 # Non-local effect/force in bilinear form

Hello all. I’m modeling a plate immersed in fluid, and for that I want to add a pressure \Delta p to the bilinear form of the Kirchhoff-Love plate equation. The pressure is non-locally dependent on the displacement \phi of the plate, i. e. pressure at one cell depends on the displacement of other cells in the cross-section, as

\int_\Omega \Delta p~w ~\mathrm{d}x = \int_{\Omega_k} \sum_{j=1}^N A (j, k) ~\phi(x_j)~w ~\mathrm{d}x_k.

The A-matrix comes from the fundamental solution method to the Navier Stokes Equations.

Does someone have an idea on how to implement this non-local dependent pressure? Or even if it is possible to do that with Fenics?

Ideally, it should look something like this

import fenics as fe
mesh = fe.UnitIntervalMesh(10)
v_element = fe.FiniteElement("Lagrange", mesh.ufl_cell(), 2)
v_space = fe.FunctionSpace(mesh, v_element)
phi = fe.TrialFunction(v_space)
w = fe.TestFunction(v_space)
for k in range(1):
for j in range(1):
xj = fe.Point(0.5)
a_pressure = 1e2*phi(xj)*w*dx(1)


where the dx(1) should be a subdomain of the mesh.

Thanks for any input!