量化血泵不确定性的多项式混沌多级预处理方法
Multilevel preconditioner of Polynomial Chaos Method for quantifying uncertainties in a blood pump
摘要：
全世界有超过2300万人患有心力衰竭。尽管现代移植手术已经建立，但缺乏心脏捐献成为移植频率的一个很大限制。关于这个问题，心室辅助装置（VAD）可以在等待期间和手术后在支持患者方面发挥重要作用。此外，它已被证明，VAD的血泵的方式有优势，在不同的条件下工作。虽然已经做了大量的工作对血液泵的功能建模，但量化的数值模型中的不确定性是一项具有挑战性的任务。我们考虑多项式混沌（PC）方法，它是由维纳提出的建模高斯分布的随机过程。Galerkin投影，广义多项式混沌（gPC）的侵入式版本，已被密集研究和应用于各种问题。该方法可以直接用多项式混沌级数展开来表示随机过程，因此与传统的非侵入方法相比，它可以优化计算量。在我们以前的工作中，我们比较了不同的预处理技术的血泵配置的稳态模拟，比较表明，一个不精确的多级预处理器有一个有前途的性能。在这项工作中，我们显示了一个不稳定的血流通过FDA血泵配置与Galerkin投影法，这是在我们的开源有限元库Hiflow3实现。考虑了入流边界条件、转子角速度和动粘滞系数三个不确定性源，在超级计算机上对3000多万个自由度的数值结果进行了验证。
英文摘要：
More than 23 million people are suffered by Heart failure 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.
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