Data-Informed Decomposition for LocalizedUncertainty Quantification of Dynamical Systems

Waad Subber˚, Sayan Ghosh, Piyush Pandita, Yiming Zhang, Liping Wang

Probabilistic Design and Optimization Group,GE Research,

1 Research Circle, Niskayuna, NY 12309, USA

Abstract

Industrial dynamical systems often exhibit multi-scale response due to materialheterogeneities, operation conditions and complex environmental loadings. Insuch problems, it is the case that the smallest length-scale of the systems dynam-ics controls the numerical resolution required to effectively resolve the embeddedphysics. In practice however, high numerical resolutions is only required in aconfined region of the system where fast dynamics or localized material vari-ability are exhibited, whereas a coarser discretization can be sufficient in therest majority of the system. To this end, a unified computational scheme withuniform spatio-temporal resolutions for uncertainty quantification can be verycomputationally demanding. Partitioning the complex dynamical system intosmaller easier-to-solve problems based of the localized dynamics and materialvariability can reduce the overall computational cost. However, identifying theregion of interest for high-resolution and intensive uncertainty quantificationcan be a problem dependent. The region of interest can be specified based onthe localization features of the solution, user interest, and correlation lengthof the random material properties. For problems where a region of interest isnot evident, Bayesian inference can provide a feasible solution. In this work,we employ a Bayesian framework to update our prior knowledge on the local-ized region of interest using measurements and system response. To addressthe computational cost of the Bayesian inference, we construct a Gaussian pro-cess surrogate for the forward model. Once, the localized region of interest isidentified, we use polynomial chaos expansion to propagate the localization un-certainty. We demonstrate our framework through numerical experiments on athree-dimensional elastodynamic problem.

Keywords: Bayesian inference; Machine Learning ;Uncertainty Quantification;Dynamical Systems; Inverse Problem; System Identification ; Gaussian processregression; Polynomial chaos.

˚Corresponding authorEmail address: Waad.Subber@ge.com (Waad Subber)

Preprint submitted to August 18, 2020

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1. Introduction

With the increase in demand for high-performance and highly-efficiency sys-tems, the complexity of industrial design and manufacturing process is increas-ing proportionally; exposing many opportunities for novel technologies as wellas many associated technical challenges. For example, advancements in thedesign of composite structures allows us to reduce weight, advancements inadditive manufacturing enables us to reduce cost, and optimal computationalmaterial design pushes the boundary in the discovery of new alloys with de-sirable electro-mechanical properties. Introducing a new technology typicallyhappens at the lowest level of the systems hierarchy (e.g., at the parts or sub-component levels). Extending the new technologies to the system level requiresrigorous testing. For example, in 1980s composite material was only used forlimited components of an aircraft (i.e,. the wing and tail [1]). Recently however,after multiple test flights, about 50% of the materials used in the Boeing 787Dreamliner are composite materials [2].

In the industrial setting, the process of adaptation of a new technology canbe accelerated by a proper assessment of uncertainty at various aspects of theproduct ’s life cycle spanning the design, manufacturing and maintenance stages.For example, at the design stage of an aircraft wing rib, it is crucial to considerthe effect of uncertainty in the material and operation conditions on the safetyfactor and aeroelastic dynamics of the wing [3]. At the manufacturing stage, itis important to consider the impact of manufacturing uncertainties on qualitycontrol [4] and non-destructive testing [5]. The maintenance stage requires aholistic assessment of the effect of measurement uncertainty on the static anddynamic responses of the wing during structural health monitoring [6].

Quantifying uncertainty at the system level often requires a physics-basedcomputational model for the entire structure. However, in structures such as anaircraft wing, traditional computational models may become too complex andcostly for simulating the multi-scale dynamical response especially due to mate-rial heterogenity at the sub-component level. The effect of the sub-componenton the entire structure depends on the size, location and loading conditions ofthe part. It is therefore, necessary to consider a different level of fidelity forthe analysis of the sub-components in order to reduce the cost and complexityof uncertainty quantification. To this end, the concept of localized uncertaintypropagation for dynamical systems having muti spatio-temporal scales can beutilized to address such issues [7, 8, 9].

In this work, we consider assessing the effect of localized uncertainty in a re-gion of interest within the entire structure. For structures composed of distinctparts that can be clearly identified, the localized region for uncertainty propa-gation may become obvious. When the distinguished components of a structureare not clear, measurement data can be utilized within Bayesian frameworkto identify the localized region of interest. The Bayesian paradigm integratesthe domain-knowledge, physics-based computational models and observationaldata in one framework to update the current state of knowledge [10, 11]. TheBayesian methods offer two major advantages namely: a) allow quantification

2

of epistemic uncertainty under limited-data, and b) retain physical sense for theparameters and the quantity-of-interest. Conditioning apriori physics beliefs onthe available data, Bayes rule provides aposteriori distribution on the modelparameters. A robust method to estimate the posterior distribution in theBayesian inference (i.e. sampling values of the model parameters from the pos-terior probability) is Markov Chain Monte Carlo (MCMC) [12, 13]. Estimatingthe posterior probability density function in the Bayesian method requires solv-ing the forward model many times, which may become challenging for limitedcomputing resources. This issue is often addressed by building a data-drivenprobabilistic surrogate model using Gaussian process (GP) regression [14]. Con-structing a GP model requires executing the forward model only few numberof times. The GP models are non-parametric and Bayesian in nature, andthey provide uncertainty bound on their predictions. Nevertheless, for problemswith stochastic field representation of the variability in the propagation media,uncertainty quantification using GP models may become challenging for gen-eral non-Gaussian description of the underlying random variability. PolynomialChaos (PC), on the other hand, provides an effective framework to representand propagate an arbitrary random variable through complex computationalmodels [15, 16]. In PC, the response of the physical model is represented asspectral expansion in a polynomial series with basis function being orthogo-nal with respect to the probability density function of the underlying randomvariables of the propagation media.

The rest of this work is organized as follows: in Section 2, we provide theproblem statement and the associated mathematical formulations. Our numer-ical demonstrations for the mathematical framework are provided in Section 3.We provide the conclusions of the current work in Section 4.

2. Methodology

In this section, we present the mathematical framework of our approach fordata-driven partitioning scheme for localized uncertainty quantification. In par-ticular, in Subsection 2.1, we introduce the problem statement in the Bayesiansetting. As mentioned previously, for problems where the localized region ofinterest is not defined explicitly, we rely on measurement data of a responsequantity (aided by a computational model) to infer the localized region of in-terest. The Bayesian framework requires a computational model (the forwardproblem) to estimate the response of the system for a given set of the inputparameters that to be inferred. Consequently, in Subsection 2.2 we discuss thestochastic elastodynamic problem and its finite element discretization. Esti-mating the localized region of interest in the Bayesian setting necessitates manysolutions to the stochastic elastodynamic problem which can become compu-tationally demanding. Furthermore, it is worth noting that for the Bayesiancalculation the entire solution field of the stochastic elastodynamic problem isnot required, only a realization at the measurement location is needed. Thus,a surrogate model for the realization of the response can be used to reduce thecomputational cost of the Bayesian framework. In Subsection 2.3, we present

3

the Gaussian process surrogate to emulate the solution to the stochastic elas-todynamic problem with less cost. Once the localized region of interest is esti-mated, a confined uncertainty representation of the material properties withinthe region of interest can be performed. The localized uncertainty is propagatedforward through the model in order to estimate its effect on the variability ofthe response. For this task, we use the polynomial chose expansion for efficientassessment of uncertainty with less cost. The polynomial chose expansion isreviewed in Subsection 2.4.

2.1. Bayesian Inference

In Bayesian inference, the prior knowledge is updated to posterior usingnoisy measurements and the response of a physical model [10, 11]. The updateis based on the Bayes’ rule defined as

ppθ|dq “ppθqppd|θq

ppdq, (1)

where θ is the uncertain parameters to be estimated, d is the measurement ofan observable quantity, ppθ|dq is the posterior probability density function, ppθqis the prior probability density function, and ppd|θq denotes the likelihood ofthe observations given the parameter. We assume that the measured data d isgenerated from a statistical model composed of a physical model Mpθq plus anadditive measurements noise ε as

d “Mpθq ` ε. (2)

Here we represent the measurement noise as a Gaussian random variable withunknown variance ε „ N p0, σ2

nq. For a Gaussian noise, the likelihood functionbecomes

ppd|θq “1

p2πσ2nq´N2

exp

˜

´1

2σ2n

Nÿ

i

rdi ´Mpθiqs2¸

. (3)

The task in hand is to utilize the measurement d and the physical model Mpθq toestimate the system parameters θ that best satisfy Eq.(1). The process requiresmany executions to the physical model Mpθq, which can be computationallyexpensive. It is often, the expensive computational model is emulated by asimpler easy-to-evaluate model that can estimate the response with a quantifiedaccuracy as:

Mpθq »Mpθq, (4)

where Mpθq denotes the surrogate model that is constructed using a limited runsof the physical model Mpθq. In our work, we represent Mpθq as Gaussian processsurrogate model [17]. Once we constructed and validate the surrogate model,the parameterization of the localized features θ is estimated using Bayes’ ruleevaluated by Markov Chain Monte Carlo (MCMC) sampling technique. Havingthe localized region of interest identified, a localized uncertainty quantificationof the confined variability of the material properties can be performed efficientlyusing polynomial chaos expansion [15].

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2.2. The Forward Problem

In this section, we give a brief summary of the mathematical formulation tothe linear stochastic dynamical system considered in this work. The frameworkis presented for localized uncertainty propagation work-flow, whereby the de-composition of the physical domain is based on the variability of the materialproperties. Consequently, we consider an arbitrary physical domain Ω P Rd withBΩ being its boundary as shown in Fig. (1-a), and define the following problem:

Find a random function upx, t, ξq : Ω ˆ r0, Tf s ˆ Ξ Ñ R, such that thefollowing equations hold

ρpξq:upx, t, ξq “ ∇ ¨ σ ` b in Ω ˆ r0, Tf s ˆ Ξ,upx, t, ξq “ u on BΩu ˆ r0, Tf s ˆ Ξ,

σ ¨ n “ t on BΩt ˆ r0, Tf s ˆ Ξ,upx, 0, ξq “ u0 in Ω ˆ Ξ,9upx, 0, ξq “ 9u0 in Ω ˆ Ξ,

(5)

where ρpξq is the mass density, σ is the stress tensor, u is the displacement field,b is the body force per unit volume, u is the prescribed displacement on BΩu,t is the prescribed traction on BΩt, n is a unit normal to the surface, and u0

and 9u0 are the initial displacement and velocity, respectively. Here, we definethe stochastic space by (Θ,Σ, P q, where Θ denoting the sample space, Σ beingthe σ-algebra of Θ, and P representing an appropriate probability measure.The stochastic space is paramatrized by a finite set of standardized identicallydistributed random variables ξ “ tξipθqu

Mi“1, where θ P Θ. The support of

the random variables is defined as Ξ “ Ξ1 ˆ Ξ2 ˆ ¨ ¨ ¨ΞM P RM with a jointprobability density function given as ppξq “ p1pξ1q ¨ p2pξ2q ¨ ¨ ¨ pM pξM q.

For linear isotropic elastic martial, the constitutive relation between thestress and strain tensors is given by:

σ “ λpξqtrpεqI` 2µpξqε, (6)

where λpξq and µpξq are the Lemae’s parameters, I is an identity tensor and εis the symmetric strain tensor defined as

ε “1

2

`

∇u`∇uT˘

. (7)

For a random Young’s modulus Epx, ξq and deterministic Poisson’s ratio ν , theLemae’s parameters can be expressed as

λpξq “Epx, ξqν

p1` νqp1´ 2νq, µpξq “

Epx, ξq

2p1` νq. (8)

We consider the case that uncertainty steams from a localized variabilityin a confined region within the physical domain. For example as shown inFig. (1-b), the variability in the quantity of interest can be attributed to thematerial random properties within the subdomain Ω2. The artificial martialboundaries shown in Fig.( 1-b) for subdomain Ω2 is estimated using Bayesian

5

inference. Localizing random variability in the neighborhood of the quantity ofinterest reduces the computational cost of uncertainty propagation in problemswhere a region of interest can be specified. Depending on the interest in theregion, each subdomain can have its local uncertainty representation and thecorresponding mesh and time resolutions. As a results the Asynchronous Space-Time Domain Decomposition Method with Localized Uncertainty Quantifica-tion (PASTA-DDM-UQ) [7, 8, 9] can be utilized. In PASTA-DDM-UQ, spatial,temporal and material decompositions are considered. In this work however,we only consider material decomposition and apply non-intrusive approach foruncertainty propagation.

Consequently, let the physical domain Ω be partitioned based on the martialvariability into ns non-overlapping subdomains Ωs, 1 ď s ď ns as shown inFig. (1-b) and such that:

Ω “nsď

s“1

Ωs, Ωsč

Ωr “ H for s ‰ r, Γ “nsď

s“1

Γs, Γs “ BΩszBΩ. (9)

b

∂Ωu

∂Ωt

Ω

E(x, ξ)

(a) Spatial domain

@u

@t

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E2(x, )

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E1(x)

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E3(x)

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(b) Domain decomposition.

Figure 1: An arbitrary computational domain Ω with a random material property (i.e.,Epx, ξq) and its partitioning into non-overlapping subdomains. The partitioning is based onmaterial variability.

According to the decomposition in Eq.(9), the stochastic dynamical problemin Eq.(5) can be transformed into the following minimization problem:

Find a random function upx, t, ξq : Ωˆ r0, Tf s ˆ Ξ Ñ R, such that

Lpu, 9uq “nsÿ

s“1

pTsp 9uq ´ Vspuqq Ñ min, s “ 1, ¨ ¨ ¨ , ns, (10)

where Lpu, 9uq is the Lagrangian of the system, Tsp 9uq denotes the subdomainkinetic energy and Vspuq is the subdomain potential energy defined as:

6

Tsp 9uq “

ż

Ξ

ż

Ωs

1

2ρspξq 9u ¨ 9u dΩdΞ, (11)

Vspuq “ż

Ξ

ˆż

Ωs

1

2ε : σs dΩ`

ż

Ωs

u ¨ bs dΩ`

ż

BΩt

u ¨ ts dΓ

˙

dΞ, (12)

The Hamilton’s principle with a dissipation term reads

ż Tf

0

ˆ

δL´ BQB 9ε

: δε

˙

dt “ 0, (13)

where δL is the first variation of the augmented Lagrangian defined as

δL “nsÿ

s“1

ż

Ξ

ˆż

Ωs

ρspξqδ 9u ¨ 9u dΩ´

ż

Ωs

δε : Dspξq : εdΩ`

ż

Ωs

δu ¨ bs dΩ`

ż

BΩt

δu ¨ ts dΓ

˙

dΞ, (14)

here we define Dspξq as the uncertain linear elasticity tensor. The dissipationfunction Qp 9uq in the Hamilton is defined as

Qp 9uq “nsÿ

s“1

1

2

ż

Ξ

ż

Ωs

9ε : pDs : 9ε dΩdΞ, s “ 1, ¨ ¨ ¨ , ns, (15)

where pDs is the damping tensor assumed to be deterministic. SubstitutingEqs. p15 and 14q into the Hamilton’s principle Eq. (13) gives the followingstochastic equation of motion for a typical subdomain Ωsż

Ξ

ż

Ωs

ρspξq:u ¨ δu dΩ dΞ`

ż

Ξ

ż

Ωs

9ε : pDs : δε dΩ dΞ`

ż

Ξ

ż

Ωs

ε : Dspξq : δε dΩ dΞ

(16)

“

ż

Ξ

ż

Ωs

δu ¨ bs dΩ dΞ`

ż

Ξ

ż

BΩt

δu ¨ ts dΓ dΞ.

In the next section, we describe the finite element discretization of the weakform Eq.(16).

2.2.1. Spatial and Temporal Discretizations

Let the spatial domain Ω be triangulated with finite elements of size h andlet the associated finite element subspace be defined as Xh Ă H1

0 pΩq, then anapproximate deterministic finite element solution can be expressed as

uh “niÿ

i

Nipxquiptq. (17)

7

Substituting the discrete field, Eq.(17) in the weak form Eq.(16) gives the fol-lowing semi-discretized stochastic equation of motion :

ż

Ξ

pM:uptq `C 9uptq `Kuptqq dΞ “

ż

Ξ

FptqdΞ. (18)

We drop the nodal finite element marks (tilde) for brevity of the representationand define the following matrices:

M “

nsÿ

s“1

ż

Ωs

ρsNTNdΩ, C “

nsÿ

s“1

ż

Ωs

BTpDsBdΩ,

K “

nsÿ

s“1

ż

Ωs

BTDisBdΩ, Fptq “

nsÿ

s“1

ˆż

Ωs

bTs NdΩ`

ż

BΩs

tTs NdΓ

˙

.

Here, B is the displacement-strain matrix. For time discretization, we use theNewmark time integration scheme to advance the stochastic system one timestep as

9uk`1 “ 9uk ` p1´ γq∆t:uk ` γ∆t:uk`1, (19)

uk`1 “ uk `∆t 9uk `

ˆ

1

2´ β

˙

∆t2 :uk ` β∆t2 :uk`1, (20)

where γ and β are the integration parameters, and ∆t “Tf´T0

nt. Substituting

he Newmark scheme into the semi-discretized stochastic equation of motionEq.(18), gives the following fully discretized linear system for a give realizationof the random vector ξ:

ApξqUk`1pξq “ Fk`1

´GUkpξq (21)

where for compact representation, we define

Apξq “

»

–

Mpξq C Kpξq´γ∆T I I 0´β∆T 2I 0 I

fi

fl , G “

»

–

0 0 0´p1´ γq∆T I ´I 0´p 1

2 ´ βq∆T2I ´∆T I ´I

fi

fl ,

Upξq “

$

&

%

:upξq9upξqupξq

,

.

-

, F “

$

&

%

f00

,

.

-

.

For the data-driven decomposition approach, many solutions to the forwardproblem Eq. (21) are required in estimating the appropriate decomposition forlocalized uncertainty propagation. To mitigate the computational cost involvedwith identifying the underlying localized region of interest, a Gaussian Process(GP) surrogate model is utilized as explained in the next section.

8

2.3. Surrogate Modeling

The Gaussian Process (GP) surrogate model is widely used for engineeringproblems as a cost-effective alternative to costly computer simulator [18]. In theauthors’ previous work [19], a fully-Bayesian industrial-level implementation forGP-based metamodeling and model calibration has been exhaustively covered.This implementation, called GE’s Bayesian hybird modeling (GEBHM), hasbeen rigorously tested and validated on numerous benchmark problems and theimpact of using Bayesian surrogate modeling has been demonstrated on severalchallenging industrial problems. In GP for dynamical systems, we considerD “ tpxi,yiq | i “ 1, 2, ¨ ¨ ¨ , Nu to be a set of training data consists of N samples,where xi P Rd represents the input sample i, and yi is the corresponding outputvector of size nT . For time-series data, the output is observed at a sequence oftime steps tj P rt1, t2, ¨ ¨ ¨ , tnT

s. We concatenate all the input and output intothe design matrix X and the corresponding observation matrix Y, respectivelyas:

X “

»

—

—

—

—

—

—

—

—

—

—

—

–

t1 x1

......

tnTx1

......

t1 xN...

...tnT

xN

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

, Y “

»

—

—

—

—

—

—

—

—

—

—

—

–

y11...

y1nT

...yN1...

yNnT

fi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

ffi

fl

, (22)

where yij is the response at time tj for the input parameters xi. The sizes ofthe design matrix X and the observation matrix Y are pN ˆ nT q ˆ pd` 1q andpN ˆ nT q ˆ 1, respectively. In compact form, the training dataset (X , Y ) canbe rewritten as:

X ““

1N bT Xb 1nT

‰

, Y “ vecpYq, (23)

where 1N is an identity vector of size N , X “ rx1, ¨ ¨ ¨ ,xN sT , T “ rt1, ¨ ¨ ¨ tnT

sT ,1nT

is an identity vector of size nT , Y ““

y1 ¨ ¨ ¨ yN‰

and yi “ ryi1, ¨ ¨ ¨ yinTsT .

Here the symbols b and vecp‚q represent Kronecker product and vectoriza-tion operators, respectively. Consequently, a general regression model for time-dependent data can be expressed as a function fpX q that maps the input Xto time-series observation Y. In GP regression, the goal is to infer the func-tion fpX q from noisy observation of the the output Y. To this end, the func-tion fpX q is viewed as a random realization of Gaussian processes fpX q „GP pµpX q,KpX ,X 1qq, where µpX q and KpX ,X 1q are the mean and covariancematrix of the process, respectively. Training the GP model can be performedby finding the optimal values to the covariance parameters. Systematically, thisis done by maximizing the evidence or the marginal likelihood with respect tothe hyperparameter parameters of the kernel. The prediction of the GP for

9

a new input x˚, is a Gaussian process with the following posterior mean andcovariance

µpx˚q “ kpx˚,X qrKpX ,X 1q ` σ2nIs´1Y, (24)

σ2px˚q “ kpx˚,x˚q ´ kpx˚,X qrKpX ,X 1q ` σ2nIs´1kpX ,x˚q (25)

The covariance function in the GP framework encodes the smoothness andit measures the similarity of the process between two points. The covariancefunction also encodes the prior belief over the regression function to model themeasurements. The prior belief can be on the level of the function smoothness,or behavior and trend such as periodicity, for example. Selecting the right co-variance kernel can be challenging for time-dependent data and may require acomposition of several covariance functions together to model the right behav-ior of the data. On the other hand, for problems where the training data isgiven in the form as in Eq.(22), the size of the data may grow exponentiallydemanding large computational budged. In this case a scaleable framework forthe GP regression of large dataset can be exploited to efficiently address thecomputational cost [20].

In this work, the ultimate goal of the GP model is to serve as a surrogate tothe costly simulation code in the Bayesian inference. Thus, we follow a simplifiedapproach to reduce the cost of building the surrogate [21]. For the case whenthe time index of measurement is set a priori and prediction at intermediatetime instant is not required, the inter correlation between the time steps can berelaxed. Specifically, the prediction of the model in this case is always set atthe location of the measured data, and the model only considers the correlationamong the input variables xi. Thus the GP can be constructed on on the subsetof the data (X,Y) instead of (X , Y ) as GPpµpXq,KpX,X1qq, where

X “

»

—

–

x1

...xN

fi

ffi

fl

, Y “

»

—

–

y11 , . . . , y1

nT

...y1N , . . . , yNnT

fi

ffi

fl

(26)

2.4. Polynomial Chaos

The Polynomial Chaos (PC) expansion is based on the spectral decom-position of a stochastic process into deterministic coefficients scaling randomfunctions. In particular, the PC approximates a stochastic process as a linearcombination of stochastic orthogonal basis functions as

upt, ξq “Nÿ

j“0

Ψjpξqujptq, (27)

where Ψjpξq are a set of multivariate orthogonal random polynomials and ujptq,are the deterministic projection coefficients. The PC coefficients can be esti-mated non-intrusively as

10

ujptq “

ş

Ξupt, ξqΨjpξqdΞş

ΞΨ2j pξqdΞ

, (28)

whereş

Ξp‚q dΞ denotes the expectation operator with respect to the probability

density function of the underlying random variables. The expectation integralcan be estimated using random sampling or deterministic quadrature rule [22]

3. Numerical Example

For the numerical demonstration, we consider the problem of detecting thedesired geometry (e.g. localized features) for a given specimen from noisy mea-surements of its dynamical response. We paramatrize the geometry by thedimensions of the inner section (the inner length li and radius ri) as shown inFig. (2). The inner dimensions are inferred from noisy measurement of the beamdeflection at the mid-span. Once the dimensions are estimated, we perform alocalized uncertainty propagation of the material parameters of the inner core.

3.1. The Forward Problem

We consider a 3-D Aluminum beam with mean elastic properties of E “ 70GPa, ν “ 0.3 and ρ “ 26.25 kN/m3. For the damping representation, weconsider Rayleigh damping whereby the damping is assumed as C “ ηmM `

ηkK. In the numerical implementation, we consider the proportion constantsηm “ 0 and ηk “ 0.001, and we use stiffness K based on the mean properties.We utilize FEniCS for the forward finite element simulations. [23]. Fig. (2)shows a 2D projection of the beam geometry, whereby we parameterize theinner cylinder by (length li and radius ri), and the outer cylinder by (lengthlo and radius ro). For the reference case the inner and outer dimensions are(li “ 0.45 m, ri “ 0.025 m) and (ro “ 0.05 m and, lo “ 1.0 m), respectively. Thebeam is subjected to an impact force defined as:

F pt,xq “ r0, 0, F0ttcδpt´ tcqsT, (29)

where F0 “ ´5.0 GN and the ramp time tc “ 0.5 ms. The beam is fixed at bothends and subjected to zero initial displacement and velocity. The dynamics isintegrated up to 0.01 s.

11

ri

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2ro

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li

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lo

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Figure 2: Schematic showing a 2D projection of a typical beam. For the reference case theinner and outer dimensions are (li “ 0.45 m, ri “ 0.025 m) and (ro “ 0.05 m and, lo “ 1.0 m),respectively.

We consider the vertical deflection at the mid-span to be the quantity ofinterest (QoI) in identifying the underlying beam geometry. Fig. (3) shows hemid-span displacement and velocity for a the reference case.

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.02

−0.01

0.00

dis

pla

cem

ent

[m]

(a) displacement

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.005

0.000

0.005

0.010

velo

city

[m/s

]

(b) velocity

Figure 3: The displacement and velocity at the mid-span of the reference caseP p0.5, 0.0, 0.0qrms and using the mean material properties E “ 70 GPa, ν “ 0.3 and ρ “ 26.25kN/m3

3.2. The Surrogate Model

In order to infer the beam geometry from the QoI, many runs of the forwardmodel (the 3D finite element dynamical code are required. This may causea computational burden when the computational budget is limited. Thus, asurrogate model can overcome this issue by utilizing a limited number of aprespecified runs. The design of computer experiments concept can be used tooptimally select the required runs [24, 25, 26]. For multi-fidelity simulations,where a high-cost high-accuracy and a low-cost low-accuracy simulators areavailable, a balance between the cost and accuracy can be achieved in designingthe numerical simulations experiments [27].

The surrogate model is constructed based on samples that can representthe variability in the beam geometry due to different values of the inner di-mensions. We define the variability of the inner dimensions by assigning a uni-form random distribution with a specified bounds as li „ Up0.25, 0.75qm and

12

ri „ Up0.01, 0.05qm. Using Latin hypercube sampling technique [28], we gener-ate 50 independent samples for the inner dimensions. Using these samples, wegenerate the geometry of the beam followed by constructing the correspondingfinite element mesh, and executing the forward model to calculate the mid-span deflection (QoI). Samples of the training geometries are shown in Fig. (4).Clearly, the samples span a wide range of the probable geometries of the beam.The corresponding scatter of the mid-span vertical displacement of the 50 sam-ples are shown in Fig. (5). Of course, the variability of the inner dimensions notonly affect the geometry, but also the location and magnitude of the bouncingdeflection at around time t “ 0.002s and t “ 0.005s.

(a) (b)

(c) (d)

Figure 4: Four samples showing the variability in the beam geometry due to different valuesof the inner dimensions (li, ri).

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.02

0.00

dis

pla

cem

ent

[m]

Figure 5: The mid-span vertical displacement of the 50 samples.

We randomly split the 50 samples into 40 samples for training and 10 fortesting. For practical numerical implementation and to reflect the reality ofthe real world, we add a Gaussian random noise of strength 10´3 ˆ maxpuqto the deflection measurements u. The GP surrogate model is trained on thetraining samples and used to predict the held-out testing samples. Fig.( 6)shows samples of observed and predicted responses for different values of theinner dimensions. The maximum and minutemen values of the mean squarederror between the prediction and the observed response are 2.10 ˆ 10´7 and5.35 ˆ 10´9, respectively. Given the fact that the testing samples are not seenby the model during the training phase, the GP model can predict the unseendata within the given accuracy.

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0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.03

−0.02

−0.01

0.00d

isp

lace

men

t[m

]

observed

predicted

(a)

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.03

−0.02

−0.01

0.00

dis

pla

cem

ent

[m]

observed

predicted

(b)

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.03

−0.02

−0.01

0.00

0.01

dis

pla

cem

ent

[m]

observed

predicted

(c)

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.03

−0.02

−0.01

0.00

0.01

dis

pla

cem

ent

[m]

observed

predicted

(d)

Figure 6: Observed and predicted QoI for different testing samples. The test samples are notpart of the training set.

To summarize the quality of the prediction, in Fig. (7), we show the L2-norm of the observed and predicted QoI. The observed/predicted validationplot indicates that the coefficient of determination between the prediction andobservation is 0.98, and the corresponding mean squared error is 2.53 ˆ 10´6.These statistical metrics indicate that the GP model can estimate the unseengeometry from a noisy measurement of the QoI within a given accuracy.

0.13 0.14 0.15 0.16 0.17 0.18observed

0.13

0.14

0.15

0.16

0.17

0.18

obse

rved

/pre

dic

ted

observed

predicted

Figure 7: The observed/predicted validation plot showing the norm of the observed (test data)and the corresponding model predictions.

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Once the GP model is validated, it can be deployed as a low-cost surrogatefor the 3D finite element analysis code. The prediction of GP model takes onlya fraction of the time that is needed by the finite element code to estimate theQoI with a fair accuracy.

3.3. The Backward Problem

In the backward problem, we try to estimate the inner dimensions (li, ri) ofthe beam from noisy measurements of the QoI. To this end, we utilize the GPsurrogate model constructed in the previous subsection as a substitute for theforward model within the Bayesian framework.

We assume that a noisy measurement for the QoI is available as shown inFig. (8). The synthetic data is generated using inner dimension li “ 0.313 mand ri “ 0.055 m plus (σn “ 0.1 ˆ maxpuq) Gaussian noise to mimic a realexperiment setting.

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.03

−0.02

−0.01

0.00

dis

pla

cem

ent

[m]

Figure 8: Noisy measurement of the QoI

For the Bayesian calculation, we use non-informative prior for both the pa-rameters θ “ rli, ris and utilize an adaptive MCMC method (DRAM) [29, 30]to estimate the posterior density. In Fig. (9), we show the estimated posteriordensity of the parameters θ “ rli, ris . We also show the prior density and thetrue value of the parameters. Note that the true parameters where not partof either the training nor the testing datasets. This highlights the robustnessof the framework. The mean of the estimated values are li “ 0.310 ˘ 0.048 mand ri “ 0.054 ˘ 0.004 m (the confidence bounds are based on two standarddeviation).

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0.28 0.30 0.32 0.34 0.36length [m]

0

10

20

30

40d

ensi

ty[1

/m]

posterior

prior

true

(a) the inner length li

0.052 0.054 0.056 0.058hight [m]

0

100

200

300

400

den

sity

[1/m

]

posterior

prior

true

(b) the inner radius ri

Figure 9: The estimated posterior density function of the inner dimensions θ “ rli, ris. Thesold line is the posterior PDF, the dotted line is the prior PDF and the bullet dot representsthe true value li “ 0.313 m and ri “ 0.055 m.

Next, the uncertainty in the parameter estimation represented by the pos-terior density in Fig. (9) is propagated forward through the surrogate modelto estimate a confidence bound on the prediction of the QoI. In Fig. (10), weshow the model prediction and the 95% confidence interval as well as the truemeasured response. The L2 for the discrepancy between the mean model pre-diction and the measured data is 0.005 m. This conforms that the response dueto the estimated parameters uncertainty agrees reasonably well with the trueresponse.

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.03

−0.02

−0.01

0.00

dis

pla

cem

ent

[m]

Model

Data

CI

Figure 10: The prediction of the surrogate model and its confidence interval due to uncertaintypropagation of the variability in the estimated inner dimensions.

3.4. Localized Uncertainty Propagation

The QoI is confined within the core cylinder defined by inner dimensionsθ “ rli, ris. Once these dimensions are available, the effect of the randomvariability in the material properties of the inner subdomain can be estimatedusing PC expansion. Without loss of generality, here we assume that for theinner cylinder, the Young’s modulus and material density are random quantities,

16

while Poisson’s ratio is deterministic as

Epx, ξ1q “

#

E0p1` σEξ1q, for x P Ω2

E0, otherwise(30)

and

ρpx, ξ2q “

#

ρ0p1` σρ ξ2q, for x P Ω2

ρ0, otherwise(31)

where the artificial boundary for Ω2 are defined by the Maximum A Posteriori(MAP) estimation of the inner dimensions θ “ rli, ris, E0 “ 70 GPa, ρ0 “ 26.25kN/m3, σE “ 0.25 and σρ “ 0.15 and ξ1, ξ2 are standard normal randomvariables. Note that, not only the solution over Ω2 is stochastic, but also over allthe whole domain since the spatial finite element and stochastic basis functionsare continuous across the domains interfaces. We use second order PC expansionto propagate the localized uncertainty due to the random Young’s modulus andmaterial density as shown in Fig. (11). The uncertainty bounds follow thetrend of the response, with a higher value near the shock location. Althoughnot explored here, high spatio-temporal resolution solver can be directed towardthe region of interest, while a less resolution alternative can be assigned to theregions away from the QoI. As demonstrated in [7, 8, 9], PASTA-DDM-UQapproach leads to a customized solver for localized uncertainty propagationwith less computational cost.

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.04

−0.03

−0.02

−0.01

0.00

dis

pla

cem

ent

[m]

(a) displacement

0.000 0.002 0.004 0.006 0.008 0.010time [s]

−0.01

0.00

0.01

velo

city

[m/s

]

(b) velocity

Figure 11: The PC prediction of the displacement and velocity at the mid-span. The uncer-tainty bounds represent two standard deviation.

4. Conclusion

We present a data-based partitioning scheme for localized uncertainty quan-tification in elastodynamic system. The localized region of interest is identifiedusing Bayesian inference framework. Measurement of the system response atone location in conjunction with a physics-based computational model is used

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to infer the localized features of the region of interested. A data-based surrogatemodel for the physics-based simulator is constructed using Gaussian process re-gression in order to reduce the computational cost of the Bayesian inversion.Material uncertainty in the region of interest is propagated through the sys-tem using polynomial chaos. We exercise our framework on a three-dimensionalbeam with localized feature and subjected to an impact load. The presentedframework can facilitate quantifying the effect of the sub-component uncertaintyon the system-level. Proper assessment of uncertainty at various level can ac-celerate the adaptation process of a new component introduced to an existingsystem.

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