基于脆弱点法的心脏单域模型的无网格实现
Meshfree implementation of the cardiac monodomain model through the Fragile Points Method
摘要：
用于心脏电生理学的计算机建模和仿真的无网格方法越来越受欢迎。这些方法不需要网格，并且比有限元法（FEM）更适合于模拟复杂几何结构（如人类心脏）的活动。然而，诸如数值积分精度和时间效率的挑战仍然存在，并且限制了它们的适用性。最近，脆弱点方法（FPM）已被引入无网格方法家族。它使用局部的、简单的、多项式的、不连续的函数来构造Galerkin弱形式的试函数和检验函数。这允许精确积分和提高效率，同时能够像FEM中那样施加基本和自然的边界条件。在这项工作中，我们考虑应用FPM心脏电生理模拟。我们推导出心脏单域模型使用FPM制定，我们解决了几个基准问题，在2D和3D。我们表明，FPM导致类似的精度和效率的解决方案，有限元，同时减轻了需要一个网格。此外，FPM表现出更好的收敛性比FEM在所考虑的基准。
英文摘要：
Meshfree methods for in silico modelling and simulation of cardiac electrophysiology are gaining more and more popularity. These methods do not require a mesh and are more suitable than the Finite Element Method (FEM) to simulate the activity of complex geometrical structures like the human heart. However, challenges such as numerical integration accuracy and time efficiency remain and limit their applicability. Recently, the Fragile Points Method (FPM) has been introduced in the meshfree methods family. It uses local, simple, polynomial, discontinuous functions to construct trial and test functions in the Galerkin weak form. This allows for accurate integration and improved efficiency while enabling the imposition of essential and natural boundary conditions as in the FEM. In this work, we consider the application of FPM for cardiac electrophysiology simulation. We derive the cardiac monodomain model using the FPM formulation and we solve several benchmark problems in 2D and 3D. We show that FPM leads to solutions of similar accuracy and efficiency with FEM while alleviating the need for a mesh. Additionally, FPM demonstrates better convergence than FEM in the considered benchmarks.
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