A partition of unity approach to fluid mechanics and fluid-structure interaction

Abstract

For problems involving large deformations of thin structures, simulating fluid-structure interaction (FSI) remains a computationally expensive endeavour which continues to drive interest in the development of novel approaches. Overlapping domain techniques have been introduced as a way to combine the fluid-solid mesh conformity, seen in moving-mesh methods, without the need for mesh smoothing or re-meshing, which is a core characteristic of fixed mesh approaches. In this work, we introduce a novel overlapping domain method based on a partition of unity approach. Unified function spaces are defined as a weighted sum of fields given on two overlapping meshes. The method is shown to achieve optimal convergence rates and to be stable for steady-state Stokes, Navier-Stokes, and ALE Navier-Stokes problems. Finally, we present results for FSI in the case of 2D flow past an elastic beam simulation. These initial results point to the potential applicability of the method to a wide range of FSI applications, enabling boundary layer refinement and large deformations without the need for re-meshing or user-defined stabilization.

Keywords: Finite element methods; Fluid–structure interaction; Overlapping domains; Partition of unity.

Fig. 1. The problem domain in three instances: continuous, classical, single mesh setup and PUFEM setup.

In the later, the discrete domain is subdivided into the embedded and background overlapping meshes. Both meshes contain a set of constrained nodes used to avoid ill-conditioning. Here we mark them with circles.

Fig. 2. (Left) An example of the embedded mesh (1191 elements) with marked fluid–fluid and fluid–solid interfaces. (Centre) The intersection mesh (742 elements): to replicate a realistic application, we used a background grid that is four levels coarser than the embedded one. (Right) The resulting weighting field (ψ^h).

Fig. 3. The intersection of one background element and two embedded elements at different stages of computing the PUFEM weak form.

(A) Identifying the intersection polygons. (B) Creating a new tessellation for the overlap area. (C) Defining new set quadrature points for the sub-elements and re-evaluating the basis functions.

Fig. 4. Domain used in the steady-state Stokes and Navier–Stokes problems

Fig. 5. Example of solution fields for the Stokes problem obtained using the PUFEM and classic approaches.

The PUFEM figures include the background and embedded components, as well as the total (resulting from the weighted sum).

Fig. 6. Plot showing the errors in the velocity and pressure fields for the Stokes flow problem obtained with the classical and PUFEM approaches.

The dotted line indicates the optimal convergence rate.

Fig. 7. Example of solution fields for the Navier–Stokes problem at Re 100 obtained using the PUFEM and classic approaches.

The PUFEM figures include the background and embedded components, as well as the total (resulting from the weighted sum).

Fig. 8. The error plots for steady state incompressible Navier–Stokes.

Both classic (C) and PUFEM (PU) methods are run at Reynolds number of 30 and 100.

Fig. 9. The domain of the Schäfer–Turek benchmark problem with boundary labels as well as example meshes used in PUFEM and classic approaches.

Fig. 10. Compiled results for the Turek benchmark.

(Top) A breakdown of the PUFEM solution at peak cl (9.57 s in simulation time) into background, embedded and total fields for each component of the solution. The classic result is introduced for comparison. (Bottom) Velocity magnitude over a cycle in both PUFEM and classical simulations. Contour lines where added to aid comparison.

Fig. 11. Compiled results for the ALE problem.

(Top) A breakdown of the PUFEM solution into background(global), embedded and total. Classical result is introduced for comparison. (Middle) The x component of the velocity as it evolves between the two points of maximum displacement from the centre. Contour lines were added to aid comparison. (Bottom) Estimated drag force and pressure difference across the cylinder for the PUFEM and classical (BF1) methods.

Fig. 12. An example of transient fixed nodes on the background mesh.

The red arrows point to free nodes, while the blue ones point to their constrained counterpart at different time. The element colour coding indicates the following: element cut by ${\Gamma }_{ff}^h$ (deep grey), element completely weighed out by ψ^h (light grey). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 13. (Left) Illustration of the fluid and beam domains used in the FSI test. A summary of the domain dimensions is found in Table 4. (Right) The meshes used in the classic and PUFEM approaches. For visualization purposes the solid mesh was excluded.

Fig. 14. (A) Six snapshots of the velocity magnitude as computed by the PUFEM and classical approaches. (B) The PUFEM and ALE mesh setups at the medium and minimum mesh deflection. At minimum deflection, the ALE mesh reaches a point of almost maximum deterioration. (C) The distribution in time of elements as a function of quality for both approaches.

Fig. 15. The x and y displacements of the beam tip as it evolved in the classic and PUFEM approaches.

The reference nodes where chosen such that they have identical coordinates at t = 0 s.