Piezoelectric PDEs

Linear Algebra
1

hello everyone please I have a Problem I’m new to fenics programming and I’d like to solve the linear equation of a piezoelectric

Cture
Cture661×533 24.8 KB

2

Hi,
transform it into weak form and use a mixed function space for the displacement u and electric potential. You can start by the quasistatic version and then adapt any elastodynamic demo to this case quite easily…

1 Like
3

Hi Jeremi,
thank you for your answer.My difficulty resides exactly in the quasistatic version case because for the elastodynamic case I already have an idea. Please can you give me if it’s possible a document or a type that illustrates the quasistatic version case?

4

well then if you are OK for the elastodynamic case, what is your problem ? If you don’t post your code with a specific issue, people will not be likely to help you

5

Hi,
please, I still have a problem, I have changed my expression as you asked me. My difficulty lies in the fact that I don’t know where to start in order to implement this. I would like to know if there are any examples or tipps that you can give me

weakform
weakform899×233 23.5 KB

6

Hi,
I assume that B is the symetrized gradient and Bbar the gradient operators. Here is some partial code for the quasistatic part of the bilinear weak form:

Ue = VectorElement("CG", mesh.ufl_cell(), 1)
Ve = FiniteElement("CG", mesh.ufl_cell(), 1)
W = FunctionSpace(mesh, MixedElement([Ue, Ve]))
u_, v_ = TestFunctions(W)
du, dv = TrialFunctions(W)
def sigma(u):
    eps = sym(grad(u))
    return lmbda*tr(eps)*Identity(u.geometric_dimension()) + 2*mu*esp
a = inner(sigma(du), eps(u_))*dx + inner(sigma(du), dot(e, grad(v_)))*dx \
    + inner(sigma(u_), dot(e, grad(dv)))*dx - dot(grad(dv), dot(eps_S, grad(v_)))*dx

I assumed isotropic elasticity for the mechanical part. You will also need to define the third-rank tensor e for coupling and the second-rank tensor eps_S for the electrical part.
For the extension to dynamics, you just need to adapt elastodynamics demos e.g. https://fenicsproject.org/docs/dolfin/latest/python/demos/elastodynamics/demo_elastodynamics.py.html