Thick cylinder buckling#
Buckling of a thick cylinder
This is a three-dimensional simulation
Degrees of freedom#
Displacement: u
pressure: p
chemical potential: mu
concentration: c
Units#
Length: mm
Mass: kg
Time: s
Mass density: kg/mm^3
Force: milliN
Stress: kPa
Energy: microJ
Temperature: K
Amount of substance: mol
Species concentration: mol/mm^3
Chemical potential: milliJ/mol
Molar volume: mm^3/mol
Species diffusivity: mm^2/s
Gas constant: microJ/(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 ,\
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#
# Thick cylinder parameters in mm
H0 = 1.2
D0 = 4.636
T0 = 0.309
# Read in the 3D mesh and cell tags
with XDMFFile(MPI.COMM_WORLD,"meshes/3D_thick_cyl_mesh.xdmf",'r') as infile:
domain = infile.read_mesh(name="Grid",xpath="/Xdmf/Domain")
cell_tags = infile.read_meshtags(domain,name="Grid")
domain.topology.create_connectivity(domain.topology.dim, domain.topology.dim-1)
x = ufl.SpatialCoordinate(domain)
Identify boundaries of the domain
# Identify the planar boundaries of the domain
def xBot(x):
return np.isclose(x[0], 0)
def yBot(x):
return np.isclose(x[1], 0)
def zBot(x):
return np.isclose(x[2], 0)
def zTop(x):
return np.isclose(x[2], H0)
# Mark the sub-domains
boundaries = [(1,xBot),(2,yBot),(3,zBot),(4,zTop)]
# 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(2) # index map of 2D entities in domain (facets)
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 indices
# Surface numbering:
#boundaries = [(1,xBot),(2,yBot),(3,zBot),(4,zTop)]
[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)
# Surface numbering:
# boundaries = [(1,xBot),(2,yBot),(3,zBot),(4,zTop)]
# Add colored 2D surfaces for the named surfaces
xBot_surf = pyvista.UnstructuredGrid(*vtk_mesh(domain, domain.topology.dim-1,facet_tags.indices[facet_tags.values==1]) )
yBot_surf = pyvista.UnstructuredGrid(*vtk_mesh(domain, domain.topology.dim-1,facet_tags.indices[facet_tags.values==2]) )
zBot_surf = pyvista.UnstructuredGrid(*vtk_mesh(domain, domain.topology.dim-1,facet_tags.indices[facet_tags.values==3]) )
zTop_surf = pyvista.UnstructuredGrid(*vtk_mesh(domain, domain.topology.dim-1,facet_tags.indices[facet_tags.values==4]) )
#
actor = plotter.add_mesh(xBot_surf, show_edges=True,color="blue") # xBot face is blue
actor2 = plotter.add_mesh(yBot_surf, show_edges=True,color="red") # yBot is red
actor3 = plotter.add_mesh(zBot_surf, show_edges=True,color="green") # zBot is green
actor3 = plotter.add_mesh(zTop_surf, show_edges=True,color="magenta") # zTop is yellow
labels = dict(xlabel='X', ylabel='Y',zlabel='Z')
plotter.add_axes(**labels)
plotter.screenshot("results/thick_cylinder_mesh.png")
from IPython.display import Image
Image(filename='results/thick_cylinder_mesh.png')
# #Use the following commands for a zoom-able view
# if not pyvista.OFF_SCREEN:
# plotter.show()
# else:
# plotter.screenshot("thick_cylinder_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})
# Define facet normal
# n = ufl.FacetNormal(domain)
n = ufl.FacetNormal(domain)
Material parameters#
# Set the locking stretch to a large number to model a Neo-Hookean material
#
Gshear_0= Constant(domain,PETSc.ScalarType(1000.0)) # Shear modulus, kPa
lambdaL = Constant(domain,PETSc.ScalarType(100)) # Locking stretch, Neo-Hookean material
Kbulk = Constant(domain,PETSc.ScalarType(1000*Gshear_0)) # Bulk modulus, kPa
Omega = Constant(domain,PETSc.ScalarType(1.00e5)) # Molar volume of fluid
D = Constant(domain,PETSc.ScalarType(5.00e-3)) # Diffusivity
chi = Constant(domain,PETSc.ScalarType(0.1)) # Flory-Huggins mixing parameter
theta0 = Constant(domain,PETSc.ScalarType(298) ) # Reference temperature
R_gas = Constant(domain,PETSc.ScalarType(8.3145e6)) # Gas constant
RT = Constant(domain,PETSc.ScalarType(8.3145e6*theta0))
#
phi0 = Constant(domain,PETSc.ScalarType(0.999)) # Initial polymer volume fraction
mu0 = Constant(domain,PETSc.ScalarType(ln(1.0-phi0) + phi0 + chi*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=(3,)) # For displacement
P1 = element("Lagrange", domain.basix_cell(), 1) # For pressure, chemical potential and species concentration
#
TH = mixed_element([U2, 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 = split(w) # displacement u, pressure p, chemical potential mu, and concentration c
# 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 = split(w_old)
# Define test functions
u_test, p_test, mu_test, c_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 \(\phi_0 = 0.999\). 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))
Subroutines for kinematics and constitutive equations#
#---------------------------------------------------
# Deformation gradient
#---------------------------------------------------
def F_calc(u):
Id = Identity(3) # 3D Identity tensor
F = Id + grad(u) # 3D Deformation gradient
return F
#---------------------------------------------------
# Effective stretch lambdaBar
#---------------------------------------------------
def lambdaBar_calc(u):
F = F_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
#---------------------------------------------------
# Subroutine for calculating the elastic jacobian Je
#---------------------------------------------------
def Je_calc(u,c):
F = F_calc(u)
detF = det(F)
#
detFs = 1.0 + c # = Js
Je = (detF/detFs) # = Je
return Je
#----------------------------------------------
# Subroutine for calculating the Piola stress
#----------------------------------------------
def Piola_calc(u,p):
F = F_calc(u)
zeta = zeta_calc(u)
zeta0 = zeta0_calc()
Piola = (zeta*F - zeta0*inv(F.T) ) - J*p*inv(F.T)/Gshear_0
return Piola
#--------------------------------------------------------------
# Subroutine for calculating the normalized species flux
#--------------------------------------------------------------
def Flux_calc(u, mu, c):
F = F_calc(u)
#
Cinv = inv(F.T*F)
#
Mob = (D*c)/(Omega*RT)*Cinv
#
Jmat = - RT* Mob * grad(mu)
return Jmat
Evaluate kinematics and constitutive relations#
# Kinematics
F = F_calc(u)
J = det(F) # Total volumetric jacobian
#
lambdaBar = lambdaBar_calc(u)
#
# Elastic volumetric Jacobian
Je = Je_calc(u,c)
Je_old = Je_calc(u_old,c_old)
# Normalized Piola stress
Piola = Piola_calc(u, p)
# Normalized species flux
Jmat = Flux_calc(u, mu, c)
Weak forms#
# Residuals:
# Res_0: Balance of forces (test fxn: u)
# Res_1: Pressure variable (test fxn: p)
# Res_2: Balance of mass (test fxn: mu)
# Res_3: Auxiliary variable (test fxn: c)
# Time step field, constant within body
dk = Constant(domain, PETSc.ScalarType(dt))
# The weak form for the equilibrium equation
Res_0 = inner(Piola, grad(u_test) )*dx
# The weak form for the auxiliary pressure variable definition
Res_1 = dot((p*Je/Kbulk + ln(Je)) , p_test)*dx
# The weak form for the mass balance of solvent
Res_2 = dot((c - c_old)/dk, mu_test)*dx \
- Omega*dot(Jmat , grad(mu_test) )*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 = - (1./2.) * (Omega/(Kbulk*RT)) * ((p*Je)**2.0) # This works
fac4 = fac1 + fac2 + fac3
#
Res_3 = dot(fac4, c_test)*dx
# Total weak form
Res = Res_0 + Res_1 + Res_2 + Res_3
# Automatic differentiation tangent:
a = derivative(Res, w, dw)
Set-up output files#
# results file name
results_name = "gel_thick_cylinder_swell"
# Function space for projection of results
U1 = element("DG", domain.basix_cell(), 1, shape=(3,)) # 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"
# 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())
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())
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, p_vis, mu_vis, c_vis, phi_vis, J_vis, P11, P22, P33,
lambdaBar_vis,Mises_vis,
],
engine="BP4",
)
def writeResults(t):
# 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))
phi_vis.interpolate(phi_expr)
J_vis.interpolate(J_expr)
P11.interpolate(P11_expr)
P22.interpolate(P22_expr)
P33.interpolate(P33_expr)
lambdaBar_vis.interpolate(lambdaBar_expr)
Mises_vis.interpolate(Mises_expr)
# Write output fields
file_results.write(t)
Analysis Step#
# Give the step a descriptive name
step = "Swell"
Boundary conditions#
# Constant for applied chemical potential
mu_cons = Constant(domain,PETSc.ScalarType(muRamp(0)))
# Recall the sub-domains names and numbers
#boundaries = [(1,xBot),(2,yBot),(3,zBot),(4,zTop)]
# 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(2))
#
zBot_u1_dofs = fem.locate_dofs_topological(ME.sub(0).sub(0), facet_tags.dim, facet_tags.find(3))
zBot_u2_dofs = fem.locate_dofs_topological(ME.sub(0).sub(1), facet_tags.dim, facet_tags.find(3))
zBot_u3_dofs = fem.locate_dofs_topological(ME.sub(0).sub(2), facet_tags.dim, facet_tags.find(3))
#
zTop_mu_dofs = fem.locate_dofs_topological(ME.sub(2), facet_tags.dim, facet_tags.find(4))
# Dirichlet BCs for displacement
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(0.0, zBot_u1_dofs, ME.sub(0).sub(0)) # u1 fix - zBot
bcs_4 = dirichletbc(0.0, zBot_u2_dofs, ME.sub(0).sub(1)) # u1 fix - zBot
bcs_5 = dirichletbc(0.0, zBot_u3_dofs, ME.sub(0).sub(2)) # u1 fix - zBot
#
# Dirichlet BCs for chemical potential
bcs_6 = dirichletbc(mu_cons, zTop_mu_dofs, ME.sub(2)) # mu_cons - zTop
# Complete set of Dirichlet bcs
bcs = [bcs_1, bcs_2, bcs_3, bcs_4, bcs_5, bcs_6]
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])
# #
# timeHist3[0] = mu0 # Initialize the chemical potential
# 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):
# # smaller time-step needed for initiation of buckling
# if (t>=750 and t<=1200):
# dt = 1
# dk = Constant(domain, PETSc.ScalarType(dt))
# else:
# dt = 5
# dk = Constant(domain, PETSc.ScalarType(dt))
# increment time
t += dt
# increment counter
ii += 1
# update time variables in time-dependent BCs
mu_cons.value = float(muRamp(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
# 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: 900.0 s of 900 s
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End computation
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Elapsed real time: 0:05:39.156422
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