Multilevel Preconditioning of Polynomial Chaos Method for Quantifying Uncertainties in a Blood Pump

Abstract

Abstract. Heart failure (HF) is a severe cardiovascular disease, it happens when the heart muscle is so weakened such that it can not provide sufficient blood as body needs. More than 23 million people are suffered by HF worldwide. Despite the modern transplant operation is well established, the lack of heart donations becomes a big restriction on transplantation frequency. With respect to this matter, ventricular assist devices (VADs) can play an important role in supporting patients during waiting period and after the surgery.Moreover, it has been shown that VADs by means of blood pump have advantages for working under different conditions. While a lot of work has been done on modeling the functionality of the blood pump, but quantifying uncertainties in a numerical model is a challenging task. We consider the Polynomial Chaos (PC) method, which is introduced by Wiener for modeling stochastic process with Gaussian distribution. The Galerkin projection, the intrusive version of the generalized Polynomial Chaos (gPC), has been densely studied and applied for various problems. The intrusive Galerkin approach could represent stochastic process directly at once with Polynomial Chaos series expansions, it would therefore optimize the total computing effort comparing with classical non-intrusive methods. We compared different preconditioning techniques for a steady state simulation of a blood pump configuration in our previous work, the comparison shows that an inexact multilevel preconditioner has a promising performance. In this work, we show an instationary blood flow through a FDA blood pump configuration with Galerkin Projection method, which is implemented in our open source Finite Element library Hiflow3. Three uncertainty sources are considered : inflow boundary condition, rotor angular speed and dynamic viscosity, the numerical results are demonstrated with more than 30 Million degrees of freedom by using supercomputer.

Uncertainty Assessment of the Blood Damage in a Fda Blood Pump

“…This work is a following work of [27], which was focused on the solving techniques, i.e. the Multilevel preconditioner, for dealing with the coupled large stochastic system.…”

“…According to [27], we consider three different sources of input uncertainty in our model: the inflow boundary condition g, the angular speed ω and the dynamic viscosity µ. We employ the generalized Polynomial Chaos Expansion (gPCE) [28,25,2] technique to model the uncertainty propagation in the input parameters.…”

“…The last publication [27] in 2017 presented our first unsteady blood flow simulation on a full blood pump based on the residual-based Variational Multiscale method (VMS) [16,5,4,17]. The blood pump geometry is provided by the U.S. Food and Drug Administration (FDA) under the framework of Critical Path Initiative (CPI).…”

“…We here continue to apply the stochastic finite element method (SFEM) [15] for the blood pump modelization. The shear layer update approach, which is introduced in [27], is employed in this work for acquiring the moving mesh. The intrusive stochastic Galerkin approach is implemented for quantifying the uncertainty introduced by three uncertain parameters, by means of inflow velocity, rotation speed and dynamic viscosity.…”

A stochastic Galerkin method with adaptive time-stepping for the Navier–Stokes equations

“…Some aspects of the gPC methods for time-dependent problems were studied in literature see, e.g., [16,41,42]. In particular, long-term integration was addressed by Gerritsma et al [10], Heuveline, Schick and Song [19,34,36], Wilkins [38], Özen nad Bal [29,30], and most recently by Esquivel et al [9], among others. Methods for flows exhibiting uncertain periodic dynamics were proposed, e.g., by Bonnaire et al [2], Lacour et al [22] and Schick et al [33].…”