Download reduced order model based multiobjective optimal control of fluids

XXIV ICTAM, 21-26 August 2016, Montreal, Canada
REDUCED ORDER MODEL BASED MULTIOBJECTIVE OPTIMAL CONTROL OF FLUIDS
1
Sebastian Peitz∗1 , Sina Ober-Blöbaum2 , and Michael Dellnitz1
Department of Mathematics, University of Paderborn, Warburger Str. 100, D-33098 Paderborn
2
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
Summary In this article, we show how to combine reduced order modeling and multiobjective optimal control techniques in order to efficiently solve multiobjective optimal control problems constrained by PDEs. Using Galerkin projection and Proper Orthogonal Decomposition (POD), the underlying PDE is replaced by a system of ordinary differential equations, thereby drastically reducing the computational
effort. We develop a gradient based algorithm by using an adjoint approach in combination with a scalarization method and the numerical
results are illustrated with the example of the two-dimensional incompressible flow around a cylinder.
INTRODUCTION
In a wide range of applications it is desirable to optimally control a dynamical system with respect to concurrent, potentially competing goals. This gives rise to a multiobjective optimal control problem where, instead of computing a single
optimal solution, the set of optimal compromises, the so-called Pareto set, has to be approximated. When the problem under
consideration is described by a partial differential equation (PDE), as is the case for fluid flow, the computational cost rapidly
increases and makes its direct treatment infeasible. Reduced order modeling (ROM, see e.g. [1]) is a very popular method to
reduce the computational cost, in particular in a multi query context such as uncertainty quantification, parameter estimation
or optimization. In the following, we present a method for the solution of multiobjective optimal flow control problems. The
velocity field U (x, t) is described by the 2D incompressible Navier Stokes equations within the domain Ω (Fig. 2). Dirichlet
boundary conditions (BC) with a constant horizontal velocity are imposed on the inflow as well as on the upper and lower
boundary, whereas a standard no-shear Neuman BC is imposed on the outflow boundary. On the cylinder we prescribe a time
dependent Dirichlet BC such that it performs a rotation with the angular velocity γ(t). The system is controlled by γ(t) and
the aim is to compute the Pareto set of optimal compromises for the conflicting objectives minimization of flow field fluctuaRT
tions u(x, t) = U (x, t) − hU (x)i around the mean flow field hU (x)i = T1 0 U (x, t) dt and control cost for the flow around
a cylinder at Re = 200:
R te
ku(·, t)k2L2 dt
b γ) = min
t0
.
(1)
min J(U,
U,γ
U,γ
kγk2L2
A more detailed description and derivation of the algorithms can be found in [2].
REDUCED ORDER MODELING
Reduced order models obtained by Galerkin projection and POD [3] have been very successful in the past for the analysis
of coherent structures in fluid flow. In the context of flow control, ROMs have been investigated by many researchers (see [2]
for an overview). Here, we follow the procedure in [4] to derive a ROM for a system with non-homogeneous, time-dependent
BC. The flow field is decomposed into the mean value hU (x)i, a so-called control function Uc (x) and a fluctuating field
u(x, t) in such a manner that u(x, t) possesses homogeneous BC. The Galerkin Ansatz U (x, t) ≈ hU (x)i + γ(t)Uc (x) +
Pl
i=1 αi (t)Ψi (x) is inserted into the weak formulation of the Navier-Stokes equations, resulting in the following optimality
system (see [2] for details):
α̇ = A + Bα + C(α) + Dv + (E + Fα)γ + Gγ 2 ,
T
∂J
∂C(α)
λ̇ = −
− B+
+ Fγ
λ,
∂α
∂α
γ̇ = v,
µ̇ = −
∂J
T
− (E + Fα + 2Gγ) λ,
∂γ
∂J
+ DT λ + µ,
∂v
α(0) = α0 , λ(T ) = 0, µ(0) = 0, µ(T ) = 0,
0=
(2)
with ordinary differential equations for the states α and γ as well as for the adjoint variables λ and µ. Note that by introducing
a new control function v = dγ/dt (v is the time derivative of the angular velocity), the actual control γ is now formally a
state variable. The optimality condition (2) is only satisfied by an optimal control function, otherwise it provides a descent
direction for the optimization algorithm.
0.8
J2
.(t)
40
20
J1
J1
J1
J1
J1
J1
0.4
=
:
:
:
:
:
74:57;
75; J2
80; J2
85; J2
90; J2
95; J2
J2 = 52:28
= 33:59
= 7:68
= 1:84
= 0:17
= 0:0061
0
0
75
80
85
90
95
-0.4
0
2.5
5
J1
(a)
7.5
10
t
(b)
(c)
Figure 1: (a) Scalarization of (1) using target points T . (b) Pareto set of optimal compromises between fluctuation energy (J1 )
and control cost (J2 ). (c) Comparison of different Pareto optimal controls.
(a)
(b)
(c)
Figure 2: (a) Solution for γ(t) = 0: von Kármán vortex street. (b) Solution corresponding to maximal stabilization after
T = 10 seconds. (c) Solution corresponding to maximal stabilization after T = 60 seconds using a periodic control [5].
RESULTS
Problem (1) can now be expressed in terms of the reduced model (J1 =
R te
R te R
t0
Ω
k
Pl
i=1
αi (t)ψi (x)k22 dx dt and J2 =
l t0 γ 2 (t) dt, where l is the dimension of the ROM) and transformed into a scalar optimization problem with the objective to
minimize the euclidean distance between a feasible point (J1 , J2 ) and an infeasible target point (T1 , T2 ) in image space (cf.
Fig. 1a) such that the optimal solution (J1∗ , J2∗ ) is on the boundary of the feasible set in image space and hence Pareto optimal
(i.e. there does not exist a feasible point J with J1 ≤ J1∗ and J2 ≤ J2∗ ).
The resulting Pareto set and a selection of Pareto optimal controls are depicted in Fig. 1 (b, c), indicating that a tradeoff
between flow stabilization and control cost is almost exclusively achieved by varying the amplitude whereas the oscillatory
behavior is changed only marginally. The reason for this is very intuitive: the control has to counteract the natural oscillations
of the von Kármán vortex street (cf. Fig. 2a). The uncontrolled solution and the solution corresponding to a strong stabilization
are shown in Fig. 2. Note that due to the relatively short time interval, the effect is restricted to the immediate wake region
behind the cylinder. For longer time intervals, the effect becomes more evident [5].
For future work, we intend to investigate error estimates for multiobjective optimal control problems constrained by PDEs.
Moreover, we will extend our method to systems of coupled nonlinear PDEs, for instance the coupling of the Navier-Stokes
and scalar transport equations, e.g. for the temperature, in order to solve multiobjective optimal control problems for HVAC
applications.
References
[1] Kunisch, K., Volkwein, S. Galerkin Proper Orthogonal Decomposition Methods for a General Equation in Fluid Dynamics. SIAM Journal on Numerical
Analysis 40(2):492-515. 2002.
[2] Peitz S., Ober-Blöbaum, S., Dellnitz, M. Multiobjective Optimal Control Methods for Fluid Flow Using Model Order Reduction. arXiv:1510.05819
[3] Holmes, P., Lumley, J. L., Berkooz, G. Turbulence, coherent structures, dynamical systems and symmetry. Cambridge University Press. 1998.
[4] Bergmann, M., Cordier, L., Brancher, J. P. Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model.
Physics of Fluids 17(097101):1-21. 2005.
[5] Peitz, S., Dellnitz, M. Multiobjective Optimization of the Flow Around a Cylinder Using Model Order Reduction. PAMM15:613-614. 2015.
∗ Corresponding
author. Email: [email protected]