How does the reisz representation of shape derivatives work?


I am trying to understand how shape optimization works but am getting confused about how the Reisz representation of the shape derivative is used. For some shape functional
J = ...
I understand UFL can compute the shape derivative as

import dolfin as dfn
import ufl

mesh = dfn.UnitSquareMesh(10, 10)
W = dfn.VectorFunctionSpace(mesh, 'CG1')

x = ufl.MeshCoordinate(mesh)
dJ = dfn.derivative(J, x)
gradJ = dfn.Function(W)

where dJ is the shape derivative.

From reading some past papers ( my understanding is that dJ is a linear functional that needs a Reisz representation (stored in gradJ here) in the mesh’s CG1 space so that you can then use that representation to deform the mesh.

My understanding is that the straightforward one is simply using
dfn.assemble(dJ, tensor=gradJ.vector())
which I guess corresponds to the Reisz representation with respect to the L2 inner product.

In this paper they use the Reisz representation from the laplacian operator by

trial = dfn.TrialFunction(W)
test = dfn.TestFunction(W)
A_reisz = dfn.assemble(ufl.inner(ufl.grad(trial), ufl.grad(test))) * dx
dfn.solve(A_reisz, gradJ, dJ)

What I’m confused about is whether gradJ.vector() is then the vector you would output to an optimization algorithm? Specifically in optimization algorithms for a given shape x you would need a function like the below

def obj_and_grad(x):
    dfn.ALE.move(mesh, x)

    obj = dfn.assemble(J)

    ## use one of the above methods to compute `gradJ`
    grad = gradJ.vector()
    return (obj, grad)

which returns the objective function value and gradient. When using the laplacian type riesz representation, I’m confused because I think gradJ in this case doesn’t correspond to the correct gradient of obj_and_grad because of the different inner product in the Riesz representation, i.e:

# `obj_and_grad(x+dx)[0] - obj_and_grad(x)[0]`
# would not converge to
# `, dx)`
# for small dx if `gradJ` is computed from a Laplacian type Riesz representation?

It seems like this is what was done in

Am I missing something or misunderstanding how the shape derivative works?

Thanks very much for any help!