In this paper, we introduce a fully-monolithic, implicit finite element method designed for investigating fluid–structure interaction problems within a fully Eulerian framework. Our approach employs a coupled Navier–Stokes Cahn–Hilliard phase-field model, recently developed by Mokbel et al. (2018). This model adeptly addresses significant challenges such as large solid deformations, topology changes, stable incorporation of surface tensions, and eliminates the need for remeshing methods. While the original model was primarily tested for axisymmetric problems, our work extends its application to encompass a range of two- and three-dimensional verification tests. Additionally, we advance the model to handle multi-solid–fluid interaction scenarios through the integration of a multi-body contact algorithm. Assuming both the solid and fluid to be incompressible, we describe them using Navier–Stokes equations. For the solid, a hyperelastic neo-Hookean material is assumed, and the elastic solid stress is computed based on the left Cauchy–Green deformation tensor, which is governed by an Oldroyd-B like equation. We employ a residual-based variational multiscale method for solving the full Navier–Stokes equations, a stabilized Galerkin finite element method using Streamline-Upwind/Petrov–Galerkin (SUPG) stabilization for solving the Oldroyd-B equation, and a mixed finite element splitting scheme for the Cahn–Hilliard equation. The system of partial differential equations is solved using an implicit, monolithic scheme based on the generalized-α time integration method. Our approach is verified through two-dimensional numerical examples, including the deformation of an elastic wall by flow, the deformation and motion of a solid disk in a lid-driven cavity flow, and the bouncing of an elastic ball, showcasing the method’s ability to handle solid-wall contact. Furthermore, we extend the application to multi-body contact problems and verify the model’s accuracy by solving three-dimensional benchmark problems, such as the motion of an elastic solid sphere in lid-driven cavity flow and the falling of an elastic sphere onto an elastic block.

中文翻译：

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用于全欧拉流固耦合相场建模的整体有限元方法


在本文中，我们介绍了一种完全整体的隐式有限元方法，旨在研究完全欧拉框架内的流-固耦合问题。我们的方法采用了 Navier-Stokes Cahn-Hilliard 耦合相场模型，该模型最近由 Mokbel 等人（2018 年）开发。该模型巧妙地解决了重大挑战，例如大型实体变形、拓扑变化、表面张力的稳定结合，并且无需重新划分网格。虽然原始模型主要针对轴对称问题进行测试，但我们的工作将其应用扩展到包括一系列二维和三维验证测试。此外，我们通过集成多体接触算法来改进模型以处理多固体-流体相互作用场景。假设固体和流体都是不可压缩的，我们用 Navier-Stokes 方程来描述它们。对于固体，假设采用超弹性新胡克材料，弹性固体应力基于左柯西-格林变形张量计算，该张量由类似 Oldroyd-B 的方程控制。我们采用基于残差的变分多尺度方法来求解完整的 Navier-Stokes 方程，使用流线-逆风/彼得罗夫-加辽金 （SUPG） 稳定性的稳定伽辽金有限元方法求解 Oldroyd-B 方程，以及混合有限元分裂方案用于 Cahn-Hilliard 方程。偏微分方程组使用基于广义α时间积分法的隐式整体方案进行求解。 我们的方法通过二维数值示例进行了验证，包括弹性壁随流动的变形、固体盘在盖子驱动的空腔流中的变形和运动以及弹性球的弹跳，展示了该方法处理固体壁接触的能力。此外，我们将应用扩展到多体接触问题，并通过求解三维基准问题来验证模型的准确性，例如弹性固体球体在盖子驱动的空腔流中的运动以及弹性球体落到弹性块上。