view
# if not pyvista.OFF_SCREEN:
#     plotter.show()
# else:
#     plotter.screenshot("packer_mesh.png")

../_images/a04ad149519f5a1cc0df66ef6bec5da94665eb81de4515aad9d4f651404cd2c8.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#

# Set the locking stretch to a large number to model a Neo-Hookean material
#
k_B      = Constant(domain,PETSc.ScalarType(1.38E-20))              # Boltzmann's constant
R_gas    = Constant(domain,PETSc.ScalarType(8.3145E3))              # Gas constant
theta0   = Constant(domain,PETSc.ScalarType(298))                   # Initial temperature
Gshear_0 = Constant(domain,PETSc.ScalarType(0.3))                   # 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(5.2))                   # Locking stretch
Kbulk    = Constant(domain,PETSc.ScalarType(1000.0*Gshear_0))       # Bulk modulus
#
Omega    = Constant(domain,PETSc.ScalarType(1.0E5))                 # Molar volume of fluid
D        = Constant(domain,PETSc.ScalarType(5.0E-2))                # Diffusivity
chi_L    = Constant(domain,PETSc.ScalarType(0.7))                   # Flory-Huggins mixing parameter low value
chi_H    = Constant(domain,PETSc.ScalarType(0.9))                   # Flory-Huggins mixing parameter high value
# chi_L    = Constant(domain,PETSc.ScalarType(0.1))                   # Flory-Huggins mixing parameter low value
# chi_H    = Constant(domain,PETSc.ScalarType(0.7))                   # Flory-Huggins mixing parameter high value
theta_T  = Constant(domain,PETSc.ScalarType(307))                   # Transition temperature
Delta_T  = Constant(domain,PETSc.ScalarType(5.0))                   # Transition temperature width
alpha    = Constant(domain,PETSc.ScalarType(70.0E-6))               # Coefficient of thermal expansion
c_v      = Constant(domain,PETSc.ScalarType(4.18))                  # Specific heat
k_therm  = Constant(domain,PETSc.ScalarType(0.53))                  # Thermal conductivity
#
phi0    = Constant(domain,PETSc.ScalarType(0.999))                # Initial polymer volume fraction
mu0     = Constant(domain,PETSc.ScalarType(ln(1.0-phi0) + phi0 )) # Initial chemical potential
c0      = Constant(domain,PETSc.ScalarType((1/phi0) - 1))         # Initial concentration

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, chemical potential,
                                                 # species concentration, and temperature
#                                      
TH = mixed_element([U2, P1, P1, 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, mu, c, theta = split(w)  # displacement u, pressure p, chemical potential mu,
                               # concentration c, and temperature theta

# A copy of functions to store values in the previous step for time-stepping
w_old = Function(ME)
u_old,  p_old, mu_old, c_old, theta_old = split(w_old)   

# Define test functions    
u_test, p_test,  mu_test, c_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 conditions for \(\mu=\mu_0\) and \(\hat{c} = \hat{c}_0\) which correspond to \(\phi_0 = 0.999\), as well as the initial temperature \(\vartheta=\vartheta_0\). This is done below.

# Assign initial  normalized chemical potential  mu0 to the domain
w.sub(2).interpolate(lambda x: np.full((x.shape[1],),  mu0))
w_old.sub(2).interpolate(lambda x: np.full((x.shape[1],), mu0))  

# Assign initial  value of normalized concentration  c0 to the domain
w.sub(3).interpolate(lambda x: np.full((x.shape[1],),  c0))
w_old.sub(3).interpolate(lambda x: np.full((x.shape[1],), c0))     

# Assign initial  value of tempearture  theta0 to the domain
w.sub(4).interpolate(lambda x: np.full((x.shape[1],),  theta0))
w_old.sub(4).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

#  Elastic deformation gradient Fe
def Fe_calc(u,c):
    F = F_axi_calc(u)      
    J = det(F)             
    #
    Js = 1.0 + c          
    Fs = Js**(1/3)*Identity(3) 
    #
    Fe = F*inv(Fs)
    return   Fe   

def lambdaBar_calc(u):
    F = F_axi_calc(u)
    C = F.T*F
    I1 = tr(C)
    lambdaBar = sqrt(I1/3.0)
    return lambdaBar

# Calculate zeta
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

# Calculate zeta0
def zeta0_calc():
    # Use Pade approximation of Langevin inverse (A. Cohen, 1991)
    z    = 1/lambdaL
    z    = conditional(gt(z,0.95), 0.95, z) # Keep from blowing up
    beta0 = z*(3.0 - z**2.0)/(1.0 - z**2.0)
    zeta0 = (lambdaL/3)*beta0
    return zeta0

# chi-parameter
def chi_calc(theta):
     chi = 0.5*(chi_L + chi_H)- 0.5*(chi_L - chi_H)* tanh((theta-theta_T)/Delta_T)
     return chi
 
# Stress-temperature modulus
def MH1_calc(u,c):
    Id = Identity(3)
    Fe = Fe_calc(u,c)
    Je = det(Fe)
    Ce = Fe.T*Fe
    Js = 1 + c
    zeta = zeta_calc(u)
    zeta0 = zeta0_calc()
    fac1 = N_R * k_B * ( zeta * Js**(2/3) * Id - zeta0 * inv(Ce) )
    fac2 = (3*Kbulk*alpha/Je) * inv(Ce)
    MH1 = 0.5*(fac1 +fac2)
    return MH1

# Chemical potential-temperature modulus
def MH2_calc(u,c):
    Id = Identity(3)
    F = F_axi_calc(u)
    C = F.T*F
    Js = 1 + c
    zeta  = zeta_calc(u)
    zeta0 = zeta0_calc()
    chi   = chi_calc(theta)
    #
    fac1 = R_gas*( ln(c/Js) + 1/Js + chi/Js**2 )
    fac2 = (Omega*N_R*k_B/Js)*( zeta*tr(C)/3 - zeta0 )
    MH2 = fac1+ fac2
    return MH2

# Normalized Piola stress 
def Piola_calc(u, p, theta):
    F = F_axi_calc(u)
    #
    J = det(F)
    #
    zeta = zeta_calc(u)
    #
    zeta0 = zeta0_calc()
    #
    Gshear0  = N_R * k_B * theta
    #
    Piola = (zeta*F - zeta0*inv(F.T) ) - (J * p/Gshear0) * inv(F.T) 
    return Piola

# Species flux
def Flux_calc(u, mu, c, theta):
    F = F_axi_calc(u) 
    #
    Cinv = inv(F.T*F) 
    #
    RT = R_gas * theta
    #
    Mob = (D*c)/(Omega*RT)*Cinv # Mobility tensor
    #
    Jmat = - RT* Mob * axi_grad_scalar(mu)
    return Jmat

#  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
# Macaulay bracket function
def ppos(x):
    return (x+abs(x))/2.

Evaluate kinematics and constitutive relations#

# Kinematics
F = F_axi_calc(u)
J = det(F)  

# lambdaBar
lambdaBar = lambdaBar_calc(u)

# Fe 
Fe     = Fe_calc(u,c)                    
Fe_old = Fe_calc(u_old,c_old)

# Je
Je = det(Fe)

# Ce
Ce = Fe.T * Fe
Ce_old = Fe_old.T * Fe_old

#  Piola stress
Piola = Piola_calc(u, p, theta)

# Species flux
Jmat = Flux_calc(u, mu, c, theta)

# Heat flux
Qmat = Heat_flux_calc(u, theta)

# Heat-coupling terms
MH1 = MH1_calc(u,c)
MH2 = MH2_calc(u,c)

# RT
RT      = R_gas*theta

# chi-parameter
chi = chi_calc(theta)

Weak forms#

# Residuals:
# Res_0: Balance of forces (test fxn: u)
# Res_1: Pressure (test fxn: p)
# Res_2: Balance of mass   (test fxn: mu)
# Res_3: Concentration     (test fxn:c)
# Res_4: Tempearture       (test fxn:theta)

# 
# boundaries = [(1, xBot),(2,xTop),(3,yBot),(4,yTop)]

# The weak form for the equilibrium equation
Res_0 = inner(Piola , axi_grad_vector(u_test) )*x[0]*dx 

# Penalty weak form for the contact at the x[0] = r_wall surface
#
# The mathematical form of the penalty force is a simple linear relationship,
#
# f = - k * < \delta u >,
#
# where f is a horizontal contact force,
#       k is a penalty stiffness parameter, and
#       \delta u is the amount by which the material surface coordinates 
#                penetrate the contact plane.
#       < x > is the Macaulay bracket, which returns: 0 if x < 0
#                                                     x if x > 0
#             (so that f=0 when contact isn't occurring.)
#
l_pen       = r_wall # x-coord of contact plane
k_pen       = 1.0e3 # penalty stiffness, mN/mm
f_pen       = - k_pen*ppos(  x[0] + u[0] - l_pen )  # penalty force, scalar
Res_contact = - dot( f_pen, u_test[0] )*x[0]*ds(2)  # weak form contribution

# The weak form for the  pressure
fac_p =   (ln(Je) - 3*alpha*(theta-theta0))
#
Res_1 = dot((p*Je/Kbulk + fac_p) , p_test)*x[0]*dx 
      
      
# The weak form for the mass balance of solvent      
Res_2 = dot((c - c_old)/dk, mu_test)*x[0]*dx \
        -  Omega*dot(Jmat , axi_grad_scalar(mu_test) )*x[0]*dx    
        

# The weak form for the concentration
fac = 1/(1+c)
#
fac1 =  mu - ( ln(1.0-fac)+ fac + chi*fac*fac )
#
fac2 = -(Omega*Je/RT)*p  
#
fac3 = - (Omega/RT)* ((1/2) * ((Je*p)**2.0)/Kbulk - 3*Kbulk*alpha*(theta-theta0)*ln(Je))
#
fac4 = fac1 + fac2   + fac3
#
Res_3 = dot(fac4, c_test)*x[0]*dx 


# The weak form for the the heat equation
tfac1 = theta * inner(MH1, (Ce-Ce_old))
tfac2 = theta * MH2*(c - c_old)/Omega
tfac3 = dk * RT * inner(Jmat, axi_grad_scalar(mu))
tfac4 = tfac1 + tfac2 - tfac3
#
Res_4 = dot( c_v*(theta - theta_old), theta_test)*x[0]*dx \
        -  dk* dot(Qmat , axi_grad_scalar(theta_test) )*x[0]*dx   \
        -   dot(tfac4, theta_test)*x[0]*dx 
        
# Total weak form
Res = Res_0 + Res_1 + Res_2 + Res_3 + Res_4 + Res_contact

# Automatic differentiation tangent:
a = derivative(Res, w, dw)

Set-up output files#

# Set up projection problem for fixing visualization issues
# of fields in  the axisymmetric simulation
#
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 = "thermogel_axi_packer"

# 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, chemical potential, and concentration 
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"

p_vis = Function(V1)
p_vis.name = "p"

mu_vis = Function(V1)
mu_vis.name = "mu"

c_vis = Function(V1)
c_vis.name = "c"

theta_vis = Function(V1)
theta_vis.name = "theta"

# # calculated fields to write to output file
# phi = 1/(1+c)
# phi_vis = Function(V1)
# phi_vis.name = "phi"
# phi_expr = Expression(phi,V1.element.interpolation_points())

# Project the polymer volume fraction  for visualization
#
phi = 1/(1+c)
phi_projection = setup_projection(phi, V1)
phi_vis        = phi_projection.solve()
phi_vis.name   = "phi"

# Project the volumetric Jacobian J for visualization
#
J_projection = setup_projection(J, V1)
J_vis        = J_projection.solve()
J_vis.name   = "J"

# Project the effective stretch lambdabar for visualization
#
lambdaBar_projection = setup_projection(lambdaBar, V1)
lambdaBar_vis        = lambdaBar_projection.solve()
lambdaBar_vis.name   = "lambdaBar"

# 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, mu_vis, c_vis, theta_vis, phi_vis, J_vis, P11, P22, P33, 
        lambdaBar_vis,Mises_vis,
    ],
    engine="BP4",
)

def writeResults(t):
       
       # Re-project some fields. 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()
       phi_vis        = phi_projection.solve()
       J_vis          = J_projection.solve()
       lambdaBar_vis  = lambdaBar_projection.solve()
       

       # Output field interpolation 
       u_vis.interpolate(w.sub(0))
       p_vis.interpolate(w.sub(1))
       mu_vis.interpolate(w.sub(2))
       c_vis.interpolate(w.sub(3))
       theta_vis.interpolate(w.sub(4))
       #phi_vis.interpolate(phi_expr)
       P11.interpolate(P11_expr)
       P22.interpolate(P22_expr)
       P33.interpolate(P33_expr)

       # Write output fields
       file_results.write(t) 
        

Infrastructure for pulling out time history data (displacement, force, etc.)#

# Identify point for reporting temperature
pointForDisp = np.array([r_out,L0/2,0.0])

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

# Form for evaluating the reaction force at each step
ReactionForce = fem.form(2*np.pi*dot(Gshear_0*Piola*n, n)*x[0]*ds(2))

Analysis Step#

# Give the step a descriptive name
step = "Swell-deswell-reswell"

Boundary conditions#

# Boundary chemical potential history:
#
def muRamp(t):
    return mu0*exp(-t/ttd)

# Constant for chemical potential history
#
mu_cons = Constant(domain,PETSc.ScalarType(muRamp(0)))

# t          = 0.0        # initialization of time
# ttd        = 300        # time for increasing mu Note ttd= 300s does not work for axi-symmetric simulation
# tRamp      = 3600*2     # time to increase or decrease tempearture

# time1      = 3600*2   
# # 
# time2      = 3600*4      
# #
# time3      = 3600*5    
# #
# time4      = 3600*7 
# #
# time5      = 3600*8    
# #
# Ttot       = time5 


# Boundary temperature history:
#
# Temperature increase deltaTheta =25K
#
deltaTheta = Constant(domain,PETSc.ScalarType(25))
# 
def thetaRamp(t):
    if t<= time1:
       temperature = theta0  
       
    elif time1 < t <= time2:
       temperature = theta0 + deltaTheta*(t-time1)/tRamp  
        
    elif time2 < t <= time3:
        temperature = (theta0 + deltaTheta)  
        
    elif time3 < t <= time4 :
        temperature = (theta0 + deltaTheta) - deltaTheta*(t-time3)/tRamp  
         
    else:
        temperature = theta0
        
    return temperature

# Constant for temperature history
theta_cons = Constant(domain,PETSc.ScalarType(thetaRamp(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))
xBot_u2_dofs = fem.locate_dofs_topological(ME.sub(0).sub(1), facet_tags.dim, facet_tags.find(1))
#
xTop_mu_dofs = fem.locate_dofs_topological(ME.sub(2), facet_tags.dim, facet_tags.find(2))
yBot_mu_dofs = fem.locate_dofs_topological(ME.sub(2), facet_tags.dim, facet_tags.find(3))
yTop_mu_dofs = fem.locate_dofs_topological(ME.sub(2), facet_tags.dim, facet_tags.find(4))
#
xTop_theta_dofs = fem.locate_dofs_topological(ME.sub(4), facet_tags.dim, facet_tags.find(2))
yBot_theta_dofs = fem.locate_dofs_topological(ME.sub(4), facet_tags.dim, facet_tags.find(3))
yTop_theta_dofs = fem.locate_dofs_topological(ME.sub(4), facet_tags.dim, facet_tags.find(4))

# Build 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(mu_cons, xTop_mu_dofs, ME.sub(2))          # mu_cons - xTop
bcs_4 = dirichletbc(mu_cons, yBot_mu_dofs, ME.sub(2))          # mu_cons - yBot
bcs_5 = dirichletbc(mu_cons, yTop_mu_dofs, ME.sub(2))          # mu_cons - yTop
#
bcs_6 = dirichletbc(theta_cons, xTop_theta_dofs, ME.sub(4))    # theta_cons- xTop
bcs_7 = dirichletbc(theta_cons, yBot_theta_dofs, ME.sub(4))    # theta_cons - yBot
bcs_8 = dirichletbc(theta_cons, yTop_theta_dofs, ME.sub(4))    # theta_cons - yTop

# # Zero displacement bc for left bottom node
# V0, submap = ME.sub(0).collapse()
# fixed_u_dofs = fem.locate_dofs_geometrical((ME.sub(0), V0), ground)
# fixed_disp = Function(V0)
# fixed_disp.interpolate(lambda x: np.stack(( np.zeros(x.shape[1]), np.zeros(x.shape[1]) ) ) )
# #
# bcs_9 = dirichletbc(fixed_disp, fixed_u_dofs, ME.sub(0))        # u fix -  bottom left node

bcs = [bcs_1, bcs_2, bcs_3, bcs_4, bcs_5, bcs_6, bcs_7, bcs_8]

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])
# #
timeHist1[0] = theta0 # Initialize the tempearture

# Initialize a counter for reporting data
ii = 0

# Write initial state to file
writeResults(t=0.0) 

Start calculation loop#

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 
    mu_cons.value    = float(muRamp(t))
    theta_cons.value = float(thetaRamp(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 time history variables at this time  
    timeHist0[ii] = t # current time
    #
    timeHist1[ii] =  thetaRamp(t) # time history of applied tempearture
    #
    timeHist2[ii] =  domain.comm.gather(fem.assemble_scalar(ReactionForce))[0] # time history of Recation force
   
    
   
    # 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
------------------------------------
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      Simulation Time: 28800.0 s  of  28800 s

-----------------------------------------
End computation
------------------------------------------
Elapsed real time:  0:01:22.974807
------------------------------------------

Plot results#

# set plot font to size 14
font = {'size'   : 14}
plt.rc('font', **font)

# Get array of default plot colors
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)

#
fig = plt.figure() 
#fig.set_size_inches(7,4)
ax=fig.gca()  
#--------------------------------------------------------------
plt.plot(timeHist0[0:ind]/3600, timeHist1[0:ind], linewidth=2.0,\
         color='b')
#-------------------------------------------------------------
plt.xlim([0,8])
plt.ylim([296,325])
#plt.axis('tight')
plt.xlabel(r'Time (h)')
plt.ylabel(r'Temperature (K)')
plt.grid(linestyle="--", linewidth=0.5, color='b')
#-------------------------------------------------------------
#ax.set_title("Stress dispalcement  curve", size=14, weight='normal')
from matplotlib.ticker import AutoMinorLocator,FormatStrFormatter
# ax.xaxis.set_minor_locator(AutoMinorLocator())
# ax.yaxis.set_minor_locator(AutoMinorLocator())
#plt.show()

fig = plt.gcf()
fig.set_size_inches(7,5)
plt.tight_layout()
plt.savefig("results/packer_thermal_history.png", dpi=600)

../_images/9d0795c1196a8314b6887497111a46497bebce95b806b13125a751bd6067e87b.png

# set plot font to size 14
font = {'size'   : 14}
plt.rc('font', **font)

# Get array of default plot colors
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)

#
fig = plt.figure() 
#fig.set_size_inches(7,4)
ax=fig.gca()  
#--------------------------------------------------------------
plt.plot(timeHist0[0:ind]/3600, -timeHist2[0:ind], linewidth=2.0,\
         color='b')
#-------------------------------------------------------------
# plt.xlim([0,8])
# plt.ylim([0,15])
# plt.axis('tight')
plt.xlabel(r'Time (h)')
plt.ylabel(r'Reaction force (N)')
plt.grid(linestyle="--", linewidth=0.5, color='b')
#-------------------------------------------------------------
#ax.set_title("Stress dispalcement  curve", size=14, weight='normal')
from matplotlib.ticker import AutoMinorLocator,FormatStrFormatter
# ax.xaxis.set_minor_locator(AutoMinorLocator())
# ax.yaxis.set_minor_locator(AutoMinorLocator())
#plt.show()

fig = plt.gcf()
fig.set_size_inches(7,5)
plt.tight_layout()
plt.savefig("results/packer_reaction_force_history.png", dpi=600)

../_images/9575fa2b9a263ec376d87423fe95eb8be2b24871e8ce22d27e2986c0c2318b7a.png