Dual-time smoothed particle hydrodynamics for incompressible fluid simulation
Introduction
The Smoothed Particle Hydrodynamics (SPH) method originated with the work of Gingold and Monaghan [1] and Lucy [2] as a method to simulate astrophysical problems. The method is grid-free and Lagrangian in nature. It has since become a very general purpose technique and applied to a variety of problems including incompressible fluid flow[3], [4], [5], solid mechanics [6], and fluid-structure interaction [7], [8].
There are several SPH schemes for simulating incompressible and weakly compressible fluid flows. The original weakly compressible SPH scheme (WCSPH) was proposed by Monaghan [3]. This scheme treats the fluid as weakly compressible with an artificial sound speed and a stiff equation of state. This allows the scheme to utilize a hyperbolic system of equations and integrate them in time. There are many significant variants of this scheme including a Transport Velocity Formulation (TVF) [9] which introduces a transport velocity to ensure particle homogeneity. The original WCSPH and their derivatives generally suffer from a large amount of pressure oscillations and the -SPH scheme [10], [11] reduces these oscillations by introducing a dissipation into the continuity equation. Similarly, an Entropically Damped Artificial Compressibility SPH scheme (EDAC-SPH)[12] has been proposed which introduces entropy by diffusing the pressure. This approach is quite similar to the -SPH scheme and both schemes produce superior pressure distributions. All of these schemes employ an artificial sound speed and this places severe time step limitations due to stability considerations.
Cummins and Rudman [4] proposed a family of projection based schemes for incompressible fluids. Shao and Lo [5] and Hu and Adams [13] proposed incompressible SPH (ISPH) schemes which satisfy incompressibility by solving a pressure-Poisson equation. These approaches eliminate the need for evolving the pressure at the sound speed and this significantly increases the allowed time steps. The difficulty with the projection and incompressible schemes is the requirement to solve a linear system of equations which can be time consuming and involved. Recently, a Predictive-Corrective ISPH (PCISPH) [14] has been proposed for use in the graphics community for rapid simulation of incompressible fluids. A more accurate and efficient scheme has been proposed called the Implicit-Incompressible SPH (IISPH) [15]. The IISPH is matrix-free, and very efficient. It has been shown to be close to an order of magnitude faster than traditional schemes. However, the IISPH can be more involved to implement than many of the traditional WCSPH-based schemes. Recently, another fast and matrix-free implementation of the ISPH method has been proposed [16].
In this paper we propose a new scheme for weakly-compressible fluid flows. Our paper takes inspiration from the Artificial Compressiblility-based Incompressible SPH (ACISPH) scheme proposed by Rouzbahani and Hejranfar [17]. Our scheme uses a different formulation that is also very efficient. The original scheme was not noted in particular for its efficiency. We propose an original derivation and suggest many improvements that make the proposed scheme efficient. The performance is significantly better than that of the traditional WCSPH schemes and comparable to that of the ISPH schemes without sacrificing any accuracy or being unduly hard to implement. Our approach employs the classic artificial compressibility of Chorin [18] in a dual-time stepped framework. We call the resulting scheme, DTSPH for Dual-Time stepped SPH.
Fatehi et al. [19] propose density-based dual-timestepping schemes for the SPH method. They propose two different formulations in order to update the pressure in pseudo-time. They perform an accurate discretization of the derivatives. While their method is in principle similar to the present work, it does not demonstrate any significant performance advantage over the traditional WCSPH schemes. Moreover, they do not demonstrate their method with any free-surface problems. Our proposed implementation and method is however significantly faster than the WCSPH scheme and has been demonstrated to work well for free-surface and three-dimensional problems. Zhang et al. [20] on the other hand propose a dual-criteria timestepping strategy that is quite different from the method proposed here or in [17], [19]. They use an “acoustic timestep” to relax the pressure and an “advective timestep” to update the verlet lists used for neighbor computation. This optimization results in a performance improvement without significant complexity albeit at the cost of increased memory. The present method demonstrates much improved performance and does not involve a significant memory penalty.
The new scheme is designed to be implemented as an extension of the classic weakly-compressible schemes. The advantage of this is that these are relatively easy to implement, boundary conditions may be easily enforced, and there are several well-established schemes that may be used. Although we have not done so in this manuscript, it is important to note that this approach may also be employed for solid mechanics problems where the artificial speed of sound is usually very large.
The new scheme can be adapted to any WCSPH formulation which uses a density or pressure evolution equation based on a continuity equation. We demonstrate the scheme with the EDAC scheme [12]. We note that it can be easily applied to other schemes like the -SPH. We show how our scheme can be used to obtain steady state solutions, although this is not particular to the new scheme and can be easily performed for a variety of other schemes. Obtaining steady state solutions in the context of SPH simulations is useful in different contexts. For example, in the case of the impulsively started flow past a complex geometry, an initial potential flow solution is useful and this may be easily obtained using this approach.
In this manuscript we provide a new formulation as compared to the work of [17], explore several important details for the implementation of the scheme, and more importantly provide a high-performance, open source implementation of the scheme. Our implementation uses the open source PySPH framework [21], [22] and all the code related to the manuscript is available at https://gitlab.com/pypr/dtsph. In order to facilitate reproducible research, this entire manuscript is completely reproducible and every figure in this paper is automatically generated using automan [23].
In the next section we discuss the proposed DTSPH scheme in a general setting and then discuss the SPH discretization. We briefly discuss the stability requirements of the scheme. We show how the resulting scheme is efficient and then proceed to simulate various standard benchmark problems. We perform comparisons with the TVF [9], the -SPH scheme [10], and the EDAC scheme [12] where relevant to demonstrate the accuracy and efficiency of the proposed scheme.
Section snippets
The dual-time SPH method
In dual-time stepping schemes, a new time dimension called the “dual-time”, denoted by the variable , is introduced. We have the following important considerations to keep in mind. If is the position vector of a particle, then the real velocity of the particle is defined as, . On the other hand, if the particle were to move in pseudo-time, we define the velocity in pseudo-time as .
If we consider a property of the particle, , then we can define a material derivative in
SPH discretization
The basic scheme discussed in the previous section should work for any particular SPH discretization of the momentum and pressure equations. In the following, we use a WCSPH formulation for the SPH discretization. We keep density fixed as per the original problem. We consider two different cases one where we move the particles in pseudo-time and the other where the particles are fixed in pseudo-time.
Results and discussion
In this section we perform various numerical experiments using standard benchmark problems. We explore the following specific questions using the Taylor-Green problem which has an exact solution:
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Is it worth moving the particles in pseudo-time or can we freeze the particles? This has significant performance implications.
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What possible values of can be used?
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What suitable values of the tolerance can be chosen and what does this imply for accuracy?
Once these are explored, a suite of test
Conclusions
In this paper we propose a scheme called Dual-Time stepped SPH (DTSPH) that employs a dual-time stepping approach for incompressible fluid flow simulations. The method has been demonstrated with the EDAC formulation [12]. There are a few recent developments in the literature that employ a similar approach [17], [19] but these implementations are not efficient despite the ability to use much smaller timesteps than the WCSPH scheme. We find that by not moving the particles in pseudo-time we are
CRediT authorship contribution statement
Prabhu Ramachandran: Conceptualization, Formal analysis, Investigation, Methodology, Software, Supervision, Visualization, Writing - original draft, Writing - review & editing. Abhinav Muta: Investigation, Methodology, Software, Visualization, Validation, Writing - original draft, Writing - review & editing. M. Ramakrishna: Conceptualization, Formal analysis, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
We are grateful to the anonymous reviewers for their comments that have improved the quality of this manuscript.
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