High-order remapping using MOOD paradigm (Multi-dimensional Optimal Order Detection)
` 1 R. Loubere 1 Institut
´ de Mathematique de Toulouse (IMT) and CNRS, Toulouse, France http://loubere.free.fr
Joint work with M.Kuchaˇr´ık (CVUT Prague), S.Diot (LANL), R.Poncet, J.-P. Braeunig (CEA-DAM) ECCOMAS’14 Barcelona
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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Symmary
1
Quick review on MOOD (Eulerian scheme) : philosophy, detection criteria and example
2
High accurate remapping using a posteriori MOOD paradigm. Intersection, intergration and representation. Algorithm.
3
Numerical tests in 1D in Lagrange+Remap context
4
Conlusions and perspectives
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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MOOD : Multi-dimensional Optimal Order Detection Philosophy for Eulerian schemes High accurate FV schemes need some sort of limitation MUSCL (Kolgan, Van Leer. . .) : piecewise linear reconstruction + limiter to enforce bounds (W)ENO (Shu, Osher. . .) : arbitrary polynomial reconstruction + non-linear combination of polynomials to get ENO behavior Slope limiter – An a priori process based on 1
’Worst case scenario’ – Over/undershoot of reconstructions are premisses of oscillations !
2
’Precautional principle’ – I preventively must act to avoid creation of new extrema
3
’Prediction capability’ – Given solution at t n , I KNOW when/how my scheme does misbehave
However 1
’Worst case scenario’ – Increase of mean value does not imply occurence of an oscillation
2
’Precautional principle’ – Is it always/often legitimate to act ?
3
’Prediction capability’ – Can I trully predict the scheme’s reaction to NL behaviors ?
→ Why not ? Try, (possibly locally) Fail then a posteriori Detect and Repair → MOOD ` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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MOOD : Multi-dimensional Optimal Order Detection MOOD a posteriori detection criteria — #1 Convection equation Transport of a scalar quantity ∂t u + ∇ · Vu = 0,
with V = V (x) ∈ Rm , u ∈ R
If ∇ · V = 0 then solution fulfills a maximum principle. u ? numerical solution given by highest polynomial degrees =⇒ unlimited ! DETECTION PROCESS Discrete Maximum Principle (DMP) as 1st filter U2 as 2nd filter → Discont. vs smooth extrema Check if ui? fulfills the DMP A solution violating the DMP is nonetheless acceptable if n n ? n n min (ui , uj ) ≤ ui ≤ max (ui , uj ) j∈ν(i)
j∈ν(i)
2nd -order
Wimin Wimax > 0 (Non oscillatory) and min Wi /Wimax > 1 − ε = 1/2 (Smoothness)
If detection is strict on {DMP} ⇒ error at smooth extrema ⇒ DMP violation must be allowed at smooth extrema to reach higher orders ! where Wimin=minj∈ν(i) Wj , Wi (idem for max) Need a second filter for these possibly with Wi a rough estimation of local curvature problematic cells. computed with reconstructed polynomials. ` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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Results : Solid Body Rotation in 2D (5190 triangles)
Exact
UNLIM-P3
UNLIM-P5
MUSCL
MOOD-P3
MOOD-P5
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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Results : Solid Body Rotation in 2D (5190 triangles)
MOOD-P3 DMP
MOOD-P3 U2
UNLIM-P3
MOOD-P5 DMP
MOOD-P5 U2
UNLIM-P5
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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Remapping using MOOD paradigm Towards an effective a posteriori high-accurate remapping Remapping of given profiles demands 1
Exact or approximate intersection btw starting and target meshes ←− geometrical error
2
Reconstruction of variables of interest
←− representation error
3
Exact/Numerical integration of reconstructions over overlays
←− integration error
“Classical 2D remapping techniques” involve 1
Exact intersection (no geometrical error), or swept or hybrid methods (2nd order error)
2
Piecewise-Linear (PL) reconstruction using limiters (2nd order error on smooth profile)
3
Exact integration of reconstructions (no integration error)
(Very) high-accurate remapping technique demands Exact intersection and exact integration High-accurate reconstructions, as instance polynomials of degree 1 to 5 Limiter for polynomials ? Huh !. . . a` la MOOD by reducing the polynomial degrees. Note : a similar a posteriori treatment (reduction of gradient in PL reconstr. to fulfill DMP) can be found in P. Hoch. An ALE strategy to solve compressible fluid flows. HAL, 2009. hal.archives-ouvertes.fr/hal-00366858 ` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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High-accurate remapping a posteriori MOOD Exact mesh intersection S e f=S Ω Starting mesh : M = i Ωi , and target mesh : M j j. Intersection : [ [ ej = ej = Ω Ωi ∩ Ω ωij , i
∼ Ωj Ωa
Ωi ωij
ωaj
i
ωbj
ωcj
Ωb
ωij : intersection polygon (may be empty) Note : in practice intersection-based remap in flux form on same topologies
Ωc
Numerical integration P Integration of polynomial A(X ) = dk =0 ak X k over ωij using Gauss quadrature of appropriate accuracy 4
1 2
Z A(X )dX = ωij
V [ O=I
` R. Loubere (IMT and CNRS)
Z A(X )dX = TO
V X G X
ω ij
3
V I
II
wgO A(X O g)
III
IV
O=I g=1
MOOD
ECCOMAS’14 Barcelona
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High-accurate remapping a posteriori MOOD Polynomial reconstruction : Piecewise constant data per cell Ai Polynomial of degree d centered on cell Ωi ! Z X 1 A(X ) = Aj + ak (X − X j )k − (X − X j )k dX |Ωj | Ωj
d=3,4,5
d=1,2
1≤|k |≤d
(d+1)(d+2)
In 2D need 2 d = 1, 2, 3, 4, 5)
neighbors (2, 5, 9, 14, 29 for
Ωj
Neighborhoods are fixed. Centered neighborhood ! ak defined in the least square sense with respect to neighbor mean values Overdetermined system (QR decomp.), matrix precomputation Boundary conditions and symmetry preservation BCs : This is difficult even with ghost cells. ’Exact’ symmetry preservation also difficult (with non symmetric mesh due to ill-conditionned system) → expect high resolution to help ` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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High-accurate remapping a posteriori MOOD High-accurate MOOD paradigm
∼
Ωj
0 Set polynomial degrees dc = d max in all old cells for all variables
da=3
1 Reconstruct polynomials (degree dc ). Remap without limitation
3: 2
di 2
~ (d) 1: u
2 Validity of cell remapped solution given detection criteria D ? 3 Decrement for unvalid cells, dc = dc − 1 4 Back to 1 or exit (validity or dc = 0
∀c)
2
2
2: ~ u valid?
dc db
Advantages Difficulty of defining a priori limiters for polynomials Pk is discarded User can choose the sequence of decrementing (i.e 5, 3, 2, 0, or, 5, 2, 1lim) and the parachute final scheme (i.e P0 or P1 +limiter) Worst case : dc = 0 and the parachute scheme is used for every cell and every variable Difficulties Detection criteria set D is the key and D-friendly parachute scheme needed Iterative scheme : only bad cells are remapped several times. ` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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High-accurate remapping a posteriori MOOD. Advection equation. High-accurate MOOD remapping of passive scalar : Detection criteria D = f (nature(u))
Non smooth
∂t u + ∇.(Vu) = 0 Smooth
Physical-exact bounds (MP) 0 ≤ ui ≤ 1 Numerical bounds (DMP) : min(uj ) ≤ ui ≤ max(uj ) j∈νi
j∈νi
Numerical smoothness and non-oscillation detection (using P2 reconstruction) : u2
smooth/nonsmooth extrema ?
O (h 2)
h
Numerical tests for remapping of a passive scalar 1D cyclic remapping of profiles using Pk , k = 0, 1, 2, 3, 4, 5 and MOOD decrementing. We test several detection criteria D (D = ∅ means unlimited). ` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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MUSCL
I Cyclic remap — 128 cells/640 remaps — P0 , MUSCL, P3 /P5 MOOD with U2
MOOD-P3 U2
MOOD-P5 U2
P0
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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` R. Loubere (IMT and CNRS)
MOOD-P5 U2
MOOD-P5 GLB
UNLIM.P5
MOOD-P5 DMP
II Cyclic remap — 128 cells/640 remaps — Unlim P5 , P5 MOOD with DMP, GLB, U2
MOOD
ECCOMAS’14 Barcelona
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High-accurate remapping a posteriori MOOD. Euler equation. ALE = CC-Lagrangian (PL+lim) + Rezone + Remap,
Euler = CC-Lagrangian + Remap
What’s the point of high-accurate (MOOD) remapping ? Lagrangian scheme of order greater than 2 are on the way Even if the order of convergence is nominally 2 what matters is the effective error at maximal available resolution
err
Building a high-accurate remapping procedure is tempting. Classical Lagrangian schemes are nominally of 2nd order (space/time).
optimal maximal mesh
err better
Black-box remapper of high-accurate may be usefull to switch from code to code MOOD : one polynomial reconstruction per cell/variable. 6= from (W)ENO limitation Cheng, Shu in Applied Numerical Mathematics 58, 2008 ` R. Loubere (IMT and CNRS)
MOOD
FUTURE
Although the order of convergence is nominally 2 it is at most ∼ 1 (3D, disc.). Discussing the order of convergence alone is incongruous. lowest err better
given mesh
N=C/h ECCOMAS’14 Barcelona
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High-accurate remapping a posteriori MOOD. Euler equation.
Remapping of interleaved variables Primitive variables are (ρ, U = (u, v ), ε) and conservative ones are (m, Q = mU, E) with 1 E = m(ε + kUk2 ), and p = P(ρ, ε), and ρ > 0, ε > 0 2 f Sketch of remapping of conservative variables from M onto M Primitive (ρc , U c , εc )
−→ −→
Conservative (mc , Q c , Ec )
REMAP −→
Conservative ec, E ec ) ec, Q (m
Conservation (global/local) and Admissibility X X X X . M= e c = Q, e e e c = M, mc = m Q= mc U c = Q c
c
c
c
ec > 0, e c /V . ρec = m
e c /m e c k = kQ e c k < c light , kU
easy
never checked
` R. Loubere (IMT and CNRS)
MOOD
−→ −→
E=
X c
Primitive e c , εec ) (e ρc , U
Ec =
X
ec = E e E
c
1 e2 ec /m ec − Q e c + ∆U c > 0 εec = E /m 2 c difficult to handle
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High-accurate remapping a posteriori MOOD. Euler equation. Classical nominally 2nd order remapper (PL reconstruction with limitation of variables) Remap mass, momentum and energy (total or kinetic/internal). Expect density ρc in numerical bounds, velocity components U c in bounds, internal energy εc positive. What if εc < 0 ? High-accurate remapper with a posteriori MOOD — Mimick MOOD finite volume scheme Remap with polynomial reconstruction with degree dc → remapped candidate solution. Detection criteria D Physical Admissible Detection (PAD) : 0 ≤ ρec , and, 0 ≤ εec (or pressure) Numer. Adm. Detect. (NAD) on ρ 1 DMP violation : minj∈νc (e ρj ) ≤ ρec ≤ maxj∈νc (e ρj ) ? → Possible problem 2 Numer. smoothness and non-oscillation detection so-called U2 → dc decrementing u~ u~
u~
u
u
c−1 decrementing
c
c+1 decrementing
` R. Loubere (IMT and CNRS)
u
c−1
c decrementing
MOOD
c+1
c−1
c
decrementing
decrementing
c+1
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High-accurate remapping a posteriori MOOD. Euler equation. 1D ALE code
1D cell-centered ALE code Cell-centered Lagrangian scheme (2nd order space/time). Goal : test the high-accurate MOOD remapping within the ALE framework. Methodology : Lagrange+Remap regime to emphasize the remapping behaviors as a function of mesh size (under- ultra-resolved), detection criteria (DMP, DMP+U2), cascade of schemes. Remappers : MUSCL : classical PL reconstruction + limitation P0 : donor cell remapper MOOD P1 : cascade P1 → P0 MOOD Pk : cascade Pk → P2 → P0 or Pk → · · · → P2 → P1 → P0 Test cases Sod tube, Collela-Woodward blastwave
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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1D Lag+Remap with high-accurate MOOD remapping Sod tube 100 cells — MUSCL, MOOD P1 , P3 , P5 0.49
1 Exact MUSCL MOOD P1 MOOD P3 MOOD P5
0.9
Exact MUSCL MOOD P1 MOOD P3 MOOD P5
0.48
0.47
0.8
0.46
0.45
0.7 0.44
0.6
0.43
0.42
0.5 0.41 0.46
0.4
0.48
0.5
0.52
0.54
0.56
0.44 Exact MUSCL MOOD P1 MOOD P3 MOOD P5
0.42
0.3 0.4
0.38
0.2
0.36
0.1
0.34
0
0.2
0.4
0.6
0.8
1 0.32
MUSCL : P1 with limiter, MOOD P1 : P1 → P0 , MOOD Pmax : Pmax → P2 → P0 , Detection criteria : DMP+U2 on ρ ` R. Loubere (IMT and CNRS)
0.3
0.28
0.26
0.24 0.64
MOOD
0.66
0.68
0.7
0.72
ECCOMAS’14 Barcelona
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0.76
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1D Lag+Remap with high-accurate MOOD remapping Sod tube 200 cells — MOOD P3 — Cell polynomial degrees
P3 → P2 → P0
` R. Loubere (IMT and CNRS)
P3 → P2 → P1 → P0
MOOD
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1D Lag+Remap with high-accurate MOOD remapping Blastwave 400 cells — MUSCL, MOOD P3 , P2 , P1 Zoom 7
6
on
left
contact
and
peak
2.5
MUSCL MOOD P1 MOOD P2 MOOD P3 MOOD P3 d-1
MUSCL MOOD P1 MOOD P2 MOOD P3 MOOD P3 d-1 2
1.5
5
1
4 0.5
3 0 0.55
2
0.56
0.57
0.58
0.59
0.6
0.61
0.62
0.63
0.64
7 MUSCL MOOD P1 MOOD P2 MOOD P3 MOOD P3 d-1
6
1 5
4
0 0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9 3
MUSCL : P1 with limiter, MOOD P1 : P1 → P0 , MOOD Pmax : Pmax → P2 → P0 , MOOD Pmax d-1 : P3 → P2 → P1 → P0 Improvement while beneath still 2nd order Lag. scheme.
2
1
0 0.7
` R. Loubere (IMT and CNRS)
MOOD
0.72
0.74
0.76
0.78
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0.82
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1D Lag+Remap with high-accurate MOOD remapping Blastwave 400 cells — MOOD P3 — Cell polynomial degrees
P3 → P2 → P0
` R. Loubere (IMT and CNRS)
P3 → P2 → P1 → P0
MOOD
ECCOMAS’14 Barcelona
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Conclusion and perspectives High-accurate MOOD like remapping Construct one conservative polynomial per degree, per cell. No choice of stencil, no a priori limitation. Unlimited remap. Detect (DMP+U2) problematic cells for which degree decrementing will occur. Iterative remap of problematic cells until validity or P0 . Detection criteria, cascade of scheme and final parachute scheme are user-choices Numerical results Static remapping of passive scalar shows improvement when using high-accurate polynomials. Though weird behaviors may occur. Dynamic remapping of system of interleaved variables (1D Euler equations in ALE regime). Cannot expect extreme improvement though nice behaviors are already observed. Next Testing in 2D Lag+Remap (simple and efficient P2 → P1 → P0 remap). Questions about which cells need decrementing ? how many cycles ? expensive versus accurate, etc.
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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Acknowledgement CNRS — www.cnrs.fr. CEA-DIF — www.cea.fr — R.Motte, R.Poncet, J.-P.Braeunig, J.-P.Perlat L’Agence Nationale pour la Recherche (ANR) — www.agence-nationale-recherche.fr/ The young investigator program JCJC has funded the project ’ALE INC(ubator) 3D’.
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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Joined effort on MOOD paradigm Participants Universidade do Minho (Braga, Portugal) : S.Clain, J.Figuereido, G.Machado, R.M.S.Pereira Elliptic solver, steady-state Los Alamos National Laboratory (Los Alamos, NM, U.S.A) : S.Diot Multi-material, comparison vs WENO Math.Department (Nantes, France) : C.Berthon, V.Desveau, G.Moebs Analyse, Entropic MOOD scheme, MPI OpenMP // CVUT (Prague, Czech Republik) : M.Kuchar´ık, and CEA-DAM-DIF (Paris, France) : R.Poncet High-accurate MOOD remapping (ALE) Dipartimento di Ingegneria Civile (Trento, Italy) : M.Dumbser, O.Zanotti MHD, MPI //, comparison vs WENO ` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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´ y por su participacion. ´ Gracias por su atencion Thank you for your attention and participation. Merci de votre attention and participation. Next meeting ? MULTIMAT’15 in Germany September 2015, be ready !
` R. Loubere (IMT and CNRS)
MOOD
ECCOMAS’14 Barcelona
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