So in order to replicate the passive mixer problem by C. S. Andreasen et al. 2009 we need adjoint passive scalar capability.

Their objective function was essentially

min \(\int_{\Gamma_{out}} (\phi - \bar{\phi})^2 d\Gamma\) (with some normalization)

s.t. \(\Delta P \leq \beta \Delta P_{ref}\)

They set \(P_{out} = 0\) implying \(\Delta P = \int_{\Gamma_{in}}p d \Gamma\)

So their workflow involves:

Solve the steady forward problem (velocity and passive scalar)
Solve an adjoint problem (velocity and passive scalar) for the objective
Solve an adjoint problem (only velocity) for the constraint

So for the passive scalar we need

adjoint_scalar_scheme :white_check_mark:
adjoint_scalar_pnpn :white_check_mark:
update adv_adjoint_dealias.f90 :white_check_mark:
update adv_adjoint_no_dealias.f90 :white_check_mark:
adjoint_scalar_convection_source_term :x:

That last one I don't know a good name for... but it's the term arising from linearizing the adjoint passive scalar convective term which then enters the adjoint velocity equation. The term looks like \(\nabla \phi \phi^\dagger\). The reason it's not fully completed with a :white_check_mark: is because I beleive if one is running with dealiased=true then this term should be evaluated on the dealiased mesh, which I currently haven't implemented.

Then for this case specifically we need

enhanced_mixing_objective_function :white_check_mark:
pressure_drop_constraint :x:
adjoint_mixing_scalar_source_term (this enters the adjoint passive scalar equation.) :white_check_mark:
adjoint_enhanced_mixing_scalar_bc (this enters the adjoint passive scalar equation.) :x:
adjoint_pressure_drop_BC This enters the adjoint velocity equation to account for the pressure drop constraint. :x:
multi_objective_problem_type :x: This will be a way of handling the additional constraint and subsequent adjoint solve that comes with it.

The adjoint BCs are, at the time of writing, very poorly handled, so implementing those last two is going to be hard, and I'll save it for another PR.

Often times one can replace an objective on a BC with a volume integral, ie, instead of of evaluated the "mixedness" on the outlet surface, measure the mixedness in a volume downstream close to the outlet. This turns the surface integral (and subsequent adjoint BCs) into a volume integral (and subsequent adjoint source terms) which may be easier to work with.

Hence, for testing, until we fix the handing of the adjoint BCs, I've implemented a adjoint_enhanced_mixing_scalar_source_term instead.

‍I also feel the design needs to enter the passive scalar equation. Probably not in the diffusive term, but I think it should show up in the convective term. ie, not

\((\mathbf{u} \cdot \nabla ) \phi\)

but \(C(\rho) (\mathbf{u} \cdot \nabla ) \phi\) such that \(C(\rho) \in [0,1]\)

I could be wrong... I mean... I suppose if the Brinkman term is doing its job correctly we should have \(\mathbf{u}= \mathbf{0}\) in the solid anyways.. So maybe it's ok.

Progress

Solver

We have basically cloned

scalar_scheme.f90
scalar_pnpn.f90

and adjust the BCs such that ‘'v’ -> 'w'` and changed the order, because in principal when you do an adjoint the order of everything reverses. Which means we'll go:

step fluid forward
step scalar forward
step adjoint scalar backward
step adjoint fluid backward

So this means the adjoint_scalar_convection_source_term is evaluated explicitly but on the correct timestep because we've stepped the scalar backwards before the adjoint fluid. So this is nice.

A little caveat worth mentioning: Due to the one way coupling, I believe one could do this a bit more efficiently in a steady context by converging only the fluid to a steady state and only then start the passive scalar, etc for the adjoint (instead of having the fluid and the scalar solved together at each timestep) However, for unsteady simulations you would want to solve it the way it's being done right now.

Convective term

I've added the adjoint convective term in

sources/adjoint/adv_adjoint_dealias.f90
sources/adjoint/adv_adjoint_no_dealias.f90
sources/adjoint/adjoint_scalar_convection_source_term.f90

Optimization

Just for this case I've implemented

to test out an unconstrained version of the mixing problem.

There's also been a change to objective_function_t such that the inputs are primal and adjoint case_t as opposed to fluid_scheme_t. This means we have access to both the fluid and the scalar.

general todo's:

obviously testing... but we don't really have a good case to test yet When I did my derivation (I'll write it up neatly at some point...)
Double check with Casper what the BC's are, it looks like he keeps drawing parabolic profiles, is there a periodic or symmetric directions?
The JSON stuff isn't finished
The adjoint BCs aren't finished
We need to look into non-dimensionalization etc more carefully, to make sure everything is consistent with the standard passive scalar. (I think it is... but I would like to double check)
boundary conditions (especially user) are not handled well for the adjoint!