A hybrid Reduced Order Model to enforce outflow pressure boundary conditions in computational haemodynamics

Abstract: This paper deals with the development of a Reduced-Order Model (ROM) to investigate haemodynamics in cardiovascular applications. It employs the use of Proper Orthogonal Decomposition (POD) for the computation of the basis functions and the Galerkin projection for the computation of the reduced coefficients. The main novelty of this work lies in the extension of the lifting function method, which typically is adopted for treating nonhomogeneous inlet velocity boundary conditions, to the handling of nonhomogeneous outlet boundary conditions for the pressure, representing a very delicate point in the numerical simulations of the cardiovascular system. Moreover, we incorporate a properly trained neural network in the ROM framework to approximate the mapping from the time parameter to the outflow pressure, which in the most general case is not available in closed form. We define our approach as "hybrid", because it merges physics-based elements with data-driven ones. At full order level, a Finite Volume method is employed for the discretization of the unsteady Navier-Stokes equations while a two-element Windkessel model is adopted to enforce a valuable estimation of the outflow pressure. Numerical results, firstly related to a 2D idealized blood vessel and then to a 3D patient-specific aortic arch, demonstrate that our ROM is able to accurately approximate the FOM with a significant reduction in the computational cost.