The control of fluid flows for the purpose of achieving some desired objective is crucial to many applications. The reduction of drag forces acting on airfoils [159] or the vorticity minimization in cardiovascular prostheses [254, 273] are just two examples from very different fields, which can be recast in the same control framework. In particular, design problems are of particular interest whenever the performances of a system can be greatly enhanced by acting on its shape, such as in aircraft, ship or automotive design, as well as in biomedical engineering.
In fluid mechanics, cost functionals are expressed in terms of flow variables (such as velocity, pressure, temperature, etc.), while constraints are represented by the PDE (advection-diffusion, potential, Euler, Stokes, Navier-Stokes or Boussinesq equations, etc.) describing the flow, and by topological constraints on the shape of the domain, whenever necessary. In the applications of our interest we will focus on steady, incompressible viscous fluid flows, governed by either Stokes or Navier-Stokes equations for laminar Newtonian flows. These equations – extensively analyzed in Sect. 3.1 – are the steady version of the classical mathematical description of Newton’s law of motion and conservation of mass for an incompressible viscous flow, and read as follows:
−ν∆v + δ(v · ∇)v + ∇p = f + uΩ in Ω ∇ · v = 0 in Ω v = uc on Γc v = vin on Γin v = 0 on Γw −pn + ν∂v ∂n = g on Γout. (1.17)
Here δ = 1 in the case of Navier-Stokes (resp. δ = 0 in the case of Stokes) equations, v is the velocity vector, p is the pressure, ν > 0 is the kinematic viscosity – defined as the ratio between the dynamic viscosity µ > 0 and the density ρ > 0 of the fluid – and Ω ⊂Rd for d = 2, 3 is
the domain occupied by the fluid. The Dirichlet portion ΓD of the boundary is further divided
into an inlet Γinwhere we prescribe an inflow condition, a wall Γwwhere we prescribe a no-slip
boundary condition and a portion Γc where we prescribe a boundary control uΓc. Furthermore,
we prescribe the value of the normal stressT(v, p)n at the outlet Γout6= ∅, whereT(v, p) is the
stress tensor defined by
T(v, p) = −pI + 2νσ(v), σ(v) =1
2(∇v + ∇
T
v);
σ(v) is called strain rate tensor. The control u can be applied at the boundary, i.e. u = uc ∈
Uc⊆ (H1/2(Γc))d, or can be distributed in the domain, i.e. u = uΩ∈ UΩ⊂ (L2(Ω))d, as a source
term. Typical problems are related with the reduction of the vorticity ∇ × v, e.g. in turbulent flows, for which a model problem can be stated as follows:
ˆ
u = arg min
u∈Uad
J (u) s.t. (v, p) = (v, p)(u) is the solution of (1.17) associated with u, (1.18)
where J (u) = ν 2 Z Ω |∇ × v(u)|2dΩ +ε 2 Z Ω |u|2dΩ
is the cost functional related to a distributed observation of the vorticity and a distributed control
u = uΩ. The second integral appearing in the cost functional plays the role of a regularization
In the case of a boundary (Dirichlet) control u = uc, J (u) takes the following form: J (u) = ν 2 Z Ω |∇ × v(u)|2dΩ +ε 2 Z Γc |u|2dΩ.
Another cost functional which can be used is the following energy functional
J (u) = ν 2 Z Ω |∇v(u)|2dΩ +ε 2 Z Ω |u|2dΩ,
if we are interested in the minimization of the viscous energy dissipation ν|∇v|2 (referred also as
strain rate) for flow regularization. Other examples will be discussed for instance in Sect. 5.2.4.
In the case of optimal design problems, we act instead on a portion Γc ⊆ ∂Ωo of the boundary
(also denoted as free-boundary), in order to minimize a shape functional depending on velocity v and/or pressure p, which – following the same notation as before – solve:
−ν∆v + δ(v · ∇)v + ∇p = f in Ωo ∇ · v = 0 in Ωo v = vin on Γin v = 0 on Γc∪ Γw −pn + ν∂v ∂n = g on Γout. (1.19)
Thus, a model problem can be stated as follows: ˆ
Ωo= arg min
Ωo∈Oad
J (Ωo) s.t. (v, p) = (v, p)(Ωo) is the solution of (1.19) on Ωo, (1.20)
where the set of admissible shapes can be defined as Oad= {Ω ∈ O : |Ω| ≤ V, Γin∪ Γw∪ Γout is given } ,
where Ωois the fluid domain and in this case O is the set of domains Ωo⊂Rd piecewise C2with
convex corners. When dealing with internal flows, ∂Ωorepresents the external wall of the branch
Ωo whereas in the case of external flows the target is the shape of a body B embedded into a
fictitious fluid volume D, so that Ωo= D \ B and Γc⊆ ∂B.
Depending on the objective, several shape functionals can be defined; omitting for the sake of simplicity the regularization term, in case of energy or vorticity minimization, we can introduce the following functionals:
J (Ω) = ν 2 Z Ω |∇ × v(Ω)|2dΩ, J (Ω) = ν 2 Z Ω |∇v(Ω)|2dΩ,
respectively. Another very well studied problem deals with the minimization of drag forces (or resistances) on a body B in relative motion within a fluid, for which the usual functional to be minimized is given by the drag acting on the body:
J (Ω) = − Z
∂B
(T(v(Ω), p(Ω))n) · ˆv∞dΓ,
where v∞= U ˆv∞ is the horizontal component of the fluid flow acting on the body and ˆv∞ a
unit vector in the horizontal direction. An equivalent choice is the adimensional drag coefficient:
J (Ω) = − 1
q∞d
Z
∂B
where d is a characteristic length of the body and q∞= 12ρU
2. Also concerning the case of shape
optimization problems, several other examples will be introduced in Sect. 5.2.4.
Indeed, in the context of optimal flow control problems, the challenges stem from the nature of the state system: the nonlinearity of Navier-Stokes equations leads to nonconvex problems which cause major difficulties concerning existence and uniqueness analysis. The case of optimal design problems is even more difficult, due to further critical aspects related with shape regularity and deformations, and to continuity of the nonlinear state solution with respect to shape variations. Furthermore, on the computational side, applications within these contexts lead to large-scale nonlinear optimization problems, which are among the most challenging optimization problems in computational science and engineering.
Early contributions on optimal control problems associated with Navier-Stokes equations (both in the stationary and in the time-dependent case) date back to 90’s and are addressed by Abergel and Temam [2], Casas [1], a series of articles by Gunzburger, Hou, Svobodny ( [130, 131], e.g.) and Kim [132, 167], Ghattas [112], Heinkenschloss [138], Berggren [30, 31, 32], Ito and Ravindran [81, 266]. These authors are mainly concerned with important questions such as the formulation of feasible problems, existence of optimal controls, first-order necessary conditions for optimality, and discretization issues. A complete review of challenges and features related to optimal flow control problems can be found in the monograph by Gunzburger [128]. More recent contributions – mainly concerning the numerical approximation, and among a very long list including many authors – have been presented by Hintermuller, Kunisch, Volkwein [142] and Vexler [174], Agoshkov, Quarteroni and Rozza [5, 6] and Dedè [71].
Concerning optimal design problems related with fluid flows, the first theoretical contributions date back to the 70’s and are due to Pironneau [246, 247] and Glowinski [116], and subsequently to many other authors such as Simon [301, 27], Zolésio and coauthors [37, 89], Gunzburger [133] and Kim [166], and more recently by Gao, Ma and Zhuang [108, 107]. These works are mainly concerned with the formulation of feasible problems, existence of optimal shapes, regularity and differentiability of state solution and cost functionals with respect to shape and optimality conditions. Starting from a pioneering work by Bourot [42], a long list of authors have proposed many tools for tackling the numerical difficulties arising in shape optimization for fluid dynamics problems. We just mention Jameson [158, 159, 160], whose role has been fundamental for the application of shape optimization techniques to aerodynamics problems such has the optimal design of airfoils. An exhaustive survey on both theoretical and numerical aspects is provided e.g. in the book by Pironneau and Mohammadi [214] and other reviews by these authors [249, 215].
On the other hand, the first attempts to solve flow control problems by reduced order models date back to last decade and are given to Ito [153, 154] and Ravindran [267, 265]; subsequent developments in flow control problems are given by Quarteroni, Rozza and Quaini [257], Tonn and Urban [309], Dedè [73, 74]. Concerning reduced order models applied to shape optimization problems – at the best of our knowledge – the only contributions are due to Antil, Heinkenschloss and Hoppe [13, 12], beyond the works by Rozza [276, 273] and Lassila [185], which constitute the seeds from which applications and methods presented in this thesis actually stem.
Some general remarks about the use of reduced models in flow control problems have been addressed by Gunzburger in his monograph [128]. Nevertheless, whether several strategies for geometrical reduction have been fully exploited in optimal design problems, the introduction of reduced-order models is somehow very recent and largely still to be explored. In our opinion, this is due basically to (i) the lack of adequate methods for the certification of reduced-order solutions up to the last decade, and (ii) to the lack of advanced parametrization techniques within a parametrized PDEs context. We underline that efficient and rigorous a posteriori error bounds are still missing for a large class of optimal control problems: the main difficulty stands in the construction of rigorous error bounds not just for reduced state variables, but also for the reduced cost functional and the reduced control – an aspect which is not taken into account in this thesis, but represents an interesting topic for forthcoming research.
Indeed, our ultimate goal is the investigation of a reduced framework based on the reduced basis method and suitable shape parametrization techniques for optimal design problems arising in fluid dynamics. In particular, our driving applications deal with haemodynamics, such as design of cardiovascular prostheses and assessment of pathological risks through inverse identification of flow and shape features. To do this, we will face both optimal control and shape optimization, as well as with more general inverse identification problems. Clinical motivations, modelling features and detailed results will be addressed in Part III.
Nevertheless, techniques and methods developed and discussed in this thesis prove to be useful also in facing other optimization problems arising in computational fluid dynamics and, more generally, in science and engineering contexts modelled by PDEs.