This document discusses optimization problems on dynamical domains with non-matching meshes. It motivates the problem using tidal turbines and general PDE-constrained optimization. It presents an approach using overlapping meshes with Nitsche's method for interface conditions. It discusses the discretization and derivation of adjoint equations to efficiently compute gradients for optimization. Numerical results are shown for an optimal Poisson problem and investigating the gradient of a goal functional for a heat equation on a moving domain. Further work is outlined on extending software to use overlapping meshes for optimization of tidal turbine farm layouts.

1. Optimization problems on dynamical
domains with non-matching meshes
Jørgen S. Dokken, Simon W. Funke, August Johansson
Simula Research Laboratory
October 27, 2016

2. Motivation - Rotating tidal-turbines
Figure: The Tidal-stream turbine named Seaflow.

3. Motivation - General optimization problem
Optimization problems with PDE’s as constraint
min
u,m
J(u, m), (1)
subject to F(u, m) = 0 (2)
Costly to solve PDE
Gradient Based Optimization
One functional of interest, large number of design parameters
Adjoint Equations

4. Representation of Dynamical Domains
Figure: Turbine rotating with
prescribed velocity with smoothing
of mesh movements
Figure: Turbine rotating with
prescribed velocity with overlapping
meshes

5. Overlapping meshes
Our approach
Nitsche method for interface
conditions on the boundary
of meshes meshes
Challenge
Detect where the two
meshes intersect
Figure: Illustration of overlapping
meshes.

6. Coupling between overlapping domains and meshes
Figure: How to create a computational domain for two meshes.

7. Heat-equation with Backward Euler in time
Discretization in time and
interface-conditions
un − un−1
∆t
− 2
un
= f(tn
) in Ω0 ∪ Ω1
[[un
]] = 0
[[n · un
]] = 0
Notation
un
= (un
Ω0
, un
Ω1
)
[[v]] = v0|Γ − v1|Γ
v =
v0 + v1
2
Figure: Domain of
interest.

8. Variational form of the problem using Nitsches
method
Modify the standard variational form by
integration of the jump-condition for u on Γ.
For symmetry and coercivity two extra terms are also added.
ah(Un
, v) = L(v), ∀v ∈ V h
,
ah(Un
, v) : =
2
i=1
( Un
i , vi)Ωi +
1
∆t
(Un
i , vi)Ωi
− ([[Un
]], n · vh )Γ − ([[vh
]], n · Un
)Γ
+ (γh−1
[[Un
]], [[v]])Γ,
L(v) =
2
i=1
(f, vi)Ωi +
1
∆t
(Un−1
i , vi)Ωi .

9. Adjoint equations
Optimization problem
min
u,m
J(u, m),
subject to F(u, m) = 0
Reduced Functional
ˆJ(m) = J(u(m), m)
Gradient of Functional
dJ
dm
=
∂J
∂u
du
dm
+
∂J
∂m
One goal-functional J.
Large number of
design-parameters m.
Evaluation in the space of
the design parameter is
costly.
Gradient-based optimization

10. The Adjoint equation
Gradient of Functional
dJ
dm
=
∂J
∂u
du
dm
+
∂J
∂m
Rewriting the gradient of the functional
dJ
dm
∗
=
∂F
∂m
∗
λ +
∂J
∂m
∗
The adjoint equation
∂F
∂u
∗
λ =
∂J
∂u
∗
(3)
Avoid computation of Jacobian by using the adjoint method
Solving Equation (3) require specification of functional, not of
design parameter.

11. FEniCS and dolfin-adjoint
Software for solving PDEs
MultiMesh - Implementation of overlapping meshes
Automatic derivation of the adjoint equations

12. Optimal Poisson problem
Finding the best heating/cooling f of a cook-top to get a desired
temperature profile d.
min
u,f
J(u(f), f)) = min
u,f

1
2
Ω
(u − d)2
dΩ +
α
2
Ω
f2
dΩ


subject to
− 2u = f in Ω
u = 0 on ∂Ω
Given a desired temperature profile
d =
1
2π2
sin(πx) sin(πy),
we have the analytical solution to the optimization problem
f =
sin(πx) sin(πy)
1 + 4απ4

13. Meshes used for the optimal Poisson problem
Figure: Mesh used in dolfin for
Optimization with a single mesh.
Figure: MultiMesh used in dolfin for
Optimization.

14. Results for Poisson-optimization
Figure: fanalytical projected into a the
same function-space as the
approximated solution (Piecewise
constant functions).
Figure: The resulting source-term f
for the optimization problem with
MultiMesh where f is approximated
by piecewise constant functions.

15. Results for Poisson-optimization
Computing the L2-error of the computed f and u.
Figure: The error in the MultiMesh-problem is of the same size as the
single mesh problem.

16. Heat equation with moving domain
Figure: Turbine rotating with prescribed velocity.

17. Investigation of the gradient of a goal functional
min
u0
ˆJ(u0
) =
Ω
un
(u0
)
2
dΩ, u0
intital condtion,
un
: solution of the heat equation after n-timesteps.
Figure: dJ
du0 at time T = 1.5.

18. Further Work
OptCutCell - Extending dolfin-adjoint to use overlapping
domains and extending MultiMesh-user interface.
Open-source optimization software for tidal turbine farms.
Optimal position of a set of dynamic tidal-stream turbines