MPS-DEM coupling method for interaction between fluid and thin elastic structures
Introduction
The Fluid-Structure Interaction (FSI) problems with violent free surface flows widely exist in ocean engineering field, such as sloshing in a rolling tank with elastic baffles (Idelsohn et al., 2008a), ocean waves interacting with the sea ice floes (Zhang et al., 2019) and the water impacting the ship bottom plate (Stenius et al., 2011). The fluid dynamics applied by severe flows may lead to large deformations and even fractures of structures. Therefore, it is necessary to investigate the FSI problems for the safety of the offshore structures. Although the experimental technologies have developed maturely and reliably, these still have some unavoidable drawbacks. Experiments are usually time and money consuming. In addition, sometimes the experimental model sizes are much smaller than the actual sizes of objects. Therefore, the accuracy of the experimental data may be affected due to the scale effect. The numerical technique has developed fast in the past years and gradually formed complementary relationship with experimental techniques.
According to the treatment for interface of fluids and structures, the numerical techniques for FSI problems can be divided into two main categories (Hwang et al., 2014): the monolithic and the partitioned approaches. For the former, the fluid movement and the structure responses are solved with one system of governing equations. For the latter, the fluids and the structures are treated separately and the information is exchanged through the interpolation. Although the errors exist during the interpolation procedure, the partition approaches are more simplified and flexible for the complex FSI problems, which has attracted more attention of researchers. Considering the advantages of different methods for fluid and structure solution, many coupling methods have been proposed for FSI problem.
The grid-based methods are the mainstream approaches to simulate the flow field. The high-order schemes and the parallel strategy can be easily introduced to the grid-based methods, which can enhance the accuracy and efficiency of these methods. The low accuracy and poor stability of the particle-based methods always bothered the researchers and many corrective methods have been proposed to overcome those drawbacks. Tanaka and Masunaga (2010) proposed a mixed source term for the Poisson Pressure Equation (PPE) and a gradient model of momentum conservation. Khayyer and Gotoh, 2010, Khayyer and Gotoh, 2011, Khayyer and Gotoh, 2012 did a series of work to improve the accuracy and stability of particle-based methods. High-order schemes were proposed for Laplacian model, source term of Poisson Pressure Equation (PPE) and gradient model of MPS. Considering the errors bringed by the uneven distribution and the aggregation of the particles, the optimized particle shifting scheme was also proposed by Khayyer et al. (2017). Compared with traditional grid-based methods, the particle-based method shows its advantages in capturing complex free surface, especially for fragmentations and splashing. Besides, the structures with large deformation and movement are hard to handle, which may cause the distortion of the grids. The distorted grids affect the accuracy of simulation and the mesh reconstruction is computationally expensive. In the contrast, it is easy for particle-based methods to overcome these drawbacks. There are no fixed topological relationships among lagrangian particles and the information exchange is not restricted to specific nodes. Therefore, the particle methods have the potential to be applied to the severe FSI problems.
In recent years, many structural analysis methods have been coupled with particle methods to investigate FSI problems. Yang et al. (2012) coupled Smoothed Particle Hydrodynamics (SPH) method with Finite Element Method (FEM). The accuracy of coupling method was verified by comparison with the benchmark tests, such as the sloshing with elastic baffle and flood discharge with elastic gate. Khayyer et al. (2018a)) employed projection-based Incompressible SPH (ISPH) method for simulation of fluid and developed a novel SPH-based structure model. The Fluid-Structure Acceleration-based (FSA) or the Pressure Integration (PI) coupling schemes were compared in detail, The ISPH-SPH method was applied for the water entry of elastic wedge and panel. (Khayyer et al. (2021b)) further developed an ISPH-HSPH solver for simulation of fluid interacting with composite structures. A Hamiltonian SPH (HSPH) was developed for the structure analysis. Falahaty et al. (2018) coupled Moving Least Squares (MLS) method and Dual Particle Dynamics (DPD) structure model with Moving Particle Semi-implicit (MPS) method, respectively. The MLS-MPS model and DPD-MPS model were systematically compared through a series of numerical simulations. Chen et al. (2019) developed an in-house 3-D FSI solver MPSFEM-SJTU based on MPS method and FEM. A Kernel Function-Based Interpolation (KFBI) was introduced to handle the information exchange between MPS boundary particles and FEM grids. The 3-D sloshing in the tank made of elastic material was investigated. Hu et al. (2019) coupled SPH and Smoothed Point Interpolation Method (SPIM) to simulate the sloshing in the tank with elastic baffle. There was not a consistent one-to-one match between SPIM nodes and SPH ghost particles. Physical qualities, such as velocity, displacement and force, are exchanged using interpolation. Long et al. (2016) introduced a particle-element algorithm to ISPH-FEM and SPH-FEM. Through some typical simulations, the effectiveness of this algorithm was proved. Zhang et al. (2020) used Decoupled Finite Particle Method (DFPM) and Smoothed FEM (SFEM) to study the effect of the configuration of elastic baffles to the sloshing. The virtual particle strategy was adopted to treat the interaction between the fluid and structures. Yang and Zhang (2018) used MPS method coupling with Large Eddy Simulation (LES) methods to study the interaction between the dam break flows and elastic baffles. Although both the fluid domain and elastic structures were modelled by MPS methods, the partitioned approach was adopted for the coupling.
Discrete Element Method (DEM) as another lagrangian-particle based method can be used to describe the interaction among the solid bodies and has been widely applied to the engineering and nature field, including the particle flows (Sakai et al., 2012), multi rigid body interaction (Canelas et al., 2016), deformation and fracture of structures (Wu et al., 2016; 2018). The real ocean environment is more complicated, which includes interactions among fluids, particles and structures. Compared with the solid part of afore-mentioned FSI methods, DEM has the potential to tackle more complex ocean problems. The response of elastic structures was usually simulated by DEM with Bonded Particle Model (BPM), which was firstly proposed by Potyondy and Cundall (2004). The moment induced by relative rotation between neighbouring DEM particles is calculated through Hooke's law. Jiang et al. (2005) introduced a rolling contact model to the traditional bond model, which assumed that there were countless springs in the bond between two neighbouring particles. The basic concept of rolling contact model is similar with the BPM. The rolling contact model was further modified to suit for the situation with large relative rotation by Wang (2020). Besides, the force–displacement contact law of DEM was replaced by stress–strain form. The new bond model was successfully applied to simulate the large deformation and fracture of elastic structures. Zhao (2015) adopted only two subnormal springs to transmit a moment between two neighbouring particles.
Several particle-based methods have been coupled with DEM to investigate the FSI problems. There are two main models (7-disc and 9-disc) for structures according to the arrangement of DEM particles (Owen et al., 2020). Tang et al. (2018) coupled SPH and DEM for the FSI problems. Neighbouring DEM particles was joined together by two parallel springs, which was a simplified version of rolling contact model. For the stability of the structure, DEM particles were arranged as a hexagonal scheme (7-disc). However, it is difficult to configure SPH boundary particles. Sun et al. (2019a) adopted 9-disc arrangement of DEM particles and the BPM. The MPS-DEM method was used to investigate the influence of dam break flows to a floating deformable structure with moorings. A more precise bond model containing the rolling contact model (Jiang et al., 2005) is used to simulate the structural response in MPSDEM-SJTU solver. The coupling schemes of MPS/SPH with DEM are not showed in detail in those works mentioned above. In present work, a simple coupling scheme is proposed and its stability and accuracy are checked in detail. Besides, the stress field is presented in the structure of DEM, by which the area where its structure is prone to damage can be easily detect. In addition, some marine structures are very thin such as the ship hull shell and the hydraulic turbine blade. The fluid on both sides of the thin structures will not interfere with each other. The interaction between fluid and thick structures or the fluid only acting on one side of the structure can be simulated by previous SPH-DEM and MPS-DEM models. However, the effect of thin structures to the neighbor particle search have rarely been considered or mentioned. To offset this vacancy, a Partition Technique (PT) is established and implanted into the in-house MPSDEM-SJTU solver. Due to the more accurate neighbouring particles search, the force exerted on the structures is more accurate.
The elastic structures tend to be thin and the corresponding wall particles should be small. Therefore, the total amount of particles will be large in the whole region with single-resolution. In order to improve the computational efficiency, many approaches have been introduced to particle methods. Those approaches can be classified as multi-resolution techniques, parallel technique and GPU acceleration technique. For multi-resolution techniques, the particles with small spacing are only arranged at the field that needs be focused on, which reduces computational load and meets the requirements of local accuracy. Sun et al. (2019b) investigated a violent fluid-structure dam-break problem with multi-phase δ-SPH methods and Adaptive Particle Refinement (APR). The APR region was set around the elastic structure and the coarse particles were splitted into more fine particles when they entered the refinement region. Khayyer et al., 2018b, Khayyer et al., 2021a firstly introduced a multi-resolution method into MPS and ISPH-SPH methods to solve the hydro-elastic FSI problem. The elastic baffle was simulated by fine particles while the flow field was composed of coarse particles. The key idea of parallel technique is to divide tasks into several parts and assign them to different CPU processors for processing at the same time. Marrone et al. (2012) developed a parallel 3-D SPH model, which achieves a high parallel efficiency. The wave pattern generated by a moving ship is successful captured. As a new hardware, GPU is in possession of more Arithmetic Logic Units (ALU) than CPU. Therefore, GPU can achieve high Floating Point Operations Per Second (FLOPS) and deal with large amounts of data simultaneously (Chen and Wan, 2019a). Chen and Wan (2019b) introduced GPU acceleration technique to MPS method and the speed-up ratio can be up to 25–33 times compared with single CPU. However, GPU acceleration technique is not used as widely as multi-CPU parallel technique. In this paper, the multi-CPU parallel technique is introduced to the MPS-DEM method, which also lays a foundation for the application of multi-GPU parallel technique in large-scale flow simulation.
In present study, an in-house parallel solver MPSDEM-SJTU based on MPS method and DEM is developed to simulate FSI problems. Firstly, the improved MPS method, the DEM method, the MPS-DEM coupling scheme, the parallel strategy and Partition Technique (PT) are presented briefly. In the section of numerical simulations, an oscillating cantilever plate is simulated to validate the DEM part of developed solver. Then, several numerical examples, including the hydrostatic water column on an elastic plate, the flood discharge with an elastic gate, sloshing flows in a rolling tank with a thin elastic plate and dam-break with an elastic plate, are carried out to test the accuracy and performance of the MPSDEM-SJTU solver. The parallel efficiency, the convergency and the energy conservation properties of the MPS-DEM solver are tested in the hydrostatic case. In the cases with thin elastic structures, the superiority of the PT for the neighbouring particles search near the thin structures is presented. Finally, the MPSDEM-SJTU solver is successfully extended to simulate the FSI with fracture.
Section snippets
MPS method for fluid simulation
MPS method was firstly proposed by Koshizuka and Oka (1996) for simulation of viscous incompressible fluid. The governing equations of the fluid consist of mass conservation equation and momentum conservation equation, given by,where the , , , , and are the velocity vector, time, fluid density, pressure, kinematic viscosity of the fluid and gravity acceleration vector, respectively.
The interaction among MPS particles controlled by kernel function. In order to
Validation of DEM solver for structure dynamic analysis
The DEM solver is firstly used to simulate an oscillating cantilever plate. Fig. 9 shows the sketch of the model. The cantilever plate is 0.2 m long and 0.02 m thick. At the beginning, the plate without deformation is in the equilibrium position and the initial vertical velocity distribution of the plate is given by,where is the initial vertical velocity at the free end of the plate.
Conclusions
In this paper, the MPS-DEM coupling method is developed to solve FSI problems. The fluid field is modelled by MPS while the deformations of the structures are simulated by DEM. The particles of the structures have characteristics of MPS and DEM. Information, such as force, velocity and displacement, exchanging between MPS and DEM are achieved by interpolation.
In order to enhance the computational efficiency, the parallel technique is introduced to the MPS-DEM method. The misjudgment of neighbor
CRediT authorship contribution statement
Fengze Xie: Data curation, Writing – original draft, Visualization, Investigation, Software, Validation. Weiwen Zhao: Software, Data curation, Visualization, Investigation, Validation. Decheng Wan: Supervision, Conceptualization, Methodology, Investigation, Writing – review & editing.
Declaration of competing interest
The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Decheng Wan reports financial support was provided by Shanghai Jiao Tong University.
Acknowledgements
This work is supported by the National Key Research and Development Program of China (2019YFC0312401 and 2019YFB1704200), National Natural Science Foundation of China (51879159), to which the authors are most grateful.
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