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Partitioning strategies for the interaction of a fluid with a poroelastic material based on a Nitsche's coupling approach

机译:基于Nitsche耦合方法的流体与多孔弹性材料相互作用的分配策略

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We develop a computational model to study the interaction of a fluid with a poroelastic material. The coupling of Stokes and Biot equations represents a prototype problem for these phenomena, which feature multiple facets. On one hand, it shares common traits with fluid-structure interaction. On the other hand it resembles the Stokes-Darcy coupling. For these reasons, the numerical simulation of the Stokes-Biot coupled system is a challenging task. The need of large memory storage and the difficulty to characterize appropriate solvers and related preconditioners for the equations at hand are typical shortcomings of classical discretization methods applied to this problem, such as the finite element method for spatial discretization and finite differences for time stepping. The application of loosely coupled time advancing schemes mitigates these issues, because it allows to solve each equation of the system independently with respect to the others, at each time step. In this work, we develop and thoroughly analyze a loosely coupled scheme for Stokes-Biot equations. The scheme is based on Nitsche's method for enforcing interface conditions. Once the interface operators corresponding to the interface conditions have been defined, time lagging allows us to build up a loosely coupled scheme with good stability properties. The stability of the scheme is guaranteed provided that appropriate stabilization operators are introduced into the variational formulation of each subproblem. The error of the resulting method is also analyzed, showing that splitting the equations pollutes the optimal approximation properties of the underlying discretization schemes. In order to restore good approximation properties, while maintaining the computational efficiency of the loosely coupled approach, we consider the application of the loosely coupled scheme as a preconditioner for the monolithic approach. Both theoretical insight and numerical results confirm that this is a promising way to develop efficient solvers for the problem at hand. (C) 2014 Elsevier B.V. All rights reserved.
机译:我们开发了一个计算模型来研究流体与多孔弹性材料的相互作用。 Stokes和Biot方程的耦合代表了这些现象的原型问题,这些现象具有多个方面。一方面,它具有流固耦合的共同特征。另一方面,它类似于Stokes-Darcy耦合。由于这些原因,Stokes-Biot耦合系统的数值模拟是一项艰巨的任务。需要大容量存储空间以及难以为手边的方程式描述合适的求解器和相关前置条件的困难是应用于该问题的经典离散化方法的典型缺点,例如用于空间离散化的有限元方法和用于时间步长的有限差分。松散耦合的时间提前方案的应用减轻了这些问题,因为它允许在每个时间步上独立于系统求解方程。在这项工作中,我们开发并彻底分析了Stokes-Biot方程的松散耦合方案。该方案基于Nitsche的用于强制执行接口条件的方法。一旦定义了与接口条件相对应的接口运算符,就可以通过时间滞后建立具有良好稳定性的松散耦合方案。只要将适当的稳定算子引入每个子问题的变式中,就可以保证该方案的稳定性。还分析了所得方法的误差,表明分解方程会污染基础离散方案的最佳逼近性质。为了恢复良好的近似性能,同时保持松耦合方法的计算效率,我们考虑将松耦合方案的应用作为整体方法的前提。理论上的洞察力和数值结果都证实,这是开发有效解决手头问题的有前途的方法。 (C)2014 Elsevier B.V.保留所有权利。

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