Classical Smoothed Particle Hydrodynamics Research Articles

Smoothed particle hydrodynamics implementation of the standard viscous–plastic sea-ice model and validation in simple idealized experiments

Abstract. The viscous–plastic (VP) rheology with an elliptical yield curve and normal flow rule is implemented in a Lagrangian modelling framework using the smoothed particle hydrodynamics (SPH) meshfree method. Results show, from a perturbation analysis of SPH sea-ice dynamic equations, that the classical SPH particle density formulation expressed as a function of sea-ice concentration and mean ice thickness leads to incorrect plastic wave speed. We propose a new formulation for particle density that gives a plastic wave speed in line with theory. In all cases, the plastic wave in the SPH framework is dispersive and depends on the smoothing length (i.e., the spatial resolution) and on the SPH kernel employed in contrast to its finite-difference method (FDM) implementation counterpart. The steady-state solution for the simple 1D ridging experiment is in agreement with the analytical solution within an error of 1 %. SPH is also able to simulate a stable upstream ice arch in an idealized domain representing the Nares Strait in a low-wind regime (5.3 m s−1) with an ellipse aspect ratio of 2, an average thickness of 1 m and free-slip boundary conditions in opposition to the FDM implementation that requires higher shear strength to simulate it. In higher-wind regimes (7.5 m s−1) no stable ice arches are simulated – unless the thickness is increased – and the ice arch formation showed no dependence on the size of particles, in contrast to what is observed in the discrete-element framework. Finally, the SPH framework is explicit, can take full advantage of parallel processing capabilities and shows potential for pan-Arctic climate simulations.

Improved MPS models for simulating free surface flows

Two improved Laplacian models are introduced in this study for more accurate simulation of free surface flows in the context of the moving particle semi-implicit (MPS) method, a well-known mesh-less approach. The Euler equations are the governing equations of these flows without considering viscous forces which are solved in the Lagrangian frame by the MPS and two-step projection methods for spatial and temporal discretization, respectively. Considering the similarity between the MPS and SPH (smoothed particle hydrodynamics) methods, one of the introduced Laplacian models is formulated by a corrective matrix firstly proposed in an improved SPH method [28]. The other modified Laplacian model is also derived from the divergence of a newly developed MPS gradient model proposed for solving elasticity problems [33]. The higher accuracy of these methods compared to the classic SPH, MPS and even three modified SPH and MPS Laplacian models [7, 34, 44] is verified by solving 2D Poisson equations. Moreover, the excellent performance of the Laplacian models and the improved gradient model in obtaining smooth pressure field is validated by solving three benchmark free surface flows. Furthermore, the problem of wave damping in the original MPS computations can be resolved by using the applied models.

An efficient non-iterative smoothed particle hydrodynamics fluid simulation method with variable smoothing length

In classical smoothed particle hydrodynamics (SPH) fluid simulation approaches, the smoothing length of Lagrangian particles is typically constant. One major disadvantage is the lack of adaptiveness, which may compromise accuracy in fluid regions such as splashes and surfaces. Attempts to address this problem used variable smoothing lengths. Yet the existing methods are computationally complex and non-efficient, because the smoothing length is typically calculated using iterative optimization. Here, we propose an efficient non-iterative SPH fluid simulation method with variable smoothing length (VSLSPH). VSLSPH correlates the smoothing length to the density change, and adaptively adjusts the smoothing length of particles with high accuracy and low computational cost, enabling large time steps. Our experimental results demonstrate the advantages of the VSLSPH approach in terms of its simulation accuracy and efficiency.

Development of time-space adaptive SPH for large deformation and impact

<p indent="0mm">A spatially adaptive smoothed particle hydrodynamics (ASPH) method, where the particles can split and merge according to the adaptive criteria, was developed to improve the computational efficiency of the classical smoothed particle hydrodynamics (SPH) method. Recently, the ASPH algorithm has been widely applied to solve complex problems in mechanics. However, the splitting pattern in the current ASPH algorithm may lead to particle disorder when simulating large deformation and impact problems, and a smaller time step should be performed after particle splitting. An adaptive splitting pattern that is based on a relative position vector and velocity interpolation is introduced in this study. Subsequently, this numerical method is combined with the Runge-Kutta Chebyshev method to achieve adaptive time integration. Thus, the time-space adaptive smoothed particle hydrodynamics method (T-S ASPH) is proposed in this study. This algorithm can increase the computational accuracy by eliminating the particle disorder when simulating the large deformation and impact problems while improving the computational efficiency and time integration accuracy. Typical examples are simulated by the classical SPH, current ASPH and developed T-S ASPH methods. The results indicate that, compared with the other two methods, the proposed T-S ASPH method exhibits higher accuracy, efficiency, stability, and order of convergence for the large deformation and impact problems.

Meshless modeling of bending behavior of bi-directional functionally graded beam structures

Bi-directional Functionally Graded Materials (FGMs) is of more and more focus in special engineering application like aerospace craft at high altitude and speed. In this paper, bending analysis of this type Functionally Graded Beam (FGB) is performed by a proposed meshless Total Lagrangian (TL) Corrective Smoothed Particle Method (CSPM) which owns more capability in accuracy and stability than the classical Smoothed Particle Hydrodynamics (SPH) method. This method is validated by comparison of the bending deformation with analytical method, ABAQUS and a self-programmed finite element code. The influence of the gradient indexes and boundary conditions on the deformation behavior is also investigated. The method is proved to be more precise and stable than the semi-analytical and FE results when dealing with power-law bi-directional FGB.

Coupled SPH and Phase Field method for hydraulic fracturing

AbstractSimulation of fracture initiation and propagation using classical mesh‐based methods involves computationally expensive operations for pre‐processing and (adaptive) remeshing of complex geometries. To overcome the difficulties we propose to use a Smoothed Particle Hydrodynamics (SPH) method to model hydraulic fracturing. SPH is a meshless Lagrangian method highly suitable for large deformations [1].The present contribution discusses a numerical approach to model fractures initiation and propagation, by coupling SPH with the Phase Field method [2]. The proposed hybrid method overcomes the instability problems that can present SPH due to kernel incompleteness. We first validate the proposed model with a stationary elastic fracture and compare the results with the classical SPH to the SPH‐Phase Field approach. (© 2017 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Curvilinear smoothed particle hydrodynamics

SummaryWe suggest a new set of equations to employ smoothed particle hydrodynamics (SPH) in a curvilinear space, and we refer to it as curvSPH. In classical SPH, the horizontal and vertical resolution of discretization is supposed to be equal for fluid particles. However, curvSPH makes the horizontal and vertical resolutions independent from each other. This is performed by transformation of physical space into an appropriate computational space with a different scale in horizontal and vertical directions. Solving a problem using SPH in a curvilinear space also provides capability to model curved boundaries as straight lines. In classical SPH, special care is needed to reach a uniform mass distribution along curved boundaries; however, producing uniform mass distribution along a line using curvSPH is straight forward. Different simulations, including simulation of a flip bucket are performed to demonstrate the applicability of the proposed method. Good agreement of results with experimental data and classical SPH confirms the capabilities of curvSPH. Copyright © 2016 John Wiley &amp; Sons, Ltd.

NIFTy galaxy cluster simulations – II. Radiative models

We have simulated the formation of a massive galaxy cluster (M$_{200}^{\rm crit}$ = 1.1$\times$10$^{15}h^{-1}M_{\odot}$) in a $\Lambda$CDM universe using 10 different codes (RAMSES, 2 incarnations of AREPO and 7 of GADGET), modeling hydrodynamics with full radiative subgrid physics. These codes include Smoothed-Particle Hydrodynamics (SPH), spanning traditional and advanced SPH schemes, adaptive mesh and moving mesh codes. Our goal is to study the consistency between simulated clusters modeled with different radiative physical implementations - such as cooling, star formation and AGN feedback. We compare images of the cluster at $z=0$, global properties such as mass, and radial profiles of various dynamical and thermodynamical quantities. We find that, with respect to non-radiative simulations, dark matter is more centrally concentrated, the extent not simply depending on the presence/absence of AGN feedback. The scatter in global quantities is substantially higher than for non-radiative runs. Intriguingly, adding radiative physics seems to have washed away the marked code-based differences present in the entropy profile seen for non-radiative simulations in Sembolini et al. (2015): radiative physics + classic SPH can produce entropy cores. Furthermore, the inclusion/absence of AGN feedback is not the dividing line -as in the case of describing the stellar content- for whether a code produces an unrealistic temperature inversion and a falling central entropy profile. However, AGN feedback does strongly affect the overall stellar distribution, limiting the effect of overcooling and reducing sensibly the stellar fraction.

Geometrically nonlinear analysis of two-dimensional structures using an improved smoothed particle hydrodynamics method

Purpose – The purpose of this paper is to present an efficient smoothed particle hydrodynamics (SPH) method particularly adapted for the geometrically nonlinear analysis of structures. Design/methodology/approach – In order to resolve the inconsistency phenomenon which systematically occurs in the standard SPH method at the domain’s boundaries of the studied structure, the classical kernel function and its spatial derivatives were modified by the use of Taylor series expansion. The well-known tensile instabilities inherent to the Eulerian SPH formulation were attenuated by the use of the Total Lagrangian Formulation (TLF). Findings – In order to demonstrate the effectiveness of the present improved SPH method, several numerical applications involving geometrically nonlinear behaviors were carried out using the explicit dynamics scheme for the time integration of the PDEs. Comparisons of the obtained results using the present SPH model with analytical reference solutions and with those obtained using ABAQUS finite element (FE) commercial software, show its good accuracy and robustness. Practical implications – An additional application including a multilayered composite structure and involving buckling and delamination was investigated using the present improved SPH model and the results are compared to the FE results, they confirmed both the efficiency and the accuracy of the proposed method. Originality/value – An efficient 2D-continuum SPH model for the geometrically nonlinear analysis of thin and thick structures is proposed. Contrarily to the classical SPH approaches, here the constitutive material relations are used to link naturally the stresses and strains. The Total Lagrangian approach is investigated to alleviate the tensile instabilities problem, allowing at the same time to avoid the updating procedure of the neighboring particles search and therefore reducing CPU usage. The proposed approach is valid for isotropic and multilayered composites structures undergoing large transformations. CPU time savings and better results with the new 2D-continuum SPH formulation compared to the classical continuum SPH. The explicit dynamic scheme was used for time integration allowing a fast resolution algorithm even for highly nonlinear problems.

Smoothed Particle Hydrodynamics for water wave propagation in a channel

In this paper, Smoothed Particle Hydrodynamics (SPH) is used to simulate the propagation of waves in an intermediate depth water channel. The major advantage of using SPH is that no special treatment of the free surface is required, which is advantageous for simulating highly nonlinear flows with possible wave breaking. The SPH method has an option of different formulations with their own advantages and drawbacks to be implemented. Here, we apply the classical and Arbitrary Lagrange–Euler (ALE) formulation for wave propagation in a water channel. The classical SPH should come with an artificial viscosity which stabilizes the numerical algorithm and increases the accuracy. Here, we will show that the use of classical SPH with an artificial viscosity may cause the waves in the channel to decay. On the other hand, we will show that using the ALE-SPH algorithm with a Riemann solver is more stable, and in addition to producing the pressure fields with much less numerical noise, the waves propagate in the channel without dissipation.

Efficient smoothed particle hydrodynamics method for the analysis of planar structures undergoing geometric nonlinearities

In the present investigation, a new modified smoothed particle hydrodynamics (SPH) method particularly adapted for the analysis of planar structures undergoing geometric nonlinearities is introduced and discussed. The problem of inconsistency which is often encountered in the classical SPH method is alleviated by introducing modifications of the kernel function and its derivatives using an explicit polynomial feature representation based on the Taylor series expansion. Tensile instabilities arising in the Eulerian-based SPH formulation which appear due to large deformations are attenuated by the introduction of a total Lagrangian formulation which is robust for large displacements/rotations. The resulting nonlinear problem is solved using the explicit dynamics time integration scheme. The validity of the proposed approach is demonstrated through two numerical applications involving geometrical nonlinearities, where the obtained results are compared to those obtained using the standard finite element method.

A scalable consistent second-order SPH solver for unsteady low Reynolds number flows

Abstract Smoothed Particle Hydrodynamics (SPH) has successfully been used to study a variety of cases involving nearly inviscid flows where conservation properties allow for good physical approximation despite poor theoretical approximation properties of differential operators. When used to study unsteady low Reynolds number flow with large dissipation, conservation alone cannot ensure quality of approximation and the traditional approach is inconsistent. An alternative formulation has recently become popular making use of an approximate splitting scheme to ensure a divergence-free velocity field. However, this scheme relies on an inconsistent discretization of the Laplacian that diverges as particles become disordered under flow. We present an incremental pressure correction scheme and combination of existing differential operator renormalizations that are able to achieve second order accuracy in time and space. A brief review of SPH approximation theory is provided to highlight the necessity of these renormalizations in implementing an approximate factorization scheme. We demonstrate that when fast algebraic multigrid preconditioners are used to solve the resulting linear systems, the scheme results in a consistent approximation that is scalable and amenable to parallelization. Several validation cases are presented for which a speedup of several orders of magnitude is achieved over traditional SPH approaches. Finally, an interface has been developed between the particle library LAMMPS and the sparse linear algebra libraries in Trilinos providing a massively parallel 3D SPH capability. Scaling results for up to 134 million particles on 32,768 cores are presented along with a demonstration of the capability to simulate complex 3D geometries. These results show that the added complexity of applying the necessary consistency corrections actually provides a factor of four speed-up per linear solver iteration versus the uncorrected case, despite the additional cost of constructing the corrections. The resulting library provides a method that is consistent, efficient, and second order in both space and time while maintaining the flexibility of classical SPH for single phase flows.

Evaluation of Accuracy and Stability of the Classical SPH Method Under Uniaxial Compression

The accuracy and stability of the classical formulation of the smoothed particle hydrodynamics (SPH) method for modelling compression of elastic solids is studied to assess its suitability for predicting solid deformation. SPH has natural advantages for simulating problems involving compression of deformable solids arising from its ability to handle large deformation without re-meshing, complex free surface behaviour and tracking of multiple material interfaces. The `classical SPH method', as originally proposed by Monaghan (in Ann Rev Astron 30:543---574, 1992, Rep Prog Phys 68:1703---1759, 2005), has become broadly established as a robust method in different areas, especially involving fluid flows. However, limited attention has been paid to understanding of its numerical performance for elastic deformation problems. To address this, we evaluate the classical SPH method to explore its stability, accuracy and convergence and the effect of numerical parameters on elastic solutions using a generic uniaxial stress test. Short term transient and long term uniform state SPH solutions agree well with those from the finite element method (FEM). The SPH elastic deformation solution showed good convergence with increasing particle resolution. The tensile instability stabilisation method was found to have little impact on the solution, except for higher values of the correction factor which then produce small amplitude benign artificial banded stress patterns. The use of artificial viscosity is able to eliminate the instability and improve the accuracy of the solutions. Overall, the classical SPH method appears to be robust and suitable for accurate modelling of elastic solids under compression.

Efficient meshless SPH method for the numerical modeling of thick shell structures undergoing large deformations

The objective of this paper is to present an extension of the Lagrangian Smoothed Particle Hydrodynamics (SPH) method to solve three-dimensional shell-like structures undergoing large deformations. The present method is an enhancement of the classical stabilized SPH commonly used for 3D continua, by introducing a Reissner–Mindlin shell formulation, allowing the modeling of moderately thin structure using only one layer of particles in the shell mid-surface. The proposed Shell-based SPH method is efficient and very fast compared to the classical continuum SPH method. The Total Lagrangian Formulation valid for large deformations is adopted using a strong formulation of the differential equilibrium equations based on the principle of collocation. The resulting non-linear dynamic problem is solved incrementally using the explicit time integration scheme, suited to highly dynamic applications. To validate the reliability and accuracy of the proposed Shell-based SPH method in solving shell-like structure problems, several numerical applications including geometrically non-linear behavior are performed and the results are compared with analytical solutions when available and also with numerical reference solutions available in the literature or obtained using the Finite Element method by means of ABAQUS© commercial software.

The formation of entropy cores in non-radiative galaxy cluster simulations: smoothed particle hydrodynamics versus adaptive mesh refinement

Abridged: We simulate a massive galaxy cluster in a LCDM Universe using three different approaches to solving the equations of non-radiative hydrodynamics: `classic' Smoothed Particle Hydrodynamics (SPH); a novel SPH with a higher order dissipation switch (SPHS); and adaptive mesh refinement (AMR). We find that SPHS and AMR are in excellent agreement, with both forming a well-defined entropy core that rapidly converges with increasing mass and force resolution. By contrast, SPH exhibits rather different behaviour. At low redshift, entropy decreases systematically with decreasing cluster-centric radius, converging on ever lower central values with increasing resolution. At higher redshift, SPH is in better agreement with SPHS and AMR but shows much poorer numerical convergence. We trace these discrepancies to artificial surface tension in SPH at phase boundaries. At early times, the passage of massive substructures close to the cluster centre stirs and shocks gas to build an entropy core. At later times, artificial surface tension causes low entropy gas to sink artificially to the centre of the cluster. We use SPHS to study the contribution of numerical versus physical dissipation on the entropy core, and argue that numerical dissipation is required to ensure single-valued fluid quantities in converging flows. However, provided this dissipation occurs only at the resolution limit, and provided that it does not propagate errors to larger scales, its effect is benign. There is no requirement to build `sub-grid' models of unresolved turbulence for galaxy cluster simulations. We conclude that entropy cores in non-radiative simulations of galaxy clusters are physical, resulting from entropy generation in shocked gas during cluster assembly, putting to rest the long-standing puzzle of cluster entropy cores in AMR simulations versus their apparent absence in classic SPH simulations.

Geometrically nonlinear analysis of thin-walled structures using efficient Shell-based SPH method

Abstract In this paper a shell-based meshless formulation is proposed for the geometrically nonlinear analysis of thin-walled structures using an explicit dynamics scheme based on the Smoothed Particle Hydrodynamics (SPH) method. In the present investigation the SPH method is modified to deal with shell-like structures, while keeping its character of a strong formulation based on the principle of collocation directly applied on the differential equilibrium equations. The current SPH formulation is an extension of the continuum-improved and stabilized SPH method allowing a thin structure to be modeled using only one layer of particles to represent the shell mid-surface. Application of the present Shell-based SPH (SSPH) formulation for the analysis of several benchmarks including geometrically nonlinear behavior shows its validity and its potential especially in terms of CPU time saving while keeping a very good level of accuracy compared to the classical SPH method and to the Finite Element method.

SPHysics – development of a free-surface fluid solver – Part 1: Theory and formulations

A free-surface fluid solver called SPHysics is presented. Part 1 provides a description of the governing equations based on Smoothed Particle Hydrodynamics (SPH) theory. The paper describes the formulations implemented in the code including the classical SPH formulation along with enhancements like density filtering, arbitrary Lagrange–Euler (ALE) schemes and the incorporation of Riemann solvers for particle–particle interactions. Second-order time stepping schemes are presented along with the boundary conditions employed which can handle floating objects to study fluid–structure interaction. In addition, the model implementation is briefly described. This information will be used in Part 2, where the efficiency of the code is discussed, along with several study cases.

Improvement on stress instability in smoothed particle hydrodynamics

Smoothed particle hydrodynamics (SPH) is a Lagrangian meshfree particle method. It has special advantages in modeling large deformation and free surface flow, and has been widely applied to different problems in engineering and science. However, the classical SPH suffers from stress instability which resticts its further development and applications. The fundamental reason of stress instability is that the stress state and the kernel do not match each other. For frequently used bell-shaped kernel function, in tensile state the attraction between particles increases as particle spacing decreases, thereby leading to tensile instability. In a compressible state, the repulsive force between particles increases, and then decreases as particle spacing decreases, thereby leading to compressible instability. In this paper is presented an approach to removing stress instability in SPH by proposing a new kernel function and a modified SPH discrete form. In the modified SPH, the force between particles is always repulsive and it increases as particle spacing decreases. Two numerical examples are given to test the proposed approachs, and the obtained numerical results clearly demonstrate that the new approach can eliminate stress instability effectively.

Moving least-squares corrections for smoothed particle hydrodynamics

First-order moving least-squares are typically used in conjunction with smoothed particle hydrodynamics in the form of post-processing filters for density fields, to smooth out noise that develops in most applications of smoothed particle hydrodynamics. We show how an approach based on higher-order moving least-squares can be used to correct some of the main limitations in gradient and second-order derivative computation in classic smoothed particle hydrodynamics formulations. With a small increase in computational cost, we manage to achieve smooth density distributions without the need for post-processing and with higher accuracy in the computation of the viscous term of the Navier–Stokes equations, thereby reducing the formation of spurious shockwaves or other streaming effects in the evolution of fluid flow. Numerical tests on a classic two-dimensional dam-break problem confirm the improvement of the new approach.

State-of-the-art of classical SPH for free-surface flows

Smoothed Particle Hydrodynamics (SPH) is the most widely established mesh-free method which has been used in several fields as astrophysics, solids mechanics and fluid dynamics. In the particular case of computational fluid dynamics, the model is beginning to reach a maturity that allows carrying out detailed quantitative comparisons with laboratory experiments. Here the state-of-the-art of the classical SPH formulation for free-surface flow problems is described in detail. This is demonstrated using dam-break simulations in 2-D and 3-D. The foundations of the method will be presented using different derivations based on the method of interpolants and on the moving least-squares approach. Different methods to improve the classic SPH approach such as the use of density filters and the corrections of the kernel function and its gradient are examined and tested on some laboratory cases