Coupling edge-based smoothed finite element method with smoothed particle hydrodynamics for fluid structure interaction problems

Highlights

• Coupling edge-based smoothed finite element method (ES-FEM) with SPH is proposed for solving FSI problems.

• The coupling strategy is based on the automatically generated virtual particles.

• The gradient smoothing technique in ES-FEM overcomes the ‘‘overly-stiff’’ effect in conventional FEM.

• The SPH method with integrated algorithms is effective and robust in modeling fluid flows.

• ES-FEM-SPH is demonstrated to be effective in modeling diversified FSI problems.

Abstract

Numerical simulation of fluid structure interaction (FSI) problems is one of the most challenging topics in computational fluid dynamics. In this paper, coupling edge-based smoothed finite element method (ES-FEM) and smoothed particle hydrodynamics (SPH) method (ES-FEM-SPH) is proposed for solving FSI problems, where the edge-based smoothed finite element method is used to model the movement and deformation of structures, and the smoothed particle hydrodynamics is used to model the fluid flow. In ES-FEM, the gradient smoothing technique is applied over the smoothing domain and it can effectively overcome the ‘‘overly-stiff’’ effect in conventional FEM model. Some correction algorithms including density correction, kernel gradient correction and particle shift technique are integrated into the SPH method to improve computational stability and accuracy. A virtual particle coupling scheme is used to implement the coupling of ES-FEM and SPH with complex geometry interface. As ES-FEM is more accurate than conventional FEM, and it is expected that this ES-FEM-SPH coupling approach should be superior than existing FEM-SPH coupling approaches. A number of test examples with FSI are investigated with the presented ES-FEM-SPH, and compared with results from other approaches including FEM-SPH. From the obtained numerical results, we can conclude that the ES-FEM-SPH coupling approach is effective to simulate FSI problems.

Introduction

FSI is a significant field in aerospace, marine engineering, biomechanics and other applications (Bennet and Edwards, 1998; Van Loon and Van de Vosse, 2010; Taylor and Figueroa, 2009). In recent decades, a plenty of numeric methods have been proposed for FSI problems, among which the grid-based method occupies the dominant position in numerical research. The Lagrangian grid-based methods, such as the Finite Element Method (FEM) (Strang et al., 1973), are commonly used for structural modeling. While the Eulerian grid-based methods, such as the Finite Difference Method (FDM) (Brandimarte, 1960), the Finite Volume Method (FVM) (Versteeg and Malalasekera, 1995) and Finite Element Method (FEM) (Strang et al., 1973), are classically used for fluid modeling. There are some ways for coupling Eulerian grid and Lagrangian grid, such as the Immersed Boundary Method (IBM) (Peskin, 2002) and Arbitrary Lagrangian Eulerian (ALE) method (Donea et al., 1982; Hirt et al., 1974). However, these methods are time-consuming for tracking moving boundaries or interfaces (Donea et al., 1982; Hirt et al., 1974; Morinishi and Fukui, 2012; Peskin, 2002). It is very attractive to solve the FSI problem under the unified Lagrangian description, because it can deal with the moving interface naturally. Recently, the FSI problem is solved by coupling the finite element method with mesh free methods (Fourey et al., 2010; Groenenboom and Cartwright, 2009; Hu et al., 2014; Long et al., 2017; Vuyst et al., 2005; Yang et al., 2012; Zhang and Wan, 2018), and the coupling approaches leverage the advantages of convenience of mesh free methods to simulate hydrodynamics and robustness of FEM to solve structural dynamics.

The FEM has become a powerful and effective technique for numeric simulation in science and engineering. However, some shortcomings also exist in the conventional FEM. It is well-known that the ‘‘overly-stiff’’ characteristic of the conventional finite element models often leads to a significant reduction in the accuracy of numerical solutions (Liu and Quek, 2003). Certainly, the FEM-SPH approaches have the ‘‘overly-stiff’’ characteristic because of conventional finite element model. Thus, it is important to develop an effective numerical technique for dealing with this issue. The smoothed finite element method (S-FEM) was proposed by Liu et al. (2007a), which is based on the gradient smoothing technique of mesh-less method (Chen et al., 2001). The gradient smoothing technique provides a good way to reduce the ‘‘over-stiffness’’ (Liu, 2008). A new class of smoothing methods have been developed that can effectively “soften” the numerical model, such as cell based S-FEM (CS-FEM) (Liu et al., 2007b), nodal based S-FEM (NS-FEM) (Liu et al., 2009b), edge based S-FEM (ES-FEM) (Liu et al., 2009a) and face based S-FEM (FS-FEM) (Nguyen-Thoi et al., 2009). In addition, thanks to the gradient smoothing technique, smoothed point interpolation methods (S-PIMs) have been proposed (Gui-rong and Gui-yong, 2013; Liu and Zhang, 2008), and S-PIMs are developed for FSI problems (Wang et al., 2017, 2018; Zhang et al., 2018). Furthermore, S-FEM is also utilized to FSI problems. Zhang et al. proposed Immersed S-FEM (IS-FEM) to solve FSI problems (Zhang et al., 2012). Recently, Jiang et al. proposed a sharp-interface IS-FEM for interactions between large deformation structures and incompressible flows, the method has great application prospects for FSI problems(Jiang et al., 2018). Since the ES-FEM provides more precise solutions than the traditional linear FEM (Liu et al., 2009a). In this paper, the ES-FEM is used to model the structure domain.

The SPH approach is a purely Lagrangian mesh-free approach, which was originally developed by Lucy (1977) and Monaghan and Gingold (Gingold and Monaghan, 1977; Monaghan and Gingold, 1983). It is widely applied in astrophysics, impact dynamics, fluid mechanics and other fields. Recently, Han and Hu used a uniform SPH frame to modeling fluid structure interaction problems (Han and Hu, 2018). Yang et al. proposed an improved SPH-EBG (Element bending group) method for interactions between free surface flows and flexible structures (Yang et al., 2016). Zhang et al. reviewed the latest developments in SPH methods and their typical applications in FSI problems (Zhang et al., 2017). Asai et al. proposed a stabilized Incompressible SPH to simulate the free surface flow problems (Asai et al., 2012). He et al. (2020) used a finite particle method to modeling the solitary wave breaking over a slope. In this article, the SPH approach is applied to solve the fluid domain. The meshless property of SPH approach eliminates the troubles caused by large deformations. Therefore, SPH is quite suitable to solve fluid flows with moving interfaces and free surfaces.

In the FEM-SPH model, the SPH is used to solve the fluid domain and the FEM is applied to solve the solid domain, where the coupling strategy usually utilizes a pure Lagrangian description in both media. The FEM-SPH coupling approach was firstly developed by Attaway et al. (1994) to investigate impact problems. Vuyst et al. (2005) investigated fluid structure impact problems using an FEM-SPH model. The FEM-SPH coupling approach can also be applied to interaction between elastic structures and free surface flow (Fourey et al., 2010; Groenenboom and Cartwright, 2009; Hu et al., 2014; Yang et al., 2012). The master-slave schemes are utilized in most FEM-SPH models. And the contact forces are used to prevent particles from entering the element mesh (Attaway et al., 1994; Groenenboom and Cartwright, 2009; Hu et al., 2014). Each coupling approach has its own specific feature. For the modeling of complex FSI problems, some FEM-SPH coupling approaches without contact algorithm are also proposed (Fourey et al., 2010; Long et al., 2017; Yang et al., 2012). Fourrey et al. (Fourey et al., 2010) utilized the virtual particles coupling strategy to realize the FEM-SPH model. Yang et al. (2012) used Monaghan type repulsive force to prevent the particles penetrate the element meshes. Long et al. (2017) developed a virtual particle coupling scheme for FEM-SPH model with complex geometry interface, in which the virtual particles are produced through discretizing the truncated region of support domain of kernel function. Recently, Fourey et al. (2017) proposed an efficient coupling scheme between SPH and FEM for violent fluid–structure interaction modeling.

In this paper, the coupling method of ES-FEM and SPH is proposed for simulating FSI problems, and the virtual particles coupling scheme is used for the ES-FEM-SPH model. Three-node triangular elements are used to discretize the structure domain, which is simple and flexible. The particles are used to discretize the fluid domain and the FSI in ES-FEM-SPH model is implemented through deploying virtual particles within ES-FEM elements. Since ES-FEM provides a more precise solution than traditional linear FEM using the same constant strain triangular element, it can be expected that ES-FEM-SPH coupling model will be applicable to FSI problems. Comparisons between the ES-FEM-SPH model and the FEM-SPH coupling approach are given for several numerical examples.

The organization of this paper is as follows. Firstly, the ES-FEM to solve the solid domain is concisely introduced. Secondly, the SPH approach to solve hydrodynamics is briefly introduced. Thirdly, the coupled ES-FEM-SPH model is briefly described. Finally, numerical examples are given to illustrate the effectiveness of the ES-FEM-SPH model.