An integrated particle model for fluid–particle–structure interaction problems with free-surface flow and structural failure

Abstract

Discrete Element Method (DEM) and Smoothed Particles Hydrodynamics (SPH) are integrated to investigate the macroscopic dynamics of fluid–particle–structure interaction (FPSI) problems. With SPH the fluid phase is represented by a set of particle elements moving in accordance with the Navier–Stokes equations. The solid phase consists of physical particle(s) and deformable solid structure(s) which are represented by DEM using a linear contact model and a linear parallel contact model to account for the interaction between particle elements, respectively. To couple the fluid phase and solid particles, a local volume fraction and a weighted average algorithm are proposed to reformulate the governing equations and the interaction forces. The structure is coupled with the fluid phase by incorporating the structure’s particle elements in SPH algorithm. The interaction forces between the solid particles and the structure are computed using the linear contact model in DEM. The proposed model is capable of simulating simultaneously fluid–structure interaction (FSI), particle–particle interaction and fluid–particle interaction (FPI), with good agreement between complicated hybrid numerical methods and experimental results being achieved. Finally, a specific test is carried out to demonstrate the capability of the integrated particle model for simulating FPSI problems with the occurrence of structural failure.

Introduction

Fluid–particle–structure interaction problems have been frequently encountered in the flooding events with the collapse of infrastructures (e.g. buildings and bridges), where the particles could be soil, sediment and/or debris. Particularly, stone bridges which are one of the most common masonry bridges in the UK were widely built in the past due to the availability of stone and easy construction, and many of those historic and listed masonry bridges are still in service in the UK. Masonry bridges were built through the application of rock blocks with high compressive strength to transmit the loads to the ground. In fact, masonry bridges cannot resist a high amount of the shearing load in comparison with modern concrete bridges, therefore they are at risk of being damaged and even collapsed due to the occurrence of flooding, which imposes enormous impacts on local transportation, and it is costly to get them repaired/rebuilt. Preventing or mitigating such unexpected accidents could be attained through proactive reinforcing or strengthening techniques which are preferred in order to make the bridges more resistant to scouring and buoyancy effects caused by flooding. To address this challenging problem, a combination of interdisciplinary knowledge of geotechnical, hydraulic and structural engineering are required to better understand the complicated interaction mechanism among bridges, flood water and soil/sediment/debris. This also raises a demand for a robust and reliable computer model to fulfil the requirement of large-scale simulation in order to predict the simultaneous interaction between soil/sediment/debris, flood and bridges/buildings. Up to now, there are various computational or numerical models for fluid–structure interaction (FSI) De Hart et al. (2003), Souli et al. (2000), Wall et al. (2006) or fluid–particle interaction (FPI) Génevaux et al. (0000), Adeniji-Fashola and Chen (1990), Sarkar et al. (2009), and they have been extensively studied in terms of problem scales and numerical methods. However, to the authors’ best knowledge, computational models that are capable of handling the simultaneous interaction between fluids, particles and structures are rarely reported.

One of the challenging issues involved in FPSI problems is the contact detection and subsequent collision and separation between two particles or between a particle and a structure/boundary. It becomes even more complicated when a fracture of the structure is allowed to create new surfaces which may interact with the particles and fluids. Therefore an explicit Lagrangian method to capture the movement of individual particles is required. Although both Eulerian and Lagrangian methods have been well developed for fluid flow and structural analysis, but to integrate particles with fluid and structure a single Lagrangian computational framework would usually be preferred.

When simulating a discontinuous system of particles, discrete element method (DEM) is usually considered due to its simplicity and capability of handling the contact and interaction between particles. The interaction forces at the contacts are governed by a force–displacement law driven and used to determine the movement of each individual particle according to the Newton’s Second Law. In addition, DEM can model the deformation (and failure) of a structure by simply adding a bond at the contact between a pair of particles to represent the material properties (elasticity and strength) of a structure. Comprehensive applications of DEM have been reported in modelling mixing processes of particles Ketterhagen et al. (2009), Rhodes et al. (2001) and fracture of various engineering materials and structures such as rock (Zhuang et al., 2012), ceramics (Tan et al., 2009), concrete (Cundall and Strack, 1979) and composites (Yang et al., 2010), etc.

For the Lagrangian simulations of fluid flow, there are two widely-used mesh-free methods, e.g. Smoothed Particles Hydrodynamics (SPH) (Monaghan, 1994) and Moving Particle Simulation (MPS) (Koshizuka and Oka, 1996). In these two methods, Navier–Stokes equations, which are partial differential equations (PDEs), are transformed into ordinary differential equations (ODEs) through kernel approximation and particle approximation respectively, and the fluid domain is consequently dissolved into discrete particles with certain particle spacing. Both SPH and MPS provide approximations for partial differential equations (e.g. Navier–Stokes equations), but a weighted averaging process applied in MPS is different from taking the gradient of the kernel function in SPH. It should be noted that another meshfree but Eulerian method, Lattice Boltzmann method (LBM) (Chen and Doolen, 1998) solves Newtonian fluid flow with collision and separation models on a fixed space grid/lattice. As SPH and MPS methods are intended to approximate mathematical equations in the domain only by nodes without being connected by meshes, each discrete particles move continuously in accordance with surrounding particles, thus complex boundary flow and free surface flow can be easily accounted for. Due to this benefit, they have been popular in hydraulic engineering, for example, coastal erosion (Dalrymple and Rogers, 2006), sedimentation (Falappi et al., 2007), sloshing and flooding (Shen and Vassalos, 2011).

In this paper, SPH and DEM are coupled together to form an integrated particle model to simulate the interactions among fluid, particles and structure. As SPH and DEM are both meshfree particle methods under the Lagrangian scheme, the identification of free surfaces, moving interfaces and deformable boundaries can be handled straightforwardly (Liu and Liu, 2003). Coupled SPH-DEM models have been developed and applied to multiphase flow problems with FPI in Sun et al. (2013), Robinson et al. (2014) and Karunasena et al. (0000) and FSI problems in Wu et al. (2016). Other similarly coupled particle models in the Lagrangian framework such as SPH–SPH (Antoci et al., 2007) and MPS–MPS (Shakibaeinia and Jin, 2011) have also been applied in either FSI or FPI problems, but the kernel functions used in SPH or MPS for particles and structures lack physical representations of particle–particle contact and structural failure. In other mesh-based coupled models for either FSI or FPI in Eulerian-Lagrangian scheme (e.g. CFD–FEM model Ahmed et al. (2009), Kim et al. (2013), Peksen (2014), Peksen et al. ( 2011) and the Arbitrary Lagrangian-Eulerian method Donea et al. (1982), Hu et al. (2001)) and Eulerian-Eulerian scheme (e.g. Finite volume method (Kalteh et al., 2011)), the accuracy of the solution is generally limited by large translation and rotation of the solid particles or significant deformation of the structure, consequently the mesh cells for fluid elements in those mesh-dependent models tend to become ill-shaped. Therefore remedies such as mesh regeneration and adaptive meshing have to be adopted to improve the mesh quality at the expense of sharply increased computational cost.

When dealing with the interface between fluid and particles, two approaches have been developed so far. One is the direct numerical simulation (DNS) (Shakibaeinia and Jin, 2011) and the other one is locally averaged Navier–Stokes equation associated with local volume fraction (Robinson et al., 2014). In DNS approach, the drag force acting on particle phase is directly computed from the Navier–Stokes equations with assigned dynamic viscosities of the fluid and the particle, but when the same theory is applied to compute the interaction forces between particle phases it lacks physical representation of the collisions between particles. Whilst in the second approach, an empirical equation subjected to specific problems (e.g. the transport of sediment-induced by the movement of fluid flow) is required to evaluate the drag force, and the interaction forces between particle phases can be independent of the Navier–Stokes equations.

In this study, an improved integrated particle model coupling SPH and DEM with a local averaging technique is proposed for the fluid–particle–structureinteraction problems. In our previous study (Wu et al., 2016), the integrated model only dealt with fluid–structure interaction with the failure of the structure. As a further model improvement, the solid particle has been integrated into the current model to consider more complex engineering problems with fluid–particle–structure interaction. Validation tests for fluid–structure interaction have been carried out in our previous work (e.g. fluid–structure interaction) (Wu et al., 2016) and validation tests for fluid–particle interaction are validated (e.g. fluid–particle interaction and particle–particle interaction) in the current study. Finally, a special case with the free-surface flow and structural failure is used to demonstrate the capability of the newly developed model in modelling fluid–particle–structure interaction (FPSI) problems.