An efficient MPS refined technique with adaptive variable-size particles

Highlights

• The shape and position of the high-resolution domains were able to dynamically trace the moving/deformable body with mass and momentum conservation.

• The particle size is not unique but a range for each resolution domains, and the principle of resolution selection between neighboring domains is optimized.

• New splitting/coalescing criterion is proposed to avoid chain split/coalesce reaction, and to improve the convergence near the multi-resolution boundaries.

• The AVSP-MPS method sharply reduces the particle number and the computational time.

Abstract

For the single-resolution particle method, large computation demand with high accuracy commonly requires enormous number of particles, which becomes greatly time-consuming. The moving particle semi-implicit (MPS) method with variable-size particles (VSP) has been proposed to refine particles in local domain. However, in VSP-MPS method, the refined area is fixed in the calculation, which makes it difficult to precisely capture flow near moving and deformable boundaries. This study develops a MPS method with adaptive variable-size particles model, called AVSP-MPS. In this method, by capturing moving and deformable boundaries, the objective computational domain can be dynamically refined. A new criterion, that splitting and coalescing only happen when the entire particle crosses the refined interface, is proposed to avoid unnecessary chain reactions of particle splitting and coalescing. The stability of the method is therefore improved. The district resolution is not a specific value, but a range in VSP model. An inappropriate selection of resolution range would cause repetitive splitting and coalescing, leading to low efficiency of the computation. In this research, the new maximum and minimum volumes in the region with different resolutions, as an optimized result, are obtained to improve convergence near the multi-resolution boundaries. Several cases as dam break and water entry are verified. The numerical results show that the AVSP-MPS method has an efficient, flexible and stable refinement technique in treating complex flow with large deformation.

Introduction

In conventional mesh-based computational methods, variable resolutions can be easily achieved by refined structured/unstructured meshes to improve accuracy in specific computational domains. When analyzing unsteady flow involving large deformations, particle-based methods as the moving particle semi-implicit (MPS) [[1], [2]] method can avoid mesh reconstruction, and have been used extensively in engineering applications. However, single-resolution computation requires a large number of particles with small size to satisfy increasingly computation requirements, which is time consuming. Multi-resolution particle methods have therefore been developed to reduce computational cost.

In contrast to MPS method, incompressible flow in smoothed particle hydrodynamics (SPH) [[3], [4]] is usually solved by introducing a weakly compressible scheme (WCSPH [5]). To increase the accuracy of local calculation, many researchers have proposed initial arrangements of particles with different resolutions. Omidvar [6] generated the initial particle distribution with different sizes to simulate the test of waves with cylinder, where fine particles are closer to the cylinder while coarse particles are farther away. Khayyer [7] proposed a multi-resolution ISPH-SPH fluid–structure interaction (FSI) solver in which fine particles are used to simulate fluid motion and coarse particles are used to simulate structure. Other researchers dynamically changed particle size in the calculation process to improve efficiency. Feldman [8] proposed a method to dynamically refine particles in which candidate particles split into several “daughter” particles according to a given refinement pattern. Though energy and mass are easily conserved in particle splitting, their conservation in particle coalescing encounters difficulties, which is not discussed in the article. Chiron [9] presented the basis of an adaptive particle refinement (APR [10]) technique that was inspired by adaptive mesh refinement (AMR [[11], [12]]) in mesh-based methods. This approach includes both particle splitting and coalescing processes, so the change of the resolution is more flexible. However, the background mesh used in this research remains limitation in simulating more complex flows. Sun [[13], [14]] implemented a particle shifting technique (PST) in the framework of δ⁺-SPH combined with APR and used it to realize tensile instability control (TIC). High computational cost and numerical tensile instability are avoided in the δ⁺-SPH scheme. However, mass and momentum are not always conserved due to particle deletion in the coalescing process. Vacondio [15] dynamically changed particle size by means of particle splitting and coalescing with conservation of both mass and momentum. The relevant simulations showed that the particle refinement procedure could increase efficiency while maintaining the same level of accuracy as a uniform distribution with high resolution.

Tanaka [16] developed a multi-resolution technique for the MPS method in two dimensions. However, the formulation was derived for the classical MPS method, and its accuracy and convergence need to be improved. Tang [17] improved its stability by changing kernel functions and cutoff radii for neighboring particles of different sizes. Khayyer [18] proposed a multi-resolution MPS-based FSI solver for efficient and accurate simulations of incompressible fluid flows interacting with elastic structures, and the scheme can accurately simulate both the pressure of the fluid and the stress of the solid. In contrast to making initial arrangements for particles with different resolutions, some researchers have sought to improve computational efficiency by dynamic particle refinement technology. Shibata [[19], [20]] developed a multi-resolution technique called the overlapping particle technique (OPT). In this method, the entire simulation domain is divided into overlapping sub-domains, and all of them have their own spatial resolutions and particle sizes. Nevertheless, the total mass of the algorithm is not conserved in the OPT. Tanaka [21] developed a multi-resolution MPS method with various particle sizes by using dynamic particle splitting and coalescing algorithm. In this method, the surface detection algorithm is modified for multi-resolution simulations to reduce the fluctuation in pressure caused by the incorrect identification of surface particles. To improve the stability of calculations, Chen [22] developed a multi-resolution MPS method with variable-size particles (VSP-MPS) that is based on an improved MPS method with no surface detection (NSD-MPS) [23]. In this method, there is no need to detect surface particles. A gradient model associated with different particle size is used, and the effective radii of all particles remain the same to ensure the Newton's third law.

In this study, we propose the MPS method with adaptive variable-size particles, called AVSP-MPS. First, for most variable-resolution MPS methods, domains with different resolution are set in the preprocessing, and their locations and shapes are fixed in the calculation, which makes it difficult to simulate problems involving complex flow, such as flow with moving or deformable boundaries [[24], [25]]. A new adaptive refined region is proposed in AVSP-MPS method. The distribution of the high-resolution region in this method changes dynamically according to the movement or deformation of the objective computational domain, and the spatial resolution changes adaptively during the calculation. Second, particles with different size exist near the coarse/fine interface decreasing the stability of the simulation. In this research, unnecessary chain reactions of particle splitting and coalescing are avoided to increase the stability of the calculation because splitting and coalescing only happen when the entire particle crosses the interface of domains with different resolution. Third, as the resolution of the refined area is variable in VSP model, an inappropriate selection of resolution range would cause repetitive splitting and coalescing, leading to low efficiency of the computation. In this research, the detailed conditions for optimizing resolution range are proposed theoretically. The new maximum and minimum volumes in the region with different resolutions are obtained. Several cases as conventional dam-break, dam-break with elastic structure and water entry are simulated using AVSP-MPS method.