Full text of DOI:10.1155/2002/540189

Shock and Vibration 9 (2002) 105–121
IOS Press
105
In-situ residual tracking in reduced order
modelling
a,∗
b
b
Joseph C. Slater , Chris L. Pettit and Philip S. Beran
a
Wright State University, Dayton, OH 45435, USA
Wright-Patterson Air Force Base, OH 45433, USA
b
Received 22 August 2001
Revised 17 September 2001
Abstract: Proper orthogonal decomposition (POD) based reduced-order modelling is demonstrated to be a weighted residual
technique similar to Galerkin’s method. Estimates of weighted residuals of neglected modes are used to determine relative
importance of neglected modes to the model. The cumulative effects of neglected modes can be used to estimate error in the
reduced order model. Thus, once the snapshots have been obtained under prescribed training conditions, the need to perform
full-order simulations for comparison is eliminates. This has the potential to allow the analyst to initiate further training when
the reduced modes are no longer sufficient to accurately represent the predominant phenomenon of interest. The response of a
fluid moving at Mach 1.2 above a panel to a forced localized oscillation of the panel at and away from the training operating
conditions is used to demonstrate the evaluation method.
1
. Introduction
efficient for 2-D configurations, but has not been ex-
tended to treat 3-D configurations owing to the fully
implicit nature of the general procedure.
Computationaldeterminationof flutter boundariesin
Recently, Karhunen-Lo e` ve(K-L) analysis, or proper
orthogonal decomposition (POD), has been used to ac-
celerate greatly the time integration of aeroelastic con-
figurations by reducing system order [9,19]. POD ap-
plications to aeroelastic analysis have been limited to
time-linearized subsonic and transonic analyses of 2-D
configurations. Reduced-order modeling (ROM) with
POD is also being applied to unsteady flows for the
purpose of developing control models of these sys-
tems [16,18]. Other viable ROM strategies proposed
and/or tested for aeroelastic analysis and stability pre-
diction include eigenmode techniques [4,5], multireso-
lution modeling [12] and transition matrix analysis [1].
Pettit and Beran extended POD to a fully nonlinear
representation of the discrete Euler equations, and ap-
plied the resulting reduced-order model to the simula-
tion of unsteady flow about an oscillating bump [17].
This work followed previous developmentsof a general
framework for studying nonlinear systems with POD,
which was tested through the successful reduced-order
simulation of limit-cycle oscillation (LCO) onset in a
the transonic regime is an especially demanding prob-
lem owing to essential nonlinearities in the aerody-
namics. To properly capture the effects of aerody-
namic nonlinearities on flutter onset, time-integration
methods based on the transonic small-disturbance [6],
Euler [15], and Navier-Stokes [7,8,21] equations have
been developed to simulate the behavior of aeroelastic
systems. Direct methods based on Hopf bifurcation
theory also have been developed to compute critical
flutter onset speeds of the discrete aeroelastic equations
without time integration [3,14]. The former class of
techniques generally require large computation times
due to the time-accurate nature of the calculations and
the large integration times required to establish flow
stability properties. The latter class of methods is very
∗
Corresponding author: J.C. Slater, Department of Mechanical
and Materials Engineering, Wright State University, Dayton, OH
4
5435, USA. Tel.: +1 937 775 5040; Fax: +1 937 775 5009; E-
mail:joseph.slater@wright.edu.
ISSN 1070-9622/02/$8.00  2002 – IOS Press. All rights reserved

1
06
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
simple system [2]. The current paper extends these
results by demonstrating an alternative way to evaluate
the importance of selected modes to simulation results
in-situ. In-situ monitoring promises to allow the an-
alyst to adjust the ROM during simulation in an ap-
propriate fashion to optimize for simulation speed and
accuracy.
3. Karhunen-Lo e` ve analysis
The mode vectors, φm = 1, 2, 3, . . . , n, can be,
and often are, obtained using Karhunen-Lo e` veanalysis
(also referred to as proper orthogonal decomposition,or
(
i)
POD) [11,19]. A set of snapshot vectors, w , are gen-
erally obtained through integration of the full system
Eq. (1) for some predefined conditions similar to the
current analysis. Choice of the predefined conditions is
based upon “average” operating conditions. The hope
is that the space covered by the training simulation is
sufficient to cover the space within which the second
simulation, the reduced-order simulation, should oper-
ate. If not, then a full-order simulation can/should be
performed under the perturbed operating conditions in
order to generate snapshots more useful over the new
local domain.The vectors are combined into a snapshot
2
. Galerkin’s method applied to a set of 1st order
ODEs
Consider a set of nonlinear first order state space
equations of the form
w˙ = R(w; α)
(1)
where R is a vector of nonlinear functions and α is a
parameter, or list of parameters, upon which the system
depends. Assume a solution of the form
ꢅ
ꢆ
(1) (2) (3)
(i)
matrix S = w
w
w
. . . w . A reduced set
of vectors, Φ, are obtained from these snapshots repre-
senting the space spanned by S in an optimal fashion.
N
ꢀ
ꢃ
ꢄ
T
w(t) =
wˆ n(t)φn
(2)
Considering the term Φ Φ in Eq. (8). Substituting
Φ = SV yields
n=1
ꢃ
ꢄ
T
T
T
where wˆ n(t) represent so-called modal coordinates and
φn represent so-called mode shapes. This solution
form can also be written as
Φ Φ = V S SV
(9)
If the matrices V are chosen to be the eigenvectors
T
of S S, then
w(t) = Φ wˆ (t)
(3)
ꢃ
ꢄ
T
T
T
Φ Φ = V S SV = Λ
(10)
where Φ = [φ φ φ . . . φ ]. Substituting (3) into the
1
2
3
n
T
where Λ is the diagonal eigenvalue matrix of S S.
equation of motion (1) yields
T
−1
Since V is orthonormal,V = V . Thus, the solution
˙
Φ wˆ = R(Φ wˆ , α)
(4)
(5)
of the eigenvalue problem
We define the residual to be
T
S SV = V Λ
(11)
˜
˙
R = Φ wˆ − R(Φ wˆ , α)
along with the relation Φ = SV yields a set of orthog-
onal modes Φ. The magnitude of each of the eigenval-
ues Λ represents the degree to which the correspond-
ing eigenvector, vi, participates in the set of snapshots.
As a result of this analysis, the modes are weighted by
the square root of the eigenvalues λ. Modes that do
not participate “significantly” in the snapshots are trun-
cated. The more commonly used form of the reduced-
order equations is obtained by substituting Φ = SV
and Eq. (10) into Eq. (8) yielding
The nth weighted residual is then given by
ꢁ
ꢂ
T
n
T
n
˜
˙
φ R = φ Φ wˆ − R(Φ wˆ , α)
(6)
Putting the weighted residuals in vector form, and
constraining them to be equal to zero yields
ꢁ
ꢂ
T
T
˙
˜
Φ R = Φ Φ wˆ − R(Φ wˆ , α) = 0
(7)
˙
Solving for wˆ yields the reduced-order equations of
motion as
−1
T
˙
wˆ = (Λ) V SR(Φ wˆ , α)
(12)
ꢃ
ꢄ−1
Φ R(Φ wˆ , α)
˙
T
T
wˆ = Φ Φ
(8)
The shortcoming of this approach is that there is no
guarantee that the mode shapes obtained from one sim-
ulation will be sufficient to span the space of another
simulation. When performing the reduced-order sim-
ulation, it becomes necessary to truncate the vectors
Φ to only those containing sufficient energy content
which can be simulated in time, using larger time
steps [2,17,13] than the original equations of motion,
with the intention of obtaining results functionally
equivalent to those that would have been obtained from
a full-order model simulation.

J.C. Slater et al. / In-situ residual tracking in reduced order modelling
107
Deformed Panel Segment
U∞
(Shown More Deformed Than Actual)
y
-1/2
+1/2
x
Fig. 1. Schematic of panel and coordinate system.
(
significant eigenvalues, λ) [17] or else Eq. (12) can
the magnitude of the ignored weighted residuals, given
by Eq. (16), can be monitored during simulation. A
significant increase in the values as compared to their
nominal values obtained during training can be used as
an indicator of error in the reduced-order simulation.
break down due to the near singularity of the matrix Λ.
Likewise, Eq. (10) illustrates that the sensitivity exists
even when using the formulation of Eq. (8). Thus an
“
optimal” truncation method is necessary.
An alternative formulation is possible by un-
weighting the mode shapes φ by dividing them each
by the square root of their corresponding eigenvalues.
Thus the eigenvalue problem described by Eq. (10) be-
comes
5
. Test problem
1
1
The ROM/POD framework described above for an-
−
T
− 2
T
2
¯ ¯
Λ
Φ ΦΛ = Φ Φ
alyzing unsteady solution behavior is placed in a pro-
gram called RAPOD, which is designed to be problem
independent, except for evaluation of the full-system
array, R [17]. To test the RAPOD implementation for
the unsteady response of a flowfield to time-dependent
changes in geometry, we have studied the problem of
inviscid flow over a 2-D, oscillating bump using the
Euler equations.
The flowfield is assumed to occur above an infinite,
segmented panel that nominally lies in the y = 0 coor-
dinate plane, and whose surface is defined by ys(x, t)
(see Fig. 1). A deformingsegment of the panel is speci-
fied between x = −1/2 and x = 1/2 such that the seg-
ment length, c, is normalized to unit value. Away from
the deforming segment, the panel is flat (ys(x, t) = 0,
(13)
1
1
− 2
T
T
− 2
=
Λ
V S SV Λ
= I
The advantageof this formulationis that the equation
of motion Eq. (8) can be simplified to the form
˙
¯T
¯
wˆ = Φ R(Φ wˆ , α)
(14)
¯
where Φ represent the unit length mode shapes. The
elements of the eigenvaluematrixΛcanhavea variation
of multiple orders of magnitude. Thereis some concern
that this may impact stability and/or accuracy of the
simulations as a result of roundoff error, however this
hypothesis has not yet been tested.
4
. Validation of simulation results
|
x| > 1/2), while over the deforming segment, a time-
The weighted residual is given in Eq. (6) as
dependent deflection is specified:
ꢁ
ꢂ
T
n
T
n
ꢃ
−βtꢄ
¯
˜
¯
¯ ˙
¯
φ R = φ Φ wˆ − R(Φ wˆ , α)
(15)
δ(t) = δ cos(ωt) 1 − e
1
,
(17)
(18)
Recall that for n greater than some chosen value,
N, this weighted residual is not constrained to be zero
ꢃ
2ꢄ
ys(x, t) = δ 1 − 4x
(|x| ꢀ 1/2),
because some of the modes φ were truncated in the
n
where δ1 is a deflection amplitude, ω is a frequency,
and β is a modulation parameter to adjust the short-
time behavior of the system. The spatial and tempo-
ral variables are non-dimensionalized using c and the
aerodynamic characteristic time based on the far-field
velocity U∞. The instantaneous shape of the panel is
that of a parabolic arc with maximum deflection |δ(t)|
at x = 0. The large-time behavior of δ(t) is δ1 cos(ωt),
which represents simple periodic motion of the bump
height.
reduced-order analysis. From Eq. (10) it can be seen
T
that for any n greater than N, φ Φ = 0 due to the
n
orthogonality of the modes. The second part of the
residual is not necessarily zero and thus the weighted
residual for n > N is given by
ꢃ
ꢄ
T
n
T
n
¯
˜
¯
¯
φ R = φ −R(Φ wˆ , α)
(16)
It is a reasonable assumption that these truncated
modes become more significant for simulations in ad-
jacent regions of the design space (e.g. α = α0). Thus
ꢀ

1
08
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
6
4
2
0
1
Range
0.5
of Bump
0
-
1
-0.5
0
0.5
1
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
x
Fig. 2. Baseline grid.
2
Residual 5
Residual 6
Residual 7
Residual 8
1.5
1
0.5
0
-0.5
-1
-1.5
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 3. Residuals 5–8 for 4 mode simulation with δ1 = 0.005.
6
. Initial and boundary conditions
Initially, the flow is that of the freestream state and
the flowfield. To avoid potential difficulties associated
with grid deformation and the reduced-order model-
ing procedure described above, a transpiration bound-
ary condition is applied at y = 0 to model the effects
of a moving boundary [20]. For the bump problem
examined herein, the transpiration boundary condition
involves the enforcement of the exact condition of im-
permeability at the panel surface,
the panel is undeflected. After using the scales de-
fined above and non-dimensionalizing density by the
far-field value, ρ∞, and pressure by ρ∞U , the non-
dimensional farfield conditions are
2
∞
ρ → 1, u → 1, v → 0,
(19)
∂y
∂ys
s
2
p → 1/(γM ),
−u ∂ + v = ∂t
(y = ys(x, t)),
(20)
∞
x
where p is the pressure, (u, v) are velocity components
in the (x, y) coordinate directions, γ is the ratio of
specific heats, and M∞ is the freestream Mach number.
In a conventional simulation of the aerodynamic re-
sponse to an oscillating bump, a deforming, panel-
conforming grid would be used in the discretization of
at y = 0. This transfer of boundary conditions is iden-
tical to that employed in small-disturbance theory, and
assumes regularity of the computed solution and small-
ness of the deformation: y (x, t) ꢁ 1. In addition to
s
impermeability, we apply additional conditions to close
the discretized Euler equations at the panel surface:

J.C. Slater et al. / In-situ residual tracking in reduced order modelling
109
3
2
1
0
Residual 9
Residual 10
Residual 11
Residual 12
-
-
-
-
1
2
3
4
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 4. Residuals 9–12 for 4 mode simulation with δ1 = 0.005.
1.5
Residual 9
Residual 10
Residual 11
Residual 12
1
0.5
0
-0.5
-1
-1.5
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 5. Residuals 9–12 for 8 mode simulation with δ1 = 0.005.
∂
∂
u
y
∂p
∂ρ
= ∂y = 0
spect to the y-coordinate are specified to vanish, rather
than the coordinate normal to the deformed panel.
This approximation assumes smallness of deforma-
= ∂y
(y = 0)
(21)
In Eq. (21), derivatives of primitivevariables with re-

1
10
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
Residual 13
Residual 14
Residual 15
Residual 16
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 6. Residuals 13–16 for 8 mode simulation with δ1 = 0.005.
tion slopes, consistent with the prior assumption of
8. Governing equations and method of solution
ys(x, t) ꢁ 1.
The governing aerodynamic equations are the Euler
equations, cast in nondimensional form for a general
curvilinear coordinate system (ξ,η). For the grid de-
scribed above, the ξ coordinate runs in the x-coordinate
direction, whereas η is associated with the y coordi-
nate. The equations are expressed in terms of conserved
7
. Grid construction
The flow is simulated over a physical domain of
length L, centered about x = 0 and height H. The
domain is discretized using I nodes in the streamwise
direction and J nodes normal to the panel. Indices i
T
variables, U ≡ [ρ, ρu, ρv, Et] :
ˆ
ˆ ˆ
∂
U
∂E(U) ∂F(U)
+
+
= 0,
(22)
(1 ꢀ i ꢀ I) and j (1 ꢀ j ꢀ J) are used to denote grid
∂
t
∂ξ
∂η
points at which variables are evaluated. Grid points cor-
responding to j = 1 are placed on the x-coordinate line
and do not move with changes in δ. Grid points are clus-
tered in the direction normal to the panel at the panel
surface, with the minimum spacing denoted by ∆wall.
The spacing of grid points is specified to grow geo-
metrically with normal index position from the panel
boundary. In the streamwise direction, the node spac-
ing is chosen to be uniform over the deforming panel
segment, while growing geometrically upstream of the
leading edge (positioned at i = ILE) and downstream
of the trailing edge (positioned at i = iT E). A baseline
grid, shown in Fig. 2, is constructed with the following
values: I = 141, J = 71, L = 15, H = 6, ILE = 55
and IT E = 85. For the baseline grid, ∆wall ≈ 0.0137.
ˆ
ˆ
ˆ
where U = U/(ξxηy − ξyηx), and E and F are trans-
formed flux arrays [14]. The aerodynamic equations
are placed in discrete form following the upwind to-
tal variation diminishing (TVD) scheme of Harten-
Yee [10,23], with a correction for grid-point motion.
The discretization of second-order or first-order spa-
tial accuracy, depending on parameter selection, and
first-order temporal accuracy is expressed as [3]
ˆn+1
ˆn
i,j
U
− U
n
n
i,j
˜
˜
ˆ
ˆ
= F
1
2
− F
1
2
i,j−
i,j+
∆
t
(23)
n
n
˜
˜
ˆ
ˆ
+E
1
− E
1
2
,j,
i− ,j
i+
2
˜
˜
ˆ
ˆ
where the arrays E and F are modified numerical fluxes

J.C. Slater et al. / In-situ residual tracking in reduced order modelling
111
0.5
0.4
0.3
0.2
0.1
0
Residual 13
Residual 14
Residual 15
Residual 16
-0.1
-0.2
-0.3
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 7. Residuals 13–16 for 12 mode simulation with δ1 = 0.005.
0
0
0
.3
Residual 17
Residual 18
Residual 19
Residual 20
.2
.1
0
-
0.1
0.2
0.3
0.4
-
-
-
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 8. Residuals 17-20 for 12 mode simulation with δ1 = 0.005.
that implement the TVD formulation.
Conditions for boundary nodes, except on the panel
surface (y = 0), are developed and placed in first-
order evolutionary form to be consistent with Eq. (1).
Freestream conditions are enforced along the inflow
and farfield boundary (y = H) using the soft evolu-

1
12
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
0.4
0.3
0.2
0.1
0
Residual 17
Residual 18
Residual 19
Residual 20
-0.1
-0.2
-0.3
-0.4
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 9. Residuals 17–20 for 16 mode simulation with δ1 = 0.005.
Pressure
2
1
0
.5
2
0.505
0
0
.5
.5
1
.495
0.49
0
0
.485
.5
.48
0
-
1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 10. Pressures after 20 second simulation using 8 modes with δ1 = 0.005.
n+1
n
n
tionary equation (q
− q )/∆t = −(q − q∞)/∆t,
9. Results
where q is a conserved variable. Outflow condi-
tions are similarly enforced with a convective equation:
This section describes sample results from the os-
n+1
n
n
n
(q
− q )/∆t = −u(ξxq + ηxq ).
cillating bump problem for M = 1.2 and γ = 1.4.
ξ
η
∞

J.C. Slater et al. / In-situ residual tracking in reduced order modelling
113
Pressure
2
1
0
.5
2
0
0
0
.505
.5
.5
1
.495
0.49
0
0
.485
.5
.48
0
-
1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 11. Pressures after 20 second simulation using 12 modes with δ1 = 0.005.
Pressure
2
1
0
.5
2
0
0
0
.505
.5
.5
1
.495
0.49
0
0
.485
.5
0
.48
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 12. Pressures after 20 second simulation using 16 modes with δ1 = 0.005.
Snapshots were collected by integrating the full system
with ∆t = 0.01, δ1 = 0.005, ω = 1.0, and α = 3.0.
Two thousand iterations were performed using first-
order spatial accuracy and snapshots were taken every
25 iterations for a total of 80 snapshots. Although the
RAPOD algorithm was constructed to permit taking

1
14
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
Pressure
2
1
0
.5
2
0
0
0
.505
.5
.5
1
.495
0.49
0
0
.485
.5
.48
0
-
1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 13. Pressures after 20 second simulation, full model with δ1 = 0.005.
4
Residual 5
Residual 6
Residual 7
Residual 8
3
2
1
0
1
2
3
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 14. Residuals 5-8 for 4 mode simulation with δ1 = 0.01.
snapshots at a response-dependent rate, this capability
was not used in the present problem.
Modes were generated as described by Eqs (9–14)
and ranked, in the traditional manner, by decreasing
eigenvalue. The first 24 modes were kept for the anal-
yses, and four reduced-order model simulations were

J.C. Slater et al. / In-situ residual tracking in reduced order modelling
115
6
4
2
0
Residual 9
Residual 10
Residual 11
Residual 12
2
4
6
8
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 15. Residuals 9–12 for 4 mode simulation with δ1 = 0.01.
2
1
0
.5
Residual 9
Residual 10
Residual 11
Residual 12
2
.5
1
.5
0
0
1
2
.5
1
.5
2
.5
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 16. Residuals 9–12 for 8 mode simulation with δ1 = 0.01.
performed: one each using the first 4, 8, 12, and 16
modes. In addition, simulations were repeated for the
off-training-condition of δ1 = 0.01.
For the 4 mode reduced-order simulation, observing
Fig. 3 in the fully developed part of the solution, the
residuals decrease in the order of modes 5, 7, 8, then

1
16
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
1.5
Residual 13
Residual 14
Residual 15
Residual 16
1
0.5
0
0.5
1
1.5
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 17. Residuals 13-16 for 8 mode simulation with δ1 = 0.01.
1
Residual 13
Residual 14
Residual 15
Residual 16
0.5
0
0.5
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 18. Residuals 13–16 for 12 mode simulation with δ1 = 0.01.
6
, while Fig. 4 shows that two of the later modes have
modes would result in significant error as compared to
the full order solution.
Performing an 8 mode simulation results in residuals
greater residuals than that for mode 5. This seems to
indicate that choosing to perform the simulation with 8

J.C. Slater et al. / In-situ residual tracking in reduced order modelling
117
0.6
0.4
0.2
0
Residual 17
Residual 18
Residual 19
Residual 20
0.2
0.4
0.6
0.8
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 19. Residuals 17–20 for 12 mode simulation with δ1 = 0.01.
0
0
0
0
.8
Residual 17
Residual 18
Residual 19
Residual 20
.6
.4
.2
0
0.2
0.4
0.6
0.8
0
2
4
6
8
10
12
14
16
18
20
Time (sec)
Fig. 20. Residuals 17–20 for 16 mode simulation with δ1 = 0.01.
for the higher modes that are increasing with time (See
Figs 5 and 6). It is not possible from the simulation to
determine whether the simulation is unstable, or if at
some point the error will be self limiting. Instability of
reduced-ordermodels has been observedby the authors
for other systems. The phenomenon is not yet widely

1
18
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
Pressure
2
1
0
.5
2
0
0
0
.505
.5
.5
1
.495
0.49
0
0
.485
.5
0
.48
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 21. Pressures after 20 second simulation using 8 modes with δ1 = 0.01.
Pressure
2
1
0
.5
2
0
0
0
.505
.5
.5
1
.495
0.49
0
0
.485
.5
0
.48
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 22. Pressures after 20 second simulation using 12 modes with δ1 = 0.01.
understood, and is currently under investigation. A
simple example illustrating this phenomenon is given
by Slater [22]. On the contrary, the 12 mode simulation
(Figs 7 and 8) shows residuals that tend to remain small
over time, and are remaining well below 0.1, with the
exception of mode 18 which approaches 0.1 in the late

J.C. Slater et al. / In-situ residual tracking in reduced order modelling
119
Pressure
2
1
0
.5
2
0
0
0
.505
.5
.5
1
.495
0.49
0
0
.485
.5
.48
0
-
1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 23. Pressures after 20 second simulation using 16 modes with δ1 = 0.01.
Pressure
2
1
0
.5
2
0
0
0
.505
.5
.5
1
.495
0.49
0
0
.485
.5
0
.48
-1
-0.5
0
0.5
1
1.5
2
2.5
Fig. 24. Pressures after 20 second simulation, full model with δ1 = 0.01.
transient part of the response.
The 16 mode simulation (Fig. 9) shows results simi-
lar to the 12 mode simulation. The only mode of con-
cern is mode 18, which has a greater residual than all
other modes 13–20 for both the 12 and 16 mode models.
However, when compared to the 4 and 8 mode simula-

1
20
J.C. Slater et al. / In-situ residual tracking in reduced order modelling
tions, it is clear that including additional modes in the
reduced order model will have a relatively negligible
effect.
One could conclude that 12 modes are sufficient by
referring to Figs 7 and 8. In fact, referring to Figs 10–
content (eigenvalue) ranking of the modes based on a
training simulation under potentially different operat-
ing conditions or different regimes of operation (e.g.
start-up or fully developed). Hence the importance of a
mode can be evaluated more appropriately for the cur-
rent simulation, enabling a more appropriate selection
of a set of modes in the reduced-order simulation.
1
3, it is apparent that very little improvement in the
model is made by including modes 13 or higher.
The results shown in Figs 7 and 8 suggest that 12
retained modes are sufficient to capture accurately the
aerodynamic response to bump oscillation. This ob-
servation is reinforced through comparison of pressure
fields computed with 8-, 12- and 16-mode ROMs at
t = 20 (shown in Figs 10–12). In each of these contour
plots, pressure values are in close agreement with re-
sults computed using the full-system equations, repro-
ducing flow structure adjacent to, and away from, the
oscillating bump. In addition, Figs 7–9 illustrate that
the higher modes are important only for representing
the startup transient. This trait was previously noted by
Pettit and Beran [17]. However, Fig. 8 also illustrates
that mode 18 contributes to the response when the flow
is fully developed.
Acknowledgements
The first author would like to thank the Air Force
Research Laboratory for support through the Air Ve-
hicles Directorate Summer Faculty Research Program
(Dr. Beran, Research Manager).
References
[
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