Automatic Differentiation

Gudrun talks with Asher Zarth. He finished his Master thesis in
the Lattice Boltzmann Research group at the Karlsruhe Institute
for Technology (KIT) in April 2018.


Lattice Boltzmann methods (LBM) are an established method of
computational fluid dynamics. Also, the solution of
temperature-dependent problems - modeled by the Boussinesq
approximation - with LBM has been done for some time. Moreover,
LBM have been used to solve optimization problems, including
parameter identification, shape optimization and topology
optimization. Usual optimization approaches for partial
differential equations are strongly based on using the
corresponding adjoint problem. Especially since this method
provides the sensitivities of quantities in the optimization
process as well. This is very helpful. But it is also very hard
to find the adjoint problem for each new problem. This needs a
lot of experience and deep mathematical understanding.


For that, Asher uses automatic differentiation (AD) instead,
which is very flexible and user friendly. His algorithm combines
an extension of LBM to porous media models as part of the shape
optimization framework. The main idea of that framework is to use
the permeability as a geometric design parameter instead of a
rigid object which changes its shape in the iterative process.
The optimization itself is carried out with line search methods,
whereby the sensitivities are calculated by AD instead of using
the adjoint problem.


The method benefits from a straighforward and extensible
implementation as the use of AD provides a way to obtain accurate
derivatives with little knowledge of the mathematical formulation
of the problem. Furthermore, the simplicity of the AD system
allows optimization to be easily integrated into existing
simulations - for example in the software package OpenLB which
Asher used in his thesis.


One example to test the algorithm is the shape of an object under
Stokes flow such that the drag becomes minimal. It is known that
it looks like an american football ball. The new algorithm
converges fast to that shape.
References

F. Klemens e.a.: CFD- MRI: A Coupled Measurement and
Simulation Approach for Accurate Fluid Flow Characterisation and
Domain Identification. Computers & Fluids 166, 218-224, 2018.

T. Dbouk: A review about the engineering design of optimal
heat transfer systems using topology optimization. Applied
Thermal Engineering 112, pp. 841-854, 2017.

C. Geiger and C. Kanzow. Numerische Verfahren zur Lösung
unrestringierter Optimierungsaufgaben. Springer-Verlag, 2013.

M. J. Krause and V. Heuveline: Parallel fluid flow control
and optimisation with lattice Boltzmann methods and automatic
differentiation. Computers & Fluids 80, pp. 28-36, 2013.

A. Kamikawa and M. Kawahara: Optimal control of thermal fluid
flow using automatic differentiation. Computational Mechanics
43.6, pp. 839-846, 2009.

A. Griewank and A. Walther. Evaluating derivatives:
principles and techniques of algorithmic differentiation. Vol.
105. SIAM, 2008.