Purpose

When simulating fluid-structure interaction (FSI), it is often essential that the no-slip condition is accurately enforced at the wetted boundary of the structure. This paper aims to evaluate the relative strengths and limitations of the penalty and Lagrange multiplier methods, within the context of modelling FSI, through a comparative analysis.

Design/methodology/approach

In the immersed boundary method, the no-slip condition is typically imposed by augmenting the governing equations of the fluid with an artificial body force. The relative accuracy and computational time of the penalty and Lagrange multiplier formulations of this body force are evaluated by using each to solve three test problems, namely, flow through a channel, the harmonic motion of a cylinder through a stationary fluid and the vortex-induced vibration (VIV) of a cylinder.

Findings

The Lagrange multiplier formulation provided an accurate solution, especially when enforcing the no-slip condition, and was robust as it did not require “tuning” of problem specific parameters. However, these benefits came at a higher computational cost relative to the penalty formulation. The penalty formulation achieved similar levels of accuracy to the Lagrange multiplier formulation, but only if the appropriate penalty factor was selected, which was difficult to determine a priori.

Originality/value

Both the Lagrange multiplier and penalty formulations of the immersed boundary method are prominent in the literature. A systematic quantitative comparison of these two methods is presented within the same computational environment. A novel application of the Lagrange multiplier method to the modelling of VIV is also provided.

中文翻译：

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拉格朗日乘子和流固耦合建模惩罚方法的比较分析

目的

在模拟流固耦合 (FSI) 时，通常必须在结构的润湿边界处准确执行无滑移条件。本文旨在通过比较分析，在 FSI 建模的背景下，评估惩罚和拉格朗日乘数方法的相对优势和局限性。

设计/方法/方法

在浸入边界法中，无滑移条件通常是通过用人造体力增加流体的控制方程来施加的。该体力的惩罚和拉格朗日乘子公式的相对精度和计算时间通过分别解决三个测试问题进行评估，即流经通道、圆柱体通过静止流体的谐波运动和涡诱导气缸的振动 (VIV)。

发现

拉格朗日乘数公式提供了一个准确的解决方案，尤其是在强制执行无滑移条件时，并且由于不需要“调整”问题特定参数而稳健。然而，相对于惩罚公式，这些好处带来了更高的计算成本。惩罚公式实现了与拉格朗日乘子公式相似的准确度水平，但前提是选择了适当的惩罚因子，这很难先验确定。

原创性/价值

浸入边界法的拉格朗日乘数和惩罚公式在文献中都很突出。在相同的计算环境中对这两种方法进行了系统的定量比较。还提供了拉格朗日乘法器方法在 VIV 建模中的新应用。