recent developments in the flux reconstruction...
TRANSCRIPT
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Recent Developments in the Flux ReconstructionMethod
Joshua Romero, PhD candidateKartikey Asthana, PhD candidate
Jonathan Bull, Postdoctoral ScholarAntony Jameson, Professor
Aerospace Computing Laboratory, Stanford University
AFOSR Computational Math MeetingArlington, VA July 28-30 2014
J. Romero, K. Asthana, J. Bull, A. Jameson 1/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Acknowledgements
The research presented has been made possible by the support of thefollowing organizations:
Airforce Office of Scientific Research under grant FA9550-10-1-0418by Dr. Fariba Fahroo
National Science Foundation under grant 1114816 monitored by Dr.Leland Jameson
Thomas V Jones Stanford Graduate Fellowship
National Science Foundation Graduate Fellowship
J. Romero, K. Asthana, J. Bull, A. Jameson 2/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Outline1 Brief Review of Flux Reconstruction2 Direct FR Method
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
3 Spectrally-optimal FR SchemesMotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
4 High Fidelity Turbulent Flow SimulationsHiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
5 ConclusionsJ. Romero, K. Asthana, J. Bull, A. Jameson 3/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Outline1 Brief Review of Flux Reconstruction2 Direct FR Method
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
3 Spectrally-optimal FR SchemesMotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
4 High Fidelity Turbulent Flow SimulationsHiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
5 ConclusionsJ. Romero, K. Asthana, J. Bull, A. Jameson 4/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Flux Reconstruction (FR) Method
Unifying framework that recovers several popular DiscontinuousFinite Element methods:
Nodal Discontinuous Galerkin (nodal DG)Spectral Difference (SD)
Energy Stable Flux Reconstruction (ESFR) formulation proven to belinearly stable for advection-diffusion problems
J. Romero, K. Asthana, J. Bull, A. Jameson 5/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Consider a 1D scalar conservationlaw:
∂u
∂t+∂f (u)
∂x= 0
Represent solution and flux withineach element using Lagrangeinterpolating polynomials throughP + 1 interior solution points.Resulting polynomials are of orderP.
uj(r) =P+1∑n=1
unln
fj(r) =P+1∑n=1
fnln
−3 −2 −1 0 1 2 30
0.1
0.2
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0.9
1
L RL R
u
f(u)
J. Romero, K. Asthana, J. Bull, A. Jameson 6/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
From left and right solution valuesat each interface, compute acommon interface flux using anappropriate flux formulation.
f IL = fcn(uj−1,R , uj,L)
f IR = fcn(uj,R , uj+1,L)
−3 −2 −1 0 1 2 30
0.1
0.2
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L RL R
u
f(u)
J. Romero, K. Asthana, J. Bull, A. Jameson 7/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Acquire values of the discontinuousflux at interfaces using existingLagrange representation.
−3 −2 −1 0 1 2 30
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Degree P
J. Romero, K. Asthana, J. Bull, A. Jameson 8/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Acquire values of the discontinuousflux at interfaces using existingLagrange representation.
−3 −2 −1 0 1 2 30
0.1
0.2
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0.9
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Degree P
J. Romero, K. Asthana, J. Bull, A. Jameson 9/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Introduce left and rightcorrection polynomials ofdegree P + 1.
−3 −2 −1 0 1 2 3−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
gL gR
J. Romero, K. Asthana, J. Bull, A. Jameson 10/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Scale polynomials usingcomputed difference betweendesired common interfaceflux value and existingdiscontinuous flux value.
−3 −2 −1 0 1 2 3−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
J. Romero, K. Asthana, J. Bull, A. Jameson 11/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Add the scaled correctionpolynomials to the discontinuousflux to obtain a C-0 continuous fluxof order P + 1.
f cj (r) = fj(r) + (f I
L − fj(−1))gL(r)
+ (f IR − fj(+1))gR(r)
−3 −2 −1 0 1 2 30
0.1
0.2
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Degree P
J. Romero, K. Asthana, J. Bull, A. Jameson 12/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Add the scaled correctionpolynomials to the discontinuousflux to obtain a C-0 continuous fluxof order P + 1.
f cj (r) = fj(r) + (f I
L − fj(−1))gL(r)
+ (f IR − fj(+1))gR(r)
−3 −2 −1 0 1 2 30
0.1
0.2
0.3
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0.5
0.6
0.7
0.8
0.9
1
Degree P + 1
J. Romero, K. Asthana, J. Bull, A. Jameson 13/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Review of Existing Methodology
Advance solution using derivative of the continuous flux.
d
dtuj = −[Dfj + ILgL,x + IRgR,x]
J. Romero, K. Asthana, J. Bull, A. Jameson 14/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
New Developments in Flux Reconstruction
While existing FR methodology has shown much promise, activeresearch is still being conducted to characterize and improve themethod.
From this research, several exciting new developments haveemerged:
Direct FR - Simplified formulation of the FR method that recoversnodal discontinuous Galerkin (DG) methodSpectrally Optimal FR Schemes - New schemes that minimizewave propagation errors for the range of resolvable wavenumbers
J. Romero, K. Asthana, J. Bull, A. Jameson 15/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Outline1 Brief Review of Flux Reconstruction2 Direct FR Method
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
3 Spectrally-optimal FR SchemesMotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
4 High Fidelity Turbulent Flow SimulationsHiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
5 ConclusionsJ. Romero, K. Asthana, J. Bull, A. Jameson 16/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Direct FR Method
In existing FR method, reconstruction process involves severaldistinct computational steps, all aimed at applying correctionpolynomials to construct the continuous flux.
Correction polynomials introduced by Huynh to generate continuousflux of order P + 1 so that terms in conservation law are ofconsistent order P.
J. Romero, K. Asthana, J. Bull, A. Jameson 17/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Direct FR Method
If this consistency constraint isabandoned, entire reconstructionprocess can be consolidated into asingle Lagrange interpolationthrough the combined set of interiorsolution points and interface fluxpoints.
f C = f IL l0 +
P+1∑n=1
fn ln + f IR lP+2
−3 −2 −1 0 1 2 30
0.1
0.2
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1
Degree P
J. Romero, K. Asthana, J. Bull, A. Jameson 18/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Direct FR Method
If this consistency constraint isabandoned, entire reconstructionprocess can be consolidated into asingle Lagrange interpolationthrough the combined set of interiorsolution points and interface fluxpoints.
f C = f IL l0 +
P+1∑n=1
fn ln + f IR lP+2
−3 −2 −1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Degree P + 2
J. Romero, K. Asthana, J. Bull, A. Jameson 19/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Advantages of the New Formulation
Simpler - Does not require explicit definition of correction functions,implicit through selection of solution points
Cheaper - Less operations than existing FR methodology
Equivalent - Recovers nodal DG scheme if solution points arepositioned at the zeros of the Legendre polynomial of correspondingorder
Maintains linear stability properties of this scheme
Potential for new family of schemes - Numerical results indicatethat schemes derived through positioning the solution points at thezeros of Jacobi polynomials (with some constraints) are linearlystable.
J. Romero, K. Asthana, J. Bull, A. Jameson 20/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Equivalency Requirements
Solution update is only computed and applied discretely at interiorsolution points
DFR can be made to recover the nodal DG scheme if the followingcondition is met on the total continuous flux:
∂
∂r(f (r)− fDFR(r))
∣∣∣∣r=rj
= 0
for j = 1, ...,P + 1 which are the indices spanning only the interiorsolution points.
f (r) = f D(r) + f C (r)
J. Romero, K. Asthana, J. Bull, A. Jameson 21/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Equivalency Requirements
It can be found that this equivalency requirement can be castequivalently using only the correction fluxes:
∂
∂r(f C (r)− f C
DFR(r))
∣∣∣∣r=rj
= 0 (1)
for j = 1, ...,P + 1
J. Romero, K. Asthana, J. Bull, A. Jameson 22/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Proof of Equivalency to Nodal DG
Consider splitting the DFR correction flux using effective left and rightcorrection functions, gLDFR and gRDFR
f CDFR(r) = ∆fLgLDFR(r) + ∆fRgRDFR(r) (2)
and compare to the equivalent formula from the standard FR method
f C (r) = ∆fLgL(r) + ∆fRgR(r) (3)
J. Romero, K. Asthana, J. Bull, A. Jameson 23/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Proof of Equivalency to Nodal DG
Subtracting Eq.(3) and Eq.(2), taking a derivative, and substitutingresults from Eq.(1) gives
∆fL∂
∂r(gL(r)− gLDFR(r))
∣∣∣∣r=rj
+ ∆fR∂
∂r(gR(r)− gRDFR(r))
∣∣∣∣r=rj
= 0
Since ∆fL and ∆fR are arbitrary, it suffices to prove
∂
∂r(gL(r)− gLDFR(r))
∣∣∣∣r=rj
= 0 (4)
∂
∂r(gR(r)− gRDFR(r))
∣∣∣∣r=rj
= 0 (5)
to prove scheme equivalency.
J. Romero, K. Asthana, J. Bull, A. Jameson 24/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Proof of Equivalency to Nodal DG
It is known that to recover the nodal DG scheme using the standard FRmethodology requires correction functions of the form
gL =(−1)P
2(LP − LP+1) (6)
gR =1
2(LP + LP+1) (7)
which are the left and right Radau polynomials respectively.If the interior solution points using the DFR scheme are located at thezeros of the Legendre polynomial, the effective DFR correction functionsthat results from the Lagrange interpolation can be expressed in terms ofLegendre polynomials as
gLDFR =(−1)P
2(r − 1)LP+1 (8)
gRDFR =1
2(1 + r)LP+1 (9)
J. Romero, K. Asthana, J. Bull, A. Jameson 25/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Proof of Equivalency to Nodal DG
To satisfy Eq.(5), consider a residual defined as
D =∂
∂r(gR − gRDFR) =
1
2(∂
∂rLP − LP+1 − r
∂
∂rLP+1) (10)
According to A.9 in Huynh (2007)
(1− r2)∂
∂rLP+1 = (P + 1) [Lp − rLP+1] (11)
Substitution into Eq.(10) yields
D =1
2(1− r2)
[(1− r2)(
∂
∂rLP − LP+1)− (P + 1)(rLP − r2LP+1)
](12)
J. Romero, K. Asthana, J. Bull, A. Jameson 26/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Proof of Equivalency to Nodal DG
According to A.10 in Huynh (2007)
(1− r2)∂
∂rLP = (P + 1)(rLP − LP+1) (13)
Substitution into Eq.(12) yields
D = −1
2(P + 2)LP+1 (14)
At the zeros of LP+1, which correspond to the locations of the interiorsolution points, the residual is exactly equal to zero and Eq.(5) issatisfied.
J. Romero, K. Asthana, J. Bull, A. Jameson 27/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Proof of Equivalency to Nodal DG
Eq.(4) can be satisfied in similar fashion. Consider a residual defined as
D =∂
∂r(gL − gLDFR) =
(−1)P
2(∂
∂rLP − LP+1 − r
∂
∂rLP+1) (15)
Observing that the bracketed term in Eq.(15) is identical to the term inEq.(10), it can be immediately said that
D =(−1)P+1
2(P + 2)LP+1 (16)
At the zeros of LP+1, which correspond to the locations of the interiorsolution points, the residual is exactly equal to zero and Eq.(4) is alsosatisfied. This proves the equivalence of the DFR scheme to the FRformulation of the nodal DG scheme provided the solution points arelocated at the zeros of the Legendre polynomial of order P + 1.
J. Romero, K. Asthana, J. Bull, A. Jameson 28/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Comparison of Correction Functions for P = 2
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
r
gl
FRDFR
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−8
−6
−4
−2
0
2
r
∂ ∂rgl
J. Romero, K. Asthana, J. Bull, A. Jameson 29/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Comparison of Correction Functions for P = 4
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
r
gl
FRDFR
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−20
−15
−10
−5
0
5
r
∂ ∂rgl
J. Romero, K. Asthana, J. Bull, A. Jameson 30/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
2D Euler: Isentropic Vortex
Test Case Parameters:
P = 4
10K Degrees of Freedom
dt = 0.01
6000 Iterations (t = 60)
J. Romero, K. Asthana, J. Bull, A. Jameson 31/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
2D Euler: Isentropic Vortex
FR (nodal DG): Direct FR:
J. Romero, K. Asthana, J. Bull, A. Jameson 32/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
2D Euler: Isentropic Vortex
−10 −5 0 5 100.4
0.5
0.6
0.7
0.8
0.9
1
1.1Density Solution along x−axis
x
Den
sity
[kg/
m3 ]
FRDirect FR
−10 −5 0 5 100
1
2
3
4
5
6
7 x 10−13 Magnitude of Error Between Solutions
xD
iffer
ence
Scheme Walltime [s]FR 693.8270
Direct FR 592.9168Percent Reduction 14.5%
J. Romero, K. Asthana, J. Bull, A. Jameson 33/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
Recovery of Additional Stable Schemes
Beyond recovering the nodal DG scheme, numerical experimentsindicate that DFR is stable with solution points located at the zeros
of the Jacobi polynomial P(α,β)n (x), which is orthogonal on [−1, 1]
with respect to the weight:
W (x) = (1− x)α(1 + x)β
for 0 < α ≤ 0.12 and α = β
This may lead to a new family of schemes of interest.
J. Romero, K. Asthana, J. Bull, A. Jameson 34/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Outline1 Brief Review of Flux Reconstruction2 Direct FR Method
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
3 Spectrally-optimal FR SchemesMotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
4 High Fidelity Turbulent Flow SimulationsHiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
5 ConclusionsJ. Romero, K. Asthana, J. Bull, A. Jameson 35/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Spectral resolution
The fraction of resolved wavenumbers for which constituent waves wouldbe propagated with negligible numerical dispersion and dissipation
- The range of numerically resolved wavenumbers is limited by thegrid resolution:
uδ(x , t)|Ωδ =
kmax∫k=−kmax
uδ(k , ωδ)e i(kx−ωδt)dk
- For wave propogation problems, the numerical approximation ofderivatives provides the numerical dispersion relation
ωδ = ωδ(k)
J. Romero, K. Asthana, J. Bull, A. Jameson 36/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Importance of spectral resolution in fluid phenomena
High Reynolds number flows- In DNS, to accurately capture viscous dissipation at smallestscales, the numerical scheme must add minimal artificial dissipationat high wavenumbers (Moin 1997)
Aeroacoustics- The calculation of far field noise requires acoustic waves topropogate undissipated and undispersed across several acousticwavelengths (Tam 2001)
Instabilites- Numerical dissipation at low wavenumbers can damp out physicalinstabilities, while negative numerical group velocities can lead tospurious bypass transition (Sengupta 2008)
J. Romero, K. Asthana, J. Bull, A. Jameson 37/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
FR formulation for 1-D convection
Consider the linear flux f (u) = u on a uniform grid of unit spacing:
∂u
∂t+∂u
∂x= 0
Admit the fully upwinded interface flux so that the numerical updatebecomes:
d
dtuδj = −2[C0u
δj + C−1u
δj−1]
for the j th element, where C0 and C1 are discrete operators.
J. Romero, K. Asthana, J. Bull, A. Jameson 38/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Bloch waves
The governing eqn. admits analytical solutions of the form:
u(x , t) = e ik(x−t) = e ik(j−t)e ik (r+1)2
where r |Ωj = 2x−xj
xj+1−xj− 1 represents the parent domain.
Project the exponential onto a polynomial basis:
uδj (t) = e ik(j−aδ(k)t)v
Admitting the above numerical solution into the numerical update:
−2i
k
(C0 + e−ikC−1
)v = aδv
J. Romero, K. Asthana, J. Bull, A. Jameson 39/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Semi-discrete dispersion relation
The eigenvalue problem above results into P + 1 eigenmodes.
The eigenvalues relate directly to the numerical wavespeeds whichprovides for the numerical semi-discrete dispersion relation:
aδp(k) = aδpr(k) + iaδpi
(k)
The exact/analytical dispersion relation requires ar = 1, ai = 0
The numerical solution can be expressed as:
uδ(x , t) = ekaδp i
te ik(j−aδp r
t)v
providing the error terms:
Dispersion: e ik(1−aδp r
t)
Dissipation: ekaδp i
t
J. Romero, K. Asthana, J. Bull, A. Jameson 40/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Eigenmodes for DG via FR on Gauss pts. for P = 2
Real part of the numerical wavespeed Imaginary part of the numericalwavespeed
J. Romero, K. Asthana, J. Bull, A. Jameson 41/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Relative modal energies
The numerical initial condition is a projection of the exact one onto thepolynomial basis. Due to the Lagrangian nature of the basis, it is exactat solution points:
v0 = e ik (r+1)2 =
P+1∑p=1
vδpλp = VΛ
- Complex weights λp relate to thecontribution of each mode to theinitial condition.
- A measure of relative energyamong modes can be expressed as:
βp =|λp|2
P+1∑q=1|λq|2
Relative modal energies for DG, P=2on Gauss pts.
J. Romero, K. Asthana, J. Bull, A. Jameson 42/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Choice of solution points
Numerical wavespeeds are independent of the choice of solution points.However, relative energies depend directly on the eigenvectors which aredefined by the location of solution points
Relative modal energies for DG, P=2on equidistant pts.
Relative modal energies for DG, P=2on Gauss-Lobatto pts.
J. Romero, K. Asthana, J. Bull, A. Jameson 43/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Wave propagation error
Spectral analyses of finite difference schemes have lead to thedevelopment of several spectrally optimal compact schemes that tradeformal order of accuracy for better dispersion properties.
Tam (1993), Lele (1992), Ta’asan (1994), Kim (1996), Gaitonde (1997),Chu (1998), Adams (1996), Zhong (1998), Sengupta (2003)
FR schemes- essentially upwinded - both real and imaginary parts- need to specify relative weights for dissipation and dispersion- convenient to select the actual wave propagation error:
|ep(k , t)| = |uex(k , t)− uδ(k , t)|= |e ik(x−t) − e ik(x−cδt)| = |1− e ik(1−cδ)t |
J. Romero, K. Asthana, J. Bull, A. Jameson 44/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Constrained minimization problem
An objective function for the optimization process can be specified as theenergy weighted error in wave propagation measured at a characteristictime tc = 100h/c :
η =1
(P + 1)2
P+1∑p=1
(P+1)π∫k=0
|1− e i100k(1−aδp (k))|βp(k)dk
where βp(k) is the relative modal energy of the pth mode.
The optimzation problem can then be stated as follows:
Min η(g(r),P, r0)subject to aδpimag
(k) ≤ 0 ∀k ∈ [0, (P + 1)π], p = 1, 2, ...P + 1
J. Romero, K. Asthana, J. Bull, A. Jameson 45/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
O-ESFR schemes
Energy Stable Flux Reconstruction (ESFR) schemes(Vincent-Castonguay-Jameson 2011)
- Proved to be stable for linear fluxes
- Correction functions belong to a one-parameter ‘c’ family associatedwith the energy norm
- Recovers DG for c = 0, SD for c = 2P(2P+1)(P+1)(aPP!)2
? Optimal ESFR schemes (O-ESFR)- Can be obtained by optimizing over c for given P.
cDG = 0
cSD = 1.00× 10−3
cOESFR = 1.44× 10−4
J. Romero, K. Asthana, J. Bull, A. Jameson 46/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
O-FR schemes
General FR scheme
- Generalized correction function on the left boundary
gL(r) =PY
q=1
(r − ζq)
1 + ζq
(r − 1)
2
- The solution space of zeros ζ of gL is of dimension P- Linear stability is not satisfied except in special subsets that may not
form subspaces
? Optimal FR schemes (O-FR) can be obtained by optimizing over theset of P available zeros subject to the constraint that the resultingscheme is stable.→ For P = 1, the O-FR scheme recovers the O-ESFR scheme→ For P > 1, the optimization procedure converges at zeros nottraced by ESFR family.
J. Romero, K. Asthana, J. Bull, A. Jameson 47/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Spectrally optimal FR schemes on Gauss pts. for P = 5
Real part of the numerical wavespeedImaginary part of the numericalwavespeed
J. Romero, K. Asthana, J. Bull, A. Jameson 48/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Optimal correction functions for left boundary
Left boundary correction functions for P = 5
J. Romero, K. Asthana, J. Bull, A. Jameson 49/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
Numerical integration: CFL restrictions
P RK44 RK45
DG OESFR OFR c+ DG OESFR OFR c+
2 0.235 0.238 0.241 0.688 0.352 0.356 0.361 0.864
3 0.139 0.148 0.126 0.376 0.220 0.224 0.191 0.473
4 0.100 0.103 0.108 0.245 0.152 0.158 0.164 0.311
5 0.068 0.076 0.085 0.174 0.110 0.117 0.128 0.223
Limiting CFL values
- c+ is the member of the ESFR family with highest stability limit(Vincent (2011))
- RK45 is the 4th order accurate 5-stage RK scheme with enhancedstability region
J. Romero, K. Asthana, J. Bull, A. Jameson 50/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
1-D advection of a sharp Gaussian
6th order standard and optimal FR schemes with an initial conditionsharper than a single element
Exact and numerical solutions (acrosstwo elements) at the end of one period
Evolution of numerical error across oneperiod
J. Romero, K. Asthana, J. Bull, A. Jameson 51/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
1-D advection of a sharp Gaussian
6th order standard and optimal FR schemes with finer meshes toaccurately advect initial condition
Exact and numerical solutions (acrosstwo elements) at the end of one period
Scheme No. of elements
DG 61
Tridiag.implicit
compact FD
57
c+ 76
OFR 45
No. of elements required for < 5%error in peak amplitude across oneperiod
J. Romero, K. Asthana, J. Bull, A. Jameson 52/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
MotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
1-D advection of a high-wavenumber packet
6th order standard and optimal FR schemes with an initial condition as apacket centered at half of Nyquist limit
Exact and numerical solutions (acrosstwo elements) at the end of one period
Evolution of numerical error across oneperiod
J. Romero, K. Asthana, J. Bull, A. Jameson 53/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
Outline1 Brief Review of Flux Reconstruction2 Direct FR Method
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
3 Spectrally-optimal FR SchemesMotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
4 High Fidelity Turbulent Flow SimulationsHiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
5 ConclusionsJ. Romero, K. Asthana, J. Bull, A. Jameson 54/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
HiFiLES: Open Source High Fidelity Large EddySimulation Code
High-Order via Flux Reconstruction scheme
RANS/LES
GPU/CPU scalability
Shock capturing
Unstructured grids
4th order explicit time-stepping
J. Romero, K. Asthana, J. Bull, A. Jameson 55/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
Taylor Green Vortex - 64x64x64 mesh, 3rd order DG
Q criterion colored by velocity magnitude at 2.5, 5.0, 7.5 and 10.75 secondsJ. Romero, K. Asthana, J. Bull, A. Jameson 56/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
Taylor Green Vortex - Comparison to Beck and Gassner DG
Time
Dis
sip
ati
on
Ra
te -
dk
/dt
0 2 4 6 8 10 12 14 16 18 200
0.002
0.004
0.006
0.008
0.01
0.012
0.014
DNS128x264x432x816x16
Time
Dis
sip
ati
on
Ra
te -
dk
/dt
6 8 10 12
0.01
0.011
0.012
0.013 DNS128x264x432x816x16
2 = 1.5 N = 1 N = 3 N = 152563
2563
t = 8s N = 1
N = 3
N = 15
643
h
Re = 1600
2
Left: 3rd order DG via FR (Bull and Jameson 2014)Right: 3rd order filtered DG (Beck and Gassner 2012)
J. Romero, K. Asthana, J. Bull, A. Jameson 57/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
Taylor Green Vortex - Kinetic energy and dissipation rate
0 5 10 15 20Time (s)
0.0
0.2
0.4
0.6
0.8
1.0
Norm
aliz
ed tu
rbul
ent k
inet
ic e
nerg
y
SD-16x4SD-32x4SD-64x4DRP-512
0 5 10 15 20Time (s)
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
Dis
sipa
tion
Rate
SD-16x4SD-32x4SD-64x4Beck DG-64x4DRP-512
Kinetic energy k (left) and dissipation rate −dk/dt (right) using 3rd order SDvia FR on 16x16x16, 32x32x32 and 64x64x64 meshes vs. DG (Beck andGassner 2012) and DNS using dispersion relation preserving (DRP) scheme(Debonis 2013)
J. Romero, K. Asthana, J. Bull, A. Jameson 58/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
Taylor Green Vortex - Vorticity-based dissipation rate
7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)
0.0090
0.0095
0.0100
0.0105
0.0110
0.0115
0.0120
0.0125
0.0130D
issi
patio
n Ra
teDG-32x4SD-32x4OFR-32x4DRP-128DRP-512
7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)
0.0090
0.0095
0.0100
0.0105
0.0110
0.0115
0.0120
0.0125
0.0130
Dis
sipa
tion
Rate
DG-32x5SD-32x5OFR-32x5DRP-128DRP-512
7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)
0.0090
0.0095
0.0100
0.0105
0.0110
0.0115
0.0120
0.0125
0.0130
Dis
sipa
tion
Rate
DG-32x6SD-32x6OFR-32x6DRP-128DRP-512
7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0Time (s)
0.0090
0.0095
0.0100
0.0105
0.0110
0.0115
0.0120
0.0125
0.0130
Dis
sipa
tion
Rate
DG-32x7SD-32x7OFR-32x7DRP-128DRP-512
Vorticity-based dissipation rate using DG, SD and OFR schemes on 32x32x32mesh at 3rd-6th orders (Bull and Jameson 2014)
J. Romero, K. Asthana, J. Bull, A. Jameson 59/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
Taylor Green Vortex - Energy spectra
102
10−8
10−6
10−4
k
E(k
)
DG−32x6SD−32x6OFR−32x6spectral
Close-up of energy spectrum at 9 seconds computed using 5th order DG, SDand OFR on 32x32x32 mesh vs. spectral DNS (Carton de Wiart et al. 2014)
J. Romero, K. Asthana, J. Bull, A. Jameson 60/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
Flow Over a Supersonic NACA 0012 Airfoil
0.4
0.8
1.2
1.6
Density
0.299
1.88
0.4
0.8
1.2
1.6
2sensor
0.05
2.01
Density (left) and shock sensor (right) in flow over a NACA 0012 airfoil atMach 1.2 and 5 AoA using 6th order FR
J. Romero, K. Asthana, J. Bull, A. Jameson 61/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
HiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
LES of Flow Over a Square Cylinder at Re = 22, 000
3rd order SD and WALE model on tetrahedral mesh with 130k elements.Isosurface of Q criterion colored by velocity magnitude.
J. Romero, K. Asthana, J. Bull, A. Jameson 62/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Outline1 Brief Review of Flux Reconstruction2 Direct FR Method
Description of MethodAdvantages of New FormulationProof of Equivalency to Nodal DGNumerical ResultsRecovery of Additional Stable Schemes
3 Spectrally-optimal FR SchemesMotivationModal analysisOptimal Flux Reconstruction schemesNumerical results
4 High Fidelity Turbulent Flow SimulationsHiFiLES: Open Source High Fidelity Large Eddy Simulation CodeTaylor Green VortexShock CaptureLarge Eddy Simulation
5 ConclusionsJ. Romero, K. Asthana, J. Bull, A. Jameson 63/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Conclusions
DFR is a simplification of nodal DG which may lead to a new familyof schemes
Spectrally optimal FR schemes allow higher accuracy withoutcompromizing formal order, stability, or speed. This may enable theextension of FR to industrial applications in aeroacoustics andturbulence.
Initial simulations of the Taylor-Green vortex using OFR (Bull et. al.2014) have been shown to capture the inertial range better thanconventional FR schemes.
J. Romero, K. Asthana, J. Bull, A. Jameson 64/65
Brief Review of Flux ReconstructionDirect FR Method
Spectrally-optimal FR SchemesHigh Fidelity Turbulent Flow Simulations
Conclusions
Questions?
J. Romero, K. Asthana, J. Bull, A. Jameson 65/65