Smoothed Particle Hydrodynamics Approximation Research Articles

An efficient truncation scheme for Eulerian and total Lagrangian smoothed particle hydrodynamics methods

In smoothed particle hydrodynamics (SPH) method, the particle-based approximations are implemented via kernel functions, and the evaluation of performance involves two key criteria: numerical accuracy and computational efficiency. In the SPH community, the Wendland kernel reigns as the prevailing choice due to its commendable accuracy and reasonable computational efficiency. Nevertheless, there exists an urgent need to enhance computational efficiency while upholding accuracy. In this paper, we employ a truncation approach to limit the compact support of the Wendland kernel to 1.6h. This decision is based on the observation that particles within the range of 1.6h to 2h make negligible contributions to the SPH approximation. To decrease numerical errors from SPH approximation and the truncation method, we incorporate the Laguerre–Gauss kernel for particle relaxation to obtain the high-quality particle distribution with reduced residue [Wang et al., “A fourth-order kernel for improving numerical accuracy and stability in Eulerian and total Lagrangian SPH,” arXiv:2309.01581 (2023)], and the kernel gradient correction to rectify integration errors. A comprehensive set of numerical examples including fluid dynamics in Eulerian formulation and solid dynamics in total Lagrangian formulation are tested and have demonstrated that truncated and non-truncated Wendland kernels enable achieving the same level of accuracy but the former significantly increases the computational efficiency.

On the accuracy of SPH formulations with boundary integral terms

Handling boundaries in smoothed particle hydrodynamics (SPH) is believed to be one of the most critical problems in studies and applications of SPH. In the last two decades, excellent progress has been made in exploring the evaluation of the boundary integral terms directly to improve SPH approximation on the boundary. However, because of the fundamental difficulties, formulations in this regard are difficult to be made consistent when using SPH particle approximations, which significantly affect the solution accuracy. In this work, an intensive investigation is made in carefully analyzing the order of accuracy of the SPH approximation when boundary integral terms are included. Both kernel consistent and particle consistent boundary integral formulations of SPH are examined in great detail. Notably, a novel particle consistent boundary integral formulation for first-order derivatives is derived. An approximation of this new formulation, which is weakly consistent, is also discussed. The theoretical analysis and numerical experiments show that the SPH with particle consistent boundary integral formulation outperforms significantly in terms of accuracy compared to the correction factors used in traditional SPH implementations.

A computationally efficient SPH framework for unsaturated soils and its application to predicting the entire rainfall-induced slope failure process

The first and fully validated smoothed particle hydrodynamics (SPH) model is presented to tackle coupled flow–deformation problems in unsaturated porous media that undergo large deformation and post-failure behaviour. Unlike the commonly adopted double-layer SPH framework for saturated soils, this paper presents a three-phase single-layer SPH model capable of predicting anisotropic seepage flows through porous media and their complete time-dependent transition from unsaturated to saturated states, as well as their influence on the mechanical behaviour of the porous media and vice versa. The mathematical framework is developed based on Biot's mixture theory and discretised using the authors’ recently developed novel SPH approximation scheme for the second derivatives of a field quantity. The soil is modelled using a suction-dependent elastoplastic constitutive model, expressed in terms of effective stress and suction. In addition, an adaptive two-timescale scheme is proposed for the first time to address existing challenges in solving coupled-flow large-deformation problems that involve a significant difference in the timescale required for the solid and fluid phases. The capability of the proposed SPH model was demonstrated through fundamental consolidation tests and a large-scale rainfall-induced slope failure experiment. Very good agreements with theoretical solutions and experimental results are achieved, suggesting that the proposed SPH model can be readily extended to solve a wide range of large-scale geotechnical applications involving coupled unsaturated seepage–deformation problems.

On the convergence of the solutions to the integral SPH heat and advection–diffusion equations: Theoretical analysis and numerical verification

The convergence of the continuous Smoothed Particle Hydrodynamics (SPH) solution to the advection–diffusion equation is established in this paper. As a by-product of this result, the convergence of the SPH heat equation is also established. The most prevalent SPH approximations to diffusion operators are considered. The convergence is demonstrated using Fourier Analysis. First, the Fourier representation of the SPH approximation to the Hessian matrix is obtained. Next, the Fourier representations of the SPH diffusion operators are derived. Finally, the convergence of the SPH solution to the advection–diffusion equation (which includes, in particular, the case of the heat equation) is established in the Fourier space. The proof of convergence requires a certain condition over the Fourier transform of the SPH kernel to be satisfied. An example of kernel not satisfying this condition is numerically tested, leading to non-convergent solutions. To the authors’ knowledge, this is the first proof of convergence of an SPH equation involving the Laplacian operator. These analytical results are illustrated with a variety of numerical verification cases. In particular, the influence of the choice of the kernel on the nature of convergence is thoroughly studied.

A finite particle method (FPM) for Lagrangian simulation of conservative solute transport in heterogeneous porous media

When a smoothed particle hydrodynamics (SPH) method, a Lagrangian and meshfree numerical scheme, is used to solve the advection-dispersion equation (ADE), SPH solutions may not be accurate when particles are irregularly distributed, i.e., the distance between two neighbor particles varies irregularly in a simulation domain. Particle irregularity may be caused by nonuniform groundwater flow in a heterogeneous field of hydraulic conductivity. This study explores for the first time whether the Finite Particle Method (FPM) can provide more accurate ADE solutions than SPH does for irregularly distributed particles. FPM is similar to SPH in theory, but uses a modified kernel gradient to construct a SPH approximation of solute concentration gradients. Performance of SPH and FPM with irregularly distributed particles is evaluated by using two numerical cases. The first case considers only diffusive transport, and has an analytical solution for the evaluation. The second case considers both advection and dispersion, and uses a numerical solution as a reference for the evaluation. For each of the two cases, several numerical experiments are conducted using multiple sets of irregularly distributed particles with different levels of particle irregularity due to different levels of heterogeneity of hydraulic conductivity. Numerical results indicate that, for the numerical experiments of this study, FPM outperforms SPH to yield more accurate ADE solutions. However, FPM solutions are still subject to numerical errors, and the errors increase when the level of heterogeneity of hydraulic conductivity increases. Further improvement of FPM is warranted in a future study.

Smoothed particle hydrodynamics (SPH) and its applications in geomechanics: From solid fracture to granular behaviour and multiphase flows in porous media

We present the fundamental concepts of SPH with particular emphasis on its state-of-the-art applications in geomechanics and geotechnical engineering. In the first part of the paper, we focus on establishing fundamental SPH equations and discussing how they are used to solve partial differential equations (PDEs) in geomechanics. Through this process, we expect to provide readers with a better understanding of SPH formulations to avoid misuse or misinterpretation of its capacity and limitation. Discussions on several outstanding issues and recommendations for further developments are also be presented. Of particular interest through this revisit of the key SPH concepts is a new and robust SPH approximation formulation for the Laplacian, which involves the second-order derivatives of a field quantity. This new formulation is proven to outperform existing SPH formulations and achieve high accuracy. The second part of the paper focuses on demonstrating the applications of SPH in the fields of geomechanics and geotechnical engineering through various examples, ranging from the most fundamental to more complex applications involving multi-phase flows. We hope that this paper will become a useful resource to provide readers with a better understanding of SPH and its potential in solving complex problems in geomechanics and geotechnical engineering.

A normalized iterative Smoothed Particle Hydrodynamics method

In this paper we investigate on a normalized iterative approach to improve the Smoothed Particle Hydrodynamics (SPH) estimate of a function. The method iterates on the residuals of an initial SPH approximation to obtain a more accurate solution. The iterative strategy preserves the matrix-free nature of the method, does not require changes on the kernel function and it is not affected by disordered data distribution. The iterative refinement is further improved by ensuring linear approximation order to the starting iterative values. We analyze the accuracy and the convergence of the method with the standard and normalized formulation giving evidence of the enhancements obtained with both uniform and non-uniform data density. Numerical experiments in 2D domain with different data sets are presented to validate the iterative approach.

A Total Lagrangian SPH method for modelling damage and failure in solids

An algorithm is proposed to model crack initiation and propagation within the Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) framework. TLSPH avoids the tensile instability encountered in conventional Eulerian kernel-based Smoothed Particle Hydrodynamics (SPH) by making use of the Lagrangian kernel. In the present approach, the support domain of a particle is modified, where it only interacts with its immediately neighbouring particles. The gradient correction is employed to avoid the inconsistency of SPH approximation induced by insufficient neighbouring particles. A virtual link is used to define the level of interaction between each particle pair. The state of the virtual link is determined by damage law or cracking criterion. The virtual link approach allows easy and natural modelling of cracking surfaces without explicit cracking treatments such as particle splitting, field enrichment or visibility criterion. The performance of the proposed approach is demonstrated via a few numerical examples of both brittle and ductile failure under impact loading.

A consistent multiphase SPH approximation for bubble rising with moderate Reynolds numbers

Phenomena involving bubble flow have an important role in numerous applications such as mixing, separation, filtration and cooling. When two different phases are treated as liquids–gas, the surface tension and buoyancy must be taken into account. An alternative is proposed to simulate flow for two or more phases, using bubbles formation and the Navier–Stokes equations in a Lagrangian formalism via smoothed particle hydrodynamics (SPH). It is a mesh-free method useful for applications with interface flow. Therefore, we have presented a set of numerical methodologies for SPH in multiphases. Surface tension interface is modeled using the continuum surface force (CSF) method. In order to avoid tension instability and interface penetrations, a background pressure based on the initial pressure between phases is included in the formulation. This model is implemented inside the prediction–correction of time upgrade scheme. Examples of bubble rising around the fluid due to the gravitational force are rarely analyzed in the SPH literature, especially to parameterize density, viscosity, surface coefficient, particle size and boundary conditions. Thus, this work focus on the analysis of these parameters and their effects on the morphology, displacement and velocity of the bubble simulations. Finally, results demonstrate a good numerical stability and adequate multiphase description.

A generic approach to modelling flexible confined boundary conditions in SPH and its application

SummaryIn this paper, a new approach to applying confining stress to flexible boundaries in the smoothed particle hydrodynamics (SPH) method is developed to facilitate its applications in geomechanics. Unlike the conventional SPH methods that impose confining boundary conditions by creating extra boundary particles, the proposed approach makes use of kernel truncation properties of SPH approximations that occur naturally at free‐surface boundaries. Therefore, it does not require extra boundary particles and, as a consequence, can be utilised to apply confining stresses onto any boundary with arbitrary geometry without the need for tracking the curvature change during the computation. This enables more complicated problems that involve moving confining boundaries, such as confining triaxial tests, to be simulated in SPH without difficulties. To further enhance SPH applications in elasto‐plastic computations of geomaterials, a robust numerical procedure to implement Mohr‐Coulomb plasticity model in SPH is presented for the first time to avoid difficulties associated with corner singularities in Mohr‐Coulomb model. The proposed approach was first validated against two‐dimensional finite element (FE) solutions for confining biaxial compression tests to demonstrate its predictive capability at small deformation range when FE solutions are still valid. It is then further extended to three‐dimensional conditions and utilised to simulate triaxial compression experiments. Simulation results predicted by SPH show good agreement with experiments, FE solutions, and other numerical results available in the literature. This suggests that the proposed approach of imposing confining stress boundaries is promising and can handle complex problems that involve moving confining boundary conditions.

Cylindrical Smoothed Particle Hydrodynamics Simulations of Water Entry

This paper presents a smoothed particle hydrodynamics (SPH) modeling technique based on the cylindrical coordinates for axisymmetrical hydrodynamic applications, thus to avoid a full three-dimensional (3D) numerical scheme as required in the Cartesian coordinates. In this model, the governing equations are solved in an axisymmetric form and the SPH approximations are modified into a two-dimensional cylindrical space. The proposed SPH model is first validated by a dam-break flow induced by the collapse of a cylindrical column of water with different water height to semi-base ratios. Then, the model is used to two benchmark water entry problems, i.e., cylindrical disk and circular sphere entry. In both cases, the model results are favorably compared with the experimental data. The convergence of model is demonstrated by comparing with the different particle resolutions. Besides, the accuracy and efficiency of the present cylindrical SPH are also compared with a fully 3D SPH computation. Extensive discussions are made on the water surface, velocity, and pressure fields to demonstrate the robust modeling results of the cylindrical SPH.

On different calculation formulas of the pressure term in bi-phase SPH simulations

The smoothed particle hydrodynamics (SPH) method is usually expected to be an efficient numerical tool for calculating the multiphase phenomena in compressors. However, numerical instability arises around the interfaces due to the pressure inconsistency. By analyzing different formula of the pressure term under SPH approximation, we show that the frequently-used formulation (n=-1) leads to the worst performance and is thus not recommended. By contrast, we suggest the formulas n=0 and 1 in future practical SPH applications.

Smoothed Particle Hydrodynamics with Stress Points and Centroid Voronoi Tessellation (CVT) Topology Optimization

Various formulations of smoothed particle hydrodynamics (SPH) have been presented by scientists to overcome inherent numerical difficulties including instabilities and inconsistencies. Low approximation accuracy could cause a result of particle inconsistency in SPH and other meshfree methods. In this study, centroid Voronoi tessellation (CVT) topology optimization is used for rearrangement of particles so that the inconsistency due to irregular particle arrangement can be corrected. Using CVT topology optimization method, the SPH particles, which are generated randomly inside a predetermined domain, are moved to the centroids, i.e., the center of mass of the corresponding Voronoi cells based on Lloyd’s algorithm. The volume associated with each particle is determined by its Voronoi cell. On the other hand, it has been shown that particle methods with stress point integration are more stable than the ones using nodal integration. Conventional SPH approximations only use SPH particles, and it results in the so-called tensile instability. In this paper, a new approach of using stress points is introduced to assist SPH approximations and stabilize the SPH methods.

Simulation of anisotropic diffusion processes in fluids with smoothed particle hydrodynamics

SummaryAnisotropic diffusion phenomenon in fluids is simulated using smoothed particle hydrodynamics (SPH). A new SPH approximation for diffusion operator, named anisotropic SPH approximation for anisotropic diffusion (ASPHAD), is derived. Basic idea of the derivation is that anisotropic diffusion operator is first approximated by an integral in a coordinate system in which it is isotropic. The coordinate transformation is a combination of a coordinate rotation and a scaling in accordance with diffusion tensor. Then, inverse coordinate transformation and particle discretization are applied to the integral to achieve ASPHAD. Noting that weight function used in the integral approximation has anisotropic smoothing length, which becomes isotropic under the inverse transformation. ASPHAD is general and unique for both isotropic and anisotropic diffusions with either constant or variable diffusing coefficients. ASPHAD was numerically examined in some cases of isotropic and anisotropic diffusions of a contaminant in fluid, and the simulation results are very consistent with corresponding analytical solutions. Copyright © 2016 John Wiley & Sons, Ltd.

A smoothed particle hydrodynamics (SPH) model for simulating surface erosion by impacts of foreign particles

This paper presents a smoothed particle hydrodynamics modeling procedure to simulate surface erosion by impacts. The present SPH is improved based on the standard SPH formulation, by developing SPH formulations for the constitutive relations to describe plastic behavior and ductile fracture process. The improved SPH method is first applied to simulate impacts of particles on ductile targets. Two modified schemes in terms of density correction and kernel gradient correction are adopted to improve the accuracy of the SPH approximation. The present SPH model is applied to simulate the erosion process by angular particles. The results are compared with available experimental data, and good agreements have been achieved in terms of the crater profile and particle kinematic results.

Towards consistence and convergence of conservative SPH approximations

Typical conservative smoothed particle hydrodynamics (SPH) approximations of the gradient of a scalar field introduce two errors: one (smoothing error) is due to smoothing of the gradient by an integration associated with a kernel function; the other (integration error) is due to approximating the integration by summation over all particles within the kernel support. When particles are not on a uniform grid, the integration error leads to violation of zero-order consistency, i.e. the inability to reproduce a constant field. In this paper we confirm that partition of unity is the condition under which the conservative SPH approximation achieves both consistence and convergence. We show that this condition can be achieved by relaxing a particle distribution under a constant pressure field and invariant particle volume. The resulting particle distribution is very similar to that is typical for liquid molecules. We further show that with two different typical kernel functions the SPH approximation, upon satisfying the partition of unity property, is able to achieve very high-order of the integration error, which previously could be shown only with particles on a uniform grid. The background pressure used in a weakly compressible SPH simulation implies a self-relaxation mechanism, which explains that convergence with respect to increasing particle numbers could be obtained in SPH simulations, although not predicted by previous numerical analysis. Furthermore, by relating the integration error to the background pressure, we explain why the previously proposed transport-velocity formulation of SPH (S. Adami et al. (2013) [1]) is able to achieve unprecedented accuracy and stability.

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.

On the SPH Approximations in Modeling Water Waves

Abstract This paper presents an examination of approximation aspects of the Smoothed Particle Hydrodynamics (SPH) in modeling the water wave phenomenon. Close attention is paid on consistency of the SPH formulation and its relation with a correction technique applied to improve the method accuracy. The considerations are confined to flow fields within finite domains with a free surface and fixed solid boundaries with free slip boundary conditions. In spite of a wide application of the SPH method in fluid mechanics, the appropriate modeling of the boundaries is still not clear. For solid straight line boundaries, a natural way is to use additional (virtual, ghost) particles outside the boundary and take into account mirror reflection of associated field variables. Such a method leads to good results, except for a vicinity of solid horizontal bottoms where, because of the SPH approximations in the description of pressure, a stratification of the fluid material particles may occur. In order to illustrate the last phenomenon, some numerical tests have been made. These numerical experiments show that the solid fluid bottom attracts the material particles and thus, to prevent these particles from penetration into the bottom, a mutual exchange of positions of real and ghost particles has been used in a computation procedure.

Unified semi-analytical wall boundary conditions in SPH: analytical extension to 3-D

Solid wall boundary conditions have been an area of active research within the context of Smoothed Particle Hydrodynamics (SPH) for quite a while. Ferrand et al. (Int. J. Numer. Methods Fluids 71(4), 446---472, 2012) presented a novel approach using a renormalization factor in the SPH approximation. The computation of this factor depends on an integral along the boundary of the domain and in their original paper Ferrand et al. gave an analytical formulation for the 2-D case using the Wendland kernel. In this paper the formulation will be extended to 3-D, again providing analytical formulae. Due to the boundary being two dimensional a domain decomposition algorithm needs to be employed in order to obtain special integration domains. For these the analytical formulae will be presented when using the Wendland kernel. The algorithm presented within this paper is applied to several academic test-cases for which either analytical results or simulations with other methods are available. It will be shown that the present formulation produces accurate results and provides a significant improvement compared to approximative methods.

A Contact SPH Method with High-Order Limiters for Simulation of Inviscid Compressible Flows

AbstractIn this paper, we study a class of contact smoothed particle hydrodynamics (SPH) by introducing Riemann solvers and using high-order limiters. In particular, a promising concept of WENO interpolation as limiter is presented in the reconstruction process. The physical values relating interactional particles used as the initial values of the Riemann problem can be reconstructed by the Taylor series expansion. The contact solvers of the Riemann problem at contact points are incorporated in SPH approximations. In order to keep the fluid density at the wall rows to be consistent with that of the inner fluid wall boundaries, several lines of dummy particles are placed outside of the solid walls, which are assigned according to the initial configuration. At last, the method is applied to compressible flows with sharp discontinuities such as the collision of two strong shocks and the interaction of two blast waves and so on. The numerical results indicate that the method is capable of handling sharp discontinuity and efficiently reducing unphysical oscillations.