DOLFIN Function for L2 average on some elements - interpolate a defined function

Question

I want to implement Perzyna viscoplastic model on FEniCS for a plane stress problem. I need to find average stress (which is a function of both elastic and viscoplastic strains) on each element to check for yield criterion. I assumed that viscoplastic strains are constant on elements (DG 0 space). Is there any built in function to find average of a defined function (for e.g. sigma11 in the following code) not a given expression on another space or to interpolate a defined function?

def eps(v):
    return sym(grad(v))
def sigma11(u,e_vp11,e_vp22):
    return (2*mu+lmbda)*(eps(u)[0,0]-e_vp11)+lmbda*(eps(u)[1,1]-e_vp22)
def sigma22(u,e_vp11,e_vp22):
    return lmbda*(eps(u)[0,0]-e_vp11)+(2*mu+lmbda)*(eps(u)[1,1]-e_vp22)
def sigma12(u,e_vp12):
    return 2*mu*(eps(u)[0,1]-e_vp12)
V_u = VectorFunctionSpace(mesh, 'Lagrange', 1)
u = Function(V_u)
v = TestFunction(V_u)
V_vp = FunctionSpace(mesh, 'DG', 0)
e_vp11 = Function(V_vp)
e_vp22 = Function(V_vp)
e_vp12 = Function(V_vp)
f=Constant((0.0,0.0))
# Linear and bilinear forms
a= (sigma11(u,e_vp11,e_vp22)*eps(v)[0,0]+ sigma22(u,e_vp11,e_vp22)*eps(v)[1,1]+ sigma12(u,e_vp12)*eps(v)[0,1])*dx
L=dot(f,v)*dx
# bcs ...
solve(a == L, u, bcs)
Vol = assemble(Constant(1.0)*dx(), mesh=mesh)
s11_avg = Function(V_vp)
s11_avg.vector()[:] = assemble(sigma11(u,e_vp11,e_vp22)*dx())/Vol
s22_avg = Function(V_vp)
s22_avg.vector()[:] = assemble(sigma22(u,e_vp11,e_vp22)*dx())/Vol 
s12_avg = Function(V_vp)
s12_avg.vector()[:] = assemble(sigma12(u,e_vp11,e_vp22)*dx())/Vol
# Loop over elements
for k in range(mesh.num_cells()):
# calculate yield function over each element
    f = s11_avg.vector()[k]*s11_avg.vector()[k] +s22_avg.vector()[k]*s22_avg.vector()[k]+s12_avg.vector()[k]*s12_avg.vector()[k]
    if f > 0.0 :
        e_vp11.vector().array()[k] = Gamma*dt*((sign(f)/Sigma_0)**N)*(0.5*f)*s11_avg.vector()[k]

Comments

You probably need to make your question a bit more clear. From what I guess you are asking, it sounds like a typical job for a projection into a DG0 space. This can be done by either using the project function or by solving auxiliary problems (which project also does internally).

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Thanks for your reply, as you can see in the code, I want to find s11_avg (which is a function of DG 0 space) and it is computed based on the defined function sigma11(u,e_vp11,e_vp22), which is a function of both CG 1 and DG 0 spaces. That's why I defined Vol = assemble(Constant(1.0)*dx(), mesh=mesh) to compute integration of test function and then compute the average. I would appreciate if you explain how to solve the auxiliary problem or use project function to find this average.