MarcoCeze_defense

Robust hp-Adaptation Method for Discontinuous
Galerkin Discretizations Applied to Aerodynamic
Flows
Marco Ceze, PhD. Candidate
Department of Aerospace Engineering
University of Michigan
Doctoral Committee: Assistant Professor Krzysztof Fidkowski, Chair
Assistant Professor Eric Johnsen
Emeritus Professor Bram van Leer
Professor David W. Zingg, University of Toronto
Marco Ceze Doctoral Defense Robust hp-Adaptation Method for Discontinuous Galerkin Discretizations 1/52

An airline is a difﬁcult business
Proﬁt margin over the past 10 years for various airlines.
Source: http://www.ycharts.com

The impact of drag prediction
+
−1% drag translates into
−
+7% payload (revenue).
From the aircraft manufacturer perpective:
Drag under-prediction ⇒ contractual penalties.
Drag over-prediction ⇒ lower selling price.
AIAA organizes workshops for assessing the state-of-the-art.
Pressure distribution over NASA’s CRM-WB
geometry
DPW’s ﬁnest common meshes.

Motivation
Quantitatively reliable CFD solutions are still hard to obtain
Accuracy is usually assessed by uniform mesh reﬁnement studies
and even then, errors may be large (e.g. AIAA-DPW, HLPW).
Acceptable engineering solutions on representative geometries
often come at prohibitively-large grid sizes.
Robustness of the analysis process is rarely addressed.
Good news!
Most engineers are interested in only certain scalar outputs, eg.:
drag, lift, moments, etc..

Why good news?...Adjoints!
Sensitivity of the output w.r.t. residuals in the physics.
Input perturbations are converted into residual perturbations.
Advantage: residuals are generally cheap to compute.
Flow
simula+on

Post-‐processing

e.g.:
drag,
li7.

J(U)
Input
parameters

↵
R
J
2
6
6
6
4
R1(U)
R2(U)
...
Rn(U)
3
7
7
7
5
=
2
6
6
6
4
0
0
...
0
3
7
7
7
5
Adjoint
variables

U
“Mesh”
+


Flow
simula+on

Post-‐processing

e.g.:
drag,
li7.

J(U)
Input
parameters

↵
R
J
2
6
6
6
4
R1(U)
R2(U)
...
Rn(U)
3
7
7
7
5
=
2
6
6
6
4
0
0
...
0
3
7
7
7
5
Adjoint
variables

U
↵
“Mesh”
+

Sensitivity analysis

Flow
simula+on

Post-‐processing

e.g.:
drag,
li7.

J(U)
Input
parameters

↵
R
J
2
6
6
6
4
R1(U)
R2(U)
...
Rn(U)
3
7
7
7
5
=
2
6
6
6
4
0
0
...
0
3
7
7
7
5
Adjoint
variables

U
“Mesh”
+

Residual
evalua+on

in
ﬁner
space

Output error
estimation

A derivation of the adjoint equation
Consider an output J ((α), u) where u satisﬁes R((α), u) = 0 for
input parameter set α.
We form a Lagrangian L((α), u, ψ) to incorporate the constraint:
L((α), u, ψ) = J (u) + ψT
R((α), u).
We take the variation of the Lagrangian assuming R((α), u) = 0:
δL((α), u, ψ) =
∂J
∂u
T
+ ψT ∂R
∂u
= 0
Adjoint equation
δu +
∂J
∂α
T
+ ψT ∂R
∂α
= 0
δα.
Making δL = 0 selects only realizable δJ ’s.
Linear
equation!

Output Error Estimation
Output error: difference between an output computed with the
discrete system solution and that computed with the exact solution.
δJ = JH(uH) − J(u)
uH ∈ VH = approximate solution, u ∈ V = exact solution
Adjoint-based output error estimation techniques
Account for propagation effects inherent to hyperbolic problems.
Can provide a correction factor for the output.
Identify all areas of the domain that are important for an accurate
prediction of the output ⇒ Mesh adaptation!

Discontinuous Galerkin and hp-adaptation
P (u) = 0
PDE
Ri ≡
Ω
wi(x) P N
j=1 Ujφj (x) dx = 0
Discrete
MWR
Mesh:
κ1 κ2 κ3 κ4
Solution space:
κ1
φ1,1 φ1,2
κ2
φ2,1 φ2,2
κ3
φ3,1 φ3,2
κ4
φ4,1 φ4,2
hp-adapted
solution space:
κ1
φ1,1 φ1,2
κ4
φ4,1 φ4,2
h-ref.
κ2
φ2,1φ2,2
κ5
φ5,1φ5,2
κ3
p-ref.
φ3,1 φ3,2 φ3,3
How do we choose between h and p reﬁnements?

Solution process and this work
•  Ini$al
condi$on

•  Parameters

•  Mesh

Flow
solver
Converged?

Interven$on

by
user

Error

es$ma$on

Error
within

tolerance?

Finished

Mesh

adapta$on

NO

YES

YES
NO


Solver
Robustness

•  Ini$al
condi$on

•  Parameters

•  Mesh

Flow
solver
Converged?

Interven$on

by
user

Error

es$ma$on

Error
within

tolerance?

Finished

Mesh

adapta$on

NO

YES

YES
NO


hp
-‐
Adapta(on

Solver
Robustness

•  Ini$al
condi$on

•  Parameters

•  Mesh

Flow
solver
Converged?

Interven$on

by
user

Error

es$ma$on

Error
within

tolerance?

Finished

Mesh

adapta$on

NO

YES

YES
NO


Overview of this work
High-order DG-FEM for spatial discretization.
Exact Jacobian with line-Jacobi preconditioner and GMRES linear
solver.
MPI parallelization.
Focus on steady problems relevant to the aeronautical industry.
Oliver’s modiﬁcations to the Spalart-Allmaras turbulence model.
Quadrilateral and hexahedral meshes.
Constrained pseudo-transient continuation (CPTC) for time
integration.
Output-based anisotropic hp-adaptation.
Node-edge weighted mesh partitioning.

Constrained pseudo-transient continuation

Problem statement
Consider the semi-discrete flow equations:
Ut = −R(U).
The discrete solution U is used to approximate the state, u, as a
field uH,p(t, x). In finite elements:
uH,p
(t, x) =
j
Uj(t)φH,p
j (x).
The field uH,p(t, x) is subject to physical realizability constraints:
p(uH,p(t, x))
p∞
> 0,
ρ(uH,p(t, x))
ρ∞
> 0 and
ν(uH,p) + νt (uH,p)
ν(uH,p)
> 0.
These realizability constraints are violated
sometimes.

Marching to steady-state
M
1
∆t
+
∂R
∂U Uk
A
∆Uk
= −R(Uk
)
Pseudo-transient continuation
Multiply left-hand side by its transpose:
∆UT
AT
A∆U = −∆UT
AT
R(U)
∂f
∂U
≥ 0.
∆U is a descent direction for:
f(˜U) = |Rt (˜U)|2
L2
= |M
1
∆t
(˜U − Uk
) + R(˜U)|2
L2
.
No direct mechanism to avoid
physicality constraints!

Handling physicality constraints
We propose augmenting the residual with a penalty vector:
Rp(U) = R(U) + P(U).
For a repelling effect w.r.t the constraints, it is sufﬁcient to make
P(U)T R(U) > 0. We choose:
P = Φ R,
where Φ is a diagonal matrix with elemental penalties:
PκH (UκH (t), µ) = µ
Ni
i
Nj
j
wj
ci(uH,p(t, xj))
.
where µ is the penalty factor, ci are the constraints, and xj are
interrogation points.

Handling physicality constraints
Apply PTC to Rp and the equation for the state update becomes:





(I + Φ)−1 M
∆t
a
+
∂R
∂U
+ (I + Φ)−1 ∂Φ
∂U
R(U)
b





∆U = −R(U).
The elemental ∆t gets ampliﬁed by a factor (1 + PκH ).
In the limit ∆t → ∞, the solution seeks a minimum of |Rp|L2
.
As the state in an element approaches a non-physical condition,
term "a" vanishes locally while "b" remains due to the derivative of
the inverse barrier function.

Constrained PTC example
Solve a simple nonlinear algebraic system:
sin(4πx1x2) − 2x2 − x1 = 0
4π − 1
4π
(e2x1
− e) + 4ex2
2 − 2ex1 = 0
−1 ≤ x1 ≤ 1
−1 ≤ x2 ≤ 1
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x2
x1
Prohibited
PTC
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
x2
x1
Prohibited
Constrained PTC

Solution update methods
Uk+1 = Uk + ωk ∆Uk , for ωk ≤ 1
Maximum Primitive Change (MPC)
Limit ωk such that changes in ρ, p, and ˜ν are within ηmax = 10%
for all κH.
Line Search (LS)
Compute ωphys such that changes in ρ, p, and ˜ν are within
ηmax = 10% for κH with ∆U|κH ·
∂PκH
∂U
> 0.
Backtrack from ωk = ωphys until |Rt (˜U)|L2
< |R(Uk )|L2
.
Line Search + Greedy (LS+G)
If LS exits with ωk = ωphys < 1, amplify ωk while
|Rt (˜U)|L2
< |R(Uk )|L2
and ωk < 1.

CFL evolution strategies
If (ωk ≤ ωmin) ⇒ Repeat step “k” with smaller CFL =
λmax,κH ∆tκH
LκH
Exponential progression (EXP)
CFLk+1
=



β · CFLk
for β > 1 if ωk
= 1
CFLk
if ωmin < ωk
< 1
κ · CFLk
for κ < 1 if ωk
< ωmin
Switched Evolution Relaxation (SER)
CFLk
= min CFLk−1 |Rk−1
|L2
|Rk |L2
, CFLmax
Residual Difference Method (RDM - mRDM)
CFLk
= min

CFLk−1
· β
|Rk−1|L2
−|Rk |L2
|Rk−1|L2 , CFLmax

 for β > 1

Let’s test this!
Combine the PTC and CPTC with the solution update methods
and CFL strategies.
Test cases range from intermediate to difficult.
Residual convergence is 9 orders of magnitude reduction.
We compare the methods under same GMRES and discretization
parameters.
Rules of the game:
A maximum run time is fixed for all runs within a case.
A maximum number of iterations is fixed for all runs within a case.
Performance of the converged runs is measured in terms of number
of iterations and wall time.

MDA 30p30n - M∞ = 0.2, Re = 9 × 106
, α = 16◦
, p = 1
Quartic mesh generated by agglomerating linear cells.
y+ ≈ 3 × 103 based on a ﬂat-plate correlation.
16 CPU’s, 8 hours, maximum of 10k iterations.
Quartic mesh (4070 elements) Mach number contours

MDA 30p30n - M∞ = 0.2, Re = 9 × 106
, α = 16◦
, p = 1
Success of all runs:
→ converged,
→ timeout or max iterations,
→ CFL below minimum or non-physical.
PTC CPTC
MPC LS LS+G MPC LS LS+G
EXP run 1.1.1 run 1.2.1 run 1.3.1 run 2.1.1 run 2.2.1 run 2.3.1
SER run 1.1.2 run 1.2.2 run 1.3.2 run 2.1.2 run 2.2.2 run 2.3.2
RDM run 1.1.3 run 1.2.3 run 1.3.3 run 2.1.3 run 2.2.3 run 2.3.3
mRDM run 1.1.4 run 1.2.4 run 1.3.4 run 2.1.4 run 2.2.4 run 2.3.4
Performance of converged runs.
Run ID Nonlinear iterations GMRES iterations Wall time (seconds)
1.3.1 1.000 (1412) 1.000 (608770) 1.000 (5.085 × 103)
1.3.4 4.750 0.935 1.005
2.2.4 5.135 1.346 1.059
2.3.1 1.161 0.881 0.920

NACA 0012 - M∞ = 0.8, Re = 6.5 × 106
, α = 0◦
, p = 2
No shock-capturing ⇒ oscillations.
y+ ≈ 2 × 103 based on a ﬂat-plate correlation.
Quartic mesh (1740 elements) Mach number contours

NACA 0012 - M∞ = 0.8, Re = 6.5 × 106
, α = 0◦
, p = 2
PTC CPTC
1.1.4 1.000 (3707) 1.000 (34671) 1.000 (1.347 × 104s)
1.2.1 0.281 0.456 0.327
1.2.4 0.454 0.557 0.477
1.3.1 0.0968 0.316 0.170
1.3.4 0.179 0.301 0.229
2.1.1 0.401 0.440 0.410
2.1.4 0.308 0.537 0.367
2.2.1 0.250 0.474 0.309
2.2.4 0.386 0.542 0.422
2.3.1 0.127 0.335 0.200
2.3.4 0.136 0.529 0.275

DPW 3 W1 - M∞ = 0.76, Re = 5 × 106
, α = 0.5◦
, p = 1
Cubic mesh generated by agglomerating linear cells.
y+ ≈ 1 based on a ﬂat-plate correlation.
Mesh and pressure contours (29310 elements) Mach number and ρ˜ν contours

DPW 3 W1 - M∞ = 0.76, Re = 5 × 106
, α = 0.5◦
, p = 1
→ converged,
→ timeout or max iterations,
→ CFL below minimum or non-physical.
PTC CPTC
2.1.1 1.000 (1255) 1.000 (68253) 1.000 (5.694 × 103s)
2.1.4 0.897 0.935 0.845
2.2.1 0.880 0.944 1.021
2.2.4 0.751 1.002 1.127

Output-based hp-adaptation

Error estimate and adaptive indicator
uH,p will generally not satisfy the original PDE: R(uH,p, w) = 0
Instead, it satisﬁes the weak form:
R(u, w) + δR(w) = 0 where δR(w) = −R(uH,p
, w).
ψ ∈ V relates the residual perturbation to an output perturbation:
δJ = J(uH,p
) − J(u) ≈ −R(uH,p
, ψ)
We approximate ψ in a higher order space VH,p+1 ⊃ VH,p and
estimate the error as:
δJ ≈ −
κH ∈TH
RκH (uH,p
, ψH,p+1
− ψH,p
),
We assign an adaptive indicator to κH based on its contribution to
the error estimate:
ηκH = RκH (uH,p
, ψH,p+1
− ψH,p
) .

Output-based hp adaptation
Select a fraction fadapt of the elements with largest ηκH .
A set of discrete refinement options is considered, e.g.:
pp
(a) x
p
p
(b) y
p
p p
p
(c) xy
p+1
(d) p
Rank the refinement options based on a merit function:
m(i) =
b(i)
c(i)
c(i) is a measure of the computational cost of refinement option i.
b(i) measures the gain in accuracy due to the refinement option i.
Balance between high-cost-low-error and low-cost-high-error
options
Choose the option with the highest m(i)

The merit function is computed on local sub-problems that involve
the element to be reﬁned and its neighbors.
We compare two measures of computational cost:
Degrees of freedom
cDOF(i) =
κh∈κH
(pκh (i) + 1)dim
,
Non-zeros in the Jacobian
cNZ(i) =
κh∈κH
(pκh (i) + 1)2·dim
+
∂κh∂D[(pκh (i) + 1) · (p−
κh (i) + 1)]dim .

The benefit is defined as an output sensitivity to a local residual
perturbation due to a refinement i.
b(i) =
κh∈κH
|Rκh (U(i))| · |Ψκh (i)|
where Ψκh is the coarse adjoint injected in the semi-refined space.
Observations:
The benefit as defined is machine-zero if computed before refining
the central element.
In the limit of the discrete solution representing the exact solution
to residual tolerance, εR, the benefit will also be ∼ εR.
The refinement option with the largest b(i) is expected to be the
option that produces the largest change in the output of interest.

DPW3 W1 - M∞ = 0.76, α = 0.5o
, Re = 5 × 106
Drag-based hp adaptation results.
p = 1 baseline solution.
Initial cubic (q = 3) mesh generated by agglomerating 3 linear
elements in each direction.
Spalart-Allmaras model with Oliver’s modiﬁcation [MIT PhD.
Thesis - 2008].
Initial pressure contours
(29310 cubic elements, p = 1).
Pressure contours on the 1st level of uniform
h-reﬁnement (234480 cubic elements, p = 1).

DPW3 W1 - M∞ = 0.76, α = 0.5o
, Re = 5 × 106
Results computed with 180 Harpertown 8-core nodes.
We limited the total wall time to 72 hours.
cDOF cNZ
Adaptation step iso-h sc-h dc-h iso-p iso-h sc-h dc-h iso-p
1 0.0 99.3 0.0 0.7 0.0 100.0 0.0 0.0
2 0.0 97.3 0.0 2.7 0.0 99.9 0.0 0.1
3 0.0 94.9 0.0 5.1 0.0 99.8 0.0 0.2
4 0.0 91.8 0.4 7.8 0.0 99.1 0.3 0.6
5 0.0 90.6 0.3 9.1 0.0 98.7 0.5 0.8
6 – – – – 0.0 98.6 0.5 0.9
7 – – – – 0.0 98.6 0.4 1.0
10
5
10
6
10
7
0.02
0.021
0.022
0.023
0.024
0.025
0.026
0.027
Dragcoefficient
Number of degrees of freedom
hp − cnz
hp − cdof
Uniform h
10
6
0.0206
0.0208
0.021
0.0212
0.0214
0.0216
10
4
10
5
10
6
0.02
0.021
0.022
0.023
0.024
0.025
0.026
0.027
Dragcoefficient
CPU wall−time (seconds)
10
5
0.0206
0.0208
0.021
0.0212
0.0214
0.0216

DPW3 W1 - M∞ = 0.76, α = 0.5o
, Re = 5 × 106
Final hp-adapted meshes with pressure contours (p = 1 → 5).
Pressure contours on the 5th drag-adapted
mesh using cDOF (59503 cubic elements).
Pressure contours on the 7th drag-adapted
mesh using cNZ (85377 cubic elements).

NACA 0012, M∞ = 0.15, Re = 6 × 106
, drag polar
Drag-based anisotropic h-adaptation with p = 2.
Validation of Oliver’s SA modiﬁcations against Ladson’s
experimental data.
α = 0◦, 2◦, 4◦, 6◦, 8◦, 10◦, 12◦, and 15◦
Farﬁeld at 500-chord-lengths.
Initial mesh for α = 10◦ (720 quartic elements)

NACA 0012, M∞ = 0.15, Re = 6 × 106
, drag polar
Drag convergence:
Continuous lines: drag output.
Dashed lines: drag corrected by error estimate.
Shaded region: sum of adaptive indicators.
Largest ﬁnal error estimate: 3 counts (∼ 3%) in the α = 15o case.
Final DOF count: ∼12k.
α = 10◦
α = 15◦

NACA 0012, M∞ = 0.15, Re = 6 × 106
, drag polar
Comparison with experimental results with transition at
∼ 5%-chord location.
Our results are within 3% of CFL3D results.
NASA’s Turbulence Modeling Resource spread of results is 4%.
CFL3D computed on a ﬁne, ∼230k element, structured grid.
−5 0 5 10 15 20
−0.5
0
0.5
1
1.5
2
CL
Ladson 80grit
Ladson 120grit
Ladson 180grit
CFL3D
XFlow (drag adapted)
−0.5 0 0.5 1 1.5 2
0.005
0.01
0.015
0.02
0.025
C
L
C
D
Ladson 80grit
Ladson 120grit
Ladson 180grit
CFL3D
XFlow (drag adapted)
XFlow + error estimate

NACA 0012, M∞ = 0.15, Re = 6 × 106
, drag polar
The adjoint solution shows regions of the computational domain
where discretization errors affect the output of interest.
Final mesh and ˜ν contours for α = 10o x-mom. drag adjoint on ﬁnal mesh for α = 10o

DPW5 CRM - M∞ = 0.85, Re = 5 × 106
, CL = 0.5
Cubic (q = 3) mesh generated by agglomerating 3 linear cells in
each direction.
Drag-driven anisotropic h-adaptation at ﬁxed lift with p = 1.
y+ ≈ 100 based on a ﬂat-plate correlation for Cf .
Linear mesh (1218375 elements). Agglomerated cubic mesh (45125 elements).

DPW5 CRM - M∞ = 0.85, Re = 5 × 106
, CL = 0.5
Adjoint-based parameter correction.
General framework for computing sensitivities using adjoints.
Fixed-lift adds an extra term to error estimate.
Mesh
Initial Conditions
Jtarget, εtol, αguess
Solve
Solve
R(α, U) = 0
|J − Jtarget| ≤ εtol Finished
True
False
∂R
∂U
T
Ψ = −
∂J
∂U
Compute
δR = R(α + δα, U)
Update
α ⇐ α +
(J − Jtarget)δα
ΨT δR

DPW5 CRM - M∞ = 0.85, Re = 5 × 106
, CL = 0.5
Could not achieve CL = 0.5 on the initial mesh.
Gray shaded region: range of DPW5 data computed on "ﬁne"
mesh (∼ 50M cells).
Our last adapted mesh has a number of degrees of freedom close
to the coarse meshes from the workshop.

DPW5 CRM - M∞ = 0.85, Re = 5 × 106
, CL = 0.5
Mach contours at 37% of the span.
Separation appeared on coarse mesh due to lack of spatial
resolution.
Initial mesh (α = 2.8◦) 1st drag-adapted mesh (α = 2.675◦)

DPW5 CRM - M∞ = 0.85, Re = 5 × 106
, CL = 0.5
Mach contours at 37% of the span.
Smaller differences in the Mach contours after the ﬁrst adaptive
step.
1st drag-adapted mesh (α = 2.675◦) 5th drag-adapted mesh (α = 2.1598◦)

DPW5 CRM - M∞ = 0.85, Re = 5 × 106
, CL = 0.5
Cp comparison with NASA’s experimental data.
13% of the reference span
X/Chord
-Cp
0 0.2 0.4 0.6 0.8 1
-0.5
0
0.5
Initial mesh
1st
2nd
3rd
4th
5th
Exp. data
50% of the reference span
X/Chord
-Cp
0 0.2 0.4 0.6 0.8 1
-0.5
0
0.5
Initial mesh
1st
2nd
3rd
4th
5th
Cp

Weighted mesh partitioning

The mesh is represented as an irregular graph where elements
are nodes and interior faces are edges.
The sets in red represent lines of the line-Jacobi preconditioner.
The inter-domain communication stores the data in one layer of
ﬁctitious elements neighboring each inter-domain boundary.
We use the k-way partitioning algorithm implemented in
ParMETIS.

Node weights are based on the number of non-zeros in the
self-blocks of the residual Jacobian:
ωκH = (pκH + 1)2·dim
.
The edge weights are computed in the following sequence:
1 Loop through edges of the graph (faces of the mesh) and compute:
ω∂κH ∂D = (p+
κH + 1)dim
+ (p−
κH + 1)dim
2 Loop through lines of the preconditioner and augment ω∂κH ∂D
based on the valence (connections per node) vκH :
ω∂κH ∂D ⇐ ω∂κH ∂D · max(v+
κH , v−
κH ).
Step 1 assigns weights proportional to the amount of data transfer
and step 2 increases the weight of connections between elements
that are strongly coupled.

Is it worth doing this?...YES!
NACA 0012 - M∞ = 0.5, Re = 5 × 103, α = 1◦, hp-adaptation on
8 CPU’s.
Primal solution time
1400 1600 1800 2000 2200 2400 2600 2800
0
5
10
15
20
25
30
Primalsolvetime(seconds)
c
NZ
unweighted
cNZ
weighted
c
DOF
unweighted
c
DOF
weighted
Adjoint solution time
1400 1600 1800 2000 2200 2400 2600 2800
0
0.5
1
1.5
2
2.5
Adjointsolvetime(seconds)

Is it worth doing this?...YES!
8 CPU’s.
Number of GMRES iterations is directly related to using the
preconditioner lines in the partitioning.
Adaptation time
1400 1600 1800 2000 2200 2400 2600 2800
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Adaptationtime(seconds)
cNZ
unweighted
c
NZ
weighted
cDOF
unweighted
c
DOF
weighted
GMRES iterations for primal and dual solves
1400 1600 1800 2000 2200 2400 2600 2800
1800
2000
2200
2400
2600
2800
3000
NumberofGMRESiterations

Partition map
8 CPU’s.
Red: global preconditioner lines; Black dotted: partition boundary.
Unweighted Weighted

What about 3D?...Larger savings!
DPW III Wing 1 - M∞ = 0.76, Re = 5 × 106, α = 0.5◦,
hp-adaptation on 720 CPU’s.
Challenge:what to do with empty partitions?
Primal solution time
2 2.5 3 3.5 4 4.5
x 10
5
0
0.5
1
1.5
2
x 10
4
Primalsolvetime(seconds)
cNZ
unweighted
cNZ
weighted
Adjoint solution time
2 2.5 3 3.5 4 4.5
x 10
5
0
1000
2000
3000
4000
5000
Adjointsolvetime(seconds)

What about 3D?...Larger savings!
DPW III Wing 1 - M∞ = 0.76, Re = 5 × 106, α = 0.5◦,
hp-adaptation on 720 CPU’s.
Number of GMRES iterations is directly related to using the
preconditioner lines in the partitioning.
Adaptation time
2 2.5 3 3.5 4 4.5
x 10
5
200
300
400
500
600
700
800
900
Adaptationtime(seconds)
cNZ
unweighted
cNZ
weighted
GMRES iterations for primal and dual solves
2 2.5 3 3.5 4 4.5
x 10
5
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
x 10
5
NumberofGMRESiterations

Conclusions
RANS is still very challenging with DG (specially in 3D), inclusion
of physicality constraints and better scaling of ˜ν helps in achieving
residual convergence.
Adjoints not only can be used to localize bad cells in the mesh but
also can guide directional mesh refinement.
Our hp-adaptation method is effective in achieving output
convergence as demonstrated in challenging problems in the
aeronautical industry.
Significant savings in CPU time and degrees of freedom are
observed with hp-adaptation.
The proposed mesh partitioning algorithm significantly improves
the parallel performance of both primal and dual solves.
Many challenges still exist but we hope this work helps with
increasing the presence of adaptive, high-order methods in
industrial environments.

Some ideas for the future
Better flow initialization strategies: CPTC + line-search is not
bullet-proof.
h and p coarsening: adaptation at ∼ fixed solution cost.
Adaptive node movement: reduce dependence on the initial
mesh topology.
Other element types: simplices are better for more complicated
geometries.
µ-evolution in the constrained solver: other strategies may
further improve the robustness of the solver.
Extension to other DG methods: there are more cost-efficient
DG discretizations.
Weighted partitioning: not clear what is the best way to handle
empty partitions.

Thank you!...
“Burgers are meant to be eaten, not solved!”

MarcoCeze_defense

Recommended

Recommended

More Related Content

What's hot

What's hot (13)

Similar to MarcoCeze_defense

Similar to MarcoCeze_defense (20)

MarcoCeze_defense