# Investigations on Projection-Based Reduced Order Model Development for
Rotating Detonation Engine

Ryan Camacho 111Doctoral Student, Aerospace Engineering, University of Kansas
and Cheng Huang222Assistant Professor, Aerospace Engineering, University of
Kansas University of Kansas, Lawrence, KS

###### Abstract

The current study aims to evaluate and investigate the development of
projection-based reduced-order models (ROMs) for efficient and accurate RDE
simulations. Specifically, we focus on assessing the projection-based ROM
construction utilizing three different approaches: the linear static basis,
nonlinear quadratic basis, and an adaptive model order reduction (MOR)
formulation. First, an a priori analysis is performed to evaluate the
effectiveness of the linear static and nonlinear quadratic bases in
representing the detonation-wave dynamics. The a priori analysis reveals that
compared to the linear basis, the nonlinear quadratic basis provides
significantly improved representation of detonation-wave dynamics within the
training regime. However, it exhibits limited capabilities in representing the
dynamics beyond the training regime, either in the future state or under a
different operating parameter (i.e., inlet velocity). Second, the
investigations proceed to the adaptive MOR formulation, which constructs an
online adaptive ROM with a small amount of offline training data. It is
demonstrated that the adaptive ROM can provide significantly enhanced
predictive capabilities in modeling the RDE dynamics in the future state, and
subject to parametric variations. More importantly, the adaptive ROM is shown
to be capable of capturing the initial transience in establishing the
detonation wave.

##  1 Introduction

Pressure gain combustion is recognized as a viable alternative to traditional
combustion in propulsion applications due to its theoretical performance
improvements [1, 2, 3]. As one of the most prominent realizations based on
this concept, the rotating detonation engine (RDE) offers several advantages
for application in propulsion or land-based power generation, including
mechanical simplification, broad operability limits [4, 5], potential to
increase thermal efficiency [6, 7], and the reduction of propellant pumping
requirements [8, 9]. Therefore, RDEs have garnered significant interest for
both experimental [10, 11, 12, 13, 14, 13] and numerical [15, 16, 17, 18, 19,
20, 21] investigations to better understand the underlying physics. However,
the practical applications of RDEs require a considerable number of parametric
evaluations over a range of operating conditions and geometric designs.
Conducting such a large number of experiments consumes significant resources
and provides restricted access to quantify and understand the underlying
physics. On the other hand, high-fidelity simulations, such as large-eddy
simulations (LES), can provide valuable physical insights for RDEs. Even with
the modern exascale computing capabilities [22], however, due to the inherent
physics complexity, these simulations remain prohibitively expensive for
practical engineering applications. For example, Lietz et al. [17] illustrated
that a three-dimensional (3D) LES of an RDE was feasible, but even with a
subset of the experimental geometry, approximately 0.18 million CPU hours were
required to simulate 2ms of engine run-time. While engineering designs often
require hundreds of evaluations through these simulations, they are not
affordable for practical RDE applications.

To address this gap, reduced models, either physics-based [23] or data-driven
[24], have been developed in the literature to provide efficient and
reasonably accurate predictions of performance metrics. Physics-based reduced
models typically seek to reduce the computational cost through simplification
of the physical models, and are referred to as reduced-fidelity models (RFMs).
A common approach for RDE applications is to model a 3D configuration as an
unrolled two-dimensional (2D) domain with periodic boundary conditions, which
has been widely applied and demonstrated effective for premixed RDE
applications [25, 26, 27, 19]. Additional reductions in modeling fidelity can
further increase computational efficiency. Examples include 1) solving 2D
Euler instead of 2D Navier-Stokes equations [28] ; 2) solving 1D Euler
equations with a modeled source term to approximate second-dimension
injection, mixing, and exhaustion as a lump volume [29]; 3) developing a two-
equation phenomenological modeling framework for the detonation wave dynamics
based on experimental observations [30]; or 4) constructing a simple
thermodynamic model to predict the quantities of interest [23].

Data-driven reduced models, on the other hand, seek to inform reduced-order
models (ROMs) using either simulation and experiment data instead of relying
on physics simplifications. Such methods can be classified as either machine
learning (ML) or model-order reduction (MOR). ML is a pure data-driven black-
box approach that estimates associations between system inputs, outputs, and
parameters using a limited number of observations [31], leading to a low entry
barrier for implementations. For example, Koch [24] trains neural ordinary
differential equations from high-speed camera footage to approximate the
Arrhenius kinetics and the mixing models in the two-equation phenomenological
modeling framework [32]. Mendible et al. [33] demonstrated the feasibility of
constructing ROMs for RDE directly from experimental data. However, one
challenge is that ML methods lacks generalizability [34, 35, 36] \- i.e., the
trained ML models often need to be specially trained and are only applicable
to a narrow range of applications. As a result, broadly applicable ML models
would require copious amounts of data for training, resulting in large data
storage requirements. Additionally, ML models will not always satisfy physical
laws (e.g., conservation of mass) as this method only seeks to create a black
box that relates the inputs and outputs. While there has been recent research
into incorporating physical laws in ML [37], its applicability to physically
complex problems such as turbulent reacting flows remains an open question.

The second data-driven approach, MOR (specifically, projection-based MOR)
derives ROMs to represent the evolution of a high-dimensional nonlinear PDE by
projecting the PDE onto a low-dimensional subspace [38]. the core factor that
distinguishes MOR from ML is the direct integration of PDE (i.e., physics)
into the reduced model, ensuring that the MOR-ROM respects the model physics
and produces physically accurate results. The importance of the integration of
PDE has been extensively documented [39, 40, 41, 42, 43]. MOR has been used
successfully to build ROMs for a broad range of applications, including flow
control [44, 45], aeroelasticity [46, 47], hypersonics [48], and combustion
[49, 50]. More recently, Farcas et al. [51, 52] successfully used a non-
intrusive MOR method to model a coarsened RDE problem. Typically, the
construction of a projection based ROM consists of two stages:

  * •

An offline stage that performs full-order-model (FOM) simulations of the
target systems to construct low-dimensional subspaces to represent the high
dimensional solution data, typically through proper orthogonal decomposition
(POD) [53].

  * •

And online stage that executes ROM through evolving the dynamics on the low-
dimensional spaces by projecting the FOM onto the lower dimensional subspace.

However, projection-based ROM development for RDE applications present one
major challenge, Detonation-wave dynamics feature convection-dominated multi-
scale physics and exhibit a slow decay of Kolmogorov N-width [54] that cannot
be effectively approximated using low-dimensional subspaces (e.g., those by
POD). Thus, these problems are typically not amenable to data reduction. This
specific issue is referred to as the Kolmogorov barrier. As a result, when
greater accuracy is desired, it becomes necessary to greatly increase the
dimensionality of the low dimensional subspace, increasing the computational
expense of the resulting ROM.

In order to break the Kolmogorov barrier, different schools of thoughts have
arisen. The first strategy seeks to construct multiple local subspaces to
describe the high-dimensional convection-dominated dynamics, as opposed to a
single global subspace [55, 56]. In this method the ROM can be adapted to
local dynamics rather than trying to resolve the entire domain, allowing for
decreased dimensionality and better performance. The second strategy uses
nonlinear basis (or manifolds) to construct the low-dimensional subspaces. One
class of popular approaches leverages subspace/snapshot transformation to
construct such basis, with the goal of recovering low-rank structures with
rapid-decay Kolmogorov N-widths. These approaches include the method of
freezing [57], shifted POD [58, 59, 60], transported subspaces [61] or
snapshots [62] , and implicit-feature tracking [63]. Other researchers
directly compute nonlinear basis using convolutional autoencoders [64, 65] or
quadratic approximation [66, 67]. The third strategy is to use adaptive MOR
[68, 69, 70] to break the Kolmogorov barrier by updating the ROM in the online
stage to either satisfy a posteriori error estimation or seek an optimal
representation of the target problems’ evaluated dynamics. For example,
Peherstorfer [71] adapts the affine approximation space by exploiting the
locality in space and time of propagating coherent structures evaluated from
the sampled FOM solutions. Ramezanian et al. [72] proposed an on-the-fly ROM
method for reactive-species transport by deriving evolution equations for a
low-dimensional time-dependent manifold through a variational principle.

Therefore, the objective of the current paper is to investigate the
projection-based ROM development for RDE applications. Specifically, we
consider three approaches to construct the projection-based ROM: 1) the linear
static basis, 2) the nonlinear quadratic basis [66], and 3) adaptive MOR [73].
Detailed assessment of the three ROM methods are performed using a benchmark
2D RDE problem, which exhibits distinct detonation wave dynamics, featuring
slow decay of Kolmogorov N-width, subject to variations in operating
conditions. The remainder of the paper is organized as follows. Section 2
presents the FOM formulation. Section 3 presents the ROM formulation,
including the construction of the linear and nonlinear quadratic bases, and a
recently developed adaptive ROM formulation. Section 4 discusses the
computational configuration for the RDE problem. Section 5 presents the
numerical results and investigations based on the 2D RDE configuration
including the a priori analysis for assessment of the linear and nonlinear
quadratic bases and the evaluations of the adaptive ROM. In section 6, we
provide concluding remarks and future directions for the work.

##  2 Full-Order Model

The governing equations of the full-order model (FOM) used for all the
numerical results in the current paper can be expressed in a generic semi-
discrete form

$\frac{\text{d}\mathbf{q}\left(\mathbf{q}_{p}\right)}{\text{dt}}=\mathbf{f}\left(\mathbf{q}_{p}\right)\
\ \ \text{with}\ \ \
\mathbf{q}_{p}\left(\text{t}=0\right)=\mathbf{q}_{p}^{0},$ (1)

where $\text{t}\in\left[0,t_{f}\right]$ is the solution time, which spans the
time interval from $0$ to $t_{f}$, $\mathbf{q}_{p}\in{\mathbb{R}^{N}}$ is the
vector of solution (or state) variables,
$\mathbf{q}_{p}^{0}\in{\mathbb{R}^{N}}$ is the vector of states to be
specified as the initial conditions at $t=0$,
$\mathbf{q}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ and
$\mathbf{f}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$ are nonlinear functions
of $\mathbf{q}_{p}$. $N$ is the total number of degrees of freedom in the
system. The function $\mathbf{q}$ represents the conservative state in the
case of a FOM based on conservation laws, and the function $\mathbf{f}$
represents surface fluxes, source terms, and body forces resulting from the
spatial discretization of the governing equations. In addition, different
time-discretization methods can be introduced to solve Eq. 1, such as linear
multi-step, or Runge–Kutta methods [74]. For the analysis performed in this
paper, a second order implicit scheme was used for both FOM and ROM
calculations

$\mathbf{r}\left(\mathbf{q}_{p}^{n}\right)\triangleq\frac{3}{2}\mathbf{q}\left(\mathbf{q}_{p}^{n}\right)-2\mathbf{q}\left(\mathbf{q}_{p}^{n-1}\right)+\frac{1}{2}\mathbf{q}\left(\mathbf{q}_{p}^{n-2}\right)-\Delta\text{t}\mathbf{f}\left(\mathbf{q}_{p}^{n}\right)=0.$
(2)

where $\Delta\text{t}\in\mathbb{R}^{+}$ is the physical time step for the
numerical solution, and $\mathbf{r}:\mathbb{R}^{N}\rightarrow\mathbb{R}^{N}$
is defined as the FOM equation residual. The state variables,
$\mathbf{q}_{p}^{n}$, are solved for at each time step so that
$\mathbf{r}\left(\mathbf{q}_{p}^{n}\right)=\mathbf{0}$. the details of the
discretization can be found in Ref. [49].

##  3 Reduced-Order Model

As discussed in section 1, the ROM construction for problems featuring multi-
scale phenomena with strong convection and nonlinear effects is well-
recognized to be a major challenge due to the slow decay of Kolmogorov
N-width. Therefore, we investigate two methods to address this challenge: 1.
using nonlinear quadratic basis to construct low-dimensional subspace [66,
67], and 2. an adaptive ROM formulation developed by Huang and Duraisamy [73].
In this section, we first briefly describe the low-dimensional subspace
construction via linear basis and then introduce the two methods.

###  3.1 Construction of Low-dimensional Subspace for Solution Variables using
Linear Basis

The state variables, $\mathbf{q}_{p}$ in Eq. 1, are approximated using a set
of low-dimensional subspace (or trial basis),
${\mathbf{{V}}_{p}}\in\mathbb{R}^{N\times n_{p}}$

$\mathbf{{H}}\left(\widetilde{\mathbf{q}}_{p}-\mathbf{q}_{p,\text{ref}}\right)={\mathbf{{V}}_{p}}\mathbf{q}_{r}.$
(3)

where $\widetilde{\mathbf{q}}_{p}$ represents an approximation of the state
variable (i.e. $\widetilde{\mathbf{q}}_{p}\approx\mathbf{q}_{p}$).
$\mathbf{q}_{\text{ref}}$ represents a reference state, which is selected to
be the initial FOM solution
$\mathbf{q}_{p,\text{ref}}=\mathbf{q}_{p}(\text{t}=\text{t}_{0})$ in the
training data for the current investigations. The trial
basis${\mathbf{{V}}_{p}}$ is computed via POD [75, 76] using the singular
value decomposition (SVD). $\mathbf{q}_{r}\in\mathbb{R}^{n_{p}}$ is the
reduced state with $n_{p}$ representing the number of trial basis.
$\mathbf{{H}}\in{\mathbb{R}^{N\times N}}$ represents a scaling matrix,
introduced to keep all variables of similar orders of magnitude when computing
the trial basis. In the current study, all quantities are normalized by their
$L^{2}$-norm, as proposed by Lumley and Poje [53]

$\mathbf{{H}}=diag\left(\mathbf{{H}}_{1},\ldots,\mathbf{{H}}_{i},\ldots,\mathbf{{H}}_{N_{elem}}\right),$
(4)

where
$\mathbf{{H}}_{i}=diag\left(\phi^{-1}_{1,norm},\ldots,\phi^{-1}_{N_{var},norm}\right)$.
Here, $\phi_{v,norm}$ represents the $v^{th}$ state variable in
$\mathbf{q}_{p}$ and

$\phi_{v,norm}={\frac{1}{\text{t}_{1}-\text{t}_{0}}\int^{\text{t}_{1}}_{\text{t}_{0}}\frac{1}{\Omega}\int_{\Omega}\phi^{\prime
2}_{v}(\mathbf{x},\text{t})\;\text{d}\mathbf{x}\;\text{dt}}.$ (5)

###  3.2 Construction of Low-dimensional Subspace for Solution Variables using
Nonlinear Quadratic Basis

Alternatively, nonlinear basis is pursued to construct the low-dimensional
subspace for improved approximation of the state variables, $\mathbf{q}_{p}$.
Specifically, in the current paper, we seek to construct such nonlinear
subspace by introducing an additional quadratic structure to Eq. 3 following
the work by Geelen et al. [66]

$\mathbf{{H}}\left(\widetilde{\mathbf{q}}_{p}-\mathbf{q}_{p,\text{ref}}\right)={\mathbf{{V}}_{p}}\mathbf{q}_{r}+\bar{\mathbf{V}}_{p}(\mathbf{q}_{r}\otimes\mathbf{q}_{r}).$
(6)

where $\bar{\mathbf{V}}_{p}\in{\mathbb{R}^{N\times n^{2}_{p}}}$ represents a
quadratic mapping operator and $\otimes$ denotes the Kronecker product. The
quadratic operator $\bar{\mathbf{V}}_{p}$ is computed from the following
optimization problem

$\bar{\mathbf{V}}_{p}\triangleq\operatorname*{arg\,min}_{\bar{\mathbf{V}}_{p}\in{\mathbb{R}^{N\times
n^{2}_{p}}}}\left\lVert\mathbf{{H}}\left(\mathbf{q}_{p}-\mathbf{q}_{p,\text{ref}}\right)-{\mathbf{{V}}_{p}}\mathbf{q}_{r}+\bar{\mathbf{V}}_{p}(\mathbf{q}_{r}\otimes\mathbf{q}_{r})\right\rVert_{2}^{2}.$
(7)

In addition, as reported by Barnett and Farhat [67] and Geelen et al. [66],
directly solving $\bar{\mathbf{V}}_{p}$ from Eq. 7 is prone to overfitting the
noise in the training data and it is necessary to introduce regularization to
the least-square minimization problem above. Therefore, we follow the approach
by Geelen et al. [66] to introduce a regularization term to Eq. 7

$\bar{\mathbf{V}}_{p}\triangleq\operatorname*{arg\,min}_{\bar{\mathbf{V}}_{p}\in{\mathbb{R}^{N\times
n^{2}_{p}}}}\left\lVert\mathbf{{H}}\left(\mathbf{q}_{p}-\mathbf{q}_{p,\text{ref}}\right)-{\mathbf{{V}}_{p}}\mathbf{q}_{r}+\bar{\mathbf{V}}_{p}(\mathbf{q}_{r}\otimes\mathbf{q}_{r})\right\rVert_{2}^{2}+\lambda\left\lVert\bar{\mathbf{V}}_{p}\right\rVert_{2}^{2}.$
(8)

where $\lambda$ is a scalar regularization parameter. We refer the readers to
the work by Geelen et al. [66] for details on computing
$\bar{\mathbf{V}}_{p}$. It is remarked that this quadratic nonlinear subspace
has been proposed and demonstrated to provide significant improvements on ROM
performance over the linear subspace in model order reduction for both non-
intrusive method by Geelen et al. [66] and intrusive method by Barnett and
Farhat [67]. On the other hand, we also remark the dimension of the operator
$\bar{\mathbf{V}}_{p}$ scales quadratically with the number of trial basis,
which incurs significant additional cost in offline and online computations
compared to its counterpart linear basis. In the current study, we focus on
evaluating the characteristics of the quadratic nonlinear basis in
representing the detonation wave dynamics in RDEs based on an offline a prior
assessment, which is discussed in section 5.2.

###  3.3 Adaptive ROM Formulation

Different from the conventional offline-online ROM methods with static trial
basis, the adaptve ROM method aims to update both the trial basis,
${\mathbf{{V}}_{p}}$, during the online ROM calculation. Following the work by
Huang and Duraisamy [73], the following minimization problem is posed to
construct the adaptive ROM

$\\{\mathbf{q}_{r}^{n},{\mathbf{{V}}^{n}_{p}},\mathbf{{S}}^{n}\\}\triangleq\operatorname*{arg\,min}_{\mathbf{q}_{r}\in\mathbb{R}^{n_{p}},{\mathbf{{V}}^{n}_{p}}\in\mathbb{R}^{N\times
n_{p}},\mathbf{{S}}^{T}_{n}\in\mathbb{S}^{N\times
n_{s}}}\left\lVert{\mathbf{{V}}^{n}_{p}}\left[\mathbf{{S}}^{T}_{n}{\mathbf{{V}}^{n}_{p}}\right]^{+}\mathbf{{S}}^{T}_{n}\mathbf{{P}}\mathbf{r}\left(\widetilde{\mathbf{q}}_{p}^{n}\right)\right\rVert_{2}^{2}$
(9)

where
$\widetilde{\mathbf{q}}_{p}^{n}=\mathbf{q}_{p,\text{ref}}+\mathbf{{H}}^{-1}{\mathbf{{V}}^{n}_{p}}\mathbf{q}_{r}^{n}$
and $\mathbf{r}$ is the nonlinear FOM equation residual defined in Eq. 2. To
circumvent the computational bottleneck in evaluating the nonlinear function
$\mathbf{r}$, hyper-reduction [77] is applied so that a full reconstruction of
$\mathbf{r}$ can be obtained using a small number of $n_{s}$ sparse samples
(i.e., $n_{s}<<N$),
$\mathbf{\overline{r}}^{n}\approx{\mathbf{{V}}^{n}_{p}}\left(\mathbf{{S}}^{T}_{n}{\mathbf{{V}}^{n}_{p}}\right)^{+}\mathbf{{S}}^{T}_{n}\mathbf{r}^{n}$.
$\mathbf{{S}}^{T}\in\mathbb{S}^{n_{s}\times N}$ represents a selection
operator that belongs to a class of matrices with $n_{s}$ columns (i.e.,
sampling points) of the identity matrix, $\mathbf{I}\in\mathbb{I}^{N\times N}$
and $\mathbf{{S}}$ is also constantly updated in adaptive ROM. However,
directly solving Eq. 9 is computationally intractable and therefore a
decoupled approach based on a predictor-corrector idea is utilized instead in
the following three steps.

First, $\mathbf{q}_{r}^{n}$ is selected as the solution to the minimization
problem below following a model-form preserving least-squares with variable
transformation (MP-LSVT) formulation developed by Huang et al. [49]

$\widetilde{\mathbf{q}}_{p}^{n}\triangleq\operatorname*{arg\,min}_{\widetilde{\mathbf{q}}_{p}^{n}\in\textrm{Range}({\mathbf{{V}}_{p}})}\left\lVert{\mathbf{{V}}^{n-1}_{p}}\left(\mathbf{{S}}^{T}_{n-1}{\mathbf{{V}}^{n-1}_{p}}\right)^{+}\mathbf{{S}}^{T}_{n-1}\mathbf{{P}}\mathbf{r}\left(\widetilde{\mathbf{q}}_{p}^{n}\right)\right\rVert_{2}^{2},$
(10)

with
$\widetilde{\mathbf{q}}_{p}^{n}=\mathbf{q}_{p,\text{ref}}+\mathbf{{H}}^{-1}{\mathbf{{V}}^{n-1}_{p}}\mathbf{q}_{r}^{n}$,
leading to a reduced non-linear system of dimension $n_{p}$ which can be
viewed as the result of a Petrov-Galerkin projection

$({\mathbf{\overline{{W}}}}_{{p}}^{n})^{T}{\mathbf{{V}}^{n-1}_{p}}\left(\mathbf{{S}}^{T}_{n-1}{\mathbf{{V}}^{n-1}_{p}}\right)^{+}\mathbf{{S}}^{T}_{n-1}\mathbf{{P}}\mathbf{r}\left(\widetilde{\mathbf{q}}_{p}^{n}\right)=\mathbf{0}.$
(11)

where ${\mathbf{\overline{{W}}}}_{{p}}^{n}$ is the resulting test basis

${\mathbf{\overline{{W}}}}_{{p}}^{n}={\mathbf{{V}}^{n-1}_{p}}\left(\mathbf{{S}}^{T}_{n-1}{\mathbf{{V}}^{n-1}_{p}}\right)^{+}\mathbf{{S}}^{T}_{n-1}{\mathbf{{W}}}_{{p}}^{n}.$
(12)

Second, the trial basis, ${\mathbf{{V}}_{p}}$, is updated by solving the
minimization problem

${\mathbf{{V}}^{n}_{p}}\triangleq\operatorname*{arg\,min}_{{\mathbf{{V}}^{n}_{p}}\in\mathbb{R}^{N\times
n_{p}}}\left\lVert\mathbf{{P}}\mathbf{r}\left(\widetilde{\mathbf{q}}_{p}^{n}\right)\right\rVert_{2}^{2}.$
(13)

As proved by Huang and Duraisamy [73], Eq. 13 can be solved exactly via the
update

${\mathbf{{V}}^{n}_{p}}={\mathbf{{V}}^{n-1}_{p}}+\delta{\mathbf{{V}}_{p}},$
(14)

through an increment, $\delta{\mathbf{{V}}_{p}}\in\mathbb{R}^{N\times n_{p}}$,
given by

$\delta{\mathbf{{V}}_{p}}=\frac{\left(\hat{\mathbf{q}}_{p}^{n}-\widetilde{\mathbf{q}}_{p}^{n}\right)(\mathbf{q}_{r}^{n})^{T}}{||\mathbf{q}_{r}^{n}||_{2}^{2}},$
(15)

where $\hat{\mathbf{q}}_{p}^{n}\in\mathbb{R}^{N}$ represents the full-state
information evaluated based on the FOM equation residual as follows

$\mathbf{r}\left(\hat{\mathbf{q}}_{p}^{n}\right)=\frac{3}{2}\mathbf{q}\left(\hat{\mathbf{q}}_{p}^{n}\right)-2\mathbf{q}\left(\widetilde{\mathbf{q}}_{p}^{n-1}\right)+\frac{1}{2}\mathbf{q}\left(\widetilde{\mathbf{q}}_{p}^{n-2}\right)-\Delta\text{t}\mathbf{f}\left(\hat{\mathbf{q}}_{p}^{n},\text{t}^{n}\right)=0,$
(16)

Here, an alternate formulation to the one in Ref. [71] is adopted by updating
the basis based on the full-state information, $\hat{\mathbf{q}}_{p}^{n}$,
evaluated at the current time step, $n$ instead of collecting at multiple time
steps, which is similar to the work done by Zimmermann et al. [78]. This
formulation is referred to as the one-step adaptive-basis approach. It is
worth pointing out that the time step of basis adaptation, denoted as $z_{b}$,
is empirically determined for the target applications and in the current work,
the basis is chosen to be adapted at each time step (i.e., $z_{b}=1$).

Third, the sampling points, $\mathbf{{S}}$, are adapted after the basis
adaptation (Eq. 14) to maximize the target quantity of interest
$\mathbf{\Theta}$

$\mathbf{{S}}^{n}\triangleq\operatorname*{arg\,max}_{\mathbf{{S}}\in\mathbb{S}^{N\times
n_{s}}}\left\lVert\mathbf{\Theta}\right\rVert_{2}^{2},$ (17)

where $\mathbf{\Theta}=\left\lVert\nabla p\right\rVert_{2}$ is selected to be
the magnitude of the pressure gradients. In practice, the sampling points are
updated and selected at the first $n_{s}$ points corresponding to the highest
magnitudes of $\mathbf{\Theta}$. It should be noted that the sampling points
adaptation (Eq. 17) requires the evaluation of full-state information,
$\hat{\mathbf{q}}_{p}^{n}$, at all the points, leading to a computationally
expensive step. Therefore, to achieve significant gains in computational
efficiency for the adaptive ROM formulation, the time step of updating the
sampling points denoted as $z_{s}$ is required to be set as larger value and
when the sampling points are not being updated, the basis is updated only at
the sampled points

$\mathbf{{S}}^{T}_{n-1}{\mathbf{{V}}^{n}_{p}}=\mathbf{{S}}^{T}_{n-1}{\mathbf{{V}}^{n-1}_{p}}+\mathbf{{S}}^{T}_{n-1}\delta{\mathbf{{V}}_{p}},$
(18)

which only requires the full-state information to be evaluated at the sampled
points, significantly reducing the computational cost

$\mathbf{{S}}^{T}_{n-1}\mathbf{r}\left(\hat{\mathbf{q}}_{p}^{n}\right)=\mathbf{{S}}^{T}_{n-1}\left[\frac{3}{2}\mathbf{q}\left(\hat{\mathbf{q}}_{p}^{n}\right)-2\mathbf{q}\left(\widetilde{\mathbf{q}}_{p}^{n-1}\right)+\frac{1}{2}\mathbf{q}\left(\widetilde{\mathbf{q}}_{p}^{n-2}\right)-\Delta\text{t}\mathbf{f}\left(\hat{\mathbf{q}}_{p}^{n},\text{t}^{n}\right)\right]=0,$
(19)

If the sampling points are being updated, the basis is updated at all the
points following Eq. 14 with the full-state information evaluated at the
sampled points, $\mathbf{{S}}_{n-1}$, using Eq. 19 while the full-state
information evaluated at the unsampled points, $\mathbf{{S}}^{*}_{n-1}$, are
evaluated using a larger time step (i.e. $z_{s}\Delta\text{t}$) to incorporate
the non-local in basis adaptation, which has been concluded as a crucial step
in the adaptive ROM development by Huang and Duraisamy [73].

$\mathbf{{S}}^{*T}_{n-1}\mathbf{r}\left(\hat{\mathbf{q}}_{p}^{n}\right)=\mathbf{{S}}^{*T}_{n-1}\left[\frac{3}{2}\mathbf{q}\left(\hat{\mathbf{q}}_{p}^{n}\right)-2\mathbf{q}\left(\widetilde{\mathbf{q}}_{p}^{n-z_{s}}\right)+\frac{1}{2}\mathbf{q}\left(\widetilde{\mathbf{q}}_{p}^{n-2z_{s}}\right)-z_{s}\Delta\text{t}\mathbf{f}\left(\hat{\mathbf{q}}_{p}^{n},\text{t}^{n}\right)\right]=0.$
(20)

Similar to $z_{b}$, the determination of $z_{s}$ is empirical and can impact
the prediction accuracy of the resulting adaptive ROMs as well. It needs to be
pointed out that compared to the conventional projection-based ROM methods
that leverage _linear static_ basis, the adaptive ROM formulation requires
additional computational operations (e.g., Eq. 16) to update the basis and
sampling points, possible resulting in fewer gains in computational
efficiency. However, the adaptive ROM formulation also presents several major
advantages over other methods: 1) it imposes much lower requirements in
_offline_ training, significantly reducing the computational costs to
construct ROMs for complex fluid problems, the high-fidelity simulations of
which are often expensive; 2) it usually requires small number of basis modes
and inherently addresses the restrictions of slow _Kolmogorov N-width_ decay
on the _linear static_ basis; and 3) more importantly, it significantly
improves the predictive capabilities of the ROM as demonstrated by the current
author [79, 73], especially in enabling accurate predictions of parametric
variations and transient dynamics, which are challenging for other ROM
methods.

##  4 Computational Configuration and Model

Following the earlier work by Schwer et al. [27], an unrolled two-dimensional
(2D) configuration is established to approximate the three-dimensional (3D)
RDE geometry as shown in Fig. 1. Both the FOM and ROM calculations are
performed using this 2D configuration in the current study. The computational
domain has a dimension of 282.7 mm in the azimuthal direction with 1414
uniformly distributed cells ($\Delta x=0.2mm$) and 100 mm in the axial
direction with 125 exponentially distributed cells ($\Delta y_{min}=0.2mm$ and
$\Delta y_{max}=.8mm$). Periodic boundary conditions are imposed on the left
and right side of the domain in order to simulate the traveling of the
detonation wave around the annulus as in the 3D configuration. Following
similar approach by Schwer and Kailasanath [80], to accommodate for the
effects of the inlet injection plane from a RDE, a variable inflow condition
is applied at the inlet based on the injector conditions $(p,T,v)$ computed
assuming isentropic expansion through the injector nozzle into the combustion
chamber

$\begin{cases}\text{if $p>p_{cr}$, no flow at the inlet and boundary is
treated as a wall}\\\ \text{else, constant velocity, $V_{in}$, and
temperature, $T_{in}$, are applied at the inlet}\end{cases}$ (21)

where $T_{in}$ is set to be 300 K and the critical pressure, $p_{cr}$, is
defined in terms of the stagnation pressure and reactant specific heat ratio,
$\gamma_{R}$

$p_{cr}=p_{0}(\frac{2}{\gamma_{R}+1})^{\frac{\gamma_{R}}{\gamma_{R}-1}}$ (22)

with $p_{0}$ as the stagnation pressure. A back pressure of 1 atm is applied
at the outlet of the computational domain. The computational infrastructure
used for the FOM and ROM solves conservation equations for mass, momentum,
energy and species transport in a fully coupled way using the in-house CFD
code, the General Mesh and Equation Solver (GEMS), which has been used to
model a variety of complex, practical reacting flow problems [81, 50]. More
details of the FOM equations can be found in A. Numerical simulations are
performed at five different inlet velocities, $V_{in}$ = 100, 125, 150, 175,
and 200 m/s, and are used as the test cases to assess the ROM’s capabilities
in predicting dynamics changes due to parametric variations.

Figure 1: Computational domain of the 2D RDE simulation.

The combustion process in the 2D RDE problem is modeled using a simplified
single-step reaction, two-species reaction
($\textbf{R}\rightarrow\textbf{P}$), which treats both the reactant (R) and
product (P) species as calorically perfect gases with identical molecular
weights. The simplified reaction model is used in the current study to make
the 2D RDE model more accessible to a broader community of researchers in
data-driven model development, while maintaining the challenging physics.
Here, the thermal properties of the reactant R and the product P are selected
to represent a hydrogen-air mixture with equivalence ratio of one and the
combustion products (nitrogen oxides and water vapor) to mimic the chemical
reaction model in the work by Schwer and Kailasanath [80]. The chemical
reaction source term for the reactant in Eq. 33 follows in the Arrhenius form

$\dot{\omega}_{\text{Reactant}}=-MW_{\text{Reactant}}\cdot
A\text{exp}\left(\frac{-E_{A}/R_{u}}{T}\right)\left[\frac{\rho
Y_{\text{Reactant}}}{MW_{\text{Reactant}}}\right]^{a},$ (23)

with the pre-exponential factor $A=1.6\times 10^{11}$, the activation energy
$E_{A}/R_{u}=24{,}358$ K, and the concentration exponent $a=1.0$, which is
designed to model the conversion of perfectly-mixed reactants to completely-
burned products to generate a detonation wave at the nominal flow conditions.
All the numerical simulations are computed using the second-order accurate
backwards differentiation formula with dual time-stepping, and a constant
physical time step size of $\Delta\text{t}=2$ns using 10 compute nodes (Dual
Intel Xeon 6442Y CPUs) with 256GB 4800MT/s DDR5 memory.

##  5 Numerical Results and Analysis

The baseline unsteady FOM simulation of the 2D RDE configuration is first
conducted at $V_{in}=150m/s$, advancing from 0 to 1ms to establish a stable
rotating detonation wave in the domain as shown in Figure 2. Initialized from
the solution at 1ms with $V_{in}=150m/s$, a parametric dataset is then
generated by conducting FOM simulations at five different equally spaced inlet
velocities ($V_{in}=100,125,150,175,\text{and }200\ m/s$), which leads to
distinct detonation-wave dynamics. Figure 3 compares the local pressure traces
at different inlet velocities, which are measured at the bottom-left corner of
the domain in Fig. 1 and show clear lifts in pressure levels as the inlet
velocity increases, indicating the amplification of the detonation waves as
more reactant is injected into the domain. The effects of inlet velocity on
detonation-wave dynamics are also reflected by comparing the instantaneous
temperature contours as shown in Fig. 4, which exhibits noticeable weakening
of the detonation waves with the decrease of inlet velocity, resulting in a
shorter and more curved wave front. A decrease in the length of the contact
surface is observed on the inlet side of the wave and a decrease in the
inclination of the oblique shock and shear layer are observed as well. This
parametric dataset establishes an ideal test bed to evaluate the ROM
performance in modeling RDE dynamics.

Moreover, due to the fact that all simulations are initialized from the 150
$m/s$ case at 1 ms, this dataset also contains transient dynamics at the
initial establishment of the detonation waves at the corresponding conditions
and thus presents additional challenges for ROM development. We remark that
this dataset is purposely configured to mimic the possible flow-condition
variations during RDE operations, and is used to evaluate the capabilities of
the nonlinear quadratic basis and adaptive ROM in capturing RDE dynamics under
practical parametric variations. The solution snapshots from 1 to 1.7ms are
collected and used for all the numerical analyses and evaluations in the
current paper, which spans a total of 5 cycles (5 complete annular motions of
the detonation waves) and each cycle contains 7,000 snapshots.

Figure 2: Representative instantaneous snapshots of pressure (top) and
temperature (bottom) from FOM simulation of the 2D RDE at $t=1\ ms$. Figure 3:
Local pressure trace results for FOM test cases.

(a) $V_{in}=100\ m/s$

(b) $V_{in}=125\ m/s$

(c) $V_{in}=150\ m/s$

(d) $V_{in}=175\ m/s$

(e) $V_{in}=200\ m/s$

Figure 4: Representative instantaneous snapshots of temperature from FOM
simulation of the 2D RDE at $t=1.6\ ms$.

###  5.1 POD Characteristics

The POD characteristics of the 2D RDE are first investigated to understand how
well the POD trial basis represents the detonation wave dynamics in FOM
dataset as shown in Fig. 5, which is evaluated using the POD residual energy

$\text{POD Residual
Energy}(n_{p}),\;\%=\left(1-\frac{\sum^{n_{p}}_{i=1}\sigma^{2}_{i}}{\sum^{n_{p,total}}_{i=1}\sigma^{2}_{i}}\right)\times
100,$ (24)

where $\sigma_{i}$ is the $i^{\text{th}}$ singular value used to compute the
trial basis $\mathbf{{V}}_{p}$. The singular values are arranged in descending
order. $n_{p}$ is the number of vectors retained in the POD trial basis, and
$n_{p,total}$ is the total number of modes in the training dataset. An in
depth a priori projection analysis of the performance of linear static basis
for different training data sets and included mode amounts can be found in B.
The results of this analysis informed the decision to utilize a training set
composed of 1 cycle of data (7000 time snapshots) each for $V_{in}$ = 100,
150, and 200 $m/s$ (21,000 time snapshots total). Figure 5 highlights the
problem of slow Kolmogorov N-width decay, illustrating that as further
accuracy is desired for this convection and shock dominated problem, many more
POD modes will need to be included in the trial basis. For example, it can be
seen that in order to capture 97% of the POD residual energy, 40 modes are
needed. However, in order to capture 99% of the residual energy, 200 modes are
required, resulting in a much larger POD basis and, therefore, a much more
computationally expensive ROM. While in order to reach 99.9% of the residual
energy (a common practice in the literature [82]), at least 1,180 modes are
required, which becomes unaffordable for efficient ROM construction.

Figure 5: POD Residual energy versus the number of POD modes included.

###  5.2 Assessment of Linear and Nonlinear Quadratic Bases

First, we evaluate and compare the performance of the linear and nonlinear
quadratic bases in representing the detonation-wave dynamics via an a priori
analysis by computing the projected FOM solution($\bar{\mathbf{q}}^{n}_{p}$
and $\bar{\mathbf{q}}^{n}_{p,\text{quadratic}}$)

$\bar{\mathbf{q}}_{p}=\mathbf{q}_{p,\text{ref}}+\mathbf{{H}}^{-1}{\mathbf{{V}}_{p}}\mathbf{q}_{r}.$
(25) $\displaystyle\bar{\mathbf{q}}^{n}_{p,\text{quadratic}}$
$\displaystyle=\mathbf{q}_{p,\text{ref}}+\mathbf{{H}}^{-1}\left[{\mathbf{{V}}_{p}}\hat{\mathbf{q}}^{n}_{r}+\bar{\mathbf{V}}_{p}(\hat{\mathbf{q}}^{n}_{r}\otimes\hat{\mathbf{q}}^{n}_{r})\right]$
(26)

where
$\hat{\mathbf{q}}^{n}_{r}={\mathbf{{V}}_{p}}^{T}\left(\mathbf{q}^{n}_{p}-\mathbf{q}_{p,\text{ref}}\right)$
and $\mathbf{q}^{n}_{p}$ corresponds to the state solution obtained at the
$n^{th}$ FOM snapshot. It is noteworthy that the projected FOM solution
determines the upper bound of the resulting ROMs’ accuracy (i.e., the accuracy
of the ROM solution cannot outperform the corresponding projected FOM
solution). This a priori analysis based on projected FOM solutions has been
demonstrated as an effective approach to measure the feasibility and expected
performance of constructed trial basis in representing target dynamics of
interest without the need to develop an online ROM [49, 83, 43].

As mentioned previously, one cycle of FOM snapshots are collected at three
representative inlet-velocity conditions ($V_{in}=100,\ 150\ $, and $200\
m/s$), following the findings in B for the most efficient training data set.
This resulted in a total of 21,000 training snapshots, taken at every 10 time
step of the simulation. Solutions at the other two conditions (i.e.,
$V_{in}=125,\ $ and $175\ m/s$) are used as the testing snapshots to evaluate
the capabilities of the linear and nonlinear quadratic bases in representing
the changes in dynamics due to parametric variations. Specifically, the
projection errors, $\bar{\epsilon}^{n}$, are quantified for the evaluations

$\bar{\epsilon}^{n}=\frac{1}{N_{var}}\sum^{N_{var}}_{i=1}\frac{\left\lVert\mathbf{\bar{q}}_{p,i}^{n}-\mathbf{q}_{p,i}^{n}\right\rVert_{2}}{\left\lVert\mathbf{q}_{p,i}^{n}\right\rVert_{2}},$
(27)

where $N_{var}$ represents the number of included quantities of interest (i.e.
pressure, temperature), $\mathbf{\bar{q}}_{p,i}^{n}$ represents the
$i^{\text{th}}$ solution variable of the projected FOM state vector,
$\mathbf{\bar{q}}_{p}^{n}$, at time step $n$. The projection errors are
computed for each training condition ($V_{in}$) within the training FOM
snapshots using both linear and nonlinear quadratic bases with different
numbers of modes included, which are compared in Fig. 6 for $V_{in}=100m/s$.
It can readily seen that the nonlinear quadratic basis provides more than one
order of magnitude improvement in the projection error in constrast to the
linear basis. We remark that different scalar regularization parameters,
$\lambda$, in Eq. 7 are tested to compute the nonlinear quadratic basis based
on the projection errors and the results of are included in C. A value of
$\lambda=10^{-4}$ is shown to be sufficient to ensure a well-conditioned
least-square problem in Eq. 7 and is used to construct the nonlinear quadratic
basis in the current study. It should be noted that the maximum number of
modes included in the a priori analysis is set to be 200 as a result of the
significantly increased dimension of $\bar{\mathbf{V}}_{p}$, which is
$n^{2}_{p}$. Though by eliminating redundancy in the Kronecker products in Eq.
7, the dimension of $\bar{\mathbf{V}}_{p}$ can be reduced to
$n_{p}(n_{p}+1)/2$, for a linear basis with $n_{p}=200$, the resulting
nonlinear quadratic basis, $\bar{\mathbf{V}}_{p}$, would be of dimension
20,100. Such a large dimension requires significant computational resources,
especially in basis computing and storage. Therefore, though increasing the
dimension of $\bar{\mathbf{V}}_{p}$ is likely to produce more accurate
results, the associated computational expense of generating the corresponding
nonlinear quadratic basis can quickly become untenable.

Figure 6: Comparisons of the projection errors between linear static and
quadratic bases in reproducing the training dataset.

Next, we proceed to examine and compare the local pressure time traces and
instantaneous field results between the FOM and the projected FOM solutions to
assess the capabilities of the nonlinear bases in representing the detonation
wave dynamics at the training conditions ($V_{in}=100m/s$). First, The local
pressure time traces are compared in Fig. 7 using 200 modes to construct the
bases at a probe placed at the same location as in Fig. 3, which shows that
the projected FOM solutions using the nonlinear quadratic basis closely
recreate the FOM results within the training regime. However, it can be
readily seen from that the nonlinear quadratic basis fails to provide physical
representations of the future-state dynamics immediately outside the training
regime. On the other hand, though the linear basis leads to less accurate
projected solutions within the training regime, it provides reasonably
accurate representation of the pressure trace in the future state. Second, the
instantaneous temperature fields are compared between the FOM and projected
FOM solutions in Fig. 8. At the time instance within the training regime
($t=1.125ms$), consistent with Fig. 7, the projected FOM solutions using the
nonlinear basis reaches better agreement with the FOM while spurious
oscillations are present in the linear-basis results. However, neither the
linear or the nonlinear basis can provide accurate representations of the
temperature fields beyond the training ($t=1.150ms$) though the local pressure
trace in Fig. 7 seems to indicate that it is feasible to the linear basis to
represent the future-state detonation-wave dynamics. In addition, we remark
that the same investigations are performed for $V_{in}=150$ and $200m/s$ which
exhibit similar comparisons as for $V_{in}=100m/s$ in Figs. 7 and 8.

Figure 7: Comparisons of local pressure time traces between the FOM and the
projected FOM solutions using linear and nonlinear quadratic bases with 200
modes included at $V_{in}=100\ m/s$.

(a) $t=1.125\ ms$

(b) $t=1.150\ ms$

Figure 8: Comparisons of instantaneous temperature fields between the FOM and
the projected FOM solutions using linear and nonlinear quadratic bases with
200 modes included at $V_{in}=100\ m/s$

Furthermore, we continue to evaluate the capabilities of the linear and
nonlinear quadratic bases in representing changes in dynamics due to
parametric variations by computing the projected FOM solutions at the testing
conditions (i.e., $V_{in}=125$ and $175m/s$). Following the same
investigations above, both local pressure time traces and instantaneous
temperature fields are compared between the FOM and projected FOM solutions as
shown in Figs. 9 and 10, respectively. It should be noted that since the
snapshot solutions at the two inlet velocities are not included in the
training dataset for the bases construction, the projected FOM solutions are
direct indicator of the feasibility of using these bases in constructing
parametric ROMs either through intrusive [49] or non-intrusive [43]
approaches. Akin to the results shown in Figs. 7 and 8, the nonlinear
quadratic basis shows restricted capabilities in representing dynamics beyond
the training dataset and produces non-physical solutions in both local
pressure traces and instantaneous temperature fields. Similarly, in Fig. 9,
the linear basis again provides a reasonably accurate trend for the local
pressure evolution for both inlet velocities, although a bias of over or under
estimation based on the case in question is apparent. This again does not tell
the full story, however, as Fig. 10 reveals the inability of this method to
provide accurate unsteady temperature field results over the entire domain as
the simulation progresses through time. As early as .001 ms into the
simulation, nonphysical striations in the detonation wave front can be seen,
with the fidelity of the detonation wave structure continuing to decay as time
progresses.

(a) $V_{in}=125\ m/s$

(b) $V_{in}=175\ m/s$

Figure 9: Comparisons of local pressure time traces between the FOM and the
projected FOM solutions using linear and nonlinear quadratic bases with 200
modes.

(a) $V_{in}=123\ m/s$

(b) $V_{in}=175\ m/s$

Figure 10: Comparisons of instantaneous temperature fields between the FOM and
the projected FOM solutions using linear and nonlinear quadratic bases with
200 modes.

The a priori analyses conducted in the current section shows that the
nonlinear quadratic basis is able to significantly improve ROM accuracy in
reproducing the detonation-wave dynamics within the training regime compared
to the linear basis. However, it exhibits very restricted capabilities in
representing the dynamics beyond the training regime (either in the future
state or at another parameter). This indicates that the resulting ROM
constructed using the nonlinear quadratic basis will likely present limited
prediction capabilities for RDE applications. In addition, though more
flexible in representing dynamics beyond the training regime, the linear-basis
ROMs are anticipate to present similar limitations as the nonlinear-basis
ROMs.

###  5.3 Assessment of Adaptive ROM

With the a priori analysis showing restrictions in using either the linear or
nonlinear quadratic static bases to represent RDE dynamics, we then proceed to
evaluate the performance of the adaptive ROM in modeling the 2D RDE problem.
It should be noted that for the investigations in the current section, an
online adaptive ROM is constructed, instead of relying on the a priori
analysis in the previous section. Following the adaptive ROM formulation in
section 3.3, the initial basis is computed using POD with 10 training
snapshots at the condition $V_{in}=150\ m/s$ with 10 modes preserved to start
the adaptive ROM. In contrast to a total of 21,000 snapshots used for the
static bases construction, the adaptive ROM formulation significantly reduces
the required offline cost in ROM construction. In addition, 8,837 sampling
points are selected following Eq. 17 for all the adaptive ROM calculations in
the current study, which constitute 5 $\%$ of the total cells in the
computation domain and are updated every 5 time steps (i.e., $z_{s}=5$ in Eq.
20).

First, the local pressure time traces are compared in Fig. 11 between the FOM
and the adaptive ROM at five different inlet velocities and it can be readily
seen that the adaptive ROM successfully predicts the changes in pressure
dynamics due to parametric variations. Specifically, the adaptive ROM captures
the weakening of the detonation-wave intensity, indicated by the peak values
of the pressure trace, when the inlet velocity is decreased from $150m/s$ to
$125$ and $100m/s$. Moreover, it also accurately predicts the strengthening of
the detonation-wave intensity when the inlet velocity is increased from
$150m/s$ to $175$ and $200m/s$. More importantly, we remark that the adaptive
ROM captures the transient detonation wave dynamics due to the abrupt change
of the inlet-velocity at $1ms$, which is well recognized to be a challenging
phenomenon for static-basis ROMs [49, 84].

(a) $V_{in}=100\ m/s$

(b) $V_{in}=125\ m/s$

(c) $V_{in}=150\ m/s$

(d) $V_{in}=175\ m/s$

(e) $V_{in}=200\ m/s$

Figure 11: Comparisons of local pressure time traces between FOM and adaptive
ROM at different inlet-velocity conditions.

Next, the instantaneous temperature fields are compared in Fig. 12 between the
FOM and adaptive ROM at the lowest and highest inlet velocities ($V_{in}=100$
and $200m/s$) in the testing conditions and a representative time instance
near the tend of the testing regime ($1.6ms$) is selected for the comparisons.
It can be readily seen that the adaptive ROM solutions reach an excellent
agreement with the FOM, especially in capturing the distinct detonation-wave
features at different inlet velocities. Though the adaptive ROM overpredicts
the temperature in the shear layer at $V_{in}=100m/s$, it accurately predicts
the characteristics of the detonation waves, oblique shocks, and contact
surfaces at both inlet velocities. More importantly, the adaptive ROM
successfully predicts the changes in detonation-wave dynamics subject to the
increase in the inlet velocity, such as 1) the decrease in the angle ($\alpha$
in Fig. 12) between the detonation front and the inlet; and 2) the increase
inclination in the shear layer and oblique shock. Additionally, when compared
to the FOM, the adaptive MOR method demonstrated the capability to provide a
factor of 4 computational acceleration. These results, coupled with the
computational efficiency gains, highlight the truly predictive capabilities of
the adaptive ROM and illustrate its potential to construct ROMs that can be
applied to a wide range of operating conditions.

(a) $V_{in}=100\ m/s$

(b) $V_{in}=200\ m/s$

Figure 12: Comparisons of instantaneous temperature fields between FOM and
adaptive ROM at $t=1.6\ ms$ with relevant features. A) detonation wave, B)
oblique shock, C) shear layer, D) contact surface.

With the modeling capabilities of adaptive ROM demonstrated for future-state
and parametric predictions, we further demonstrate its potential in predicting
the initial transience in establishing the rotating detonation waves. As
mentioned in sec 5, the FOM simulation has been run for $1ms$ at
$V_{in}=150m/s$ to ensure a stable detonation wave, after which the training
data are collected for the investigations presented in the previous sections.
While it is necessary to properly assess the performance of the ROM methods
based on statistically stationary dynamics, for practice, the computational
cost for the FOM simulation of this initial transience (comparable to the
testing period from $1$ to $1.7ms$) is fairly significant and the potential
cost reduction via ROM for this transience’s computation can be crucial.
Therefore, we proceed to explore the potential of adaptive ROM to model the
initial transience in RDE problems. To achieve this, the FOM is simulated
first from $0$ to $0.02ms$ (10,000 time steps) to pass over the unphysical
solutions due to the artifacts of the initial conditions used to start the
simulation. After that, the simulation is switched to adaptive ROM with the
initial basis trained using 10 snapshots from $0.02ms$ using the same
parameters as in Figs. 11 and 12. The resulting local pressure trace and
instantaneous temperature fields are compared in Fig. 13 between the FOM and
adaptive ROM. It can be readily seen that the adaptive ROM successfully
captures the initial transience in establishing the detonation waves as shown
in the FOM, which indicates that it is feasible to apply the adaptive ROM
starting at a much earlier stage of the simulations to further reduce the
computational cost.

(a) Local Pressure Trace

(b) $t=.022\ ms$

(c) $t=.150\ ms$

Figure 13: Comparisons of local pressure trace and instantaneous temperature
fields between FOM and adaptive ROM at $V_{in}=150\ m/s$

While the adaptive ROM exhibits great potential and versatility in providing
accurate future-state, transient, and parametric predictions for RDE problems
with minimal offline training requirements, there remains one aspect that
requires further development. The online adaptation of basis and sampling
points require additional computation cost to access the FOM equation
residual, limiting the computational acceleration that can be achieved via
adaptive ROM as highlighted in the work by Huang and Durasaimy [73]. Moreover,
scalable implementation of the adaptive ROM algorithm for large-scale
engineering problems requires further development, especially in developing an
effective load-balancing strategy to accommodate the adapted sampling points,
which is currently not incorporated in our adaptive ROM implementation.
Therefore, the current adaptive ROM exhibits only $O(4)$ acceleration in
computational time compared to the FOM but it is anticipated that $>O(20)$
acceleration can be achieved with proper implementations of dynamic load-
balancing, which will be the focus of our future work.

##  6 Conclusions

The current work focuses on investigating the capabilities of representative
projection-based ROMs constructed utilizing conventional linear static basis,
novel nonlinear quadratic basis, and a recently developed adaptive model order
reduction (MOR) formulation, in parametric modelling of rotating detonation
engines (RDEs). RDEs exhibit strong convection-dominated shock-wave physics,
which have been shown to be challenging for ROM development. A two-dimensional
(2D) RDE configuration is presented to perform large-eddy simulations (LES) at
5 different inlet velocities, allowing for investigations into the performance
of the three methods in predicting the dynamics changes as a result of
parametric variations, as would be observed in real world applications. First,
an a priori analysis is performed to assess the capability of representative
ROMs constructed using a linear static basis and nonlinear quadratic basis in
modelling the RDE dynamics by performing a full order model (FOM) simulation.
The two bases are constructed using training data generated from three of the
five inlet velocity cases ($V_{in}=\text{100, 150, and 200 m/s}$). Evaluations
of these methods reveal that the construction utilizing nonlinear quadratic
basis exhibits improvements over linear static basis in low-rank approximation
of the RDE dynamics over the training regime, but demonstrated restricted
capabilites in accurately capturing the physics outside of the training
dataset. This restricted capability is apparent in both future state and
parametric prediction, indicating limited flexibility in constructing a truly
predictive ROM for parametric evaluations of RDEs. Second, the newly developed
adaptive MOR formulation is leveraged to construct an online ROM for 2D RDE
modelling. The adaptive MOR is shown to require a small fraction of the
training dataset for just one inlet velocity case, and demonstrates improved
low-rank approximation when compared to the linear static basis method in
training regime reconstruction, future state, and parametric prediction. This
includes capturing the transient effects related to inlet velocity variations,
a challenging characteristic for ROMs to capture. Additionally, a factor of 4
computational acceleration is shown when compared to the FOM simulation.
Third, the ROM constructed from the adaptive MOR method is applied at a very
time near the initiation of the FOM simulation, where it is shown to
accurately capture the transient RDE dynamics present prior to the
establishment of a steady detonation wave. This capability allows for a
significant reduction in the runtime of the FOM, allowing for further
computational cost savings. The future work will focus on implementing proper
dynamic load balancing based on the adapted sampling-point distributions to
lift the current bottle neck in computational efficiency gain through the
adaptive ROM. In addition, the scalability of the adaptive ROM method to
large-scale 3D RDE applications will be another focus of the future work and
an important stepping stone towards real-scale engineering applications.

##  7 Acknowledgements

This work was made possible with support from the Air Force Center of
Excellence under grant FA9550-17-1-0195, titled "Multi-Fidelity Modeling of
Rocket Combustor Dyanmics" (Technical Monitors: Fariba Fahroo, Mitat Birkan,
Ramakanth Munipalli). Additional support was also provided by the SMART
Scholarship Program through the sponsorship of the Naval Air Warfare Center -
Weapons Division, China Lake facility. High performance computing capabilities
were possible thanks to support from the Center for Research Computing at the
University of Kansas.

##  Appendix A  Governing Equations for Full Order Model

The full order model computations are carried out with an in-house CFD code,
the General Equations and Mesh Solver (GEMS), which solves the conservation
equations for mass, momentum, energy and species mass fractions in a coupled
fashion,

$\frac{\partial{Q}}{\partial{t}}+\nabla\cdot\left(\vec{F}-\vec{F_{v}}\right)=H,$
(28)

where $Q$ is the vector of conserved variables defined as,
$Q=\left(\begin{array}[]{cccccc}\rho&\rho{u}&\rho{v}&\rho{w}&\rho{h^{0}-p}&\rho{Y_{l}}\\\
\end{array}\right)^{T}$ with $\rho$ representing density, $u$, $v$ and $w$
representing velocity field, $Y_{l}$ representing the $l^{th}$ species mass
fraction and the total enthalpy $h^{0}$ is defined as,
$h^{0}=h+\frac{1}{2}(u^{2}_{i})=\sum_{l}{h_{l}{Y_{l}}}+\frac{1}{2}(u^{2}_{i})$.

The fluxes have been separated into inviscid,
$\vec{F}=F_{i}\vec{i}+F_{j}\vec{j}+F_{k}\vec{k}$ and viscous terms,
$\vec{F_{v}}=F_{v,i}\vec{i}+F_{v,j}\vec{j}+F_{v,k}\vec{k}$. And the three
inviscid fluxes are,

$F_{i}=\left(\begin{array}[]{c}\rho{u}\\\ \rho{u^{2}}+p\\\ \rho{uv}\\\
\rho{uw}\\\ \rho{uh^{0}}\\\ \rho{uY_{l}}\\\
\end{array}\right),\;F_{j}=\left(\begin{array}[]{c}\rho{v}\\\ \rho{uv}\\\
\rho{v^{2}}+p\\\ \rho{vw}\\\ \rho{vh^{0}}\\\ \rho{vY_{l}}\\\
\end{array}\right)\;\text{and}\;F_{k}=\left(\begin{array}[]{c}\rho{w}\\\
\rho{uw}\\\ \rho{vw}+p\\\ \rho{w^{2}}+p\\\ \rho{wh^{0}}\\\ \rho{wY_{l}}\\\
\end{array}\right)$ (29)

The viscous fluxes are,

$F_{v,i}=\left(\begin{array}[]{c}0\\\ \tau_{ii}\\\ \tau_{ji}\\\ \tau_{ki}\\\
u\tau_{ii}+v\tau_{ji}+w\tau_{ki}-q_{i}\\\
\rho{D_{l}}\frac{\partial{Y_{l}}}{\partial{x}}\\\
\end{array}\right),\;F_{v,j}=\left(\begin{array}[]{c}0\\\ \tau_{ij}\\\
\tau_{jj}\\\ \tau_{kj}\\\ u\tau_{ij}+v\tau_{jj}+w\tau_{kj}-q_{j}\\\
\rho{D_{l}}\frac{\partial{Y_{l}}}{\partial{y}}\\\
\end{array}\right)\;\text{and}\;F_{v,k}=\left(\begin{array}[]{c}0\\\
\tau_{ik}\\\ \tau_{jk}\\\ \tau_{kk}\\\
u\tau_{ik}+v\tau_{jk}+w\tau_{kk}-q_{k}\\\
\rho{D_{l}}\frac{\partial{Y_{l}}}{\partial{z}}\\\ \end{array}\right)$

(30)

where $D_{l}$ is defined to be the diffusion of the $l^{th}$ species into the
mixture. In practice, this is an approximation used to model the
multicomponent diffusion as the binary diffusion of each species into a
mixture.

The heat flux in the $i^{th}$ direction, $q_{i}$, is defined as,

$q_{i}=-K\frac{\partial{T}}{\partial{x_{i}}}+\rho\sum^{N}_{l=1}D_{l}\frac{\partial{Y_{l}}}{\partial{x_{i}}}h_{l}+\mathbf{Q}_{\text{source}}$
(31)

The three terms in the heat flux represent the heat transfer due to the
conduction, species diffusion and heat generation from a volumetric source
(e.g. heat radiation or external heat source) respectively.

The shear stress, $\tau$ , is also found in the viscous flux and defined in
terms of the molecular viscosity and velocity field,

$\tau_{ij}=\mu\left(\frac{\partial{u_{i}}}{\partial{x_{j}}}+\frac{\partial{u_{j}}}{\partial{x_{i}}}-\frac{2}{3}\frac{\partial{u_{m}}}{\partial{x_{m}}}\delta_{ij}\right)$
(32)

The source term, $H$ includes a single entry for each of the species equations
signifying the production or destruction of the $l^{th}$ species,
$\dot{\omega}_{l}$, which is determined by the chemical kinetics [85],

$H=\left(\begin{array}[]{cccccc}0&0&0&0&0&\dot{\omega}_{l}\\\
\end{array}\right)^{T}$ (33)

##  Appendix B A priori Analysis of Training Data Effects

Different number of FOM snapshots (beginning at 1ms) are included in the
training dataset for the case where $V_{in}=150\ m/s$ to compute a POD trial
basis, corresponding to 0.5, 1, 2, and 3 cycles of snapshots. As seen in
Figure 14, the POD residual energy exhibits slow decay, indicated by the
increasingly large number of POD modes required to recover more of the total
energy (0.5 cycle: 159 modes, 1 cycle: 316 modes, 2 cycles: 386 modes, and 3
cycles with 428 modes to recover 99.9% of the energy of the system). As
discussed in section 1, this is a direct indicator of the slow decay of
Kolmogorov N-width [54], which has been observed in convection-dominated
multi-scale problems studied previously [73, 49, 43]. Unlike the common non-
convergence characteristics of the POD residual energy observed in these
previous studies, however, the residual energy here shows a trend of
convergence when increasing the number of FOM snapshots in the training
dataset. For example, the number of POD modes required to recover 99.9% of the
total energy increases less as more cycles of snapshots are included. Such
convergence of the POD residual energy can be attributed to the dominance of
detonation wave over other physics (e.g. turbulence) in the present test
problem in Fig. 2. More importantly, it indicates that it is feasible to
represent the overall (e.g. 5 cycles) detonation wave dynamics by a POD basis
computed using the smaller number of training snapshots.

Figure 14: Residual Energy Capture for Different Numbers of Simulation Cycles

We quantify the capability of the generated POD basis in representing the
target dynamics via the projected FOM solution, $\bar{\mathbf{q}}_{p}$,
defined as follows for a conventional linear static basis

$\bar{\mathbf{q}}_{p}=\mathbf{q}_{p,\text{ref}}+\mathbf{{H}}^{-1}{\mathbf{{V}}_{p}}{\mathbf{{V}}_{p}}^{T}\mathbf{q}_{p}.$
(34)

where $\mathbf{q}_{p}$ represents the state solution obtained from the FOM. It
is noteworthy that the projected FOM solution provides a best-scenario
evaluation of the accuracy of the resulting ROM (i.e. the ROM solution cannot
produce more accurate predictions of the dynamics than the projected FOM
solution) [83].

In order to determine the viability of traditional PROMs for future state
prediction, we first investigate the effects on projected solution accuracy
for linear static bases generated with different training data sets. The
results in Fig. 15 show graphically the time evolution of pressure over the
cell located at the bottom-most left-hand side of domain for the FOM and
projected FOM solutions. Fig. 15(a) focuses on evaluating the
representativeness of the basis in the future-state prediction with mode
numbers corresponding to 99.9% POD energy in Fig. 14. Here the importance of
ensuring that adequate offline training data is used ($\geq$ 1 cycle) is
demonstrated. We see that if less than this amount of training data is used
(i.e., 0.5 cycle), the projected FOM solution does not provide useful results
outside of the space traversed in the training regime. Additionally, while
increasing the number of cycles included in the training data does improve
accuracy, the improvements for training sets larger than 1 cycle are shown to
be marginal, with the benefits arguably outweighed by the added offline
computational and storage costs. This indicates, therefore, that the most
economical training set is 1 cycle (7000 time snapshots).

Having shown that a single cycle of training data is sufficient, it is now
useful to determine to what degree the amount of included modes affects the
accuracy of the solution. For mode numbers corresponding to 97%, 99%, and
99.9% residual energy capture, the resulting projected FOM solutions are shown
in Fig. 15(b). As expected, the more POD modes are included, the more
accurately the FOM solution is recreated, with 316 modes corresponding to
99.9% energy capture. Although the energy capture is only increasing by 2.9%
across the three bases, the required modes increase from 19 to 316, again
illustrating the slow Kolmogorov N-width decay exhibited by this problem.

(a) 99.9% Residual Energy Capture

(b) 1 Cycle of Training Data

Figure 15: Comparisons of local pressure trace between FOM and the projected
FOM for $\mathbf{V_{in}}$ = 150m/s.

Although Fig. 15(b) indicates the most accurate solution is provided by the
linear basis with 316 modes as expected, all three test cases capture the
important structures of the detonation dynamics. Therefore, in order to more
rigorously determine the true number of modes required for this problem, the
flow field for both the FOM and projected FOM was generated, and can be seen
in Fig. 16. The snapshots shown were taken outside of the training regime
($t=1.16ms$) and reveal that it is indeed necessary to utilize modes
corresponding to at least 99.9 % residual energy capture.

Figure 16: Comparisons of pressure field between FOM and projected FOM at
$t=1.16ms$.

##  Appendix C Regularization Parameter

Special consideration needs to be taken when determining a proper scalar
regularization parameter, ($\lambda$), as this term is responsible for
inhibiting overfitting of the system operators to the data [43, 66]. This, in
turn, promotes stability and accuracy of the solutions generated with the
nonlinear quadratic method. In order to determine the ideal value for
$\lambda$, a range of values are tested to generate the nonlinear quadratic
basis. The resulting projection errors are averaged over the three training
cases using 27 and compared in Figure 17. It can readily be seen that the
accuracy of the quadratic basis can be affected by the selection of the
regularization parameter, and as $\lambda$ increases, the projection errors of
the quadratic basis increase as well. It can also be seen that as
$\lambda\rightarrow 0$, the projection errors begin to converge, indicating
that lower values of $\lambda$ will provide diminished improvements in
accuracy. Due to the imposed cap on the number of included linear modes being
200, and the observed trend of convergence, it was determined that
$\lambda=10^{-4}$ would be suitable for this study.

Figure 17: Comparisons of the projection errors using quadratic bases with
different regularization parameters, ($\lambda$), in reproducing the training
dataset.

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