Characteristic Aspects of SPH Solutions

Characteristic Aspects of SPH Solutions for Free Surface Problems: Source and Possible Treatment of High Frequency Numerical Oscillations of Local Loads. A. Colagrossi*, D. Le Touzé & G.Colicchio* *INSEAN - Rome, Italy Ecole Ecole Centrale Nantes - LMF, France

Framework and Aims of the Research Activity Activity: Our present SPH model (called SapPHire has been successfully applied to marine and coastal hydrodynamic problems: dam breaking (D and 3D, long-time sloshing simulations, bow breaking waves (D+t, impact problems with air-cushion effect (two-phase SPH solver. During our experience (-6 of applied SPH, we have encountered a number of unexpected behaviors little documented in literature.

Background & Motivations. Unphysical Pressure Oscillations during Impacts: High Frequency Oscillations

Background & Motivations. Unphysical Pressure Oscillations during Impacts: High Frequency Oscillations How do these numerical H.F. Oscillations change with: Number of Particles Speed of Sound

Background & Motivations. Tensile Instability in Free-Surface Problems. Example, Sloshing Flow: Roll Motion Presence of a rigid plate

Background & Motivations. Tensile Instability in Free-Surface Problems. Tensile Instability inside Vortex Cores Negative Pressure Pressure descreases inside the Vortices.

Background & Motivations. Tensile Instability in Free-Surface Problems. Tensile Instability inside Vortex Cores Pressure descreases inside the Vortices.

Background & Motivations. Tensile Instability in Free-Surface Problems. Tensile Instability inside Vortex Cores Unphysical Cavity Artificial Viscosity is proportional to: αch

Framework and Aims of the Research Activity Goal of the present work: NOT to show the ability of the SPH solver in terms of application. In-depth investigation of various numerical aspects of the SPH technique. Particular attention is paid to the Dynamic Part of the solution (crucial output in Marine and Coastal Engineering Identification of Prototype Tests not too complicated for an easy analysis, but highlighting critical aspects of the method.

Selection of Prototype Tests. Main Characteristics No solid Boundaries (the presence of these would not modify the conclusions drawn in this work. High and fast deformations (short-simulations, low CPU-Costs. No External Forces acting on the Flow: Easy check of volume, momentum and energy conservation. Availability of reference solutions, also for the dynamics (analytical or by other solvers. Possible development of Tensile Instability induced by vorticity.

Selection of Prototype Tests. Free Deformation of a Fluid Patch Du Dt 1 Governing = p ρ Equation Free Surface Boundary Conditions Dx Dt n x Ω u ( x, t p ( x, t F = V = u ( x ; Initial = p Ω n F ( x ; p( x, t = Conditions

Selection of Prototype Tests. Free Deformation of a Fluid Patch Weakly Compressible Fluid Dρ γc ρ = ρdiv(u; p = 1 c > 1Max ( u Ω Dt ρ ρ Incompressible Fluid γ div( u = ; 1 p = curl( u u : ρ u

Selection of Prototype Tests. Selection of Prototype Tests. Free Deformation of a Fluid Patch 1 st prototype problem: stretching of a free-surface circular fluid patch nd prototype problem: rotation of a free-surface square fluid patch 3 rd prototype problem: vortical evolution of a free-surface square fluid patch

1 st problem: stretching of a ball Definition Proposed by Monaghan (JCP n.11n.11,, 1994 u v curl( u ( x, p ( x, y = A y = + A = Poisson equation ( x, y = ρa I.C.: incompressible solution x y

1 st problem: stretching of a ball SapPHire solution Analytical Solution for the Pressure:

1 st problem: stretching of a ball problem: stretching of a ball Comparison to the analytical solution u C convergence rates, vb, Ek 1

1 st problem: stretching of a ball problem: stretching of a ball Influence of the choice of the initial conditions! the dynamic part of the solution is rarely considered & usually as I.C.: p= => initially not at the equilibrium

1 st problem: stretching of a ball problem: stretching of a ball Influence of the choice of the initial conditions acoustic fundamental frequency of a membrane

1 st problem: stretching of a ball Influence of the choice of the initial conditions b(t R a(t acoustic fundamental frequency of a membrane

1 st problem: stretching of a ball problem: stretching of a ball Convergence of the pressure solution order of convergence : ~1.1

1 st problem: stretching of a ball problem: stretching of a ball Convergence of the pressure solution

1 st problem: stretching of a ball problem: stretching of a ball Convergence of the pressure solution order of convergence: ~.

1 st problem: stretching of a ball Effect of the Choice of the Sound Speed Velocity u I. C. v ( x, ( x, y y = A = + A x y V MAX = A R A R M = c c = 14A R M.7 1 3 c = 3.5A R M.3 c = 14A R M c = 56A R M.7.

1 st problem: stretching of a ball problem: stretching of a ball Effect of the Choice of the Sound Speed Velocity

Present SPH solver (SapPHire( SapPHire Numerical tools crucial for the code effectiveness XSPH velocity correction: prevents particles inter-penetration and regularizes the weakly-compressible treatment of liquids. Artificial viscous terms: increase the stability properties of the numerical algorithm (alpha=.3 Monaghan (Ann. Rev. Astro. Astro. Astrophys.. n. 3, 199 Tensile stability control: the tensile instability is avoided by using a small local repulsive pressure term. Monaghan Monaghan (JCP n.159n.159,, Density re-initialization: the density field is periodically reinitialized through a higher-order integral interpolation formula.

Present SPH solver (SapPHire( SapPHire Numerical tools crucial for the code effectiveness The periodic density re-initialization was first proposed in 1 in A.Colagrossi et al.: A Lagrangian Meshless Method for Free-surface Flows, Proc. 4th Numerical Towing Tank Symposium (NuTTS1, Hamburg, Germany. More recently (3, an enhanced version of the density re-initialization has been proposed: A. Colagrossi & M. Landrini: Numerical simulation of interfacial flows by smoothed particle hydrodynamics, J. Comput. Phys. 191. ρ( x i = N j= 1 W MLS j ( x i m j

1 st problem: stretching of a ball problem: stretching of a ball Usefulness of the periodic Density Re-initialization

1 st problem: stretching of a ball problem: stretching of a ball 3D: same conclusions

u v ( x, ( x, y curl( u nd problem: rotation of a square Definition p y = + ωy = ωx = ω Poisson equation ( x, y = ρω I.C.: incompressible solution Proposed by Dilts (Jnt.. J. Numer. Meth. Engng.. 44, 1999 but Without Free Surface

nd problem: rotation of a square problem: rotation of a square Tensile Instability: Unphysical Fragmentation Standard SPH No Tensile Stability Control

nd problem: rotation of a square Influence of the numerical corrections Present Model SapPHire

nd problem: rotation of a square problem: rotation of a square Comparison to other solvers SPH compared to Mixed Eulerian-Lagrangian BEM (M. Greco

nd problem: rotation of a square problem: rotation of a square 1 st Reference Solution LFDM Solver Reference Configuration (undeformed Actual Configuration at time t apxx + bpxy + cpyy + dpx + epy = F a, b, c, d, e, F = f x, x X i j, X j xi X k p ρ = ω u : u

nd problem: rotation of a square problem: rotation of a square Comparison with 1 st Reference Solution SPH vs LFDM

nd problem: rotation of a square problem: rotation of a square nd Reference Solution wclfdm Solver Reference Configuration (undeformed Actual Configuration at time t

nd problem: rotation of a square Pressure Solution L Acoustic fundamental frequency of a squared membrane: Same Speed of Sound + Same spatial resolution c = 7ωL

nd problem: rotation of a square Pressure Solution Hybrid wclfdm X p = ( p + p W ( X dv i j j i j i j

nd problem: rotation of a square Pressure Solution Hybrid wclfdm High Amplitude of Pressure Oscillations are connected to the Lower-Order formula adopted in the SPH for the Pressure Gradient X p = ( p + p W ( X dv i j j i j i j

( ( ( / (4 / (4 4 / (4 4 / (4 3 (, (, ( L y L x L x L y ye xe L V curl e e V y x v e e V y x u + = = = u / (4 / (4 3, ( L y L x e e L V y x p = ρ 3 rd rd problem: vortical evolution of a square problem: vortical evolution of a square Comparison to another solver SPH compared to Finite Differences with Level Set algorithm (G. Colicchio

Conclusions & A family of test cases suitable to investigate the numerical properties of particle methods applied to free-surface problems has been proposed. A number of numerical aspects of the SPH, little mentioned in literature, have been outlined and discussed. In particular, the convergence of our model has been heuristically proved both for kinematic quantities and for the pressure solution.

Perspectives These tests are presently used in our code to study the interest of different higher-order variants of the SPH formulation. The conclusions obtained regarding the method are exploited in the building of our 3D model S ap PH ire.

The End Thanks for your Attention This study has been conducted within the framework of a collaboration project between INSEAN & École Centrale de Nantes Some of the results presented can be found on my PHD thesis, which you can download on the University of Rome web-site: http://padis. adis.uniroma1.it