Shape design by optimal flow control and reduced basis techniques:applications to bypass configurations in haemodynamics

The purpose of this thesis is to develop numerical methods for optimization, control and shape design in computational fluid dynamics, more precisely in haemodynamics. The application studied is related with the shape optimization of an aorto-coronaric bypass. The optimization process has to keep into account aspects which are very different and sometimes conflicting, for this reason the process has been organized in more levels dealing with a geometrical scale. Moreover we have chosen to use simplified low fidelity models during the application of the complex optimization tools and to verify in feed-back with higher fidelity models the configurations previously obtained. In our case we deal with fluid models based on Stokes and Navier-Stokes equations, for lower and higher fidelity approach respectively, also in the unsteady formulation. At an outer level of the optimization process, efficient numerical methods based on parametrized partial differential equations have been developed to get real-time and accurate information concerning the preliminary configurations, and to get a sensitivity analysis on geometrical quantities of interest and on functionals, related with fluid mechanics quantities. This approach is carried out by reduced basis methods which let us rebuild approximate solutions for parametrized equations by other solutions already computed and stored, allowing huge computational savings. At an inner level we have developed local shape optimization methods by optimal flow control theory based on adjoint approach. Two different approaches have been developed: the former is based on the local displacement of each node on the boundary, the latter is based on small perturbation theory into a reference domain. This approach is more complex but let us avoid mesh reconstruction at each iteration and study the problem into a deeper context from a theoretical point of view and do a generalization dealing with unsteady flows

Parametric free-form shape design with PDE models and reduced basis method

We present a coupling of the reduced basis methods and free-form deformations for shape optimization and design of systems modelled by elliptic PDEs. The free-form deformations give a parameterization of the shape that is independent of the mesh, the initial geometry, and the underlying PDE model. The resulting parametric PDEs are solved by reduced basis methods. An important role in our implementation is played by the recently proposed empirical interpolation method, which allows approximating the non-affinely parameterized deformations with affinely parameterized ones. These ingredients together give rise to an efficient online computational procedure for a repeated evaluation design environment like the one for shape optimization. The proposed approach is demonstrated on an airfoil inverse design problem. © 2010 Elsevier B.V

Free Form Deformation Techniques Applied to 3D Shape Optimization Problems

The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation

Reduced basis methods for Stokes equations in domains with non-affine parameter dependence

In this paper we deal with reduced basis techniques applied to Stokes equations. We consider domains with different shape, parametrized by affine and non-affine maps with respect to a reference domain. The proposed method is ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. An "empirical”, stable and inexpensive interpolation procedure has permitted to replace non-affine coefficient functions with an expansion which leads to a computational decomposition between the off-line (parameter independent) stage for reduced basis generation and the on-line (parameter dependent) approximation stage based on Galerkin projection, used to find a new solution for a new set of parameters by a combination of previously computed stored solutions. As in the affine case this computational decomposition leads us to preserve reduced basis properties: rapid and accurate convergence and computational economies. The applications and results are based on parametrized geometries describing domains with curved walls, for example a stenosed channel and a bypass configuration. This method is well suited to treat also problems in fixed domain with non-affine parameters dependence expressing varying physical coefficient

Combined parameter and model reduction of cardiovascular problems by means of active subspaces and POD-Galerkin methods

In this chapter we introduce a combined parameter and model reduction methodology and present its application to the efficient numerical estimation of a pressure drop in a set of deformed carotids. The aim is to simulate a wide range of possible occlusions after the bifurcation of the carotid. A parametric description of the admissible deformations, based on radial basis functions interpolation, is introduced. Since the parameter space may be very large, the first step in the combined reduction technique is to look for active subspaces in order to reduce the parameter space dimension. Then, we rely on model order reduction methods over the lower dimensional parameter subspace, based on a POD-Galerkin approach, to further reduce the required computational effort and enhance computational efficiency

Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics

The objective of this thesis is to develop reduced models for the numerical solution of optimal control, shape optimization and inverse problems. In all these cases suitable functionals of state variables have to be minimized. State variables are solutions of a partial differential equation (PDE), representing a constraint for the minimization problem. The solution of these problems induce large computational costs due to the numerical discretization of PDEs and to iterative procedures usually required by numerical optimization (many-query context). In order to reduce the computational complexity, we take advantage of the reduced basis (RB) approximation for parametrized PDEs, once the state problem has been reformulated in parametrized form. This method enables a rapid and reliable approximation of parametrized PDEs by constructing low-dimensional, problem-specific approximation spaces. In case of PDEs defined over domains of variable shapes (e.g. in shape optimization problems) we need to introduce suitable, low-dimensional shape parametrization techniques in order to tackle the geometrical complexity. Free-Form Deformations and Radial-Basis Functions techniques have been analyzed and successfully applied with this aim. We analyze the reduced framework built by coupling these tools and apply it to the solution of optimal control and shape optimization problems. Robust optimization problems under uncertain conditions are also taken into consideration. Moreover, both deterministic and Bayesian frameworks are set in order to tackle inverse identification problems. As state equations, we consider steady viscous flow problems described by Stokes or Navier-Stokes equations, for which we provide a detailed analysis and construction of RB approximation and a posteriori error estimation. Several numerical test cases are also illustrated to show efficacy and reliability of RB approximations. We exploit this general reduced framework to solve some optimization and inverse problems arising in haemodynamics. More specifically, we focus on the optimal design of cardiovascular prostheses, such as bypass grafts, and on inverse identification of pathological conditions or flow/shape features in realistic parametrized geometries, such as carotid artery bifurcations

AUTOMATIC OPTIMIZATION METHODS FOR PATIENT-SPECIFIC TISSUE-ENGINEERED VASCULAR GRAFTS

Surgical intervention is sometimes necessary in cases of Coarctation of the Aorta (CoA). The post-repair geometry of the aorta can result in sub-optimal hemodynamics and can have long-term health impacts. Patient-specific designs for tissue-engineered vascular grafts (TEVGs) allow greater control over post-repair geometry. This thesis proposes a method for automatically optimizing patient-specific TEVGs using computational fluid dynamics (CFD) simulations and the ANSYS Fluent adjoint solver. Our method decreases power loss in the graft by 25-60% compared to the native geometry. As patient-specific graft design can be challenging due to incomplete or uncertain flow and geometry data, this thesis also quantifies the robustness of the optimal designs with respect to CFD boundary conditions derived from imaging data. We show that using velocity conditions that deviate by more than 20% of the measured peak systolic velocity, our method produces grafts with deviations on the order of 5% in predicted power loss performance. Lastly, as one way to accelerate the optimization process, we demonstrate and compare how some established machine learning models (K Nearest Neighbors and Kernel Ridge Regression) predict reasonable starting points for an optimizer on a 2D bifurcated pipe dataset

Free Form Deformation Techniques for 3D Shape Optimization Problems

The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we choose to use the Free Form Deformation parametrization, a recent technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the technique by developing a link among different specialized softwares, in order to establish a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the problem. In particular, we have studied a bulb and a rudder of a race sailing boat as model problems

Reduced order parameterized viscous optimal flow control problems and applications in coronary artery bypass grafts with patient-specific geometrical reconstruction and data assimilation

Coronary artery bypass graft surgery is an invasive procedure performed to circumvent partial or complete blood flow blockage in coronary artery disease (CAD). In this thesis, we will construct a numerical framework combining parametrized optimal flow control and reduced order methods and will apply to real-life clinical case of triple coronary artery bypass grafts surgery. In this mathematical framework, we will propose patient-specific physiological data assimilation in the optimal flow control part, with the aim to minimize the discrepancies between the patient-specific physiological data and the computational hemodynamics. The optimal flow control paradigm proves to be a handy tool for the purpose and is being commonly used in the scientific community. However, the discrepancies between clinical measurements and computational hemodynamics modeling are usually due to unrealistic quantification of hard-to-quantify outflow conditions and computational inefficiency. In this work, we will utilize the unknown control in the optimal flow control pipeline to automatically quantify the boundary flux, specifically the outflux, required to minimize the data misfit, subject to different parametrized scenarios. Furthermore, the challenge of attaining reliable solutions in a time-efficient manner for such many-query parameter dependent problems will be addressed by reduced order methods