# Recent Advances in Uncertainty Quantification Methods for Engineering
Problems

Dinesh Kumar
Department of Mechanical Engineering
University of Bristol
Bristol BS8 1TR, UK
&Farid Ahmed
Department of Nuclear Engineering
North Carolina State University
2500 Stinson Drive Raleigh, NC, USA &Shoaib Usman
Nuclear Engineering and Radiation Science
Missouri University of Science and Technology
Rolla, MO 65409, USA
&Ayodeji Alajo
Nuclear Engineering and Radiation Science
Missouri University of Science and Technology
Rolla, MO 65409, USA
&Syed Alam
Nuclear Engineering and Radiation Science
Missouri University of Science and Technology
Rolla, MO 65409, USA

###### Abstract

In the last few decades, uncertainty quantification (UQ) methods have been
used widely to ensure the robustness of engineering designs. This chapter aims
to detail recent advances in popular uncertainty quantification methods used
in engineering applications. This chapter describes the two most popular meta-
modeling methods for uncertainty quantification suitable for engineering
applications (Polynomial Chaos Method and Gaussian Process). Further, the UQ
methods are applied to an engineering test problem under multiple
uncertainties. The test problem considered here is a supersonic nozzle under
operational uncertainties. For the deterministic solution, an open-source
computational fluid dynamics (CFD) solver SU2 is used. The UQ methods are
developed in Matlab and are further combined with SU2 for the uncertainty and
sensitivity estimates. The results are presented in terms of the mean and
standard deviation of the output quantities.

_Keywords_ Uncertainty Quantification $\cdot$ Polynomial Chaos $\cdot$
Reliable Design $\cdot$ Sensitivity analysis $\cdot$ Kriging $\cdot$ Gaussian
Process

In industrial applications, during the operation of engineering devices,
several properties and parameters of the components change with time. The
material properties of its components vary continuously due to several
operational factors. The failure point information of the week components in
an industrial application is very useful. The safety of the engineering device
under consideration should always be of utmost concern for the manufacturers
while designing the device. Using statistical information, the designer can
evaluate the safety margin or make the failure design margin smaller than
other components so that the impact of the weak component can be minimized.
With advancements in computer hardware and numerical algorithms, computational
tools are used to design advanced and high-performance engineering components
in almost all engineering fields. For example, aircraft, high-speed car
manufacturers, sports, naval ship designers, etc., use computational fluid
dynamics (CFD) to simulate fluids over the device. These computational tools
are also used for thermal and structural analysis to simulate and detect
faults, cracks, and failure of the devices by almost all industries.

Uncertainties are an inherent part of the computing systems concerning real-
world applications Oberkampf and Trucano (2002); Roy and Oberkampf (2011);
Beyer and Sendhoff (2007); Schuëller and Jensen (2008); Wiener (1938); Xiu and
Karniadakis (2002a); Smith (2013); Kumar et al. (2021a, 2020a); Kabir et al.
(2021a, b, 2020); Kumar et al. (2021b, 2020b, 2019). Two physical experiments
can never produce the same results as several of the system parameters are not
known properly and have uncertainties. When the same system is modeled using
computational tools and mathematical equations, the input and system
parameters are provided with constant values to predict the results without
dealing with the uncertainties in the input parameters. Almost all aspects of
engineering modeling and design are affected by these uncertainties. Engineers
and researchers have always encountered issues related to uncertainties in
terms of design reliability and robustness. By understanding sources of
uncertainties and quantifying them, one can estimate confidence in the system
outputs. In mathematical modeling, uncertainties are usually encountered in
initial conditions, boundary conditions, material properties, weather
conditions, and manufacturing tolerances.

Uncertainty Quantification (UQ) is the field of detecting, describing,
quantifying, and managing uncertainties in computational designs of real-world
systems. In UQ, the system response is estimated in a stochastic way by
combining the deterministic solver with statistical tools. UQ methods are
statistical tools to assess safety margins in the system responses when
computer simulations are used to design an engineering device. UQ methods
address the problems associated with incorporating system parameters
variability and stochastic behavior into systems analyses. Computer
simulations answer what happens when the system under consideration is
subjected to a set of input parameters. However, UQ expands this question and
answers what will happen when the system is subjected to a range of
variability in the input parameters. UQ combines mathematics, statistics, and
engineering. Generally speaking, UQ methods predominantly treat the system to
be studied as a closed system (like a black box), and an extensive
understanding of the system’s inner functioning is not required. The UQ
methods only need information about the input parameters of the model and
model responses to estimate probabilistic model responses. In the recent past,
uncertainty quantification and management are considered as significant
elements in risk management (the system can fail or be damaged if it does not
meet the design targets) of industrial designs Oberkampf and Trucano (2002);
Hirsch et al. (2018).

Due to the non-intrusive nature of UQ methods, these methods can be adopted
easily by researchers and industries from a wide range of engineering,
industrial and financial sectors to:

  * •

Understand the uncertainties inherent in the system.

  * •

Predict system responses concerning uncertain inputs.

  * •

Quantify confidence in the system responses.

  * •

Find optimal responses concerning a wide range of inputs.

  * •

Reduce unexpected system failures.

  * •

Implement probabilistic modeling and design processes.

  * •

Predict parametric sensitivity on the model responses.

With increasing computational power and simulation techniques, it became
possible to make accurate predictions of real-world systems. Now the
challenges in engineering designs are moved toward predicting system behaviors
with respect to uncertainties efficiently. Traditional UQ methods based on
Monte Carlo Hammersley (2013); Rubinstein and Kroese (2016) usually need a
large number of system evaluations. So these methods are restricted to
simplified test cases and for research purposes only. Monte Carlo methods are
sampling-based methods, and the convergence rate is very slow. In general,
large samples (at least $10^{4}$) are needed to predict statistical quantities
accurately. Alternatively, the literature proposes several sampling schemes
such as Latin Hypercube, Sparse sampling, clustered sampling, and stratified
sampling to accelerate the convergence. However, Monte Carlo methods have not
gained massive popularity due to their expensive computational cost. For
large-scale problems and real-world engineering applications, more recent
methods based on machine learning approaches such as Polynomial Chaos Method
(PCM) Xiu and Karniadakis (2002b); Najm (2009); Hosder et al. (2007); Ghanem
et al. (2017), Gaussian Process (or Kriging) Quinonero-Candela and Rasmussen
(2005); Bastos and O’Hagan (2009), Support Vector Machine (SVM) Awad and
Khanna (2015); Smola and Schölkopf (2004), Polynomial-Chaos-Kriging (or PC-
Kriging) Schobi et al. (2015); Wang et al. (2019) are proposed in literature
and are applied to several diverse applications. Polynomial chaos and Gaussian
process models are seen as leading approaches for stochastic and robustness
analyses of very complex engineering applications. PC-Kriging is a result of
combining polynomial chaos and Kriging methods. These approaches are discussed
in detail and are applied to an engineering application in the following
sections.

In order to understand the potential influence of input parameters on system
outputs, the Sensitivity Analysis (SA) method is used Saltelli (2002); Sudret
(2008). Various methods for global sensitivity analysis, such as linear
regression and variance-based analysis are discussed for sensitivity
estimation. A standard method of estimating system responses is using Sobol’
indices-based global sensitivity analyses. Various meta-modeling methods can
compute Sobol indexes, including Monte Carlo, graphical models, Kriging, and
support vector machine approaches. In recent years, surrogate models that
calculate Sobol’ indices have gained considerable attention. The first step in
this approach is to construct a surrogate model using the design of
experiments (DOE). Furthermore, this surrogate model is used to estimate many
model responses in order to compute Sobol’ indices. Computing model responses
from surrogate models are also called data-driven approaches, as several
combinations of input parameters are used to explore a wide range of input
domains.

##  1 Polynomial Chaos Method for UQ

In 1938, Wiener proposed the polynomial chaos method for dealing with Gaussian
distributed uncertainties Wiener (1938). Xiu and Karniadakis Xiu and
Karniadakis (2002a) demonstrated the ability to use it with any probability
distribution in a detailed analysis. In the last few years, the generalized
method has been used in a variety of engineering applications, including
computational fluid dynamics, heat transfer, nuclear reactor design, and
structural analysis Kumar et al. (2020c). Because adding uncertainties
increases the computation required to quantify them, early applications dealt
with a limited number of uncertainties. With increasing uncertainties, the
number of simulations required to quantify the uncertainty grows exponentially
using the polynomial chaos method. This is referred to as the dimensionality
curse. Numerous improvements have been proposed in literature Blatman and
Sudret (2011); Liu et al. (2020); Hosder et al. (2007); Kumar et al. (2016);
Aremu et al. (2020); Liu and Bellet (2019) to cope with the curse of dimension
and move the UQ process forward. Several researchers also proposed model
reduction algorithms (based on principal component analysis) to accelerate the
polynomial chaos method.

Numerous applications have used model reduction approaches. However, these
approaches mainly were two steps processes and usually were applied in semi-
intrusive ways. Thus, they were not very straightforward to use for
engineering applications where the models can be used as a black-box. Several
researchers proposed the idea of sparse sampling. Using sparse sampling
schemes (such as Fejer, Clenshaw-Curtis, Conrod-Patterson), the number of
simulations can be reduced to achieve the same accuracy as classical
polynomial chaos. Blatman and Sudret proposed a theory of sparse polynomial
chaos, based on least angle regression in their paper Blatman and Sudret
(2011); Bourinet (2018). Based on its principle, this method used a maximum
number of polynomial order approximations for a given number of samples and a
sparse polynomial chaos expansion (PCE) for a given system response Kumar et
al. (2020c, 2021a, 2021b, a, 2016). Several other researchers also proposed
the more or less similar idea of sparse polynomial chaos using different error
minimizing schemes. Recently, numerous applications have seen the sparse
polynomial chaos approach due to its straightforward usage and faster
convergence capability. In this section, some fundamental concepts for the
polynomial chaos approach are described Ghanem et al. (2017). It is important
addressing that the lead author developed this method and the descriptions
have been reported in different studies Kumar et al. (2020c, 2021a, 2021b, a,
2016) for a range of engineering applications.

Based on a set of orthogonal polynomial basis functions, we can write a
stochastic model response for a system under uncertainty as follows:

$\displaystyle
Y=M(\bm{\xi})=\sum_{\bm{b}\in\mathbb{N}^{n}}a_{\bm{b}}\psi_{\bm{b}}(\bm{\xi})$
(1)

where $\psi_{\bm{b}}$ is an orthogonal polynomial for multidimensional
dimensions, $\bm{b}=b_{1}\dots b_{n}$ represents an index and the terms
$a_{\bm{b}}$ are called polynomial coefficients. A PCE is given by the
Equation (1). A system of $n$-dimensional input uncertainties is represented
by $\bm{\xi}$ in the above equation. From a set of orthogonal one-dimensional
polynomials, we construct the multi-dimensional polynomials $\psi_{\bm{b}}$ as
follows Kumar et al. (2016):

$\psi_{\bm{b}}(\bm{\xi})=\psi_{b_{1}\dots
b_{n}}(\bm{\xi})=\prod_{i=1}^{n}\psi_{b_{i}}(\xi_{i})$ (2)

where $b_{i}$ is the order of the polynomial expansion for the random variable
$\xi_{i}$.

Extended polynomial expansions usually truncate to a finite number of terms
because higher-order terms are not significant in the system response after a
few terms. We truncate PCE into the following in order to achieve the
following degree $|\bm{b}|=\sum_{i=1}^{n}b_{i}$ within a given order $p$ Du
(2019):

$\displaystyle Y\simeq M_{p}(\bm{\xi})=\sum_{\bm{b}\in
A^{p,n}}a_{\bm{b}}\psi_{\bm{b}}(\bm{\xi}),~{}A^{p,n}=\\{\bm{b}\in\mathbb{N}^{n}:|\bm{b}|\leq
p\\}$ (3)

The total number of terms, $P$ (basis functions), equals $\frac{(n+p)!}{n!p!}$
when the number of input uncertainties is $n$ and the highest order of
polynomial in PCE is $p$. Polynomial coefficients can be calculated based on
the PCE order and solution samples (system responses using a deterministic
solver as black box). One can compute and construct the PCE of a stochastic
output. In the PCE, the first term (the zeroth-order term) represents the
stochastic response’s mean. In addition, one can also compute higher-order
statistical moments numerically by using these polynomial coefficients.
Computing polynomial coefficients can be done using numerical methods such as
collocation and regression Kumar et al. (2016).

Once we calculate polynomial coefficients, the mean $E(Y)$ and variance $V(Y)$
of the system output $Y$ can be computed easily as below:

$E(Y)=a_{0};V(Y)=\sum_{i=1}^{P}a_{i}^{2}\psi_{i}^{2}$ (4)

where the coefficients $a_{i}$ and $\psi_{i}$ are the same as they were
defined earlier.

##  2 Gaussian Process or Kriging for UQ

Kriging, also known as Gaussian process modeling, is a statistical method for
approximating various functions and computer experiments using Gaussian
processes. Kriging is also used as a surrogate model to establish a link
between the inputs and outputs of expensive computational models Zhang and
Apley (2016). The Gaussian process has been used for several machine learning
applications related to regression and classification in the last few decades.
Krige first developed kriging method for geostatistical applications in 1951.
Further, it was used in metamodeling and data-driven modeling for numerous
applications with noisy data. The model is known as kriging (after Krige in
geostatistics). Using Gaussian processes, Tarantola and Valette designed a
Bayesian formulation for inverse problems in geophysics Tarantola et al.
(1982). Based on the work of Williams, Neal, and Rasmussen, the model was
proposed to solve regression problems in statistics O’Hagan (1978); Hinton et
al. (1995); Williams and Rasmussen (1996) and gained popularity. Hinton et al.
(1995); Williams and Rasmussen (1996); Gibbs and MacKay (1997) provides the
Bayesian interpretation and detailed description of the model. Machine
learning was introduced to Gaussian Process in the nineties. As a result of a
detailed comparison by Rasmussen Hinton et al. (1995); Williams and Rasmussen
(1996) of the GP with the most widely used models, the GP started becoming
very popular. GP approaches outperformed other approaches in the vast majority
of cases, he showed. Using the maximum-likelihood estimation method (MLE), the
GP model’s learning process involves tuning the covariance parameters to the
data. In order to obtain the prediction and the degree of uncertainty
associated with it, given a new input and conditioned on previous
observations, one can easily calculate the mean and variance of the predictive
distribution. By using the definition of conditional probabilities, we can
easily obtain Gaussian distribution based on the GP assumption Girard (2004).

The output response of the model $M$, according to Kriging, is the realization
of a Gaussian process. Kriging metamodel $M^{K}(x)$ of the true model $M(x)$
can be described as:

$M^{K}(x)=\beta^{T}f(x)+\sigma^{2}Z(x,\omega)$ (5)

Where $\beta^{T}f(x)$ is the mean of the Gaussian process, $\sigma^{2}$ is the
variance of the process, and $Z(x,\omega)$ is a stationary Gaussian process
with a zero mean and unit variance Du (2019). The underlying probability space
($\omega$) is defined in terms of a correlation function
$R(x_{1},x_{2};\theta)$ that describes the correlation between two sample
points in the output space $x_{1}$ and $x_{2}$, as well as the hyperparameters
$\theta$.

If $y={y_{1},y_{2},y_{3},...,y_{N}}$ are the outputs of the true model $M(x)$
at sampling points $x={x_{1},x_{2},x_{3},...,x_{N}}$ , the model prediction
$M^{K}(x)$ at a new point $x$ can be estimated using Kriging metamodeling. The
gaussian process metamodeling prediction is based on the fact that the
prediction $y\textquoteright$ at the new point $x$ and the responses from the
true model $y$ make a joint Gaussian distribution as:

$\begin{Bmatrix}y^{\prime}\\\
y\end{Bmatrix}=\mathbb{N}_{N+1}\begin{pmatrix}&\begin{Bmatrix}f^{T}(x)\beta\\\
F\beta\end{Bmatrix},\sigma^{2}\begin{Bmatrix}r(x)^{T}r(x)\\\
R\end{Bmatrix}\end{pmatrix}$ (6)

In the above equation, $F$ is the observation matrix with entries
$f_{j}(x_{i})$ for $i=1,2,3...,N$ and $j=1,2,3,...,P$ where, $f_{j}(x_{i})$
are arbitrary functions at observation points $x_{i}$ and $\beta$ are
regression coefficients. The vector $r(x)$ is the cross correlations between
the new point $x$ and the known points $x_{i}$ as:

$\begin{Bmatrix}r_{1}\\\ r_{2}\\\ r_{3}\\\ .\\\ .\\\
r_{N}\end{Bmatrix}=\begin{Bmatrix}R(x,x_{1},\theta)\\\ R(x,x_{2},\theta)\\\
R(x,x_{3},\theta)\\\ .\\\ .\\\ R(x,x_{N},\theta)\\\ \end{Bmatrix}$ (7)

Where $R$ is the correlation matrix at the known points $x_{i}$ and can be
written as:

$R_{ij}=R(x_{i},x_{j};\theta)$ (8)

where $i,j=1,2,3,...N$.

or

$R=\begin{Bmatrix}R(x_{1},x_{1},\theta)&R(x_{1},x_{2},\theta)&.&.&.&R(x_{1},x_{N},\theta)\\\
R(x_{2},x_{1},\theta)&R(x_{2},x_{2},\theta)&.&.&.&R(x_{2},x_{N},\theta)\\\
R(x_{3},x_{1},\theta)&R(x_{3},x_{2},\theta)&.&.&.&R(x_{3},x_{N},\theta)\\\
.&.&.&.&.&.\\\ .&.&.&.&.&.\\\
R(x_{N},x_{1},\theta)&R(x_{N},x_{2},\theta)&.&.&.&R(x_{N},x_{N},\theta)\\\
\end{Bmatrix}$ (9)

Using the conditional distribution properties of the multivariate normal, the
mean and the variance of the predictor can be written as:

$E\\{y^{\prime}|y\\}=f^{T}\beta+r^{T}R^{-1}(y-F\beta)$ (10)

$\sigma^{2}_{y}=\sigma^{2}(1-r^{T}R^{-1}r+u^{T}(F^{T}R^{-1}F)^{-1}u)$ (11)

where the regression coefficients $\beta$ and the term $u$ are defined as:

$\beta=(F^{T}R^{-1}F)^{-1}F^{T}R^{-1}y$ (12)

$u=F^{T}R^{-1}r-f$ (13)

##  3 Polynomial Chaos Kriging for UQ

Kriging interpolates local variations in the system response $Y$ as a function
of the neighboring design points, whereas PCE closely approximates the
regional behavior of $Y$ Amini et al. (2021). It is possible to obtain more
accurate PC-Kriging metamodels by combining local and global approximation
techniques. In PC-Kriging, there is an array of orthonormal polynomials that
represent the trend and which are defined as follows:

$M^{PCK}(x)=\sum_{\bm{b}\in
A^{p,n}}a_{\bm{b}}\psi_{\bm{b}}(\bm{\xi})+\sigma^{2}Z(x,\omega)$ (14)

Where $\psi_{b}$ are multivariate orthogonal polynomials concerning the input
distributions and $a_{b}$ are the corresponding coefficients.

##  4 Uncertainty Quantification of a Supersonic Nozzle

The UQ is crucial for assuring the results produced by mathematical modeling
in engineering applications. The standard deviation or variance can be
considered a safety bound or a confidence interval around the mean values.
Hence, we apply the methods discussed in the previous sections to an
engineering application in this section. We analyze the non-ideal supersonic
compressible flow within a 2D converging-diverging nozzle shown in Figure 1
Guardone (2021). The nozzle is 0.123 m long, with a throat height of 0.0084 m
and an inlet height of 0.036 m Guardone (2021). This is a test case provided
in SU2 for deterministic CFD simulations Guardone (2021). It has a simple
geometry where the flow accelerates from subsonic to supersonic speeds. It can
be used to investigate compressible flows where simple ideal gas laws are not
enough to describe thermodynamic behavior properly. To determine the
performance of the nozzle over a wide range of inlet conditions, CFD
simulations are first used to confirm its performance. Then further CFD
simulations are combined with uncertainty quantification methods.
Additionally, the uncertainty bounds for the nozzle performance in terms of
pressure and Mach number, flow density, and temperature fields along the
centerline are evaluated with respect to input uncertainties. They are shown
with mean and standard deviation values.

Figure 1: CD Nozzle

###  4.1 Test Case Description

Octamethyltrisiloxane (MDM), a pressure-sensitive fluid, is used as a working
fluid for analyzing non-ideal supersonic compressible flow inside a
converging-diverging (CD) nozzle. Table 1 presents details about the
properties of fluid and flow conditions. This configuration results in a total
exhaust pressure ratio of 3.125, which results in a supersonic outflow at Mach
number 1.5 Guardone (2021). The static pressure applied to the test case’s
outlet is 200,000 Pa. The computational domain and mesh are depicted in Figure
2. The mesh is composed of 3,540 quadrilateral elements and 3,660 nodes
Guardone (2021). At the inlet and outlet boundaries, Riemann boundary
conditions based on characteristics are used. Symmetry boundary conditions
define symmetry boundaries. By mirroring the flow around the x-axis, the mesh
size is reduced, along with the computational cost. On the boundary of a wall,
Navier-Stokes adiabatic wall conditions are applied.

Table 1: Flow conditions for C-D Nozzle CFD simulation Parameters | Values
---|---
Working fluid | Octamethyltrisiloxane
Inlet Pressure | 904388 Pa
Inlet Temperature | 542.13 K
Turbulence model | SST
Gamma Value | 1.01767
Gas Constant | 35.17
Critical Temperature | 565.3609
Critical Pressure | 1437500
$\mu$ | 1.21409E-05
$K_{T}$ | 0.030542828

Figure 2: Computational domain and mesh

###  4.2 Deterministic Results

For deterministic simulations, the SU2 solver is run at the fixed boundary
conditions, flow conditions, and fluid properties as described earlier. The
total number of iteration was given 1000 so that all solutions and residuals
are converged nicely. We post-process the data with Paraview (a multi-
platform, open-source data analysis and visualization application). In Figure
4, solution fields for pressure, temperature, Mach number, and flow density
are exhibited for the whole computational domain. Further, these quantities
are also shown at the centerline of the nozzle for better understanding and
analysis. At the inlet of the nozzle, pressure, temperature, and density of
the fluid are at their maximum and then at their minimum near the outlet. In
an inlet, the Mach number can be viewed as a minimum, and at the exit, the
Mach number reaches a maximum value.

Figure 3: Numerical results (provided serially in vertical format): pressure,
temperature, Mach, and density fields
Figure 4: Computational results: pressure, temperature, Mach number and fluid
density at the centerline of the nozzle )
Figure 5: Solution samples for Pressure, Temperature, Mach number and fluid
density at the centerline of the nozzle )
Figure 6: Mean and standard deviation for Pressure, Temperature, Mach number
and fluid density at the centerline of the nozzle )

###  4.3 Description of Uncertainties

For the uncertainty analysis, seven input parameters; two from boundary values
(inlet temperature and inlet pressure), three from gas properties (specific
heat ratio $\gamma$, gas constant $R$ and acentric factor $\omega$), and two
from the viscosity model (molecular viscosity $\mu$ and molecular thermal
conductivity $K_{T}$) are considered as uncertain. All parameters are
considered uniformly distributed. The inlet pressure and acentric factor are
assumed to have $5\%$ variability from the mean value. Gas constant, molecular
viscosity, and molecular thermal conductivity are assumed to vary $2\%$ from
their mean values. Minor uncertainties of $1\%$ from their mean values are
given to the inlet temperature and Gamma values. All the mean values for these
parameters, their uncertainties, and their ranges of variability are described
in Table 2.

Table 2: Input uncertainties for CFD simulations Parameters | Values | Uncertainties ($\%$) | Minimum | Maximum
---|---|---|---|---
Inlet Pressure (Pa) | 904388 | 5 | 859168 | 949607
Inlet Temperature (K) | 542.13 | 1 | 536.71 | 547.55
Gamma Value | 1.01767 | 1 | 1.00749 | 1.02785
Gas Constant | 35.17 | 2 | 34.47 | 35.87
$\mu$ | 1.21409E-05 | 2 | 1.18981 | 1.23837
$K_{T}$ | 0.030542828 | 2 | 0.029931971 | 0.031153684
Acentric factor ($\omega$) | 0.524 | 5 | 0.498 | 0.550

###  4.4 Uncertainty Analysis

As described in the previous section, the PC-Kriging method is used here to
estimate the combined impact of all input uncertainties on the system
responses of the CD nozzle. Usually, in the regression-based polynomial chaos
method, a total of 240 CFD samples (for PC order 3 and 7 input uncertainties)
will be required to estimate the statistical quantities (mean and standard
deviance) of the output accurately (see Kumar et al. (2016)). Here to
construct the PC-Kriging-based surrogate model, only 100 CFD simulations are
used. For input parameters, 100 designs of experiments are constructed using
the Sobol sequence-based sampling technique. In Figure 5, the CFD solutions
for pressure, temperature, Mach number, and fluid density along the nozzle
centerline are shown for all 100 samples. It can be seen that pressure and
Mach number are not varying much with the input uncertainties. However, minor
variations can be seen in density with the input variations. The most
significant variations can be seen for the temperature field. In Figure 6, the
mean and standard deviation are shown for all these quantities. These values
are calculated from the PC-Kriging based surrogate model. The mean values
behavior is similar to the deterministic solutions. For pressure, temperature,
and Mach number, the standard deviation values are higher at the outlet. That
means the highest fluctuations are at the nozzle outlet. However, the standard
deviation for fluid density is seen lower at the nozzle outlet.

It is also important to address that this developed uncertainty method can be
applied to other domains such as nuclear engineering in terms of safety
assessment of advanced reactor system Kumar et al. (2021a). In addition, the
authors also utilized this methodology to understand the evaluate the
uncertainties in composite materials Kumar et al. (2021b).

##  5 Conclusions

In this work, first, we describe the two most popular meta-modeling methods
(Polynomial Chaos and Kriging methods) suitable for uncertainty quantification
in engineering applications. Further, to increase the efficiency, the
polynomial chaos and Kriging methods are combined and used for an engineering
test problem under multiple uncertainties. A 2D supersonic converging-
diverging nozzle is considered for the analysis where the multi-physics CFD
solver SU2 is used for deterministic solutions. The UQ methods (polynomial
chaos, Kriging, and PC-Kriging) are developed in Matlab and are further
combined with SU2 for uncertainty quantification. The standard deviation can
be considered as a safety bound or a confidence interval around the mean
values. Hence, for assurance in making crucial decisions, the results are
discussed in terms of the mean and standard deviation of the output
quantities, i.e., pressure, temperature, Mach number, and fluid density.

Future work will focus on its application in multiscale modeling of composite
accident-tolerant nuclear fuels with Sic/Sic claddings for small modular
reactor (SMR) applications.

## Acknowledgement

The computational part of this work was supported in part by the National
Science Foundation (NSF) under Grant No. OAC-1919789.

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