Model Order Reduction for Shape Optimization of Cardiovascular Geometries

The study of shape optimization problems governed by partial differential equations (PDEs) may be of interest in haemodynamics [2]: the minimization of vorticity in a cardio-vascular bypass anastomosis [3] or the sensitivity of optimal shape of a blood vessel with respect to variations of geometrical parameters representing stenotic thickenings are just two examples of problems which can be solved using shape optimization tools coupled with suitable strategies for model order reduction. Involving the evaluation of outputs expressed as functionals of the field variables, optimal control or shape optimization problems usually imply big computational efforts; for this reason looking for computational efficiency in numerical methods and algorithms is mandatory. The goal of this talk is to highlight the state of the art of shape optimization techniques in designing bypass grafts or in analyzing other cardio-vascular configurations of interest, coupled with (i) suitable shape parametrization techniques for the reduction of geometrical complexity and (ii) reduced basis methods for the solution of parametric PDEs problems and computational reduction. For an efficient model order reduction, Reduced basis (RB) methods [4], built upon a high-fidelity "truth" finite element (FE) approximation, combined with free-form deformations (FFD) techniques for efficient shape parametrization [1] are introduced, decreasing both the computational effort and the geometrical complexity. A parametrized RB approach can provide rapid and reliable results [4] in real-time and many-query contexts, such as the repetitive environment shape optimization requires; reliability is ensured by rigorous a posteriori error bounds, while rapid response is ensured by a suitable Offline-Online computational strategy. Moreover, FFD approximation is built upon a low-dimensional parametrization and enable global shape deformations by acting on a small set of control parameters, resulting in a flexible, versatile and accurate tool at the same time. Some results of haemodynamics applications are presented [2], dealing with the shape optimization of parametrized configurations, discussing computational advantages and efficiency obtained by geometrical and computational model order reduction. [1] T. Lassila, G. Rozza. Parametric free-form shape design with PDE models and reduced basis method. Comput. Methods Appl. Mech. Engrg., submitted (2009). [2] A. Manzoni, A. Quarteroni, G. Rozza, Reduced basis methods and free-form deformations techniques for parametric shape optimization in cardiovascular geometries, submitted (2010). [3] G. Rozza. Shape design by optimal flow control and reduced basis techniques: applications to bypass configurations in haemodynamics. PhD Thesis No. 3400, EPFL (2005). [4] G. Rozza, D.B.P. Huynh, A.T. Patera. Reduced Basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive PDEs. Arch. Comput. Methods Engrg. 15, pp. 229-275 (2008)