adaptive particle refinement (apr) applied to sph method :...

Adaptive Particle Re�nement (APR) applied to SPHmethod : Stability, accuracy and CPU analysis

Laurent Chiron

LHEEA Lab - Ecole Centrale Nantes

1er juillet 2015

Laurent Chiron Adaptive Particle Re�nement (APR) applied to SPH method : Stability, accuracy and CPU analysis1er juillet 2015 1 / 25

Plan

1 SPH background

2 Particle Re�nement theory

3 Particle re�nement improvementsArbitrary Lagrangian Eulerian (ALE) processThe APR conceptParallelization

4 Some resultsKleefsman dambreak


Plan

1 SPH background





Sph-�ow scopes


SPH background

Conservation law

∂tU + ∂xf(U) = 0∂xf(U)→

Finite di�erencesFinite volumesDiscontinuous GalerkinSPH

• How to calculate the term ∂xf(U) in SPH ?

Convolution product properties

f ∗ δ =∫R f(y)δ(x− y)dy = f ∇f ∗ g = f ∗ ∇g

δ is approached by a kernel function W

∇fi ≈ ∇f ∗W = f ∗ ∇W =∑jwjfj∇Wij


SPH background

Bene�ts

• Meshless (particles)

• Lagrangian method

Weacknesses

• Low spatial order

• No convergence theorem

"Convergence" conditions

• h∆x →∞ and h→ 0


Plan

1 SPH background





Particle re�nement theory

Some parameters

• Spacing parameter ε ∈ [0, 1]

• Radius ratio α ∈]0, 1]

• Switch parameter γ ∈ [0, 1]

• •

•••

•

• • •

Rm

∆xm

Re�nement

Process

• •

•••

•

• • •••••

••••

••••

••••

••••

••••

••••

••••

••••

ε∆xm

Rm

αRm


Particle re�nement theory

Switch parameter properties

• γ = 1→ particle is turned on (active particle)

• γ = 0→ particle is turned o� (passive particle)

0

1

γ

Bu�er zone Bu�er zone

γm γf

Re�ned area

Modi�ed SPH operator

• 〈∇f〉i =∑jfjωj∇Wijγj


Plan

1 SPH background





Arbitrary Lagrangian Eulerian (ALE) process

d~xidt

= ~v0i, (1)

dωi

dt= ωi

∑j

ωj(~v0j − ~v0i)∇Wij , (2)

dωiρidt

= −ωi

∑j

ωj2ρE(~vE − ~v0(~xij))∇Wij , (3)

dωiρi~vidt

= −ωi

∑j

ωj2(ρE~vE ⊗ (~vE − ~v0(~xij)) + PE

¯̄I)∇Wij + ωiρi~g, (4)

3 possibilities : ~v0i =

~vi (Lagrangian)0 (Eulerian)

~vi + ~δvi (ALE)


ALE process : bene�ts on a dambreak test case


Why using such an approach ?

Pressure instabilities at the �ne/coarse interface

• Particles move in a Lagrangian way (Streamlines)


Plan

1 SPH background





The APR concept

Prolongation

φd = φm + RDh(φ)m.(xd − xm)

•••

•

•

•

Quantity Before re�nement After re�nement

Mass mm

D∑j

mj

Kinetic energy 12mm~um. ~um

12

D∑j

mj ~uj . ~uj

Linear momentum mm ~um

D∑j

mj ~uj

Angular momentum ~xm ×mm ~um

D∑j

~xj ×mj ~uj


The APR concept

Prolongation

φd = φm + RDh(φ)m.(xd − xm)

Without prolongation With prolongation


Plan

1 SPH background





The utility of well distributed tasks between processors

• Neighbors number • Particle radius

Poor distribution of particles between processors

• Particles inside/near the bu�er zone have 2 to 4 more neighbors thanother particles


New algorithm e�ciency

Load balancing

τ = minλ(k)maxλ(k) where λ(k) is the number of interactions of processor k

• Former algorithm (cutting into anequal number of particles)

• New algorithm (cutting into anequal number of interactions)

At least of 30% CPU gain

On this test case, τ = 0.8 with the new algorithm (against τ = 0.54 before)


Plan

1 SPH background





Kleefsman dambreak : pressure �eld analysis

1.22m 1.17m

0.55m

0.45m

0.16m0.67m

0.68m 0.6m

Re�ned area

• Coarse • APR • FineLaurent Chiron Adaptive Particle Re�nement (APR) applied to SPH method : Stability, accuracy and CPU analysis1er juillet 2015 22 / 25

Kleefsman dambreak : pressure �eld analysis

1.22m 1.17m

0.55m

0.45m

0.16m0.67m

0.68m 0.6m

Re�ned area

• Coarse • APR • Fine


Kleefsman dambreak : Comparison of pressure sensors

• Sensor P1 • Sensor P1 (Zoom)


Kleefsman dambreak : Comparison of pressure sensors

• Sensor P3 • Kinetic Energy

Simulation performed CPU Time (seconds) Pro�t

Fine 21 000 -

APR 9000 57.2%


Thank you for your attention !


References

D.A. Barcarolo, G. Oger, and D. Le Touzé.Adaptive particle re�nement and dere�nement applied to Smoothed ParticleHydrodynamics method.Journal of computer physics, 273 :640�657, 2014.

J.Feldman and J.Bonet.Dynamic re�nement and boundary contact forces in SPH with applications in �uid �owproblems.Int. J. Numer. Meth. Eng, 72 :295�324, 2007.

S. Kitsionas and A. Whitworth.Development of SPH variable resolution using dynamic particle coalescing and splitting.Monthly Notices of the Royal Astronomical Society, 330 :129�136, 2002.

K.M.T. Kleefsman, G. Fekken, A.E.P. Veldman, B. Iwanowski, and B. Buchner.A volume-of-�uid based simulation method for wave impact problems.Journal of Computational Physics, 206 :363�393, 2005.

Y. R. Lopez, D. Roose, and C. R. Morfa.Dynamic particle re�nement in SPH : application to free surface �ow and non-cohesive soilsimulations.Comput Mech, 51 :731�741, 2013.

S. Marrone, A. Colagrossi, D. Le Touzé, and G. Graziani.Fast free-surface detection and level-set function de�nition in SPH solvers.Journal of Computational Physics, 229 :3652�3663, 2010.

G. Oger, S. Marrone, and D. Le Touzé.A consistent continuous particle reordering in weakly-compressible SPH through an ALEformalism.Proceedings of the 10th International SPHERIC workshop, 2015.

R. Vacondio, B. D. Rogers, P. K. Stansby, P. Mignosa, and J. Feldman.Development of SPH variable resolution using dynamic particle coalescing and splitting.Proceedings of the 7th International SPHERIC workshop, page 347�354, 2012.

R. Vacondio, B. D. Rogers, P. K. Stansby, P. Mignosa, and J. Feldman.Smoothed Particle Hydrodynamics with particle splitting, applied to self-gravitatingcollapse.Proceedings of the 7th International SPHERIC workshop, pages 347�354, 2012.


adaptive particle refinement (apr) applied to sph method :...

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