Weakly Compressible Research Articles

Discussion on “comparative study on volume conservation among various SPH models for flows of different levels of violence, coastal engineering, volume 191, August 2024, 104 521” by Wang et al

The paper by Wang et al. (2024) targets an interesting topic in the context of particle methods, namely, volume conservation. This important feature is studied by focusing on the time variation of mean water level under three different scenarios corresponding to hydrostatics, standing wave and dam break, and through consideration of a few variants of the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) fluid model. In this discussion, the discussers briefly discuss some issues that could have been further considered in the paper by Wang et al. (2024) to ensure rigorousness and comprehensiveness of the conducted comparative analysis.

On the partition noise in chosen particle weighted methods and its consequences for weakly-compressible flow models

The Meshless Finite Volume (MFV) method, which is a member of the particle weighted methods (PWM) family, is analysed in the context of the so-called partition noise problem. This issue is inherent to PWM-type discretisation which is based on compact support kernel functions and particle sampling. The partition noise exhibits itself as spurious oscillations observed in the solution; they are caused by unphysical source terms in the discretised equations and approximate local volumes computation. Unfortunately, decreasing the discrete length scale which controls the spatial resolution reduces the partition noise only weakly (and never completely). When choosing numerical schemes, a special care is required since in naive implementations the partition noise may even lead to loss of stability. However, even when stability is assured, PWM does not fulfil the assumptions of the Lax equivalence theorem (i.e. the discretised equations are not consistent), therefore convergence in the classical meaning cannot be guaranteed.As an example of a problematic use of MFV, we investigate its application to the solution of fluid flow equations at low Mach numbers, aiming to mimic truly incompressible flow conditions. Taken here as isothermal Euler equations, they are the basis of the so-called weakly-compressible (WC) flow models. Such modelling does not involve any elliptic-type equation and, due to the algorithmic and computational reasons, seems to be well-suited for particle methods. The WC flow model imposes, however, a serious challenge for PWM since: (i) the method is not gauge invariant and the disparity between the speed of sound and the convective velocity scale causes a significant spurious source of momentum; (ii) the inaccuracy in the local volumes approximation leads to errors in the computed density which are comparable to physically relevant slight variations, inherent to the WC approach. We propose specific numerical and modelling measures: controlled relaxation of initial particle distribution, particle volumes computation using an optimal kernel function, a choice of gradient limiting techniques and a carefully chosen background pressure. Although these improvements significantly increase the accuracy (however at remarkably increased computational cost), the theoretical 2nd-order convergence is achieved only at lower spatial resolutions when the truncation errors dominate. The accuracy of mesh-based methods is achieved in PWM only for the pressure field; the errors in the velocity field are much larger and convergence is at most 1st-order. We point some contradictory requirements of the model equations and the numerical method. One of the conclusions is that it is doubtful to construct a convergent truly Lagrangean method of PWM type at reasonable computational cost. The present findings also apply to conservative variants of the weakly-compressible Smoothed Particle Hydrodynamics which, being first order consistent in the best case, suffers from the partition noise even more severely than the considered MFV approach which is theoretically second order consistent. The conclusions pose a question about a rationale for the use of WC-PWM approach in chosen applications, in particular free surface and interfacial flows.

A three-dimensional high-order flux reconstruction lattice boltzmann flux solver for incompressible laminar and turbulent flows

A three-dimensional (3D) high-order flux reconstruction lattice Boltzmann flux solver (FR-LBFS) is developed in this paper for accurate and efficient simulations of incompressible laminar and turbulent flows. The original lattice Boltzmann flux solver (LBFS) is a second-order finite volume method (FVM) which combines the advantages of conventional Navier-Stokes (N-S) solvers and lattice Boltzmann equation (LBE) solvers. But it is not convenient to construct high-order FVM, and the compactness is always a bottleneck. The present work separates LBFS from FVM and combines LBFS with high-order FR scheme. Thus, the FR-LBFS inherits the superiority of LBFS and FR and shares the following attractive features: (i) a fully-explicit arbitrary order compact method, (ii) weakly compressible (WC) model for unsteady incompressible flows and (iii) evaluating inviscid and viscous fluxes simultaneously and no particular technique requirement, such as local discontinuous Galerkin method (LDG), for viscous term discretization. To validate the current method, various 3D incompressible laminar isothermal and thermal numerical experiments are presented first. Good agreements are achieved between the current results and the previous experimental and numerical data whilst the present high-order solver adopt coarse meshes. Furthermore, the ability of the proposed method to perform accurate and stable computations of incompressible decaying homogeneous isotropic turbulent flows and turbulent channel flows with different mesh resolutions and accuracy orders are investigated. The results show that the present 3D FR-LBFS is a promising tool for under-resolved or implicit large eddy simulation (ILES) of incompressible turbulent flows.

Design optimization of sloshing tank using weakly compressible mesh free model

The sloshing effects from baffle and non-baffled water tank was investigated under different external excitation frequencies and amplitudes. It was simulated and analyzed using the Weakly Compressible Moving Particle Explicit (WCMPE) method under condition of high-precision simulation data. The effects from different baffle configuration, external excitation frequency and amplitude, and the behaviour of liquid sloshing were analyzed through the conversion of simulated data and images. The results showed that half-proportional baffles resulted in a highly complex water motion, which resulted in the damping of the liquid slosh. On the other hand, the sloshing of the water in the tank at non-natural frequency resulted in mutual cancellation of the particle velocities. With greater understanding of the sloshing and damping effects, the use of half-scale baffle and liquid depth at different position were further examined. Three case scenarios were simulated and analyzed. First, the length of the baffle was kept constant with gradual lifting of the baffle from the bottom of the tank until it protrudes the water surface. Secondly, the length of the baffle was gradually increased from bottom of tank until it protrudes above the water surface to explore the optimal damping effect. Lastly, the top of the baffle is situated immediately below the water surface. Determination of the optimal ratio of baffle length and liquid depth was completed from the three hypothetical case studies.

Simulation of droplet bouncing on flexible substrate in 2D and 3D with WC-TL SPH method

Droplet impacts on super-hydrophobic surfaces play an important role in many fields. Previous studies have focused on rigid substrates and ignored the effect of substrate deformation. Simulating droplet impacts on a flexible substrate is a challenge to numerical methods due to the coupling of free surface, elasticity, large deformation and surface tension. In this study, a fully meshfree model of fluid–solid interaction is developed based on the smoothed particle hydrodynamics (SPH) method. The liquid droplet is modeled by the weakly compressible (WC) SPH method, and the flexible substrate is modeled by the total Lagrangian (TL) SPH method. Liquid surface tension is modeled by the continuum surface force (CSF) method through reconstructing the free surface of the droplet. The surface geometry reconstruction process is given for both 2D and 3D cases, enabling the model to simulate the bouncing of droplets in both 2D and 3D cases. Furthermore, in order to reduce the computational cost of 3D simulation, the TL-SPH model combined with shell theory established the TL-SPH-shell model, thus achieving the simulation of 3D droplet impact on thin plates. For verification and testing, the established WC-TL SPH model is used to simulate the process of droplet impact on various substrates, including fixed-fixed and curved beams, micro-pillar substrates, and cantilever beams. The results show that the spreading, retraction and bouncing processes of the droplets interact with the deformation and recovery of the flexible substrate, accompanied by large deformation and dynamic characteristics, which can be effectively simulated by the model.

FEATURES OF THE USE OF PARALLEL CALCULATIONS IN THE PACKAGE OF APPLICATION PROGRAMS «MIRELA+»

The article examines approaches to the optimization of structural calculations, which require a comprehensive analysis of deformation processes under the action of operational loads. The finite element method is usually used to solve such problems. Modeling of structures taking into account stress concentrators requires the use of large-sized calculation grids. In real operating conditions, most structural elements are under the influence of mechanical and thermal loads. The package of application programs «MIRELA+» is intended for solving problems of thermoelasticity of structures made of weakly compressible elastomers and composites with an elastomeric matrix. One of the stages of solving problems of thermoelasticity is the formation of stiffness matrices taking into account weak compressibility and thermal conductivity of finite elements. When forming the stiffness and thermal conductivity matrices, it is necessary to perform integration over the area of ​​the finite element. For elements of the same type, calculations are used according to the same procedure, and the number of such calculations determines the time to solve the problem. In the traditional approach, these calculations are performed sequentially. In the case of large grid sizes, the number and time of calculations increase, which requires optimization of calculations using parallel calculation algorithms. Using the OpenMP library, a parallel algorithm for forming the matrix of solving equations was built. The calculation of terms according to the quadrature scheme is included in the parallel area of ​​the algorithm. Filling of the stiffness matrix of the finite element takes place in the sequential region. The execution time of parallel calculations is determined by the slowest subtask. Using the moment scheme of finite elements, displacement and deformation are approximated by the same polynomials, which simplifies the calculation of integrals. Parallel programming algorithms for constructing solving equations of the “MIRELA+” program package have been developed. Solutions for calculation grids of different sizes have been obtained. The effect of parallelization on the calculation time was studied.

Wave generation and wave–structure impact modelling with WCSPH

The reliable prediction of plunging wave induced load, which represents a challenging task in offshore engineering, is of key relevance for the safety of existing offshore structures as well for the reliable design of new ones. This work shows the early advances in the development of a FOSS derived model, obtained by independently introducing relevant modifications in the original model SPHERA v.9.0.0 (RSE SpA) based on the Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) method. This derived model is validated on single-phase laboratory flume tests from technical literature (i.e., dam break and regular waves) and on a laboratory flume test of plunging wave impacting an offshore structure. The derived model allows modelling two-phase flows with a large density ratio, as shown by an additional two-phase dam break simulation. Therefore it seems promising for application to two-phase plunging impact to improve the prediction of impact load on the structure.

Modeling structural deformation and failure in fluid-structure interaction with WC-TLSPH

In this paper, the coupled weakly compressible (WC) and total Lagrangian (TL) smoothed particle hydrodynamics (SPH) model based on the open-source multi-physics and multi-resolution SPHinXsys library is used to simulate the structural deformation and failure problem in fluid-structure interaction (FSI). Aiming at the problem that the original TLSPH method could not simulate the cracks and their propagation, the fracture model based on TLSPH is established, and then verified by fast-loading and slow-loading cases respectively. With the fracture model in hand, the TLSPH method is coupled with the WCSPH method to simulate the breaking dam flow impacting on an elastic/elastoplastic baffle. The comparison with the literature data shows that the present coupled WC-TLSPH model can accurately simulate the FSI problem where figurative fracture process is involved, indicating the compromising potential of the established model to simulate the elastoplastic structural failure in scientific and industrial applications.

Influence of Weak Compressibility on the Hydrodynamic Performance Evaluation of Pump Turbines in the Pump Mode

In general, weak compressibility is one of the properties of liquids. That is, in actual operation of hydraulic machinery, the flow is weakly compressible. However, the influence of weak compressibility is often neglected in usual numerical simulation, which makes the simulation results different from the experimental results. Based on the Computational Fluid Dynamics (CFD) solver and model test rig, by means of mutual verification between numerical simulation and experiment, the fitting degree between numerical results and experimental results before and after considering weak compressibility is compared and analyzed in this paper; it is obtained that the numerical results is closer to the experimental results after considering the weak compressibility. In addition, velocity field of pump turbines, head loss of main components, and the change of entropy yield are analyzed and reasons for numerical value being closer to the experimental value after considering weak compressibility of fluid are summarized and analyzed. It is proved that the consideration of weak compressibility is of great significance to improve the accuracy of results in the numerical simulation of pump turbines.

An investigation into the nonlinear effects in the roll motion of 2-D bodies by SPH method

Nonlinear effects in the roll motion of a 2–D body in the free surface are investigated by utilizing the Smoothed Particle Hydrodynamics (SPH) method. The continuity and Navier–Stokes equations are solved by employing the Weakly Compressible SPH (WCSPH) approach and our in-house code, which relies on the accumulated development process through the authors’ previous works including the fluid–solid coupling modeling by viscous penalty technique, (Tofighi et al. (2015)), and a hybrid Velocity-Variance Free Surface (VFS) and Artificial Particle Displacement (APD) algorithm (Ozbulut et al., 2014, 2018, 2020; Kolukisa et al., 2020). In this work, the GPU parallelization of the code is performed, to mitigate the computational burdens of the relatively high number of particles due to the long wavelengths generated by the roll motion of the cylinder. Presently, an alternative and exact approach in reducing the equation of roll motion with quadratic damping term is introduced. In parallel to this approach, a quadratic regression equation approximating the hydrodynamic roll moment is also employed. Aside from hydrodynamic coefficients, with linear and quadratic damping coefficients, vortex flow characteristics are also presented, qualitatively and quantitatively. It is understood from the results that, with the present capability, the WCSPH approach introduced is able to disclose nonlinearities inherently exist in the roll motion of oscillatory bodies in the free surface.

Development of explicit moving particle simulation method with applications

Moving particle simulation (MPS) is a Lagrange meshless method capable of simulating complex fluid problems. To improve computational efficiency with a simple algorithm, explicit algorithms have been proposed such as the Weakly Compressible MPS (WCMPS) method. Although accuracy can be fulfilled in WCMPS, instability arises due to pressure fluctuations of the moving particles. In this study, an explicit Poisson Pressure Equation (PPE) was introduced to address particle instability by improving the pressure calculation, referred to as the Weakly Compressible Moving Particle Explicit (WCMPE) method. By predicting pressure through the equation of state for WCMPS, divergence-free condition is conserved without the implementation of small time-steps. Several numerical cases were investigated by the WCMPE method, including 2D jet imping on a solid plate case, oscillating drop, lid driven cavity, rolling tank experiment, sloshing tank simulation and jet impinging from the bottom of a tank. In general, smooth pressure distribution and improved particle stability were obtained with the incorporation of WCMPE.

A meshless Lagrangian particle-based porosity formulation for under-resolved generalised finite difference-DEM coupling in fluidised beds

An under-resolved coupling strategy for the discrete element method (DEM) and the weakly compressible (WC) generalised finite difference method (GFD) is proposed. A novel filtering technique is proposed that allows for the recovery of a continuum porosity field in an arbitrary domain from DEM information. This allows fine spherical DEM particles to be treated in an under-resolved fashion using well-established drag relations for dynamic porous media to determine the fluid forces acting on them. To handle the momentum balance between phases, an inter-phase momentum transfer scheme is proposed as well. Verification and validation of the coupling strategy is performed. This includes comparisons to a fully-resolved WCGFD scheme when the associated computational cost allows for it. This strategy's benefits are seen when simulating a fluidised bed with an evolving fluid domain. It is shown that both under-resolved and fully-resolved dynamic information can seamlessly be treated with this scheme.

A weakly compressible, diffuse interface model of two-phase flows: Numerical development and validation

We present an approach to simulation of low speed, two-phase interfacial flows. It utilises the diffuse-interface method, based on the so-called conservative Allen–Cahn equation with some further modifications. The flow description allows for weak density variations (weak compressibility) as a computationally efficient treatment of incompressible flow regime. The proposed model conserves mass and momentum. The pressure field evolves according to the energy conservation law for inviscid fluids, expressed in primitive variables. The assumption of weak compressibility leads to some modelling and numerical issues related to the need of unique interface region identification required by the one-fluid formulation. We propose a successful and efficient remedy to this problem which, importantly, is independent of the Mach number. To numerically solve the governing equations we use the classical finite volume method and explicit time integration. This makes the presented strategy easy to implement in existing numerical codes and allows for efficient execution of the simulations using parallel computing devices. The spatial discretisation needs some special care to preserve the simplicity without impairing the fundamental physical consistency. We propose the values of tunable model parameters resulting in an universal approach, suitable for a wide variety of interfacial flows. The approach is validated both qualitatively and quantitatively for a selection of well known benchmark cases in two- and three-dimensional set-ups. High computational efficiency is obtained with simple programming techniques when the code is executed on multicore CPU.

SPH simulation of hydrodynamic responses for two novel types of silt curtain under combined wave-current conditions

Traditional silt curtain is a kind of commonly-used impermeable device for short-term control of suspended solids or turbidity in the water column generated during dredging operations. It is attempted to innovatively apply the silt curtain to more violent hydrodynamic conditions, such as functioning as an isolation structure for preventing sediment transportation into channels and harbor basins in a combined wave and current environment. The Weakly Compressible Smoothed Particle Hydrodynamics (WCSPH) code of Shi and Li et al. (2018), after incorporating an overlapped particle method (OPM), was applied to simulate the responses of a flexible impermeable curtain system as a thin-walled structure fixed to the sea bottom and moving violently under the action of static water pressure heads along and combined with waves. It is found that the modeled results of tensions in the curtain of a Silt Curtain System (SCS) working in emerged mode are in good consistency with the theoretical and experimental results and the SCS in submerged mode would be subject to much smaller tensions than in emerged mode. A large influx rate of q = 0.053 m3/s/m would enhance the wave-induced component of curtain tensions in the submerged mode. When subject to waves, the SCS in emerged mode imposes much more significant influence on wave transmission than in submerged mode, while the SCS in emerged mode produces lower horizontal oscillation ratio than in submerged mode.

Boundary condition enforcement for renormalised weakly compressible meshless Lagrangian methods

This paper introduces a boundary condition scheme for weakly compressible (WC) renormalised first-order accurate meshless Lagrangian methods (MLM) by considering both solid and free surface conditions.A hybrid meshless Lagrangian method-finite difference (MLM-FD) scheme on prescribed boundary nodes is proposed to enforce Neumann boundary conditions. This is used to enforce symmetry boundary conditions and the implied Neumann pressure boundary conditions on solid boundaries in a manner consistent with the Navier-Stokes equation leading to the accurate recovery of surface pressures. The free surface boundary conditions allow all differential operators to be approximated by the same renormalised scheme while also efficiently determining free surface particles.The boundary conditions schemes are implemented for two renormalised MLMs. A WC smoothed particle hydrodynamics (SPH) solver is compared to a WC generalised finite difference (GFD) solver. Applications in both 2D and 3D are explored. A substantial performance benefit was found when comparing the WCGFD solver to the WCSPH solver with the WCGFD solver realising a maximum speedup in the range of three times over WCSPH in both 2D and 3D configurations. The solvers were implemented in C++ and used the NVIDIA CUDA 10.1 toolkit for the parallelisation of the solvers.

A stabilized TL–WC SPH approach with GPU acceleration for three-dimensional fluid–structure interaction

A coupled total Lagrangian (TL) and weakly compressible (WC) smoothed particle hydrodynamics (SPH) method is presented to model three-dimensional fluid–structure interactions (FSI) with deformable structures. In the coupled TL–WC SPH, the fluid phase is simulated using WCSPH, while the structure solver is based on TLSPH. The three main deficiencies of solid simulation using conventional SPH, i.e. inconsistency, tensile instability and hourglass mode are circumvented in the stabilized TLSPH by means of corrected kernel gradient, Lagrangian kernel function and hourglass control technique, respectively. The resulted stabilized TLSPH is stable, accurate and has almost quadratic convergence rate in solid modeling. To increase the accuracy in FSI modeling, the δ−SPH technique is employed to improve the pressure results in the fluid phase. Based on the Adami boundary condition (Adami et al., 2012), a unified framework for modeling solid boundaries and the fluid–structure interfaces is presented. Furthermore, the GPU parallelization is employed to accelerate the proposed TL–WC SPH method for higher efficiency. The coupled method is employed to simulate problems of pure fluid flow, elastic solids with large deformation and fluid–structure interaction with deformable structures. The numerical results are compared with analytical solutions and results from literature. The GPU efficiency and speed-up compared with CPU implementations are analyzed. The novelty of this work consists: (1) three-dimensional SPH modeling of FSI problems with deformable structures, (2) stabilized structure simulation free of hourglass mode, (3) a unified framework for FSI problems taking advantages of δ-SPH, TLSPH, hourglass control, and GPU acceleration. Importantly, with the hourglass control technique proposed by Ganzenmüller (2015), stresses can be captured accurately in the TL–WC SPH-based FSI simulations.

Vibroseismic protection of heavy mining machines, buildings and structures

In the article, the authors analyze main issues of protecting buildings and structures against vibroseismic effects with the help of system of elastomeric blocks, which are characterized by high vertical stiffness, low shear stiffness, high energy dissipation and ability to center vertical load; besides, they feature high reliability with no risk of sudden failure. Results of static and dynamic tests of parametric series of elastomeric blocks used for protection of residential buildings against vibrations are presented. Design of pile with anti-vibration rubber supports is described. Calculation of elastomeric blocks under static compression is presented. Components of the stress-strain state of the thin-layer elements were investigated, and rate of the thin rubber layer compression under the action of vertically applied force was calculated. In numerical calculation, weak compressibility of rubber was simulated by moment force of the finite element for weakly compressible materials, which assumes triple approximation of displacement fields, deformation components, and volume change function. The numerical solution was obtained by the finite element method for different radius-thickness ratios in case of geometrically nonlinear elastic and viscoelastic deformation.

A multiphase meshfree particle method for continuum-based modeling of dry and submerged granular flows

We develop and fully characterize a meshfree Lagrangian (particle) model for continuum-based numerical modeling of dry and submerged granular flows. The multiphase system of the granular material and the ambient fluid is treated as a multi-density multi-viscosity system in which the viscous behavior of the granular phase is predicted using a regularized viscoplastic rheological model with a pressure-dependent yield criterion. The numerical technique is based on the Weakly–Compressible Moving Particle Semi-implicit (WC–MPS) method. The required algorithms for approximation of the effective viscosity, effective pressure, and shear stress divergence are introduced. The capability of the model in dealing with the viscoplasticity is validated for the viscoplastic Poiseuille flow between parallel plates. The model is then applied and fully characterized (in respect to the various rheological and numerical parameters) for dry and submerged granular collapses with different aspect ratios. The numerical results are evaluated in comparison with the available experimental measurement from the literature as well as some complementary experimental measurements, performed in this study. The results show the capabilities of the presented model and its potential to deal with a broad range of dry and submerged granular flows. They also reveal the impotent role of the regularization, effective pressure, and shear stress divergence calculation methods on the accuracy of the results. For the case of the granular collapse the results characterize the shape, evolution, and flow regimes of granular deposit, as well as the important effect of the ambient fluid.

Numerical Simulation of the Stress-Strain State of Thin-Layer Rubber-Metal Vibration Absorber Elements Under Nonlinear Deformation

Stress-strain state components of thin-layer rubber-metal elements are investigated. The compression level of a thin rubber layer under the action of vertically applied stress is calculated. For simplified hypotheses the relationship between the rubber layer thickness reduction and its radius-thickness ratio was analytically derived. The problem was solved under linear-elastic deformation of a rubber layer vulcanized onto the metal plates. For numerical calculations, weak rubber compressibility was simulated with the moment finite element scheme for weakly compressible materials, involving the triple approximation of displacement fields, strain components, and a volume variation function. The numerical solution was obtained by the finite element method for different layer radii and thicknesses under geometrically nonlinear elastic and viscoelastic deformation of the rubber material. The geometrical nonlinearity is described by the nonlinear strain tensor. Viscoelastic rubber properties are simulated by the hereditary Boltzmann– Volterra theory with the Rabotnov relaxation kernel. Nonlinear boundary problems are solved by the modified Newton–Kantorovich method. The calculation is effected for the two cases of curing the rubber layer onto the metal elements of the structure. The first case assumes that the rubber layer is vulcanized onto the metal plates, in the second one, it can freely slide over their surface. For the first case, numerical results are compared with the analytical solution. The effect of geometrical nonlinearity and viscoelastic rubber properties on the rubber layer thickness reduction was examined.

On the Resonant Behavior of a Weakly Compressible Water Layer During Tsunamigenic Earthquakes

Tsunamigenic earthquakes trigger pressure waves in the ocean, given the weak compressibility of the sea water. For particular conditions, a resonant behavior of the water layer can occur, which influences the energy transfer from the sea-bed motion to the ocean. In this paper, the resonance conditions are explained and analyzed, focusing on the hydro-acoustic waves in the proximity of the earthquake area. A preliminary estimation of the generation parameters (sea-bed rising time, velocity) is given, by means of parametric numerical simulations for simplified conditions. The results confirm the importance of measuring, modeling, and interpreting such waves for tsunami early detection and warning.