We consider the simulation of slender structures immersed in a three-dimensional (3D) flow. By exploiting the special geometric configuration of the slender structures, this particular problem can be modeled by mixed-dimensional coupled equations. Taking advantage of the slenderness of the structure and thus considering 3D/1D coupled problems raise several challenges and difficulties. From a mathematical point of view, these include defining well-posed trace operators of co-dimension two. On the computational standpoint, the non-standard mathematical formulation makes it difficult to ensure the accuracy of the solutions obtained with the mixed-dimensional discrete formulation as compared to a fully resolved one. Here we proposed to circumvent theses issues by imposing the fluid–structure coupling conditions on the 2D fluid–structure interface but in a reduced way still taking advantage of the 1D dynamic of the slender structure. We consider the Navier–Stokes equations for the fluid and a Timoshenko beam model for the structure. We complement these models with a mixed-dimensional version of the fluid–structure interface conditions, based on the projection of kinematic coupling conditions on a finite-dimensional Fourier space on each beam cross section. Furthermore, we develop a discrete fictitious domain formulation within the framework of the finite element method, establish the energy stability of the scheme, provide extensive numerical evidence of the accuracy of the discrete formulation, notably with respect to a fully resolved (ALE based) model and a standard reduced modeling approach.

中文翻译：

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用于模拟浸没在不可压缩流中的细长结构的混合维度公式


我们考虑了浸入三维 （3D） 流中的细长结构的模拟。通过利用细长结构的特殊几何配置，这个特定问题可以通过混合维耦合方程来建模。利用结构的细长性，从而考虑 3D/1D 耦合问题会带来一些挑战和困难。从数学的角度来看，这些包括定义共维度 2 的适定轨迹运算符。从计算的角度来看，与完全解析的解相比，非标准的数学公式使得很难确保使用混合维离散公式获得的解的准确性。在这里，我们提议通过在二维流固耦合界面上施加流固耦合条件来规避这些问题，但以简化的方式仍然利用细长结构的一维动力学。我们考虑了流体的 Navier-Stokes 方程和结构的 Timoshenko 梁模型。我们用流-固耦合条件的混合维版本来补充这些模型，该条件基于运动学耦合条件在每个梁横截面上的有限维傅里叶空间上的投影。此外，我们在有限元方法的框架内开发了一个离散虚构域公式，建立了该方案的能量稳定性，为离散公式的准确性提供了广泛的数值证据，特别是关于完全解析（基于 ALE）模型和标准简化建模方法。