Problem with inflation of neo-hookean membrane

variational formulation
1

Hi all,
I have a problem simulating the inflation of a hyper-elastic membrane to increasing pressures.
Since I’m new in Fenics, I started from the demo code (https://fenicsproject.org/docs/dolfin/1.4.0/python/demo/documented/hyperelasticity/python/documentation.html) and adapt it as follows:

from dolfin import *
import numpy as np 
import pdb

# Optimization options for the form compiler
parameters["form_compiler"]["cpp_optimize"] = True
ffc_options = {"optimize": True, \
               "eliminate_zeros": True, \
               "precompute_basis_const": True, \
               "precompute_ip_const": True}

# Create mesh and define function space
mesh = BoxMesh(Point(0.0, 0.0, 0.0),Point(1.0, 0.05, 1.0), 20, 10, 20)
#UnitCubeMesh(24, 16, 16)
V = VectorFunctionSpace(mesh, "Lagrange", 1)

N = FacetNormal(mesh)

# Mark boundary subdomians
left =  CompiledSubDomain("near(x[0], side) && on_boundary", side = 0.0)
right = CompiledSubDomain("near(x[0], side) && on_boundary", side = 1.0)

top =  CompiledSubDomain("near(x[1], side) && on_boundary", side = 0.05)
bottom = CompiledSubDomain("near(x[1], side) && on_boundary", side = 0.0)

front =  CompiledSubDomain("near(x[2], side) && on_boundary", side = 1.0)
back = CompiledSubDomain("near(x[2], side) && on_boundary", side = 0.0)
# Define Dirichlet boundary (x = 0 or x = 1)
c = Expression(("0.0", "0.0", "0.0"),degree=3)


bcb = DirichletBC(V, c, bottom)
bct = DirichletBC(V, c, top)
bcl = DirichletBC(V, c, left)
bcr = DirichletBC(V, c, right)
bcf = DirichletBC(V, c, front)
bcba = DirichletBC(V, c, back)
#bcr = DirichletBC(V, r, right)
bcs = [bcl, bcr,bcf,bcba]

# Define functions
du = TrialFunction(V)            # Incremental displacement
v  = TestFunction(V)             # Test function
u  = Function(V)                 # Displacement from previous iteration
B  = Constant((0.0, 0.0, 0.0))  # Body force per unit volume
T  = Constant((0.0,  0.0, 0.0))  # Traction force on the boundary

# Kinematics
d = u.geometric_dimension()
I = Identity(d)             # Identity tensor

# Elasticity parameters
E, nu = 10.0, 0.3
mu, lmbda = Constant(E/(2*(1 + nu))), Constant(E*nu/((1 + nu)*(1 - 2*nu)))

file = File("results/displacement.pvd");

dt=0.05
p_f=1
p_0=0.01
NN=(p_f-p_0)/dt
pressures=np.linspace(p_0,p_f,int(NN))
for p in pressures:
	print()
	print('pressure: ',p)
    	T  = Constant((0.0, p, 0.0))  # Body force per unit volume
    	F = I + grad(u)             # Deformation gradient
    	C = F.T*F                   # Right Cauchy-Green tensor
    	Finv=inv(F)
    	
    	# Invariants of deformation tensors
    	Ic = tr(C)
    	J  = det(F)

    	n = J*Finv.T*N

    	# Stored strain energy density (compressible neo-Hookean model)
    	psi = (mu/2)*(Ic - 3) - mu*ln(J) + (lmbda/2)*(ln(J))**2
    	pr=Constant((p))
    	# Total potential energy
    	Pi = psi*dx - dot(B, u)*dx - dot(T, u)*ds - dot(u,pr*n)*ds
            #Pi = psi*dx - dot(B, u)*dx - dot(u,pr*n)*ds
    	# Compute first variation of Pi (directional derivative about u in the direction of v)
    	F = derivative(Pi, u, v)

    	# Compute Jacobian of F
    	J = derivative(F, u, du)
    	# Solve variational problem
    	solve(F == 0, u, bcs, J=J,
          	form_compiler_parameters=ffc_options,
          	   solver_parameters={
    						"newton_solver":{\
    							"relaxation_parameter":0.5,
    							"relative_tolerance":1e-6 ,
    							"maximum_iterations":100, 
    						} })
    	#pdb.set_trace()


    	file << (u,p);

The idea of the for loop is that by increasing the pressure progressively, a partial solution can be computed and then used as an initial point for the next step.

I am not sure how to implement the internal pressure forces. I tried two approaches:

  • Changing the value to T = Constant((0.0, p, 0.0)), where p is the pressure value. This method results in applying forces to each cell at the boundary surface:

    image
    image636×554 13.5 KB

  • As pressure can be seen as a force that is applied normal to the surface, I used the following (following implementation of neuman bc from Fenics mechancis based on Nanson’s formula ):
    Pi = psi*dx - dot(B, u)dx - dot(u,prn)*ds. This results in a weirdly deformed membrane

I would appreciate any hints on the implementation of an inflatable membrane.
Sorry if the question/solution is obvious but my understanding of finite element is limited.
Thanks

2

The approach using Nanson formula would be the way to go. Your implementation seems OK to me except for the boundary measure ds. I would expect using ds(1) or something where 1 would be the portion of the boundary on which pressure applies. Here you exert pressure on the whole boundary which certainly is the cause of your “weirdly deformed membrane”

3

Thanks for the quick reply. That works. Here is the code in case someone is having the same issue:

class bottom_b(SubDomain):
    def inside(self,x,on_boundary):
        return on_boundary and near(x[1],0.0) 

boundaries = MeshFunction("size_t", mesh,mesh.topology().dim()-1, 0)
boundaries.set_all(0)
bottom_bound=bottom_b()
bottom_bound.mark(boundaries, 1)
ds_region = Measure('ds')[boundaries]

Pi = psi*dx - dot(B, u)*dx - dot(u,pr*n)*ds_region(1)