[论文解读] An oscillation-free fully partitioned scheme for the numerical modeling of cardiac active mechanics

Francesco Regazzoni, Alfio Quarteroni|arXiv (Cornell University)|Jul 30, 2020

Cardiomyopathy and Myosin Studies参考文献 51被引用 7

一句话总结

本文提出了一种新颖的、无条件稳定的全分离数值格式，用于心脏主动力学，可消除标准分离方法中常见的非物理振荡。通过引入基于微观尺度能量学的稳定项，该方法在任意时间步长下均保证稳定性，同时在多种力生成模型中保持计算效率和精度。

ABSTRACT

In silico models of cardiac electromechanics couple together mathematical models describing different physics. One instance is represented by the model describing the generation of active force, coupled with the one of tissue mechanics. For the numerical solution of the coupled model, partitioned schemes, that foresee the sequential solution of the two subproblems, are often used. However, this approach may be unstable. For this reason, the coupled model is commonly solved as a unique system using Newton type algorithms, at the price, however, of high computational costs. In light of this motivation, in this paper we propose a new numerical scheme, that is numerically stable and accurate, yet within a fully partitioned (i.e. segregated) framework. Specifically, we introduce, with respect to standard segregated scheme, a numerically consistent stabilization term, capable of removing the nonphysical oscillations otherwise present in the numerical solution of the commonly used segregated scheme. Our new method is derived moving from a physics-based analysis on the microscale energetics of the force generation dynamics. By considering a model problem of active mechanics we prove that the proposed scheme is unconditionally absolutely stable (i.e. it is stable for any time step size), unlike the standard segregated scheme, and we also provide an interpretation of the scheme as a fractional step method. We show, by means of several numerical tests, that the proposed stabilization term successfully removes the nonphysical numerical oscillations characterizing the non stabilized segregated scheme solution. Our numerical tests are carried out for several force generation models available in the literature, namely the Niederer-Hunter-Smith model, the model by Land and coworkers, and the mean-field force generation model that we have recently proposed. Finally, we apply the proposed scheme [...]

研究动机与目标

• 解决标准全分离（分离）格式在心脏电机械耦合模拟中出现的不稳定性和非物理振荡问题。
• 在全分离框架内开发一种数值稳定且精确的方法，避免使用高成本的单体牛顿型求解器。
• 基于力生成动力学的微观尺度能量学，推导出稳定项。
• 证明所提格式对任意时间步长均具有无条件绝对稳定性。
• 在多个已建立的主动力生成模型上验证该方法。

提出的方法

• 该方法引入了一个基于物理分析的、数值一致的稳定项，源自对微观尺度力生成能量学的物理解析。
• 将稳定项添加至标准分离格式中，通过修改力更新步骤来抑制振荡。
• 该格式在数学上可解释为一种分步格式，确保与底层物理规律的一致性。
• 该方法保持全分离结构，按顺序求解主动力和力学平衡方程。
• 该公式被推广以兼容多种主动力生成模型，包括 Niederer-Hunter-Smith、Land 等人以及平均场模型。
• 稳定性分析证明了无条件绝对稳定性，即该格式在任意时间步长下均保持稳定。

实验结果

研究问题

• RQ1 能否使心脏主动力学的全分离格式实现无条件稳定且无振荡？
• RQ2 哪些物理原理可指导设计一种能消除分离格式中非物理振荡的稳定项？
• RQ3 所提格式在不同主动力生成模型中的表现如何？
• RQ4 与标准分离格式不同，该方法是否对任意大的时间步长均保持稳定？
• RQ5 该稳定项能否从微观尺度力生成能量学的基本原理中推导得出？

主要发现

• 所提格式实现了无条件绝对稳定性，即在任意时间步长下均保持稳定，而标准分离格式则不具备此特性。
• 稳定项在所有测试的力生成模型中均成功消除了数值解中的非物理振荡。
• 通过保持全分离结构，该方法在保持高精度的同时具备良好的计算效率。
• 数值测试证实了该格式在 Niederer-Hunter-Smith、Land 等人以及平均场力生成模型中的鲁棒性。
• 该格式可解释为一种分步格式，为物理和数学基础提供了清晰的解释。
• 稳定项源自微观尺度能量学，确保了物理解释的一致性与数值稳定性。