Document [original]

Citation: Drieschner, M.; Herrmann,

C.; Petryna, Y. The Data Assimilation

Approach in a Multilayered

Uncertainty Space. Modelling 2023,4,

529–547. https://doi.org/10.3390/

modelling4040030

Academic Editor: Günther Meschke

Received: 27 September 2023

Revised: 1 November 2023

Accepted: 2 November 2023

Published: 8 November 2023

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

Article

The Data Assimilation Approach in a Multilayered

Uncertainty Space

Martin Drieschner * , Clemens Herrmann and Yuri Petryna

Chair of Structural Mechanics, Technische Universität Berlin, Gustav-Meyer-Allee 25, 13355 Berlin, Germany;

[email protected] (C.H.); yuriy[email protected] (Y.P.)

*Correspondence: [email protected]

Abstract:

The simultaneous consideration of a numerical model and of different observations can

be achieved using data-assimilation methods. In this contribution, the ensemble Kalman filter

(EnKF) is applied to obtain the system-state development and also an estimation of unknown model

parameters. An extension of the Kalman filter used is presented for the case of uncertain model

parameters, which should not or cannot be estimated due to a lack of necessary measurements. It

is shown that incorrectly assumed probability density functions for present uncertainties adversely

affect the model parameter to be estimated. Therefore, the problem is embedded in a multilayered

uncertainty space consisting of the stochastic space, the interval space, and the fuzzy space. Then, we

propose classifying all present uncertainties into aleatory and epistemic ones. Aleatorically uncertain

parameters can be used directly within the EnKF without an increase in computational costs and

without the necessity of additional methods for the output evaluation. Epistemically uncertain

parameters cannot be integrated into the classical EnKF procedure, so a multilayered uncertainty

space is defined, leading to inevitable higher computational costs. Various possibilities for uncertainty

quantification based on probability and possibility theory are shown, and the influence on the results

is analyzed in an academic example. Here, uncertainties in the initial conditions are of less importance

compared to uncertainties in system parameters that continuously influence the system state and the

model parameter estimation. Finally, the proposed extension using a multilayered uncertainty space

is applied on a multi-degree-of-freedom (MDOF) laboratory structure: a beam made of stainless steel

with synthetic data or real measured data of vertical accelerations. Young’s modulus as a model

parameter can be estimated in a reasonable range, independently of the measurement data generation.

Keywords:

data assimilation; ensemble Kalman filter; aleatory and epistemic uncertainty; uncertainty

quantification; stochastic variables; interval variables; fuzzy variables

1. Introduction

Data assimilation is a widespread method that combines theory (usually in the form

of a numerical model evaluation) with observations, e.g., measurement data of some model

outputs, to optimally determine reasonable system states. The method is mainly known

for weather and climate forecasts [

], but it has also applied been in various fields like

robotics [2], economics [3], and many industrial branches [4] in recent decades.

In statistics, data assimilation can be considered as a Bayesian estimation problem

based on the Bayes’ theorem

P(H|D)=P(D|H)P(H)/P(D)

with hypothesis

and data

. Basically, it is assumed that the numerical model and the observations are subject to

uncertainty, so probability density functions are defined for the prior

P(H)

, the likelihood

function

P(D|H)

, the model evidence

P(D)

, and the posterior

P(H|D)

. If normal distribu-

tions, represented using their mean and covariance, are used for modeling the uncertainty,

Kalman filters can be applied. Kalman filters are used to continuously improve the re-

sults (system state) of time-discrete models involving observations (measurements). The

standard Kalman filter [

] is efficient for low-dimensional problems and can process only

Modelling 2023,4, 529–547. https://doi.org/10.3390/modelling4040030 https://www.mdpi.com/journal/modelling

Modelling 2023,4530

linear systems. For nonlinear problems including a large number of variables, the ensemble

Kalman filter (EnKF) [

], the ensemble square root filter (EnSRF) [

], the reduced rank

square root Kalman filter (RRSQRT) [

], or the unscented Kalman filter (UKF) [

–

]

are suitable choices. A comprehensive overview and a comparison of different Kalman

filters can be found in [15,16].

In the case of uncertainties, e.g., in system parameters, robust Kalman filters have

been developed in the past. The uncertainties can be quantified in different ways, e.g., with

interval [17,18] or stochastic variables [19].

In addition to Kalman filters using probability distributions, a fuzzy Kalman filter based

on possibility theory has been developed [

], improved [

] and applied on practical experi-

ments [

] in recent years. Also, in the case of fuzzy Kalman filters, the robustness regarding

additional uncertain system parameters have to be considered to obtain usable results.

In this contribution, diverse uncertainties in the initial conditions and/or system

parameters are integrated into the data assimilation approach by using a multilayered

uncertainty space. The numerical model is embedded in the uncertainty space in which

uncertainties based on probability and the possibility theory can be defined simultaneously.

Therefore, we propose classifying the present uncertainties into aleatory and epistemic

ones [

]. Different quantification approaches are then compared using an academic

example. Finally, the applicability of the general concept with a multilayered uncertainty

space is shown on a MDOF laboratory structure.

The remainder of this paper is organized as follows: In Section 2, a brief introduc-

tion to Kalman filters is given. Then, the ensemble Kalman filter (EnKF) as the basis of

this contribution is mathematically introduced, including the possibility of estimating

model parameters simultaneously. Next, the basic concepts for quantifying aleatory and

epistemic uncertainties are presented, leading to a multilayered and nested uncertainty

space. Furthermore, the integration of the ensemble Kalman filter into the multilayered

uncertainty space is formally developed. In Section 3, different possibilities for uncertainty

quantification are shown in an academic example. In addition, an engineering application

with synthetic or, respectively, with real measurement data is given to demonstrate the

practical applicability of the proposed extensions. Finally, the discussion of the results and

an outlook are given in Section 4.

2. Materials and Methods

After introducing the ensemble Kalman filter (EnKF) in Section 2.1, the basic con-

cepts for aleatorically and epistemically uncertain parameters are given in

Section 2.2

. In

Section 2.3

, the proposal to embed the EnKF in a multilayered uncertainty space for taking

both uncertainty types into account is formally presented.

2.1. The Ensemble Kalman Filter (EnKF)

In this contribution, the EnKF is used without loss of generality since other filters

can also be incorporated into the general scheme in Section 2.3. The ensemble Kalman

filter (EnKF) is a Monte Carlo implementation of the Bayesian estimation problem. It

is applicable to various nonlinear problems [

]. The combined state and parameter

estimation [

] or a multiplicative model parameter estimation [

] is also possible within

the EnKF. With spatial heterogeneity of parameters in the underlying model, the use of a

proper orthogonal-decomposition-based ensemble Kalman filter (POD-based EnKF) can

efficiently reduce the problem dimension [

]. Generally speaking, EnKFs represent the

distribution of the system state

by using an ensemble and replacing the covariance

matrix with the sample covariance computed from the ensemble. An ensemble consists of

qmembers and can be seen as a collection of system state vectors.

The bases are the (nonlinear) model equation and the measurement equation (lin-

earized here):

xk=f(xk−1,pk−1)+wk−1and

yk=d(xk)+vk−1=D·xk+vk−1

(1)

Modelling 2023,4531

with system state vector xk∈Rn,

system input vector pk∈Rl,

system noise vector wk∈Rn,

measurement vector yk∈Rm,

mapping matrix D∈Rm×nand

measurement noise vector vk∈Rm

(2)

at the

-th time step. The uncertainty is quantified using the uncorrelated vectors

and

with underlying normal distributions representing white noise. All means are set to

zero, and the scattering is quantified using a model noise matrix

and a measurement

noise matrix

, respectively. Both matrices provide the possibility to trust the numerical

model prediction or the observation more. This is taken into account within the correction

step by the Kalman gain Kk; see below.

2.1.1. Procedure

The procedure of the EnKF is depicted in Algorithm 1, consisting of

1. One-time initialization

2. Prediction or forecast −→ •f

3. Correction or analysis −→ •a

Algorithm 1: EnKF procedure [29].

Initialization

Create an ensemble with

members representing the probability density function

(PDF) of the initial state.

X0=xa1

0,xa2

0, . . . , xaq

0,X0∈Rn×q(3)

Forecast step

xfi

k=fxai

k−1,pk−1+wi

k−1,i=1 . . . q(4)

¯xf

k=1

∑

i=1

xfi

k(5)

Analysis step

yfi

k=D·xfi

k,i=1 . . . q(6)

¯yf

k=1

∑

i=1

yfi

k(7)

Pxy,k=1

q−1

∑

i=1xfi

k−¯xf

k·yfi

k−¯yf

kT(8)

Pyy,k=1

q−1

∑

i=1yfi

k−¯yf

k·yfi

k−¯yf

kT+Rk(9)

Kk=Pxy,k·Pyy,k−1(10)

xai

k=xfi

k+Kk·yk+vi

k−yfi

k,i=1 . . . q(11)

¯xa

k=1

∑

i=1

xai

k(12)

The matrices

Pxy,k

and

Pyy,k

in Equations

(8)

and

(9)

contain the error covariance matrix

of the state Pk∈Rn×n, well-known from the KF algorithms:

Pxy,k=Pk·DTand

Pyy,k=D·Pk·DT+Rk.(13)

Modelling 2023,4532

Note that the EnKF needs “perturbed observations”, which are achieved by

Equation (11) in order to not underestimate the error covariance matrix.

2.1.2. Estimation of Model Parameters

The EnKF can also be used to estimate model parameters simultaneously with the

prediction of the system state. For this purpose, the original state

must be extended by

the parameters

to be estimated. This results in an augmented state vector

z=(x,b)T

which can be used analogously to

in the above procedure. The development of the model

parameters

bfi

k=(1−β)bai

k−1+βbfi

k−1

can be defined with a scalar value

β∈[0, 1]

in order

to prevent the model from diverging [

]. Analogous to Equation

(4)

, a noise vector

wbk

is also added to the unknown model parameters for additional scatter. Note that only

the number of model parameters that can be estimated as independent measurements

are available.

2.2. Aleatory and Epistemic Uncertainties

Various additional uncertainties

can be present in the system, which have to be

considered in the numerical simulations. Material properties, geometrical dimensions, and

also boundary conditions can be uncertain, influencing the prediction of the system state

and the estimation of unknown model parameters

. Uncertain parameters can be

divided into two groups [

], aleatory uncertainties

calea

and epistemic uncertainties

cepis

with c=calea ∪cepis and calea ∩cepis =∅.

Aleatory uncertainties

calea

are caused by randomness and natural variability and

are therefore also referred to as irreducible uncertainties. Statistical information with a

sufficient amount of realizations is at our disposal to reliably model and quantify the

uncertainty via a random variable

. The underlying distribution can be expressed by the

probability density function (PDF)

fX(x,λ)

or the cumulative distribution function (CDF)

FX(x,λ)

, where

is the vector of shape parameters of the underlying PDF. The PDF and

CDF of Xare related via the integral

FX(x)=Zx

−∞fX(t)dt. (14)

In this study, normal distributions

Nµ,σ2

with

as the mean value and

σ2

as vari-

ance or lognormal distributions

LNµ,σ2

with

as the mean value of logarithmic

values and

σ2

as variance of logarithmic values are used for all random variables; see

Figure 1(top left/top right).

Epistemic uncertain parameters

cepis

are based on a limited number of data, subjec-

tivity, or expert knowledge, and no statistical information is given that could reduce the

uncertainty. These uncertainties can be modeled using non-stochastic variables [

], e.g., as

interval ¯

xor as fuzzy set ˜

x. An interval ¯

xis defined by a lower and upper bound

¯x=[a,b], (15)

whereas a fuzzy set ˜

xis expressed using its membership function

x={(x,µ˜

x(x)∈[0, 1])}. (16)

Note that an interval is a special case of a fuzzy set with a membership function value

of either

µ˜

1 (total membership) or

µ˜

0 (non-membership). In this study, interval

variables or fuzzy variables with a triangular membership function, called triangular fuzzy

numbers

TFNha

, with support

[a,c]

and core value

, are used for epistemic uncertain

parameters cepis; see Figure 1(bottom left/bottom right) [31].

Modelling 2023,4533

0 5 10 x1

PDF(x1)

510 x2

PDF(x2)

0 0.5 1

µ(¯

x3)

012 3

µ(˜

x4)

Figure 1.

Applied uncertainty models: (

top left

): normal distribution

x1∼ Nµ,σ2=N(5, 1)

(

top right

): lognormal distribution

x2∼ LNµ,σ2=LN(5, 1)

, (

bottom left

): interval

x3=[a,b]=[0, 1], (bottom right): triangular fuzzy number ˜

x4=TFNha,b,ci=TFNh0, 1, 3i.

Defining different uncertainty models simultaneously leads to a multilayered uncer-

tainty space; see Figure 2. The deterministic finite element model (D) or an appropriate

surrogate model (

) is embedded in the uncertainty space, which generally consists of

the stochastic space (S), the interval space (I), and the fuzzy space (F). Sometimes, some

individual uncertainty spaces are empty and do not have to be considered. The simulation

can then be simplified to the non-empty uncertainty spaces. The stochastic space is usually

embedded in the interval space, which is itself embedded in the fuzzy space, leading to

a nested analysis. The computational costs for the nested simulation can be calculated

ttot =ntot ·tD=∑nF

iF=1∑nI(iF)

iI=1∑nS(iI,iF)

iS=1tD(iF,iI,iS)

, with the number of samples in the

fuzzy space

, the number of samples in the interval space

on the

-th fuzzy point, the

number of samples in the stochastic space

on the

-th interval point, and the duration

of a single model evaluation. This approach of different uncertainty models leads to a

polymorphic uncertainty modeling [

–

], for which different solution methods for the

uncertainty propagation and evaluation methods for the assessment have to be applied.

The framework

PolyUQ

[

] has been developed in

MATLAB

for those issues. The output,

denominated as augmented state vector

zk(c)

in Section 2.1.2, is in general influenced by

the uncertain parameters c; see Section 2.3 in detail.

fuzzy space (F)

interval space (I)

stochastic space (S)

deterministic model D→˜

D

Figure 2. General scheme of a nested fuzzy-interval stochastic analysis.

2.3. The Ensemble Kalman Filter (EnKF) in a Multilayered Uncertainty Space

In this Section, the ensemble Kalman filter (EnKF) of Section 2.1 is integrated into

the multilayered uncertainty space presented in Section 2.2. Since normally distributed

Modelling 2023,4534

random variables are used for the noise in the Kalman filter in this contribution, the EnKF

is integrated directly into the stochastic space (S); see Figure 3.

fuzzy space (F)

interval space (I)

EnKF in stochastic space (S)

deterministic model D→˜

D

Figure 3. The EnKF embedded in the multilayered uncertainty space.

First, uncertain model parameters are distinguished in parameters

to be estimated

and parameters

not to be estimated within the EnKF. The parameters

are classified

as aleatorically (

calea

) or epistemically (

cepis

) uncertain, and different uncertainty models

can be applied. The output, namely the augmented state vector

zk(c)=(xk(c),bk(c))T

generally depends on all uncertain parameters c=calea ∪cepis.

Samples from

calea

are used directly in the stochastic space within the EnKF without

additional computational costs compared to the classical EnKF in Section 2.1. The output

remains a purely stochastic vector if only aleatory uncertain parameters

calea

are given. The

calculation of, e.g., the mean value or the standard deviation is then still possible without

further methods.

Otherwise, the present epistemic uncertain parameters

cepis

cannot be integrated

directly in the EnKF procedure. Additional methods are necessary, and additional com-

putational costs are inevitable to take them into account. The EnKF is now executed for

each sample from

cepis

, since the interval and/or the fuzzy space are/is non-empty. Note

that the used uncertainty models influence whether an uncertainty space is empty or not.

Consequently, the output

zkcepis

is defined in the multilayered uncertainty space. For

the output interpretation, the nested structure has to be considered. The stochastic space is

evaluated first, e.g., by determining the mean value on each sample of

cepis

. In the next

step, the non-empty interval space has to be evaluated, e.g., by determining the bounds of

the output variables on each fuzzy sample of

cepis

. Finally, the non-empty fuzzy space is

considered. The membership function and also a defuzzified value for each time step

can

be determined.

3. Results

The presented EnKF in the multilayered uncertainty space is applied to an academic

example in Section 3.1 and on a laboratory structure in Section 3.2.

3.1. Academic Example

Below, an academic example is used to demonstrate the influence of various un-

certainty models for aleatory and epistemic uncertainties on the augmented state vector

zk(c)

. The deterministic model (D) is an unloaded cantilever beam with longitudinal

time-dependent degrees of freedom for the displacement

u(t)

, for the velocity

u(t)

, and for

the acceleration

u(t)

at the end of the beam; see Figure 4. Young’s modulus

, the cross-

sectional area

, the density 2.454

, the damping ratio

, and the length

are (potentially

uncertain) system parameters. The mass

of the equivalent single-degree-of-freedom

system for the cantilever beam is then

0.4075

2.454

ρAL =ρAL

. Note that the initial

displacement u0=u(0)and the initial velocity ˙

u0=˙

u(0)can be uncertain as well.

Modelling 2023,4535

E,A, 2.454ρ,ξu(t),˙

u(t),¨

u(t)

Figure 4. Cantilever beam as a deterministic model (D).

The underlying ordinary differential equation for the displacement

u(t)

is given as

m¨

u(t)+c˙

u(t)+ku(t)=

0 with mass

m=ρAL

, stiffness

k=EA/L

, damping

c=2ξ√km

and initial conditions

and

. A time-discrete solution with time steps

k={0, 1, . . . , kmax}

and time increment

∆t

can be achieved by using the Newmark method [

]. Alterna-

tively, Equation

(1)

can be transformed into the following discrete formulation (for the

unloaded case):

Rn3xk=Adis ·xk−1+wk−1and

Rm3yk=uk+vk−1

(17)

with

2, including displacement and velocity, and

1 for the displacement measure-

ment (excluding velocity measurements for simplicity). The system transition matrix

Adis

can be derived using the state space representation:

Adis =eA∆twith

A= 0 1

−E

ρL2−2ξqE

ρL2!.(18)

Note that the numerical prediction is independent of the cross-sectional area

, since the

stiffness kand the mass mare both linearly dependent on A.

The quasi-continuous system state vector

x(t) = (u(t),˙

u(t))T

can finally be composed

using the discrete solutions

xk=(uk,˙

uk)T

on all time steps

k={

0, 1,

. . .

kmax}

. Addition-

ally, unknown model parameters

can be estimated in time within the EnKF by using the

augmented state vector

z=(x,b)T

. Here, Young’s modulus

has to be estimated. The

initial Young’s modulus E0is set to be a normally distributed random variable

E0hMN/m2i∼ NµE0,σ2

E0=N0.5, (0.10 ·0.5)2(19)

with initial mean value

µE0

and initial variance

σ2

. The following normal distributions

have been defined for the noise terms wk,vkand wE,k:

wk=wuk

w˙

uk∼

N0, 0.01 ·µuk2

N0, 0.01 ·µ˙

uk2

,

vk∼ N0, 0.0052and

wEk∼ N0, 0.001 ·µEk2.

(20)

As the basis for all calculations, the values

Ereal =1 MN/m2,

Areal =0.01 m2,

ρreal =1000 kg/m3,

ξreal =10 % ,

Lreal =1 m ,

u0,real =0.1 m and

u0,real =0.1 m/s

(21)

Modelling 2023,4536

are taken as real, and the initial state is quantified by

u0∼ Nµu0,σ2

u0=Nu0,real,(0.10 ·u0,real)2and

u0∼ Nµ˙

u0,σ2

u0=N˙

u0,real,(0.10 ·˙

u0,real)2(22)

in the following unless otherwise specified.

The presented academic example is applied to different scenarios. The ensemble size

is set to

100 at all time steps

k={

0, 1,

. . .

, 1000

}

. In Section 3.1.1, different uncer-

tainty models for the uncertain initial conditions

and

are defined, and the influence

on the system state prediction as well as on the model parameter estimation is investi-

gated. Aleatory and epistemic uncertainties in different model parameters are analyzed

Section 3.1.2.

The monotonic behavior of the augmented state vector

z=(x,b)T

can be

presumed regarding Equation

(18)

. This fact leads to an exact evaluation of the interval

and the fuzzy space by using the vertex method [

] and the reduced transformation

method [

], respectively. In the fuzzy space, eleven equidistantly distributed

-levels from

0.0 to 1.0 are used. The framework

PolyUQ

[

] is used for the numerical model evaluation,

with different uncertainty models simultaneously present.

3.1.1. Uncertain Initial Conditions (u0and ˙

u0)

First, the initial conditions are defined as normally distributed random variables

u0[m]∼ N0.12, 0.0122and

u0[m/s]∼ N0.08, 0.0082(23)

with mean values

µu0=0.12 m 6=0.1 m =u0,real

and

µ˙

u0=0.08 m/s 6=0.1 m/s =˙

u0,real

Within the classical EnKF procedure, a sample of the initial state is used, as shown in

Equation

(3)

, leading to the system state prediction and the model parameter estimation

depicted in Figure 5.

Figure 5.

Results with stochastic initial conditions: (

top left

): displacement

u(t)

; (

top right

): ve-

locity

u(t)

; (

bottom left

): Young’s modulus

E(t)

; (

bottom right

): standard deviation of Young’s

modulus σE(t).

Modelling 2023,4537

The prediction of the system state vector

x(t)=(u(t),˙

u(t))T

is practically exact, except

for deviations at the beginning resulting from incorrect and scattering initial conditions.

Despite the initial Young’s modulus with a mean value of

µE0=0.5 MN/m2

, which is half

the real value

Ereal =1 MN/m2

, Young’s modulus

E(t)

converges quickly to the real value

with a coefficient of variation of around

2 %

. The scattering is small, resulting from the

random initial conditions.

It is also possible to define the initial conditions as epistemic uncertainties leading to

non-stochastic uncertainty models, e.g., the mean values as the intervals

µu0[m]=[0.0829, 0.1571]and

µ˙

u0[m/s]=[0.0553, 0.1047](24)

with a coefficient of variation of

10 %

. The interval-stochastic results are displayed in

Figure 6. Additionally, the real values, the measurements (if present), and the results from

the classical EnKF procedure for Young’s modulus are shown.

Figure 6.

Results with interval-stochastic initial conditions: (

top left

): mean of displacement

µu(t)

(

top right

): mean of velocity

µ˙

u(t)

, (

bottom left

): mean of Young’s modulus

µE(t)

compared to

µE

Figure 5(bottom left), (

bottom right

): standard deviation of Young’s modulus

σE(t)

compared to

σE

of Figure 5(bottom right).

It can be concluded that the extension using other uncertainty models for the initial

conditions does not have a concrete benefit in this example. The computational costs have

been increased without a concrete change in the results. The system state vector is predicted

again practically exactly, except for the values at the beginning. It is worth mentioning

that models exist that are more sensitive to the initial conditions. In those cases, different

uncertainty models would affect the system state prediction and the model parameter

estimation much more.

Modelling 2023,4538

3.1.2. Uncertain Model Parameters (Land ρ)

In this section, various model parameters are defined as uncertain using the differ-

ent uncertainty models from Section 2.2. The effect on the augmented state vector

investigated each time.

Stochastic problem: uncertain beam length L

In a purely stochastic problem, the beam length is defined as a lognormally distributed

random variable

L[m]∼ LN1, 0.12. (25)

The mean value of logarithmic values

µL

is set to the real value

Lreal

, and the standard

deviation of logarithmic values

σL

is set to

0.1 m

. For each of the

members within the

ensemble, a sample of Lis used, leading to the results shown in Figure 7.

Figure 7.

Results with stochastic beam length

: (

top left

): displacement

u(t)

; (

top right

): velocity

u(t)

; (

bottom left

): Young’s modulus

E(t)

; (

bottom right

): standard deviation of Young’s modu-

lus σE(t).

It is obvious that the displacement and the velocity are predicted again quite perfectly,

but the deviation of the mean value

µE

from the real value

Ereal

is around

5 %

. This results

from the right-skewed distribution of

and the relation to

; see Equation

(18)

. The

model-parameter estimation depends (more or less) on the assumptions of other model

parameters. Since the real values are in general unknown, a parameter study has been

conducted with varying σLor µL; see Figure 8.

The best estimation of Young’s modulus can be achieved naturally by having no

scattering and the correct mean value

µL=µreal

; see the blue curve in Figure 8(left).

A variation in one of both values leads to bad or useless estimations. If exact values

are unknown, it should be avoided to assume a pdf, but it can be useful to apply other

non-stochastic uncertainty models.

Modelling 2023,4539

Interval-stochastic problem: uncertain density ¯

In this section, the density ρis assumed to be uncertain using the interval

ρhkg/m3i=[900, 1200](26)

around the real value

ρreal =1000 kg/m3

. As in the previous examples, the system state

vector

is predicted practically exactly. A value

is determined to an “associated” value

in each time step

. Since the density is now defined as an interval variable, the mean and

the standard deviation of Young’s modulus are now also intervals; see Figure 9.

Figure 8.

Parameter study within the stochastic problem: (

left

): varying

σL[m]

values for fixed

µL=1 m, (right): varying µL[m]values for fixed σL=0.1 m.

Figure 9.

Results with interval-valued density

: (

left

): mean of Young’s modulus

µE(t)

(right): standard deviation of Young’s modulus σE(t).

The real value of Young’s modulus

Ereal

is inside the bounds of the predicted interval-

valued mean

µE

, except for the first time steps. The interval-valued standard deviation

σE

also remains small with a maximum of around

0.026 MN/m2

at the final time step. If

more data about the uncertain density

were available, the epistemic uncertainty could be

reduced, leading to a smaller range of possible values for Young’s modulus

. In the next

Section, the interval-based idea is extended by defining a fuzzy-valued density ˜

ρ.

Fuzzy-stochastic problem: uncertain density ˜

The extension from intervals to fuzzy values is shown by defining the density

as a

triangular fuzzy number:

ρhkg/m3i=TFNh900, 1000, 1200i. (27)

The real value

ρreal

is assumed on the core of the fuzzy variable. If an interval is preferred

on the core, a trapezoidal fuzzy interval [

] can be used instead. The extension to

Figure 9is given in Figure 10.

Modelling 2023,4540

Figure 10.

Results with fuzzy-valued density

: (

left

): mean of Young’s modulus

µE(t)

(right): fuzzy-valued density ˜

ρ.

The envelope in Figure 10 (left) is equivalent to the solution in Figure 9(left) since the

support is defined with the same bounds as the interval in Equation

(26)

. Decreasing the

uncertainty leads to a higher membership function value in Figure 10 (right) with the exact

value for

µ=

1.0. As a consequence, the blue shading in Figure 10 (left) is darker for higher

membership-function values around the real value

Ereal

. Finally, the membership function

has been defuzzified [

] at each time step

, leading to a scalar value, here namely the

center of gravity (COG). The COG curve is also in good agreement with the real value.

Finally, it should be mentioned that it is possible to combine different uncertainty

models for the same parameter, obtaining, for example, a random variable with interval-

valued parameters

. Furthermore, interval and fuzzy variables can be simultaneously

present in the same problem. These aspects can be found, for example, in [

] but are

beyond the scope of the present paper.

3.2. Laboratory Structure

In this section, a beam made of stainless steel with synthetic or real measurement

data of vertical accelerations is investigated. The beam has a length of

L=4.0 m

between

two supports with

0.20 m

lateral overhangs on each side. The cross section is a box girder

with an outer width of

w=0.05 m

, an outer height of

h=0.03 m

, and a wall thickness of

t=0.0026 m

, leading to a cross-sectional area

3.89

−4m2

and moments of inertia

Iy=

5.56

−8m4

and

Iz=

1.27

−7m4

. The material is assumed to be homogeneous

with a density of

ρ=8000 kg/m3

. Isotropic behavior is defined via a Young’s modulus

value that is to be estimated with initial mean value

µE0

= 100,000 MN/m

and initial

standard deviation

σE0

= 10,000 MN/m

and a Poisson’s ratio of

ν=

0.3. For the damping

ratio, a triangular fuzzy number

ξ[%]=TFNh

0.00, 0.50, 1.00

is defined. Additional

overhangs with a length of

ladd =0.125 m

have been welded in the beam center on both

sides for various measurement procedures. The additional mass of

madd =0.907 kg

taken into account in the numerical model.

The numerical finite element model is defined in the

-plane and consists of

21 nodes

each one with three degrees of freedom (

φz

), leading to 63 system degrees of

freedom in total. The origin of the

coordinate system is located in the left support; see

Figure 11 (top). Consequently, the translational degrees of freedom

and

are fixed at

x=0 m

and

x=4 m

. A free vibration test (

p=pown

∀t

) is defined with the following

initial conditions:

u0=K−1·(pown +padd),

˙u0=0and

¨u0=M−1·(p0−C·˙u0−K·u0))=−M−1·padd

(28)

with M,C, and Kindicating the mass, damping, and stiffness matrices, respectively.

Modelling 2023,4541

Figure 11.

Beam made of stainless steel with initial deflection

u0,y[m]

due to dead load and additional

mass of

2 kg

in the beam center: (

top

): real structure with five accelerometers, (

bottom

): calculated

deflection of the beam with E= 100,000 MN/m2.

The initial displacement

results from the dead load of the beam (

pown

) and an

additional mass of

2 kg

in the beam center (

padd

); see Figure 11 (top). Note that the

additional mass is only present at the beginning and not during the free vibration of the

beam. Additionally,

mmeas =0.1 kg

for each of the five measurement points is considered

in the load vector

pown

and in the mass matrix

. As usual, the Rayleigh damping matrix

C=α(ξ)M+β(ξ)K

is used to consider damping effects. The Newmark method [

] is

applied for solving the differential equation system using the time steps

k={

0, 1,

. . .

kmax}

and the time increment

∆t=

/fs

(resulting from the sampling frequency of

fs=204.8 Hz

in the conducted measurements).

For the EnKF, the initial state is quantified using normal distributions with a coefficient

of variation of

10 %

, which is equal to that in Section 3.1. The ensemble size is set to

100.

The model noise terms are defined equally to Section 3.1 (

for each degree of freedom) as

wk=wuk

w˙

uk∼

N0, 0.01 ·µuk2

N0, 0.01 ·µ˙

uk2

and

wEk∼ N0, 0.001 ·µEk2.

(29)

Observations of vertical accelerations

are synthetically produced (Section 3.2.1) or mea-

sured using the MEDA measurement system on the structure (Section 3.2.2) at five locations;

see Figure 11 (top).

3.2.1. Synthetic Measurement Data

The measurement noise is presumed to be small. To show the influence of the mea-

surement noise on the system state and the parameter estimation, three different values are

chosen here:

vk∼ N0, σ2

vkwith

σvk={0.001, 0.01, 0.1}.(30)

The parameters

Ereal

= 200,000 MN/m

and

ξreal =0.17 %

are chosen as the reference in

each case. The most important results with synthetic measurement data are displayed in

Figures 12 and 13.

Modelling 2023,4542

Figure 12.

Results with synthetic measurement data with

σvk=

0.01: (

top left

): mean of vertical

acceleration in the beam center

µ¨

uy(

, (

top right

): mean of final vertical acceleration

µ¨

uy(x

, 100

)

(

bottom left

): mean of final vertical velocity

µ˙

uy(x

, 100

)

, (

bottom right

): mean of Young’s modu-

lus µE(t).

The measurement data are almost equal to the real data in Figure 12 (top left), since the

measurement noise is small. The fuzzy-valued mean of the vertical acceleration in the beam

center is estimated with only small deviations. The real vertical acceleration is enveloped

by the numerical results, and the COG curve is close to the real data; see Figure 12 (top

right) for the final system state. The vertical velocity is also estimated very well due to

the fact that

˙u0=0

(see Equation

(28)

) over the beam length, an example of which is

shown in Figure 12 (bottom left) for the final system state. The deviation between the real

vertical velocity and the COG curve of the numerics results from the symmetric triangular

fuzzy number for

with the core value of

0.5 %

instead of the real value

ξreal =0.17 %

Nevertheless, the real vertical velocity is also enveloped by the numerical values. The

mean value of Young’s modulus is depicted in Figure 12 (bottom right). The real value

Ereal

= 200,000 MN/m

is found very quickly with slight variations above and below it.

The variations in the model parameter estimation due to the model and the measurement

noise are increased due to the present uncertainty in the damping ratio

. However, the

estimation is not adversely influenced by the consideration of the fuzzy-valued parameter.

The results are naturally influenced by all selected numerical parameters. In the

following, the variation in the measurement noise is investigated, leading to the results in

Figure 13. The prediction of the acceleration in the middle of the beam

µ¨

uy(

is more

noisy for higher measurement noise

σvk

and vice versa (Figure 13 (top left/top right)).

Furthermore, it can be seen that the real value of Young’s modulus

Ereal

is reached much

more slowly the higher the noise is (Figure 13 (bottom left/bottom right)). Note that

in the case of the smallest measurement noise

σvk=

0.01, the scatter in

µE(t)

is almost

the same as in Figure 12 (bottom right) due to the unchanged parameter noise

wEk

. The

highest measurement noise

σvk=

0.1 leads to the weakest update for the Kalman gain,

since the measurement data are less trustworthy. Thus, the discrepancy between the model

Modelling 2023,4543

results and the measurement data is weighted less. In general, appropriate values for the

measurement and the model noise have to be chosen by the user.

Figure 13.

Selected results obtained with higher and smaller measurement noise: (

top left

): mean

of vertical acceleration in the beam center

µ¨

uy(

with

σvk=

0.001, (

top right

): mean of vertical

acceleration in the beam center

µ¨

uy(

with

σvk=

0.1, (

bottom left

): mean of Young’s modulus

µE(t)with σvk=0.001, (bottom right): mean of Young’s modulus µE(t)with σvk=0.1.

3.2.2. Real Measurement Data

The measurement noise was determined as the difference between the recorded

and the filtered measurement data in this section. A lowpass filter was applied with

a passband frequency of

50 Hz

. The covariance matrix

of the measurement noise was

then determined as

R=10−4·





0.6990 0 0 0 0

0.3753 0 0 0

0.9798 0 0

0.5983 0

sym. 0.8534







(31)

with a maximum standard deviation value

max(σv)=√10−4·0.9798 ≈

0.01, equivalent

to the measurement noise value in Section 3.2.1.

The results of the selected system state variables and Young’s modulus as the model

parameter to be estimated are shown in Figure 14. A total of 1000 time steps were used,

leading to a total time of ttot ≈4.88 s.

In the case of real measurement data, the vertical acceleration in the beam center is also

predicted well despite the presence of an uncertain, fuzzy-valued damping ratio. The real

curve, the measurement curve, and the numerical prediction curve of the vertical accelera-

tion in the beam center are qualitatively and quantitatively similar; see Figure 14 (top left).

For the final state, the vertical acceleration is predicted very well (Figure 14 (top right)),

whereas quantitative deviations are present for the vertical velocity (Figure 14 (bottom

left)) with a maximum of

15 %

between the real curve and the COG curve. The mean of

Modelling 2023,4544

Young’s modulus is depicted in Figure 14 (bottom right). Transient effects are present in the

conducted measurement and visible in the recorded acceleration data but are decaying after

approximately

2 s

(400 timesteps). The estimation of the real value

Ereal

= 200,000 MN/m

has been performed successfully.

Figure 14.

Results with real measurement data: (

top left

): mean of vertical acceleration in the beam

center

µ¨

uy(

; (

top right

): mean of final vertical acceleration

µ¨

uy(x

, 1000

)

, (

bottom left

): mean of

final vertical velocity µ˙

uy(x, 1000); (bottom right): mean of Young’s modulus µE(t).

4. Discussion and Conclusions

The improvement in the prediction quality through the use of numerical simulation

and measurement can be achieved through the data assimilation of the both. In the

case of uncertainties influencing the numerical prediction, the classification of aleatory

and epistemic uncertainties and the definition of a multilayered uncertainty space are

proposed in this contribution. The obtained numerical results have been compared with

the observations of system state variables, using either synthetic or real measurement data.

Furthermore, the ensemble Kalman filter (EnKF) used has been extended to simultaneously

estimate an unknown model parameter. It has been shown using an academic example

that the application of non-stochastic uncertainty models is useful for present uncertain

variables if no exact probability density functions are known. Thus, a reasonable numerical

prediction can be achieved. The overall testing of the proposed integration of the data-

assimilation approach in a multilayered uncertainty space can be considered as successful,

also for an MDOF laboratory structure, even if some quantitative deviations in results are

still present.

Besides this general conclusion, some specific conclusions can also be made as follows.

Uncertainties can be present either in initial conditions or in system parameters. In the

academic example, the influence of uncertain initial conditions on the system state estima-

tion was small and disappeared after a limited number of time steps. It is important to

mention that it is necessary for a successful estimation using the EnKF that the uncertain

initial conditions are close to the real values and that the model noise is large enough to

enable assimilation in consideration of the observations. Furthermore, the extension to an

interval-stochastic mean value of the initial conditions has been investigated. The system

Modelling 2023,4545

state and model parameter estimation is successful, but higher computational costs are

unavoidable. In the presented example, the definition of non-stochastic initial conditions is

not beneficial.

It has been demonstrated that uncertain model parameters lead to a more continuous

influence on the estimations since they are present at each time step in contrast to the

initial conditions. Bad or useless estimations appear if wrong probability density functions

are assumed. Even though non-stochastic uncertainty models lead to non-stochastic out-

comes during data assimilation, the real values can be captured in the case of properly

defined input variables. Fuzzy-valued input parameters extend the concept of intervals

through a membership function with a higher informative value at the expense of higher

computational costs.

The applicability of a multilayered uncertainty space is finally shown on a MDOF

example, independently of synthetic or real measurement data. The system state and the

model parameter estimation were successfully conducted for both cases.

For future studies, the following aspects can be considered to improve or to extend the

presented approach. Localization effects [

] would improve the numerical prediction

for a MDOF structure by separating the effects of different observations. Although the EnKF

has successfully been applied in this contribution, other Kalman filters can alternatively be

used as well, since the proposal of a multilayered uncertainty space is independent of the

used filter. In the framework of structural health monitoring, the presented approach can

also be used for the detection of system changes such as stiffness reduction due to damage.

Author Contributions:

Conceptualization, M.D.; methodology, M.D.; software, M.D. and C.H.;

validation, M.D.; formal analysis, M.D. and C.H.; investigation, M.D. and C.H.; data curation,

C.H.; writing—original draft preparation, M.D. and C.H.; writing—review and editing, M.D. and

Y.P.; project administration, Y.P.; funding acquisition, Y.P. All authors have read and agreed to the

published version of the manuscript.

Funding:

The authors acknowledge support by the German Research Foundation and the Open

Access Publication Fund of TU Berlin. Furthermore, the authors gratefully acknowledge the financial

support of the German Research Foundation within the Subproject 4 (312928137) of the Priority

Programme “Polymorphic uncertainty modelling for the numerical design of structures–SPP 1886”.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement:

The data presented in this study are available on request from the

corresponding author. The data are not publicly available.

Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations

The following abbreviations are used in this manuscript:

CDF cumulative distribution function

COG center of gravity

EnKF ensemble Kalman filter

EnSRF ensemble square root filter

MDOF multi-degree-of-freedom

PDF probability density function

POD proper orthogonal decomposition

RRSQRT reduced rank square root Kalman filter

UKF unscented Kalman filter

Modelling 2023,4546

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