Available online at www.eccomasproceedia.org
Eccomas Proceedia UNCECOMP (2019) 543-554
UNCECOMP 2019
3rd ECCOMAS Thematic Conference on
Uncertainty Quantification in Computational Sciences and Engineering
M. Papadrakakis, V. Papadopoulos, G. Stefanou (eds.)
Crete, Greece, 24-26 June 2019
STOCHASTIC NON-PARAMETRIC IDENTIFICATION IN
COMPOSITE STRUCTURES USING EXPERIMENTAL MODAL DATA
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
Chair of Vibroacoustics of Vehicles and Machines,
Department of Mechanical Engineering, Technical University of Munich,
85748 Garching b. Munich, Germany.
e-mail: sourav.chandra@tum.de.
Abstract. In the stochastic structural analysis of composite structure, the probabilistic knowledge of the uncertain parameters are essential. Variability of the manufacturing process of the
composite structure introduces the uncertainty to the elastic parameters. It is easier to identify the uncertainty of the material parameters using stochastic inverse process. An efficient
stochastic inverse identification of the elastic parameters of laminated composite plate using
generalized Polynomial Chaos (gPC) theory is presented in this paper. A data set of measured
eigen frequencies and mass density are used for stochastic inversion processes. Stochastic identification of the elastic parameters of composite plate transforms into estimation of deterministic coefficients of gPC expansion for the elastic parameters. A robust optimization technique
by minimization of the quadratic difference between statistical moments is used to estimate the
deterministic coefficients of the gPC expansion. These coefficients can effectively construct the
distributions of the uncertain elastic parameters. Evaluation of the deterministic coefficients by
higher order statistical moments minimization, can efficiently simulate the randomness of the
experimental eigen frequencies.
Keywords: Stochastic inverse identification, Elastic parameter, Generalized Polynomial Chaos,
statistical moment, Composite Structure.
ISSN:2623-3339 © 2019 The Authors. Published by Eccomas Proceedia.
Peer-review under responsibility of the organizing committee of UNCECOMP 2019.
doi: 10.7712/120219.6358.18387
543
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
1 INTRODUCTION
Knowledge of material parameters of a structural system are essential before inserting into
the forward model to asses the structural responses of the dynamic system. The values of the
material parameters of composite structure in terms of elastic moduli, shear moduli, Poisson’s
ratio and mass density are often described by the manufacturer. Prior to inserting these material
parameters into realistic forward simulation, effect of uncertainty should be accounted. A real
dynamic system consists of two fundamental uncertainties such as parameter uncertainty and
model uncertainty. Parameter uncertainty for the composite structure propagates to the system
due to inherent randomness of the elastic moduli, Poisson’s ratio and mass density. The random
fiber orientations, variation of the thickness and variation of the fabrication procedure introduce
the parameter uncertainty to the dynamic system of the composite structure. Whereas, boundary conditions in the mathematical model, using homogenized theory, to evaluate the effective
elastic moduli of the composite material are responsible for modeling uncertainty of the system.
Direct measurement of the uncertainty of the elastic parameters are not able to represents these
variabilities efficiently. The elastic parameters can be evaluated by inverse identification based
on the concept of error minimization of the experimental responses and simulated responses.
The deterministic inverse problems [1, 2, 3] are involved in identifying the elastic parameters
of the composite plate from a single experimental modal data, based on various optimization
algorithms. Single measurement is not sufficient to capture the uncertainty and variability associated with the parameters and the model. The reliable prediction of overall dynamic behavior
of the composite structure is possible by incorporating the uncertainty of the material parameters within the finite element (FE) framework.
Uncertainty of elastic parameters can be evaluated by establishing stochastic relation between uncertain elastic parameters and set of measured responses. A well suited FE based
stochastic inverse method is employed to identify the variability of the elastic parameters of
the composite structure. Rikards et al. [4] presented various methods to identify the elastic
properties of laminated material using experimental data. Lauwagie [5] discussed various optimization techniques adopted for material properties identification of laminated composite materials in inverse process. Stochastic inverse technique involved to identify the probabilistic
parameters such as mean and variance, of the elastic constants of laminated composite structure
based on probabilistic representation of modal responses. Bayesian inverse updating method
offers a wide range of flexibility to multi-parameter model identification from a sufficient number of experimental data sets. Basically, posterior distribution of the material parameters are
inferred from assumed prior by evaluating the likelihood of the elastic parameters. In recent
years, several applications [6, 7, 8] of Bayesian inference technique in inverse problem have appeared. The evaluation of the integral is the most challenging part in multi-parameter Bayesian
inverse inference. However, Markove Chain Monte Carlo (MCMC) became an efficient alternative to determine the posterior density without evaluating the integral. Various sampling
based approaches such as, Metropolis-Hasting (M-H) algorithm and Gibbs sampler [9, 10] have
developed for the improvement of MCMC algorithm. However, sampling based inverse identification often suffer due to computational efficiency. Nagel [11] discussed Bayesian inverse
problems with a direction to overcome the limitations of sampling based technique for determining the posterior probability density functions of the system parameters. In recent years, spectral stochastic formulation has been proposed in combination with the Bayesian inference [12].
Rosić et al. [13] proposed a linear Bayesian estimation of the unknown parameters in combination with the Karhunen-Loève and Polynomial Chaos expansions without using any sampling
544
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
technique such as, MCMC. This method can effectively update non-Gaussian uncertainties.
The introduction of Galerkin projection technique using generalized Polynomial Chaos (gPC)
theory [14, 15] transfer the inverse problem as a deterministic one which involves to identify
the unknown gPC coefficients instead of probabilistic parameter of the quantity. Sepahvand and
Marburg [16, 17] have efficiently estimated the elastic parameters of the orthotropic material via
stochastic inverse method using non-sampling based gPC expansion technique. Non-Gaussian
experimental modal data are used to identify elastic parameters. Literature review described the
application of the stochastic non-parametric identification for the laminated composite structure
and efficiency of the inverse algorithm to predict the wide range of uncertainty in the case of
laminated composite structure.
This paper aims to identify the uncertainty of the elastic moduli and shear modulus for laminated composite plate using experimental modal frequencies. The method involves to evaluate the deterministic coefficients of the uncertain elastic parameters through minimization of
statistical moments, calculated from measured eigen frequencies and corresponding statistical moments derived from simulated eigen frequencies, using gPC expansion method. Moreover, present paper also reported the efficiency of parameter identification by implementation of
higher order statistical moments minimization technique. The in-situ randomness of the mass
density of the composite material is determined and is considered as an input to the stochastic
inverse model.
2 FE MODEL OF LAMINATED COMPOSITE PLATE
In the present formulation of the forward model of laminated composite plate, the classical
thin plate theory [19] is assumed . The assumption neglects the effect of transverse shear deformation. The relation between stress σ ′ and strain ε′ for orthotropic layer with reference to the
principal material axes (1, 2, 3) are presented by the generalized Hooke’s law as
σ ′ = Cε′ .
(1)
Here, C is the stress-strain relationship matrix along the principal material axes of the k th
lamina. The elements of the C matrix for the k th layer is expressed as
σ1
C11 C12 0
0
0
ε1
σ2
C12 C22 0
0
0
ε2
σ12 = 0
0 C33 0
0
(2)
ε12 ,
σ23
0
0
0 C44 0 ε23
σ13 k
0
0
0
0 C55
ε13 k
where C11 = E11 /(1−ν12 ν21 ), C12 = ν21 E11 /(1−ν12 ν21 ), C22 = E22 /(1−ν12 ν21 ), C33 = G12 ,
C44 = G23 and C55 = G13 . Here, Eii , Gij and νij are the set of elastic constants such as Young’s
moduli, shear moduli and Poisson’s ratio of the laminated composite plate, respectively. The
stress σ and strain ε relationship is redefined with reference to the laminate axes (x, y, z) of the
composite plate as
σ = Qε,
(3)
Q = T −1 CT ,
(4)
in which
545
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
and T is transformation matrix [20] to relate the principal material axes and laminate axes. The
element stiffness matrix of the laminated composite plate takes the form
Z
B T DBdAe ,
(5)
Ke =
Ae
in which B is the strain-displacement matrix written as
ε̄ = Bδ.
(6)
Here, ε̄ is the strain and curvature vector and δ is the nodal displacement vector of the composite
plate. The mid-plane stress resultant σ̄ and strain ε̄ of the laminate are related by stiffness
matrix D [19] as
σ̄ = Dε̄,
(7)
Am A c 0
D = Ac Ab 0 .
0
0 As
(8)
where
In the above matrix, sub-components Am , Ac , Ab and As represent membrane stiffness, membranebending coupling stiffness, bending stiffness and shear stiffness, respectively. Here,
l Z zk
X
Ai =
(Qi )k (1, z, z 2 )dz, i = m, c, b
(9)
j=1
Ai =
zk−1
l Z
X
j=1
zk
κ(Qi )k dz,
i = s, κ = 5/6
(10)
zk−1
where κ is shear correction factor [21] and l is numbers of orthotropic layers in composite plate.
The elemental mass matrix is written as
Z
N T ρN dAe ,
(11)
Me =
Ae
where N is interpolation matrix and ρ is the inertia matrix. The global stiffness matrix K
and the global mass matrix M are developed after assembling the elemental stiffness and mass
matrices, Ke and Me , respectively. Therefore, undamped modal analysis involves the solution
of
[λ2i M + K]φi = 0,
i = 1, 2, ..., n
(12)
to extracts the modal frequency λi and mode shape φi of the laminated composite plate with n
numbers of degrees of freedom (DOF) in FE model. Generalized forward model of the composite plate can be defined as
d = G(m).
(13)
Herein, m denotes vector of elastic parameters of the model and d is set of simulated data for
ideal case. The forward model operator G predicts the model output data set d in terms of eigen
frequencies as a function of model parameters m. In the present paper, model parameters are
Eii and Gij and the forward model yield the data output in the form of modal frequency λi .
546
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
3 POLYNOMIAL CHAOS EXPANSION
Assume a probability space (Ω, U , P) in which Ω is the sample space, U is the σ-algebra
over Ω, and P is the probability measure on the sample space U . Consider a random parameter
X (ω) with random outcome ω ∈ Ω. Such random parameter can effectively be presented by
gPC expansion method by projecting it onto a stochastic space spanned by a random orthogonal
polynomials. The random parameter X :Ω → R with finite variance possess the following
representation in compact form [15]
X (ω) =
P
X
ai Ψi (ξ),
(14)
i=0
where ai are unknown deterministic coefficients and Ψi (ξ) are the multivariate orthogonal basis
functions given by the product of the corresponding univariate polynomial form. Total number
of terms in the n dimensional pth order truncated gPC expansion is (P + 1), where
P +1=
(n + p)!
.
n!p!
(15)
One-dimensional orthogonal polynomial can be represented by standard normal random vector
ξ = {ξi }, i = 1, 2, ....N in a particular sample space such, that ξi ∈ Ωi . The orthogonal
relationship of the multidimensional polynomial function Ψ = {Ψi (ξ)} is written as
E[Ψi , Ψj ] = E[Ψi2 ]δij = p2i δij ,
i, j = 0, 1, 2, ....N ,
(16)
in which δij represents Kronecker delta and p2i is the norm of the polynomial. Due to the
orthogonal properties of the gPC expansion, the unknown coefficients ai can be calculated by
projecting onto the orthogonal set of polynomial chaos, such that
ai =
hX (ω)Ψi i
,
hΨi2 i
(17)
where hΨi2 i denote the inner products in the Hilbert space in the L2 norm. This orthogonal
projection minimizes the error on the space spanned by {Ψ}Pk=0 and evaluate the deterministic coefficient ai . Once, ai is known, statistical properties of the uncertain parameters can be
evaluated. For instance, the expected value µX and variance σX2 are evaluated as
µ X = a0 ,
σX2
=
P
X
a2i p2i .
(18)
i=1
Identification of the statistical properties of the uncertain parameters require calculation of the
gPC coefficients. Therefore, stochastic inverse method is employed to obtain the orthogonal
basis function of uncertain parameters via uncertainty propagation of the measured structural
responses. The technique of inverse stochastic identification of uncertain parameters from measured modal data is discussed in the next section.
4 STOCHASTIC INVERSE MODEL
For identifying probabilistic properties of the elastic parameters of the dynamic system, statistical informations of the modal responses are known a priori. Assume that, probabilistic
547
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
measure of the system parameters m are represented by gPC expansion. Therefore, calculation of statistical properties of the elastic parameters m is transferred into evaluation the finite
set of unknown gPC coefficients mi using statistical properties of the measured modal data d.
An optimization procedure is adopted to evaluate the unknown gPC coefficients mi as a design
variable. Experimental modal frequencies are represented in the form of gPC expansion as
d(ξ) =
N
X
di Ψi (ξ).
(19)
i=0
The deterministic coefficient di of the gPC expansion is estimated by minimization of statistical
moments calculated from measured frequencies and gPC expansion. The stochastic inverse
problem can be defined with reference to the Eq. (13) as
X
N
N
X
−1
di Ψi (ξ) .
(20)
mi Ψi (ξ) = G
i=0
i=0
Here, G−1 (·) is the inverse structural operator in terms of FE model. Moreover, direct evaluation
of the inverse operator is impossible and leads the inverse problem to an optimization problem
with mi as a design variable. The optimization function F is defined as the sum of the quadratic
difference between the central moments calculated from the simulated stochastic modal data and
measured modal data as
2
neig
k
X
1X
exp 2
r
.
(21)
(µDj − µj ) +
E[Dj − µDj ] − γjr
F =
2 j=1
r=1
In this equation, µDj is the expected value of the simulated j th modal frequency, E[·]r is the rth
order central moment of the simulated modal frequency and γjr is the rth order central moment
of measured j th modal frequency. The number of eigen modes is denoted by neig.The expected
value of the j th modal frequency is described by µexp
j . The stochastic representation of the
th
simulated j eigen frequency with reference to the gPC expansion of uncertain parameters and
forward structural operator is presented as
X
N
Dj = G
mi Ψi (ξ) .
(22)
i=0
j
The Eq. (21) is rewritten with reference to the Eq. (22) as
r
2
neig
N
k X
X
1X
exp 2
(µDj − µj ) +
E G
mi Ψi (ξ) − µDj − γjr
.
F =
2 j=1
j
r=1
i=0
(23)
The optimization algorithm determines the best solution for the gPC coefficients mi of the
system parameters by the functional minimization of cost function F .
5 NUMERICAL PROCEDURE
The proposed solution procedure for inverse identification of uncertain elastic parameters
from experimental modal frequencies involves estimation of deterministic coefficients of gPC
expansions for the parameters using a stochastic inverse model. A FE model is developed to
evaluate the structural responses of the composite plate and considered as a forward structural
operator. The detailed procedure of numerical simulation is summarized herein.
548
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
• Measure the eigen frequencies for each sample of composite plate.
• Measure weight of each sample of composite plate and derive mass density of composite
material.
• Evaluate the deterministic coefficients for the measured eigen frequencies and mass density based on minimization of the error function between statistical moments of the measured data and the same is derived from the gPC expansion of the quantity.
• Construct the probability distribution functions (PDFs) of the eigen frequencies and mass
density based on gPC expansion method and compare with the measured distributions.
• Construct the truncated gPC expansion for the identifiable parameters with the initial
approximation of deterministic coefficients.
• Estimate the gPC coefficients of the eigen frequencies employing the stochastic FE forward model by using initial values for the unknown coefficients of the parameters.
• Evaluate the error function between the rth order central moments calculated from gPC
expansion coefficients of the eigen frequencies and corresponding central moments calculated from the experimental data.
• A constrained optimization procedure is adapted to update the initially approximated unknown coefficients of the identifiable parameters by minimization of the cost function.
• Construct the PDFs of elastic parameters using the updated coefficients of the gPC expansion.
6 NUMERICAL RESULTS
A set of modal frequencies of 100 numbers, 12 layers glass-fiber epoxy composite plate
with identical dimension of 250 × 125 × 2 are measured in free-free boundary condition. Each
plate is suspended using two thin elastic wires to approximate the free-free boundary condition.
The composite plate is excited by impulse hammer and responses are collected at 35 points of
each plate by an accelerometer. A post-processing software is employed to derive the modal
responses i.e., eigen frequencies, mode shapes and modal damping ratios. The weight of each
plate is measured precisely to evaluate the uncertainty of the mass density. First 4 modes of
the eigen frequencies are considered for identification of the elasticity moduli E11 and E22 and
shear modulus G12 of the composite plate. To avoid modal coupling and corresponding error
only first 4 eigen frequencies are considered for the identification process. The initial 6 rigid
modes are neglected in the analysis. The uncertainty of the mass density is also considered
in the identification process. The PDFs of the measured eigen frequencies and corresponding
stochastic representations are shown in Figure 1. Third order gPC expansion, employing one dimensional Hermite polynomial Hi , is used to construct the experimental eigen frequencies [18]
as
d(ξ) =
3
X
di Hi (ξ).
(24)
i=0
The gPC coefficients dj for first 4 eigen frequencies are estimated by minimization of the statistical moments derived from experimental data and gPC expansion via an optimization procedure
549
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
and is presented in Table 1. The reconstructed PDFs as shown in Figure 1 using gPC expansion
are fitted well with the experimental distributions. Third order gPC expansion is well suited to
represent the nature of variability of the experiential eigen frequencies. Nine set of collocation
points (0, ±0.742, ±2.334, ±1.3556, and ±2.875) are selected from the roots of the 4th order
and 5th order Hermite polynomials. To check the Gaussian nature, the best fitted normal distribution are plotted against each experimental eigen frequencies. It is observed that first and
third experimental eigen frequencies are Gaussian in nature whereas, other two measured eigen
frequencies are non-Gaussian. However, 3rd order gPC expansion using Hermite polynomial
can efficiently described the non-Gaussian nature of the eigen frequencies. Experimental mass
density is represented by 3rd order gPC expansion and shown in Figure 2. The deterministic coefficients representing 3rd order gPC expansion for the mass density are 2.1143, 0.0540, 0.0075,
and 0.0023, respectively. The deterministic coefficients for the uncertain elastic parameters
Eigen freq.
d0
λ1
115.489
λ2
144.805
λ3
275.165
λ4
395.763
d1
4.536
5.771
8.761
14.867
d2
0.366
0.231
1.083
0.465
d3
0.011
0.4828
0.338
1.311
Table 1: The gPC coefficients of the first 4 eigen frequencies (Hz) from experimental measurement
0.2
0.12
0.15
0.09
0.1
0.06
0.05
0.03
0
100 110 120 130 140
0
120 135 150 165 180
0.06
0.04
0.045
0.03
0.03
0.02
0.015
0.01
0
240 260 280 300 320
0
350 375 400 425 450
Figure 1: Stochastic representation of the experimental eigen frequencies (Hz) and comparison with the normal
distribution
are estimated by employing 3rd order gPC expansion by minimization of the cost function as
stated in Section 5. The representation of the identified material parameters m in terms of gPC
expansion is
m(ξ) =
3
X
mi Ψi (ξ),
m = {E11 , E12 , G12 }.
(25)
i=0
Two cases of optimization procedure are adopted herein. The cost functions for the two cases
are developed to minimize of errors: upto 3rd order central moment and upto 4th order central moment. The identified deterministic coefficients of the elastic moduli E11 , E22 and shear
550
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
10
7.5
5
2.5
0
1.9
2
2.1
2.2
2.3
2.4
Figure 2: Stochastic representation of the experimental mass density (gm/cm3 )
th
Upto
Upto
rd
4 order 3 order
central central
moment moment
mini- minimization mization
modulus G12 are listed in Table 2. The first coefficient m0 of the gPC expansion represents the
expected value of the elastic parameters for the epoxy based glass-fiber reinforced composite
laminated plate. The standard deviation of the elastic parameters are also calculated in Table 2
referring Eq. (18). The PDFs of the uncertain elastic parameters are constructed using gPC
expansion and are presented in Figure 3. To check the accuracy of the gPC constructed PDFs,
PDFs of the first 4 eigen frequencies are reconstructed with the application forward stochastic
model and are shown in Figure 4. The histograms of the experimental eigen frequencies are
depicted in this Figure. The reconstructed PDFs of the eigen frequencies considering identified
gPC coefficients can well represent the uncertainty of experimental eigen frequencies. Moreover, gPC coefficients calculated by minimization of error functions derived from 4th order
central moments represent better accuracy over the 3rd order moments minimization. The gPC
coefficients evaluated from the higher order statistical moment minimization technique can predict the uncertainty of the eigen frequency with reasonably higher accuracy. The non-Gaussian
nature of the 2nd and 4th eigen frequencies are well estimated by the identified coefficients using
higher order statistical error minimization technique specifically near the tail region.
Parameters
E11 (GPa)
E22 (GPa)
G12 (GPa)
E11 (GPa)
E22 (GPa)
G12 (GPa)
m0
69.398
27.141
6.117
68.714
27.401
6.122
m1
6.514
3.023
0.576
7.041
2.681
0.578
m2
0.837
0.381
0.080
0.684
0.655
0.078
m3
0.158
0.017
0.022
0.898
0.080
0.010
σ
6.632
3.071
0.589
7.440
2.843
0.589
Table 2: The gPC coefficients of the uncertain elastic parameters
7 CONCLUSIONS
The identification of stochastic behavior of the elastic parameters of the laminated composite
plate using non-sampling based stochastic inverse process is presented in this paper. Collocation based non-intrusive gPC expansion method is used to identify the randomness of the elastic
parameters. The identification of uncertainty of the elastic parameters transforms into the identification of the unknown deterministic coefficients of the elastic moduli and shear modulus for
the laminated composite plate. The experimental distribution of the first 4 eigen frequencies and
mass density of the composite plate are used as inputs for the stochastic inverse identification
algorithm. An optimization technique is adopted to estimate the deterministic coefficients of the
uncertain elastic parameters by minimization of the cost function. The cost function is devel-
551
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
0.1
0.18
0.8
0.075
0.135
0.6
0.05
0.09
0.4
0.025
0.045
0.2
0
30 60 90 120
0
15 25 35 45
0
4
6
8
10
Figure 3: PDF of the identified elastic parameters
0.14
0.12
0.105
0.09
0.07
0.06
0.035
0.03
0
100
110
120
130
0
120
140
0.06
0.05
0.045
0.0375
0.03
0.025
0.015
0.0125
0
240
260
280
300
0
350
320
135
150
165
180
375
400
425
450
Figure 4: Reconstruct the PDFs of the eigen frequencies (Hz) from the identified gPC expansions of the elastic
parameters
oped by summing up the quadratic difference between experimental statistical moments of the
eigen frequencies and the simulated statistical moments of the eigen frequencies derived using
the gPC expansion method. The use of Hermite polynomial in the gPC expansion method can
efficiently inferred the distributions of the elastic parameters from the combination of Gaussian
and non-Gaussian experimental eigen frequencies. The accuracy of the inverse identification
is increased with the incorporation of the higher order statistical moments in the optimization
552
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
process. The reconstructed PDFs of the eigen frequencies can efficiently predict the uncertainty
of the experimental eigen frequencies, specifically with the application of the higher order statistical moment optimization.
8 ACKNOWLEDGMENT
The first author gratefully acknowledge the financial support extended by the Deutscher
Akademischer Austauschdienst (DAAD).
REFERENCES
[1] C.M. Mota Soares, M. Moreira de Freias, A.L. Araújo, P. Pedersen. Identification of material properties of composite plate specimens. Composite Structures, 25(1-4), 277-285,
1993.
[2] A.K Bledzki, A. Kessler, R. Rikards, A. Chate. Determination of elastic constants of
glass/epoxy unidirectional laminates by the vibration testing of plates. Composites Science and Technology, 59(13), 2015-2024, 1999.
[3] R. Rikards, A. Chate, G. Gailis. Identification of elastic properties of laminates based on
experiment design. International Journal of Solids and Structures, 38(30-31), 5097-5115,
2001.
[4] R. Rikards, A. Chate, W. Steinchen, A. Kessler, A.K. Bledzki. Method for identification
of elastic properties of laminates based on experiment design. Composites: Part B, 30,
279-289, 1999.
[5] T. Lauwagie, Vibration-based methods for the identification of the elastic properties of
layered materials. Dissertation, Catholic University of Leuven.
[6] M. Tanaka, G.S. Dulikravich. eds. Inverse Problems in Engineering Mechanics. Elsevier,
1998.
[7] J. Kaipio, E. Somersalo. Statistical and Computational Inverse Problems. Springer, 2005.
[8] R. Aster, B. Borchers, C. Thurber. Parameter Estimation and Inverse Problems. Academic
Press, 2004.
[9] L. Tierney. Markov Chains for Exploring Posterior Distributions. The Annals of Statistics,
22, 1701-1728, 1994.
[10] W.R. Gilks, S Richardson, D.J, Spiegelhalter eds. Markov Chain Monte Carlo in Practice
. Springer-science+business media, B.V., 1996.
[11] J.B. Nagel. Bayesian techniques for inverse uncertainty quantification. PhD Thesis, University of Bonn, 2017.
[12] R.G. Ghanem, A. Doostan. On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data. Computational
Physics, 217, 63-81, 2006.
553
S. Chandra, K. Sepahvand, C.A. Geweth, F. Saati, and S. Marburg
[13] B.V. Rosić, A. Litvinenko, O. Pajonk, H.G. Matthies. Sampling-free linear Bayesian update of polynomial chaos representations. Computational Physics 37, 5761-5787, 2012.
[14] R.G. Ghanem, P.D. Spanos. Stochastic Finite Elements: A Spectral Approach. Dover Publication, 1991.
[15] K. Sepahvand, S. Marburg, H.-J. Hardtke. Uncertainty quantification in stochastic systems
using polynomial chaos expansion. Applied Mechanics, 02(02), 305353, 2010.
[16] K. Sepahvand, S. Marburg. Identification of composite uncertain material parameters from
experimental modal data. Probabilistic Engineering Mechanics, 148-153, 2014.
[17] K. Sepahvand and S. Marburg. Non-sampling inverse stochastic numerical-experimental
identification of random elastic material parameters in composite plates, Mechanical System and Signal Processing, 54-55, 172-181, 2015.
[18] K. Sepahvand, C.A. Geweth, F. Saati, M. Klaerner, L. Kroll and S. Marburg. Spectral Representation of Uncertainty in Experimental Vibration Modal Data, Advances in Acoustics
and Vibration, 9695357, 2018.
[19] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells. Boca Raton: CRC
Press, 2003.
[20] T. Kant, H. Varaiya, C. P. Arora, Finite element transient analysis of composite and sandwich plates based on a refined theory and implicit time integration schemes, Composite
structures, 36(3), 401-420, 1990.
[21] A.G. Niyogi, M.K. Laha, P.K. Sinha, Finite element vibration analysis of laminated composite folded plate structures, Shock and Vibration, 6, 273-283, 1999.
554
Steffen Marburg