Direct Numerical Simulation of bubbleswith Adaptive Mesh Refinement (AMR) with

distributed algorithmsBubble workshop 2016

Arthur TALPAERT Grégoire ALLAIRE, StéphaneDELLACHERIE, Samuel KOKH, Anouar MEKKAS

CEA, École Polytechnique

2016-12-12

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Low-Mach flow DNSDirect Numerical SimulationPhysical model: low-Mach flow

AMR strategyPatch creation algorithms

Simulation and speed-upInterface advection and speed-up

Simulation and LDCAbstract Bubble Vibration modelElliptic problem and multilevel AMRNumerical results

Navier-StokesOne-fluid incompressible Navier-StokesTwo-fluid incompressible Navier-Stokes

ConclusionPerspectivesAcknowledgments

References

Back-up slidesAMR clockwork

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 2 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Direct Numerical Simulation

System Component Local 3D Local instantaneous (DNS)

Need more effort in physical modeling

Need more power for computation

Figure: Successive scales for modelling (Jamet & Mathieu, 2014), (Faccanoni, 2008), (Bois, 2011)

The Direct Numerical Simulation is the most precise simulation of a thermal-hydraulicflow in terms of scale.Strengths:

• potential for full knowledge of what happens at centimeter scale,• no need for average-scale phenomenons as the drag force, void fraction based values...• can be used to recalculate the numerical value of some coefficients of closure laws

used in larger scale models (needs to be validated).Weaknesses:

• extremely high computational cost,• extremely high data storage cost.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 3 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Ways to reduce the computation effort

Ωl(t)

Ωg(t)

Σ(t)

uW

.wall

uE.w

all

uN.wall

uS.wall

Figure: Ω(t) = Ωl(t) ∪ Ωg(t)

Though beneficial for precision, DNS is too costly as of now. There are multiple ways toreduce the computation effort, in particular for the simulation of bubbles:

• Physics use a low-Mach number model neglecting some phenomenons,• Numerical Analysis use an Adaptive Mesh Refinement for which only the most

interesting areas are finely meshed,• Computer Science use a parallel architecture to do simultaneous calculations.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 4 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Low-Mach model for nuclear Thermal-HydraulicsIn most situations, there are two very distinct time scales: e.g. Tacoustic ≃ 10−3s andTmatter ≃ 1s. This brings specific difficulties:

1. precision in schemes (e.g. the Godunov scheme has a poor precision),2. robustness of solvers (for instance, inverting matrices with very different eigenvalues

is hard with usual iterative methods, because of the bad conditioning).In a nuclear reactor in particular, we are in low-Mach number conditions in the vastmajority of cases: M =

||u||c

≃ 10−3.We can say we can neglect shock waves and other acoustic phenomenons. Howeverthermal phenomenons do remain important, so ∇ · u = 0 although M ≪ 1.

M ≪ 1 M ≪ 1 M = O(1)|∇ · λ∇T | ≪ 1 |∇ · λ∇T | = O(1)

Incomp. Navier-Stokes ≤ Low-Mach asymptotic model < Comp. Navier-Stokes×acoustic, ×thermal ×acoustic, thermal acoustic, thermal

The Diphasic Low-Mach Number model (DLMN) takes its inspiration from previous workabout combustion (see for instance Majda, 1982). It was then proposed for NuclearReactors Thermal-Hydraulics (see Dellacherie, 2012) and its theory has been studiedfrom an analytical point of view (Dellacherie, 2007; Penel, 2012; Gittel, Günther, &Ströhmer, 2014).

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 5 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Relevance of an Adaptive Mesh Refinement (AMR)Objective: Improve the precision of the calculation without making too much additionalcostly calculation.We will adapt the mesh by refining it in the most relevant areas. Two mostly usedtechniques for multi-level AMR on cartesian grids:

1. tree-based AMR (aka cell-based), one non-conformal mesh2. patch-based AMR (aka block-based), multi-level conformal meshes

→ easy to use for parallelization

Figure: Tree-based AMR Figure: Patch-based AMR

(illustrations from Fikl, 2014)

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 6 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Different patch-creation algorithmsBerger-Rigoutsos algorithm

• Introduced by Berger & Rigoutsos (Berger &Rigoutsos, 1991) and analyzed by Livne (Livne,2006a).

• Covering the region of interest is the sole constraininggoal.

• No constraint set on geometry.

nmin – nmax algorithm• New algorithm, inspired by and similar to the Livne

algorithm (Livne, 2006b).• Also includes geometrical constraints.• Aims at improving load balance and minimizing

communication between patches.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 7 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Statistics for the relevance of patch algorithm

Series of test cases: ellipsoidalbubbles for which the ratiobetween longest and shortestradius (the dilation factor) goesfrom 1 to 6.The problem is set in a grid of300∆x× 300∆x.

Normalized standarddeviation as a function ofthe dilation factor andof the patch creationalgorithm(lower is better)

Average squarenessas a function ofthe dilation factor andof the patch creationalgorithm(higher is better)

σ =

√M(N2

i i)−M(Nii)2

max(N2i i)−min(Nii)2

γ =M(

length of shortest side of patch ilength of longest side of patch i

i)

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 8 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Interface advectionWe represent the two-phase environment with a discrete field:

Y (t,x) =

1 if x ∈ Ωg(t) (i.e. gas/vapor),0 if x ∈ Ωl(t) (i.e. liquid). (1)

We want to implement the following advection equation:

∂tY + u · ∇Y = 0 (2)

We will use the Després-Lagoutière anti-diffusive scheme since it is extremely well fittedfor interface transport (see for instance Lagoutière, 2000).

Computational implementation:Kothe-Rider test, with a periodic origin-centered advection flux. The spatial domain is aunit cube [0, 1]× [0, 1]× [0, 1] and the velocity field is periodic in time. It advects aspherical bubble back and forth. If the numerical schemes are ideal, then the initial andfinal positions of the sphere should coincide.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 9 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Evolution of a 3D advected bubble

Figure: 3D simulation of a bubble with Kothe-Rider advection

Online video: https://youtu.be/Ixgge4h6eF8.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 10 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Parallelization speed-upWe use the shared-memory technology OpenMP to set the parallelization. Let thespeed-up be defined as su = sequential computation time

parallelized computation time . We use a 200× 200× 200 gridwith one refinement level.

First test case:refinement factor = 6.

• 7113 patches for Berger-Rigoutsos• 172 patches for nmin – nmax

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 11 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Parallelization speed-up

Second test case:refinement factor = 8.

• 172 patches for nmin – nmax again• the simulation with the Berger-Rigoutsos algorithm did not even pass the iteration 0,

because of a lack of hardware memory (although we had 15.6 GiB of RAM).

Notice that this simulation is strictly equivalent to having more than four billion (4× 109)fine cells.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 12 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Abstract Bubble Vibration modelIn addition to the advection equation, we coupled it with an elliptic equation whichsolution gives the velocity field u(x, t). We represented this model, for which u(x, t) ispotential (non-linear hyperbolic/elliptic coupling), with the following system of equations:

∂tY + u · ∇Y = 0

∆ϕ = ψ(t)

(Y −

1

Ω

∫Y dx

)∇ϕ · n|∂Ω = 0u = ∇ϕ

(3)

where ψ(t) is a given function and is the modelization of a vibration phenomenon.This model had already been theoretically analyzed and implemented to a large extend(Dellacherie & Lafitte, 2005), (Penel, Dellacherie, & Lafitte, 2013), (Mekkas, 2008).

Figure: Initial condition for Y field in 2D Figure: Initial condition for Y (and ϕ) fields in 3D

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 13 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Elliptic problem and multilevel AMR with LDCIn order to compute ϕ in equation 3, we want to approximate the following equation onboth the coarse and fine level, and that the information of one level helps with theresolution of the other level: ∆ϕ = ψ(t)

(Y −

1

Ω

∫Y dx

)= f(x, t)

∇ϕ · n|∂Ω = 0(4)

The Local Defect Correction method (Hackbusch, 1984) is an efficient way to benefit fromAMR (Anthonissen, Matthe, & ten The Boonkkamp, 2003) (Barbié, Ramière, & Lebon,2014):

Figure: Representation of the LDC algorithm

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 14 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Numerical results for ABV in 2D

Figure: Y field Figure: ϕ field

Results at t = 3.12, i.e. at iteration 13, here with ψ(t) a cosine function.We displayed the location of the patches on the left figure.⇒ Online video: https://youtu.be/XyxfV3w88AQ.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 15 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Numerical results for ABV in 3D

Figure: Contour of Y field Figure: Both Y and ϕ fields

Results at t = 3.046, i.e. at iteration 13, here with ψ(t) a cosine function.We displayed the location of the patches on the left figure.⇒ Online video: https://youtu.be/-V2NmaUWAJM.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 16 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Verification and usefulness of AMRWe have a verification formula for the volume of the ABV bubble as a function of time tand of the pulsation ψ(t):

C0 = log

(Vbubble(t = 0)

VΩ − Vbubble(0)

), Ψ(t) =

∫ t′=t

t′=0ψ(t′)dt′ (5)

Vbubble(t) = VΩexp(Ψ(t) + C0)

1 + exp(Ψ(t) + C0)(6)

Figure: 30 × 30, no AMR Figure: 60 × 60, no AMRCEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 17 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Figure: 60 × 60, no AMR

1min 2 s of computation

Figure: 30 × 30, AMR of factor 2

41 s of computation

This shows that with AMR, we get a similar precision to a simulation where the wholespace is highly refined.⇒ gain of 30% of computation time, although only 2D and without parallelization

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 18 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

One-fluid incompressible Navier-StokesWe used a prediction-correction scheme (Temam, 1969; Chorin, 1967) to compute theincompressible Navier-Stokes equations. We verified our results with the lid-driven cavityproblem.

u−un

∆t+ un · ∇u− ν∇2u = − 1

ρ∇p,

u = uBC on ∂Ω(7)

− 1

ρ∆ϕn+1 = − 1

∆t∇ · u,

∇ϕn+1 · n = 0 on ∂Ω(8)

pn+1 = pn + ϕn+1 (9)un+1−u

∆t= − 1

ρ∇ϕn−1,

un+1 = uBC on ∂Ω(10)

Figure: Final state of lid-drivencavity problem

⇒ Online video: https://youtu.be/esOHN--iW4Y

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 19 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

This test case is easily verified using benchmarks from the literature, like for instance(Tilak, 2003).

Figure: ux along axis y, no AMR Figure: uy along axis x, no AMR

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 20 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

We decide to refine the grid on a square located in the upper right quadrant: where theprincipal whirlpool is located. On the fine grid, the predicted velocity u is derived as alinear interpolation of the coarse level, whereas the potential ϕ is computed using theLDC algorithm. The pressure p and the velocity u are naturally deduced algebraically onthe fine grid.

Figure: Final state of lid-driven cavity problem, computed with AMR

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 21 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Two-fluid incompressible Navier-StokesIn addition to the previous equations, we add a discontinuous volumetric mass ρ,proportional to the phase color function Y , itself advected with the downwind limiting fluxscheme. We used the BCs of the lid-driven cavity problem and the initial conditions oftwo bubbles. Our model includes gravity and surface tension.

u−un

∆t+ un · ∇u− ν∇2u = gn+1 − 1

ρn∇p,

u = uBC on ∂Ω(11)

−∇ · ( 1

ρn∇ϕn+1) = − 1

∆t∇ · u,

∇ϕn+1 · n = 0 on ∂Ω(12)

pn+1 = pn + ϕn+1 (13)un+1−u

∆t= − 1

ρn∇ϕn−1,

un+1 = uBC on ∂Ω(14)

Y n+1 − Y n

∆t+ un+1∇Y n = 0 (15)

ρn+1 = ρ1Yn+1 + ρ0(1− Y n+1) (16)

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 22 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Two-fluid incompressible Navier-Stokes

Figure: Intermediary state with two bubbles Figure: Rayleigh-Taylor instability

⇒ Online videos:• https://youtu.be/11_cdEDH-2I

• https://youtu.be/l_T0op-prqM

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 23 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

PerspectivesNext short-term objectives: test cases and applications.

• inclusion of hot bubbles in a cold liquid environment; the bubbles should shrink. In aclosed environment (pressure-cooker), the other bubbles should dilate,

• bubble column reactor, in order to compute closure laws,• sloshing test cases, to follow water fronts in simulations of reflooding.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 24 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

ConclusionWe have now set the first steps for a complete the Direct Numerical Simulation of thedilation of bubbles in a less costly manner:

• we have set some of the successive layers of how to model a bubble dilation andmovement in low-Mach conditions,

• we set the base for a successful patch-based AMR,• we obtain sizable speed-up of the computation thanks to parallelization.

Acknowledgments for supervision and funding:

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 25 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Bibliography I

Anthonissen, M., Matthe, R., & ten The Boonkkamp, J. (2003). Convergence analysis of thelocal defect correction method for diffusion equations. Numerische Mathematik,95(3), 401–425.

Barbié, L., Ramière, I., & Lebon, F. (2014). Strategies involving the local defect correctionmulti-level refinement method for solving three-dimensional linear elastic problems.Computers & Structures, 130, 73–90.

Berger, M. J., & Rigoutsos, I. (1991, Sep/Oct). An algorithm for point clustering and gridgeneration. IEEE Transactions Systems, Man and Cybernetics, 21(5), 1278–1286.

Bois, G. (2011). Heat and mass transfers at liquid/vapor interfaces with phase-change:proposal for a large-scale modeling of interfaces (Unpublished doctoraldissertation). Université de Grenoble.

Chorin, A. J. (1967). A numerical method for solving incompressible viscous flow problems.Journal of Computational Physics, 2, 12 – 26.

Dellacherie, S. (2007). Numerical resolution of a potential diphasic low Mach numbersystem. Journal of Computational Physics, 39(3), 487–514.

Dellacherie, S. (2012, March). On a low-Mach nuclear core model. In Esaim: Proceedings(Vol. 35, pp. 79–106). EDP Sciences, 17, Avenue du Hoggar, Parc d’Activité deCourtabuf, BP 112, F-91944 Les Ulis Cedex A, France: EDP Sciences, SMAI.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 26 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Bibliography IIDellacherie, S., & Lafitte, O. (2005). Existence et unicité d’une solution classique à un

modèle abstrait de vibration de bulles de type hyperbolique-elliptique (Tech. Rep.No. CRM-3200). CRM, Montréal, Canada: Centre de Recherches Mathématiques.

Faccanoni, G. (2008). Étude d’un modèle fin de changement de phase liquide-vapeur.contribution à l’étude de la crise d’ébullition (Thèse, École Polytechnique X).Retrieved from http://hal.archives-ouvertes.fr/tel-00363460(Thèse en cotutelle entre l’École Polytechnique et l’Università di Trento, Italie)

Fikl, A. (2014, October). Adaptive mesh refinement with p4est (Tech. Rep.). Digiteo Labs -bât. 565 - PC 190, CEA Saclay, 91191 Gif-sur-Yvette cedex: Sup Galilée, CEA,Maison de la Simulation.

Gittel, H.-P., Günther, M., & Ströhmer, G. (2014). Remarks on a nonlinear transportproblem. Journal of Differential Equations, 256(3), 957–988.

Hackbusch, W. (1984). Local defect correction method and domain decompositiontechniques. Computing [Suppl.], 5, 89–113.

Jamet, D., & Mathieu, B. (2014, March). Simulation numérique directe des écoulementsdiphasiques. (INSTN course Thermohydraulique diphasique dans les réacteursnucléaires)

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 27 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Bibliography IIILagoutière, F. (2000). Modélisation mathématique et résolution numérique de problèmes

de fluides compressibles à plusieurs constituants (Unpublished doctoraldissertation). Université Pierre et Marie Curie. (Organisme d’accueil :Commissariat à l’Énergie Atomique, centre de Bruyères-le-Châtel)

Livne, O. E. (2006a, January). Clustering on single refinement level: Berger-Rigoustosalgorithm (Tech. Rep. No. UUSCI-2006-001). Scientific Computing and ImagingInstitute, University of Utah, Salt Lake City, UT 84112, USA: University of Utah.

Livne, O. E. (2006b, January). Minimum and maximum patch size clustering on a singlerefinement level (Tech. Rep. No. UUSCI-2006-002). Scientific Computing andImaging Institute, University of Utah, Salt Lake City, UT 84112, USA: University ofUtah.

Majda, A. (1982). Equations for low Mach number combustion (Tech. Rep. No. 112).Berkeley, California: University of California at Berkeley.

Mekkas, A. (2008). Résolution numérique d’un modèle de vibration de bulle abstraite(Unpublished master’s thesis). SupGalilée, Centre Mathématique et Informatique,École Centrale Marseille. (CEA, ONERA)

Penel, Y. (2012). Well-posedness of a low mach number system. C. R. Acad. Sci., 1(350),51 – 55.

Penel, Y., Dellacherie, S., & Lafitte, O. (2013). Theoretical study of an abstract bubblevibration model. Zeitschrift für Analysis und Ihre Anwendungen, 32(1), 19 – 36.

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Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Bibliography IVTemam, R. (1969). Sur l’approximation de la solution des équations de navier-stokes par

la méthode des pas fractionnaires (ii). Archive for Rational Mechanics and Analysis,33(5), 377–385.

Tilak, A. S. (2003, May). A pressure based unstructured grid method for fluid flow andheat transfer. In Eleventh annual conference of the CFD Society of Canada (CFD2003). Proceedings.

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Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

ToolsWe develop and use the programming interface CDMATH. It is aimed atThermal-Hydraulicists who want to quickly develop with a higher level of abstraction.CDMATH is written in C++ and is open source.Download CDMATH on https://github.com/PROJECT-CDMATH/CDMATH.Home page: http://cdmath.jimdo.com/.

CDMATH is an abstraction based on the library medCoupling, which is part of theSALOME platform (co-developed by the CEA, EDF and OpenCascade). We included ourimprovements to AMR – in particular the nmin − nmax algorithm – to SALOME’s fileformats.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 30 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

AMR diagramAn

alyt

icin

itial

cond

ition

s

Y (ΩH , 0)

Ωh2(0)

Y (Ωh2(0), 0)

Y ∗(ΩH , 1)

Y ∗(Ωh2(0), 1)

Y ∗(ΩH⃝(0), 1)

Y ∗(ΩH2 (0), 1)

Y (ΩH , 1)

Ωh2(1)

Y (Ωh2(1) \ Ωh(0), 1)

Y (Ωh2(1) ∩ Ωh(0), 1) Y (Ωh

2(1), 1)

F

F

A

O

Or

1

RH

1

A

Ih

1

1

O

Or

itert = 0 itert = 1

Figure: Flowchart for AMR on two levels

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 31 / 32

Low-Mach flow DNS AMR strategy Simulation and speed-up Simulation and LDC Navier-Stokes Conclusion References Back-up slides

Local Defect Correction algorithmThe Local Defect Correction method (Hackbusch, 1984) is an efficient way to benefit fromAMR (Anthonissen et al., 2003) (Barbié et al., 2014) to solve elliptic problems as thefollowing:

Lϕ = s,physical BCs.

(17)

InitializationCalculation of ϕcoarse0 from Lcoarse(ϕcoarse) = scoarse.Iterations, as long as no convergenceFor all patches, coarse as well as fine:

• define matrix A for linear problem• b = s+ contrib(physical BCs)• if (fine level)

b = b + contrib(Dirichlet BCs on patch borders from coarse level)

• if (iter ≥ 1) correctioniter = Aϕiter−1 − b if on area to be refined, 0 otherwise b = b + correctioniter

• solve Aϕiter = b to get the unknown ϕiter• restriction of fine level onto the coarse level

Here we detailed the algorithm for 2 levels.Note: the local defect correction correctioniter is not a residue and does not tend tozero.

CEA Saclay, CMAP – Arthur TALPAERT – DNS of bubbles with parallel AMR – 2016-12-12 – 32 / 32