# SPH Numerical Development Working Group

## SPH Numerical Development Working Group

Damien Violeau **damien.violeau@edf.fr**

Rade Vignjevic **V.Rade@cranfield.ac.uk**

### Purpose of this Group

Today SPH is considered as a promising method but still suffers from a lack of broad recognition from the scientific community as a serious candidate to become tomorrow's numerical tool. One of the main reason of this is that SPH still has unknown properties, and many questions remain unanswered on a purely theoretical ground. Although recent progress have been done, a huge amount of work remains to be done.

Convergence, numerical stability, boundary conditions, kernel properties, time marching, existence and properties of solutions, make a short list of the key issues which should be addressed in order to make SPH a mature method. In order to progress in the knowledge of the abovementioned problems, SPHERIC has started to build a working Group on SPH numerical development, named the SPHERIC Grand Challenge Working Group (GCWG).

Several Grand Challenges (GCs) are defined by the SPHERIC Steering Committee:

**GC#1: Convergence, consistency and stability**Leaders: J.J. Monaghan, D. Violeau and R. Vignjevic

GC#2:

GC#2:

**Boundary conditions**

Leaders: A. Souto-Iglesias and J-C. Marongiu

**Leaders: B.D. Rogers and R. Vacondio**

GC#3: Adaptivity

GC#3: Adaptivity

**Leaders: D. Le Touzé and M. Neuhauser**

GC#4: Coupling to other models

GC#4: Coupling to other models

**Leaders: J-C. Marongiu and M. De Leffe**

GC#5: Applicability to industry

GC#5: Applicability to industry

Ideally, the GCWG should clarify the most important problems remaining in theses GCs, list the most significant publications on each topic (see below), keep aware of the state-of-the-art and identify active research teams and foster cooperation through a living network. Sharing knowledge and clarifying the needs will be our main motivation. So far, no clear agenda has been proposed, since the GCWG is under construction. It is open to any volunteer. All suggestions are welcome.

PhD and postdoc fellows are much welcome in this Group. Their contributions to the improvement of SPH will be of great help, and the Group aims at fostering their cooperation with other institutes.

### What can YOU do?

- Contributing to the GC sections below by adding references (contact: benedict.rogers@manchester.ac.uk)
- Contributing to the GCs with your own work
- If you do so, informing the corresponding GC leaders
- Taking advantage of the SPHERIC community and Workshops to share knowledge
- Proposing publications to be nominated for the GC Award
- Proposing new GCs?

An award, the Joe Monaghan Prize, has now been initiated by SPHERIC to recognise the contribution of papers to these Grand Challenges.

## GC#1: Convergence, consistency and stability

**Leaders:**Joe Monaghan

**joe.monaghan@monash.edu**

Damien Violeau

**damien.violeau@edf.fr**

Rade Vignjevic

**v.rade@cranfield.ac.uk**

- zero-energy modes
- time-stepping
- particle pairing
- tensile instability
- etc.

**Below is a (non-exhaustive) list of relevant key publications:**

Hicks, D.L., Swegle, J.W., Attaway, S.W. (1993), *SPH: Instabilities, wall heating, and conservative smoothing*, Proc. workshop on advances in smooth particle hydrodynamics, Los Alamos National Laboratory Report #LA-UR-93-4375, pp. 223–256.

Dyka, C.T. (1994), *Addressing tension instability in SPH methods*, NRL Report NRLlMR/6384.

Morris, J.P.A. (1994), *A study of the stability properties of SPH*, Department of Mathematics, Monash University, Clayton Victoria, Australia, Report #94/22.

Wen, Y., Hicks, D.L., Swegle, J.W. (1994), *Stabilizing SPH with Conservative Smoothing*, Sandia Report #SAND94–1932.

Balsara, D.S. (1995), *Von Neumann Stability analysis of smoothed particle hydrodynamics-suggestions for optimal algorithms*, J. Comput. Phys. 121:357–372.

Morris J.P. (1996), *A study of the stability properties of Smooth Particle Hydrodynamics*, Publ. Astron. Soc. Aust., 13:97-102.

Randles, P.W., Liberski, L.D. (1996), *Smoothed Particle Hydrodynamics: some recent improvements and applications*, Comput. Methods Appl. Mech. Engrg. 139:375–408.

Dyka, C.T., Randles, P.W., Ingel, R.P. (1997), *Stress points for tension instability in SPH*, Int. J. Num. Meth. Eng. 40:2325–2341.

Hicks D.L., Liebrock L.M. (1999), *SPH Hydrocodes Can Be Stabilized with Shape-Shifting*, Comput. Math. Appl. 38:1-16.

Monaghan J.J. (2000), *SPH without a Tensile Instability*, J. Comput. Phys. 159:290-311.

Belytschko, T., Xiao, S. (2002), *Stability analysis of particle methods with corrected derivatives*, Comput. Math. Appl. 43:329–350.

Vignjevic, R., Reveles, J., Campbell, J. (2006), *SPH in a total Lagrangian formalism*, Comput. Meth. Eng. Science 14(3):181–198.

Dehnen W., Aly H. (2012), *Improving convergence in smoothed particle hydrodynamics simulations without pairing instability*, Mon. Not. R. Astron. Soc. 425:1068-1082.

Violeau, D., Leroy, A. (2014), *On the maximum time step in weakly compressible SPH*, J. Comput. Phys. 256:388-415.

Imoto, Y. (2018), *Unique solvability and stability analysis for incompressible smoothed particle hydrodynamics method*, Comput. Part. Mech. https://doi.org/10.1007/s40571-018-0214-7.

Franz, T, Wendland, H. (2018), *Convergence of the Smoothed Particle Hydrodynamics Method for a Specific Barotropic Fluid Flow: Constructive Kernel Theory, *SIAM J. Math. Anal., 50(5):4752-4784.* https://doi.org/10.1137/17M1157696*

## GC#2: Boundary conditions

**Leaders:**

Antonio Souto Iglesias **antonio.souto@upm.es**

Jean-Christophe Marongiu **jean-christophe.marongiu@andritz.com**

In order to close the fluid dynamics equations (Euler and Navier-Stokes, both compressible or incompressible) initial (ICs) and boundary conditions (BCs) are necessary. They can be classified as:

- solid boundaries (free slip, no slip, pressure normal derivative)
- free surface,
- inlet/oulet,
- initial conditions,
- coupling with other models.

To include these boundaries in an SPH simulation, researchers use various techniques depending on the type of condition considered. Let’s try to summarize the most relevant references. This list is of course open to discussion and is also a living one, since new relevant references can be incorporated. It is relevant to mention that in the most important SPH review papers, [Monaghan, 2012, 2005a, Gomez- Gesteira et al., 2010], there are already specific review sections on BCs. There are some key issues that remain to be addressed:

- How to include BCs without loosing intrinsic SPH conservation properties?
- How to include BCs consistently?
- How to include solid wall BCs for real geometries with complex shapes (2D, 3D)?
- How to provide an initial distribution of particles which avoids the onset of shocks once the time-integration starts?
- How to treat back flows when implementing inlet/oulet boundary conditions?
- etc.

**The references follow:**

A. Amicarelli, G. Agate, and R. Guandalini. A 3D Fully Lagrangian Smoothed Particle Hydrodynamics model with both volume and surface discrete elements. International Journal for Numerical Methods in Engineering, 95:419–450, 2013. doi:10.1002/nme.4514. URL http://dx.doi.org/10.1002/nme.4514.

S. Attaway, M. Heinstein, and J. Swegle. Coupling of smooth particle hydrodynamics with the finite element method. Nuclear Engineering and Design, 150 (2–3):199–205, 1994.

F. Bierbrauer, P. Bollada, and T. Phillips. A consistent reflected image particle approach to the treatment of boundary conditions in smoothed particle hydrodynamics. Computer Methods in Applied Mechanics and Engineering, 198(41-44): 3400 – 3410, 2009. ISSN 0045-7825. doi:DOI: 10.1016/j.cma.2009.06.014. URL http://www.sciencedirect.com/science/article/ B6V29-4WN2XPR-4/2/5d58483b66bffeec7d2c4a4d8eaad63d.

A. Colagrossi and M. Landrini. Numerical simulation of interfacial flows by smoothed particle hydrodynamics. J. Comp. Phys., 191:448–475, 2003.

A. Colagrossi, G. Colicchio, and D. Le Touz´e. Enforcing boundary conditions in SPH applications involving bodies with right angles. 2007.

A. Colagrossi, M. Antuono, and D. L. Touz´e. Theoretical considerations on the free-surface role in the Smoothed-particle-hydrodynamics model. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), 79(5):056701, 2009. doi:10.1103/PhysRevE.79.056701.

A. Colagrossi, M. Antuono, A. Souto-Iglesias, and D. Le Touz´e. Theoretical analysis and numerical verification of the consistency of viscous smoothed-particlehydrodynamics formulations in simulating free-surface flows. Physical Review E, 84:26705+, 2011.

A. Colagrossi, B. Bouscasse, M. Antuono, and S. Marrone. Particle packing algorithm for SPH schemes. Computer Physics Communications, 183(2):1641–1683, 2012.

M. De Leffe, D. Le Touz´e, and B. Alessandrini. Normal flux method at the boundary for SPH. In 4th SPHERIC, pages 149–156, May 2009.

M. De Leffe, D. Le Touz´e, and B. Alessandrini. A modified no-slip condition in weakly-compressible SPH. In 6th ERCOFTAC SPHERIC workshop on SPH applications, pages 291–297, 2011.

I. Federico, S. Marrone, A. Colagrossi, F. Aristodemo, and M. Antuono. Simulating 2D open-channel flows through an SPH model. European Journal of Mechanics- B/Fluids, 34:35–46, 2012.

J. Feldman and J. Bonet. Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems. International Journal for Numerical Methods in Engineering, 72(3):295–324, 2007. doi:10.1002/nme.2010.

M. Ferrand, D. R. Laurence, B. D. Rogers, D. Violeau, and C. Kassiotis. Unified semi-analytical wall boundary conditions for inviscid, laminar or turbulent flows in the meshless SPH method. International Journal for Numerical Methods in Fluids, 71(4):446–472, 2013. ISSN 1097-0363. doi:10.1002/fld.3666. URL http://dx.doi.org/10.1002/fld.3666.

G. Fourey, D. Le Touz´e, B. Alessandrini, and G. Oger. SPH-FEM coupling to simulate fluid-structure interactions with complex free-surface flows. In 5th ERCOFTAC SPHERIC workshop on SPH applications, 2010.

M. Gomez-Gesteira, B. D. Rogers, R. A. Dalrymple, and A. J. C. Crespo. State-ofthe- art of classical SPH for free-surface flows. Journal of Hydraulic Research, 48(S1):6–27, 2010. doi:10.1080/00221686.2010.9641242. URL http://dx.doi.org/10.1080/00221686.2010.9641242.

M. Ihmsen, J. Bader, G. Akinci, and M. Teschner. Animation of air bubbles with SPH. In GRAPP, 2011.

C. Kassiotis, R. B.D., M. Ferrand, D. Violeau, S. B. K., and M. Benoit. Strong coupling between 2d sph and 1d finite difference boussinesq solvers. In 6th ERCOFTAC SPHERIC workshop on SPH applications, 2011.

E. S. Lee, C. Moulinec, R. Xu, D. Violeau, D. Laurence, and P. Stansby. Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. Journal of Computational Physics, 227(18):8417– 8436, 9/10 2008.

L. Lobovsky and P. H. L. Groenenboom. Remarks on FSI simulations using SPH. In 4th ERCOFTAC SPHERIC workshop on SPH applications, pages 378–383, May 2009.

F. Macía, M. Antuono, L. M. González, and A. Colagrossi. Theoretical analysis of the no-slip boundary condition enforcement in SPH methods. Progress of Theoretical Physics, 125(6):1091–1121, 2011. doi:10.1143/PTP.125.1091. URL http://ptp.ipap.jp/link?PTP/125/1091/.

F. Macía, L. M. González, J. L. Cercos-Pita, and A. Souto-Iglesias. A boundary integral SPH formulation. Consistency and applications to ISPH and WCSPH. Progress of Theoretical Physics, 128(3), Sept. 2012a.

F. Macía, J. M. Sánchez, A. Souto-Iglesias, and L. M. González. WCSPH viscosity diffusion processes in vortex flows. International Journal for Numerical Methods in Fluids, 69(3):509–533, May 2012b. ISSN 1097-0363.

O. Mahmood, C. Kassiotis, D. Violeau, B. Rogers, and M. Ferrand. Absorbing inlet/outlet boundary conditions for SPH 2-D turbulent free-surface flows. 2012. ISBN 88-7617-003-0.

J. C. Marongiu, F. Leboeuf, and E. Parkinson. A new treatment of solid boundaries for the SPH method. In ”SPHERIC - Smoothed Particle Hydrodynamics European Research Interest Community”., pages 165+, 2007. URL http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2007sphe.work..165M.

S. Marrone, A. Colagrossi, M. Antuono, G. Colicchio, and G. Graziani. An accurate SPH modeling of viscous flows around bodies at low and moderate reynolds numbers. Journal of Computational Physics, 245(0):456 – 475, 2013. ISSN 0021-9991. doi:10.1016/j.jcp.2013.03.011. URL http://www.sciencedirect.com/science/article/pii/S0021999113001885.

J. Monaghan. Smoothed particle hydrodynamics and its diverse applications. Annual Review of Fluid Mechanics, 44(1):323–346, 2012. doi:10.1146/annurevfluid- 120710-101220.

J. Monaghan and J. Kajtar. SPH particle boundary forces for arbitrary boundaries. Computer Physics Communications, 180(10):1811 – 1820, 2009. ISSN 0010- 4655. doi:10.1016/j.cpc.2009.05.008.

J. Monaghan and A. Kos. Solitary waves on a cretan beach. J. Waterway, Port, Coastal, and Ocean Eng., 125(3):145, 1999.

J. J. Monaghan. Smoothed particle hydrodynamics. Reports on Progress in Physics, 68:1703–1759, 2005a.

J. J. Monaghan. Smoothed Particle Hydrodynamic simulations of shear flow. Monthly Notices of the Royal Astronomical Society, 365:199–213, 2005b. doi:10.1111/j.1365-2966.2005.09783.x. URL http://dx.doi.org/10.1111/j.1365-2966.2005.09783.x.

J. Simpson and M. Wood. Classical kinetic theory simulations using smoothed particle hydrodynamics. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics, 54(2):2077–2083, 1996.

A. Souto-Iglesias, L. Delorme, L. Pérez-Rojas, and S. Abril-Pérez. Liquid moment amplitude assessment in sloshing type problems with smooth particle hydrodynamics. Ocean Engineering, 33(11-12):1462–1484, 8 2006.

A. Souto-Iglesias, F. Macía, L. M. González, and J. L. Cercos-Pita. On the consistency of MPS. Computer Physics Communications, 184(3):732–745, 2013. ISSN 0010-4655. doi:10.1016/j.cpc.2012.11.009. URL http://www. sciencedirect.com/science/article/pii/S0010465512003852?v=s5.

Q. Yang, V. Jones, and L. McCue. Free-surface flow interactions with deformable structures using an SPH-FEM model. Ocean Engineering, 55(0): 136–147, 2012. doi:10.1016/j.oceaneng.2012.06.031. URL http://www.sciencedirect.com/science/article/pii/S0029801812002557.

Sh. Khorasanizade, J. M. M. Sousa. An innovative open boundary treatment for incompressible SPH. Int. J. Num. Meth. Fluids, doi: 10.1002/fld.4074.

## GC#3: Adaptivity

**Leaders:**

Benedict Rogers **benedict.d.rogers@gmail.com**

Renato Vacondio **renato.vacondio@unipr.it**

In classical Eulerian computational models, adaptive structured or unstructured grids have been used successfully for a long time to provide variable resolution and to simulate multiscale flows while retaining computational efficiency. Many engineering applications that are ideal for SPH are still using uniformly sized particles. This is inherently inefficient and unattractive to industry.

In meshfree numerical schemes there have been some early attempts to introduce variable resolution by either remeshing, and particle insertion/removal techniques, e.g. Børve et al. (2001, 2005), Lastiwka et al. (2005). Recently within the SPH formalism, dynamic particle refinement which conserves mass and momentum has been developed and applied to both the shallow water equations and the Navier-Stokes equations in 2-D (Feldman and Bonet 2007), (Vacondio et al. 2013a&b) which has built on the theoretical work of Bonet and Rodríguez-Paz (2005) who proposed a momentum-conservative weakly compressible formulation which takes into account variable smoothing length. This has been applied first to particle splitting, and then for the first time by Vacondio et al. (2013) for particle coalescing/merging to provide a truly dynamic resolution scheme that can vary in time and space. This was both variationally consistent and ensures momentum conservation in the presence of particles with different smoothing length.

Providing adaptivity within SPH is a new issue. There are some key issues that remain to be addressed:

- Should adaptivity be performed in nested blocks or continuous variation?
- How can it be implemented efficiently?
- How does adaptivity interact with the other grand challenges: Convergence, Stability and Boundary Conditions

**The references follow:**

J. Bonet, M.X. Rodríguez-Paz (2005), Hamiltonian formulation of the variable-h SPH equations, Journal of Computational Physics, 209 (2) pp. 541–558 http://dx.doi.org/10.1016/j.jcp.2005.03.030.

S. Børve, M. Omang, J. Trulsen (2001), Regularized smoothed particle hydrodynamics: a new approach to simulating magnetohydrodynamic shocks, The Astrophysical Journal, 561 (1), p. 82 http://dx.doi.org/10.1086/323228.

S. Børve, M. Omang, J. Trulsen (2005), Regularized smoothed particle hydrodynamics with improved multi-resolution handling, Journal of Computational Physics, 208 (1), pp. 345–367 http://dx.doi.org/10.1016/j.jcp.2005.02.018.

J. Feldman, J. Bonet (2007), Dynamic refinement and boundary contact forces in SPH with applications in fluid flow problems, International journal for numerical methods in engineering, 72 (3) (2007), pp. 295–324 http://dx.doi.org/10.1002/nme.2010.

M. Lastiwka, N. Quinland, M. Basa (2005), Adaptive particle distribution for smoothed particle hydrodynamics, International Journal for Numerical Methods in Fluids, 47(10–11), pp. 1403–1409, http://dx.doi.org/10.1002/fld.891.

R. Vacondio, B.D. Rogers, P.K. Stansby (2012), Accurate particle splitting for smoothed particle hydrodynamics in shallow water with shock capturing, International Journal for Numerical Methods in Fluids, 69(8), pp. 1377–1410, http://dx.doi.org/10.1002/fld.2646.

R. Vacondio, B.D. Rogers, P.K. Stansby, P. Mignosa, J. Feldman (2013a), Variable resolution for SPH: A dynamic particle coalescing and splitting scheme, Computer Methods in Applied Mechanics and Engineering, 256, April, pp. 132–148, http://dx.doi.org/10.1016/j.cma.2012.12.014.

R. Vacondio, B.D. Rogers, P.K. Stansby, P. Mignosa (2013b), Shallow water SPH for flooding with dynamic particle coalescing and splitting, Advances in Water Resources, 58, August pp. 10–23 http://dx.doi.org/10.1016/j.advwatres.2013.04.007.

F. Spreng, D. Schnabel, A. Mueller, P. Eberhard (2014), A local adaptive discretization algorithm for Smoothed Particle Hydrodynamics, Computational Particle Mechanics, 1, 131–145, http://dx.doi.org/10.1007/s40571-014-0015-6.

D.A. Barcarolo, D. Le Touzé, G. Oger, F. de Vuyst (2014) Adaptive particle refinement and derefinement applied to the smoothed particle hydrodynamics method, Journal of Computational Physics, 273, 640–657, http://dx.doi.org/10.1016/j.jcp.2014.05.040.

Y. R. López, D. Roose, C. R. Morfa, (2013) Dynamic particle refinement in SPH: application to free surface flow and non-cohesive soil simulations, Computational Mechanics, Comput Mech (2013) 51:731–741 http://dx.doi.org/10.1007/s00466-012-0748-0.

Sh. Khorasanizade, J. M. M. Sousa, (2015) Dynamic flow-based particle splitting in smoothed particle hydrodynamics, Computational Mechanics, International Journal Numerical Methods in Engineering, online http://dx.doi.org/10.1002/nme.5128.

## GC#4: Coupling to other models

**Leaders:**David Le Touzé

**David.LeTouze@ec-nantes.fr**

Magdalena Neuhuser

**Magdalena.Neuhauser@andritz.com**

## GC#5: Applicability to industry

**Leaders:**Jean-Christophe Marongiu

**jean-christophe.marongiu@andritz.com**

Matthieu De Leffe

**matthieu.de-leffe@nextflow-software.com**

Even though the SPH method is getting more popular in the
research community, there are several important aspects to be considered to
foster its diffusion and adoption among industry:

**Pre-processing**: the creation of a suitable initial condition can be a delicate task, especially regarding the initial distribution of particles. It is usually possible to use existing tools and file formats to manipulate discretized surfaces of solid objects, often in the form of triangulated surfaces (STL, PLY, VTK), describing solid wall boundary conditions. “Particle meshers” are however required to cover the initial fluid domain (volume) with particles. Traditional meshing tools are not suitable because the initial configuration of particles should minimize integration errors for the numerical scheme employed in the solver. As an illustration, most SPH practitioners make use of an isotropic SPH kernel function with uniform smoothing length. In that case the initial particle distribution should be made of equidistant particle centers. Some numerical treatments of boundary conditions also require the filling of a specific region outside the boundary with particles. The so-called particle packing algorithm (Colagrossi et al., 2012) has brought a convenient way to optimally arrange position of particles. The burden of creating the particles in an empty arbitrarily shaped volume is usually treated in complex workflows involving several steps. Turnaround time should be kept as short as possible.

**Post-processing**: the traditional visualization tools are not well suited for the visualization of results from particle methods. It is necessary to be able to visualize several hundred million particles. Moreover, in order to visualize the free surface or the interfaces between fluids, it is necessary to carry out a reconstruction of the interface, compatible with the new techniques of automatic refinement. Similarly, making slice in the computational domain or plotting the trajectory of particles should be made possible. Many advanced processing still rely on a projection of particles data onto a grid. Development of post-processing tools natively using particle data should be encouraged.

**Ratio computational cost vs precision**: in industry, the ratio cost vs precision is the first criterion of choice for numerical methods. In order to reduce this ratio it is possible to tackle either the cost or the precision.

**Robustness and ease of use**: to achieve a stable and precise simulation, several numerical parameters (choice of kernel function, size of support, artificial viscosity, CFL, ICT, XSPH ...) may be adjusted. These parameters are case dependent. For a non-expert CFD engineer to achieve an SPH simulation, generalist methods with a minimum of parameters (eg Riemann solvers instead of artificial viscosity, adaptive particle refinement) should be developed and encouraged. Indicators helping is assessing the quality of a simulation result (local / global error estimates, convergence) would provide non-specialists with some confidence in their decisions. Graphical User Interfaces are also a standard in CFD packages available on the market. Traditional CFD users are getting used to run their complete workflows from a single interface.

**Reputation**: Decision making in industry may not always receive the time and efforts it deserves, but also alternatives showing minimized risks are often favored. Publications showing technical progress of the method with respect to other Grand Challenges and well validated applications in diverse real life fields should be encouraged and well-advertised.

**The references follow:**

M.S. Shadloo, G. Oger, D. Le Touzé, Smoothed particle hydrodynamics method for fluid flows, towards industrial applications: Motivations, current state, and challenges, In Computers & Fluids, Volume 136, 2016, Pages 11-34, ISSN 0045-7930, https://doi.org/10.1016/j.compfluid.2016.05.029.

Andrea Colagrossi, B. Bouscasse, M. Antuono, S. Marrone, Particle packing algorithm for SPH schemes, In Computer Physics Communications, Volume 183, Issue 8, 2012, Pages 1641-1653, ISSN 0010-4655, https://doi.org/10.1016/j.cpc.2012.02.032.