Coupling total Lagrangian SPH–EISPH for fluid–structure interaction with large deformed hyperelastic solid bodies

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Abstract

In this work, we propose a two-way coupling technique between a total Lagrangian smoothed particle hydrodynamics (SPH) method for Solid Mechanics and the explicit incompressible SPH (EISPH) to simulate fluid–structure interaction problems. In the solid part, the total Lagrangian framework guarantees that the particle distribution keep stable to correctly calculate the deformation gradient and thus the elastic forces. The constitutive model follows hyperelastic formulations, and the stability of the method is enforced by a Jameson–Schmidt–Turkel (JST) stabilization procedure. For the fluid part, we applied an EISPH formulation, which is a fully explicit incompressible scheme based on a projection method capable of providing accurate pressure distributions for free-surface flows, while avoiding costly linear equations. The coupling scheme follows the same manner as the fixed wall ghost particle (FWGP) approach, which was here adapted to include moving walls. In addition, the non-penetration condition is rigorously reinforced through a numerical algorithm to avoid penetration of every fluid particle, including free-surface particles. Our method for solid is then verified through a large deformed tension plate numerical test, and our coupling forces through a series floating tests and hydrostatic water column over a thin infinite plate. Then, the method is validated comparing it with experimental data of a dam break test in which the water column attacks a thin rubber plate.

Introduction

Developed by Lucy [1] and Gingold and Monaghan [2], the smoothed particle hydrodynamics (SPH) method is a mesh-free Lagrangian numerical method widely used in the field of Fluid Dynamics [3]. The most attractive feature of SPH in comparison with other methods such as finite element method (FEM) and finite difference method (FDM) is its mesh-free nature, which allows it to model complex geometries and highly deformed bodies with ease.

Among a wide variety of classifications, the explicit incompressible SPH (EISPH) is a very promising method in the simulation of Newtonian low-viscous fluids [4]. Based on the semi-implicit incompressible SPH (ISPH) [5], [6], the EISPH was developed by Barcarolo et al. [7] to avoid solving the implicit pressure Poisson equation (PPE) present in ISPH, which is remarkably costly both in terms of computational performance and memory consumption. Then, Morikawa et al. [4] combined the pressure stabilization procedure from [8] to improve its accuracy.

SPH has also been appealing to the Solid Mechanics community in recent years. Early developments of SPH for Solid Mechanics include [9] and [10], which adapt SPH for strength of materials and impact problems, respectively. Later, Monaghan [11] and Gray et al. [12] improved the SPH to deal with the well-known tensile instability problem, which is the appearance or disappearance of stresses in a non physical way as a consequence of particle clumping or separation. Such problem was completely solved by using a total Lagrangian formulation proposed by Bonet et al. [13] and later enhanced by Lee et al. [14]. In the latter, the authors propose the introduction of a globally conservative Jameson–Schmidt–Turkel (JST) stabilization procedure to eliminate spurious hourglass-like modes and non-physical pressure instabilities.

As nicely summarized by Zhang et al. [15], SPH has been widely used in fluid–structure interaction (FSI). They show an extended review on SPH applications for FSI problems including liquid sloshing [16], [17], [18], [19], [20], [21], water entry and exit of a circular cylinder [15], wave–body interaction [22], [23], and others. However, most of these works are generally in the context of fluid–rigid body interaction [16], [17], [18], [19], [22], [23]. One new contribution in this area can be seen in Asai et al. [24] where an energy tracking method is used to ensure the conservation of energy on the contact problem.

In SPH applications of FSI problems involving elastic bodies, SPH is commonly used for the fluid part and coupled with a different method for the solid part. As expected, the most popular method for the solid part is the well-recognized FEM [21], [25], [26], [27], [28], although some authors have also used other methods such as the distinct element method (DEM) [29], the element bending group (EBG) [30], and others.

Given the developments of SPH in the field of Solid Mechanics explained before, some researchers have also invested in SPH–SPH coupling for FSI problems. One of the earlier studies in this topic was conducted by Antoci et al. [31] who introduced a 2D SPH–SPH coupling using a hypoelastic constitutive model for the solid part and weakly compressible SPH (WCSPH) for the fluid part. Following developments include Hwang et al. [20] and Khayyer et al. [32] who both coupled ISPH for the fluid with SPH for the solid described with the infinitesimal strain definition. One key feature of such implementations is that they heavily rely on some numerical device to control the solid deformation. For example, Antoci et al. [31] use XSPH to smooth out the solid velocity field, while Hwang et al. [20] and Khayyer et al. [32] use an additional equation for the conservation of angular momentum to compensate for the infinitesimal strain definition. More recently, Zhan et al. [33] coupled WCSPH for the fluid with a SPH formulation for the solid part with a total Lagrangian hyperelastic constitutive model.

The current paper expands the implementation of total Lagrangian SPH for FSI problems using the remarkably accurate JST-SPH from Lee et al. [14] for Solid Mechanics and the highly efficient EISPH presented in [4] for Hydrodynamics in a two-way coupling fluid–structure interaction method. Another relevant contribution of this study is to promote a coupling mechanism between SPH formulations which takes into account a strict non-penetration condition of fluid particles into the solid body.

Adami et al. [34] developed a pressure-based wall boundary condition for the WCSPH and fixed wall ghost particles (FWGP), which was later adapted to fluid–solid interactions by several researches [18], [33], [35]. In parallel, a similar technique based on a non-penetration Neumann boundary condition was adapted to the EISPH in [4]. However, the above wall boundary conditions rely heavily on the pressure values of the fluid particles. From personal experience, this fact causes some particles to be unaffected by the non-penetration boundary condition, especially on the free-surface, which, by definition, have null pressure. As a consequence, allowing fluid particle penetration might generate pressure instabilities.

In this study, we propose to use the above-mentioned FWGP with a Neumann non-penetration boundary condition in combination with an algorithmic reinforcement of such non-penetration condition. Although being basically a numerical imposition, the developed non-penetration algorithm resembles an impact collision with zero coefficient of restitution, as discussed later. In this way, it was also adapted to the fluid–solid interaction force as an impulse-based force to ensure that the coupling technique obeys the Newton’s third law of motion.

Section snippets

The SPH method

The SPH method is a Lagrangian mesh-free numerical scheme to interpolate the value of generic functions and its spatial derivatives based on a weight function called kernel W. In the context of numerical simulations, the geometry of a given problem is discretized into a finite number of particles, which constitute the continuum body. Here in this section, we summarize some basic aspects of this method.

Fluid model

In this section, we summarize the most relevant aspects of the EISPH fluid model for this study. We follow the same formulation as in [4], so, for more details such as the implementation of eddy viscosity and computational aspects, please refer to it.

Total Lagrangian SPH formulation

To describe the SPH formulation for elastic solid bodies, we first introduce the total Lagrangian form of the conservation of linear momentum equation as DvDt=1ρ00P+g,where 0 is the material nabla operator (i.e., in relation to the reference configuration X), P is the nominal stress1 and the subscript 0 represents reference configuration.

The time derivative of the

Coupling scheme

We propose a two-way coupling technique that can take into account the non-penetration condition of fluid particles into the solid body. For that, consider the conservation of linear momentum equation for both fluid (Eq. (8)) and solid (Eq. (34)) bodies with additional coupling forces DvfDt=1ρfpf+μfρf2vf+g+1ρffsf, DvsDt=1ρ0s0Ps+g+CJST+1ρ0sffs,where, in this section, the subscripts f represent fluid variables, while s, solid variables. fsf is the applying force per unit volume of the solid

Numerical examples

In this section, we conducted a series of verification and validation tests of our proposed total Lagrangian SPH–EISPH for the fluid–large deformed solid coupling simulator. First, on Section 6.1, we verify our total Lagrangian formulation comparing it with FEM results. Then, in Section 6.2, we investigate whether the force generated by our coupling scheme has a physically acceptable value with a simple test of an elastic cube falling into a pool of water. Next, Section 6.3 is dedicated to

Conclusion

In this study, we present a coupling scheme between total Lagrangian JST-SPH and EISPH for fluid–structure interaction problems in three dimensional space. The JST-SPH is capable of simulating hyperelastic constitutive models with ease, while the EISPH is proven to be effective in dealing efficiently with free-surface incompressible flows resulting in smooth pressure distributions. This coupling technique provides a strict enforcement of the non-penetration condition, which is capable of

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The first author is supported by the Japan Society for the Promotion of Science (JSPS) through the Research Fellowship for Young Scientists.

This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 20J13114, JP-20H02418, 19H01098, and 19H00812. We also received computational environment support through the Joint Usage/Research Center for Interdisciplinary Large-scale Information Infrastructures (JHPCN) in Japan (Project ID: jh200034-NAH and

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