A mode-based meta-model for the frequency response functions of uncertain structural systems

Computers and Structures 87 (2009) 332–341

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A mode-based meta-model for the frequency response functions of uncertain structural systems L. Pichler, H.J. Pradlwarter, G.I. Schuëller * Chair of Engineering Mechanics, University of Innsbruck, Technikerstraße 13, A-6020 Innsbruck, Austria

a r t i c l e

i n f o

Article history: Received 18 July 2008 Accepted 22 December 2008 Available online 6 February 2009 Keywords: Meta-modeling Frequency response analysis Uncertainties Structural reliability

a b s t r a c t Frequency response functions (FRFs) play an important role in the assessment of the structural response of linear systems subjected to dynamic forces. In this work a functional relation (meta-model) between the uncertain structural parameters and the model properties is presented. The meta-model provides a computationally fast solution to approximate the eigenfrequencies and mode shapes needed thereafter for evaluating the FRFs. The meta-model circumvents the repeated solution of the eigenvalue problem for each set of uncertain structural input parameters. The provided relations can be helpful in the design stage to control the dynamic response within certain frequency bands. Moreover, the meta-model can be used for optimization and reliability assessments based on Monte Carlo sampling procedures. Numerical examples show the application of the method focusing mainly on the variability of the FRFs. The efficiency and accuracy of the meta-model to compute approximate FRFs is assessed by a comparison with the solution by Finite Element (FE) analyses. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The use of meta-models is essential and indispensable for many scientific and technical applications. The scope of applying metamodels is to provide a possibly simple relation between the input data and output quantities of interest at low computational cost. Such surrogate models are widely applied in computer science, chemical, physical and biological applications. Although on the one hand the rapidly growing computational power of modern computer systems and continuous development of increasingly efficient algorithms and software result in faster computations, on the other hand the (numerical) models of interest are growing in complexity; more detailed representations of the geometry, more precise descriptions of physical parameters and more details are incorporated into the analysis. Many different meta-modeling procedures are available. The most commonly used methods are least squares approximation, linear or polynomial regression, kriging and the radial basis functions. Which type of meta-model is best suited for a particular application, generally depends on the specific problem under investigation. A comparison of the performance of different meta-modeling procedures can be found e.g. in Refs. [1–3]. For many cases, certain rules could be followed in a straight forward manner to achieve results that follow certain quality standards (see e.g. [4,5], etc.). * Corresponding author. Tel.: +43 512 507 6841; fax: +43 512 507 2905. E-mail address: [email protected] (G.I. Schuëller). URL: http://mechanik.uibk.ac.at (G.I. Schuëller). 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2008.12.013

The most frequently used numerical procedure for calculation of the structural response, e.g. for industrial applications or civil structures, is the Finite Element Method (FEM). The range of application is very wide and varies from fluid dynamics to fatigue and models for crack propagation in material science (e.g. mesh-free FEM [6]) to static and dynamic structural analysis (displacements, stresses, normal modes, etc.) [7]. The assessment of civil structures such as buildings or bridges, which are subjected to dynamic loading, generally results in large FE models with many degrees of freedom and high computational efforts. In this context, virtual prototyping is becoming increasingly important for many industrial applications, especially in the automotive industry. The economical benefit is due to reducing both the cost for constructing real prototypes and the time to market. Structural parameters, even those of ‘identical’ specimen originating from the same production run show variability and are subject to uncertainty. A state-of-the-art account of the treatment of uncertainties in structural dynamics has been given in a Special Issue of the Journal of Sound and Vibration entitled Uncertainty in structural dynamics [8]. Doubtless to say, different methodologies may be applied, depending on the design phase, the knowledge and expertise of the involved analysts, but also on the required results and their level of accuracy and on the purpose of the analysis, e.g. preliminary calculation, optimization, reliability assessment, etc. Probabilistic methods are most instrumental and hence widely applied to assess uncertainty propagation. They are based on a well developed mathematical theory, which assures a logically consistent reasoning (see e.g. [9]). An issue of great practical interest is

L. Pichler et al. / Computers and Structures 87 (2009) 332–341

the improvement of the numerical model and the benefit from continuously measured experimental data. Model updating procedures, such as Bayesian statistics [10–13], can be applied in order to enhance the predictive capabilities of the numerical model [14]. Ongoing discussions and research efforts in context with validation and verification are directed towards approaches which are justified from both a physical and statistical point of view, when a limited amount of data is available (see e.g. [15,16]). The stochastic approach used to carry out the analysis should reflect a specified confidence level derived from the limited number of data. In this context the emerging number of so-called non-probabilistic approaches, which on their part provide results in a so-called possibilistic manner are mentioned (see e.g. [17–19]). A recently introduced probabilistic approach, referred to as a non-parametric model of uncertainties [20,21], is aimed at uncertainties in the mechanical model in addition to those in the parameters of the model. The procedure, based on the random matrix theory, uses the modal basis of the nominal FE model. FRFs are a valuable quantity of interest in dynamic analysis, since they provide information of the response over the frequency band. The observed variability of the FRFs arises from the variability of the uncertain parameters and unknown mechanisms of the structure. The statistics of the FRFs are of high relevance for the reliability assessment of structures under dynamical loading (see e.g. [21–23]). Mace [24] describes a procedure based on component mode synthesis (CMS) and local modal and perturbational approaches to approximate the FRFs. The FRF statistics of the systems with uncertain structural properties are modeled by assumed probability density functions for the natural frequencies. The perturbational approach effectively relates the global and local properties for structural assemblies. FRFs depend in a non-linear manner on the modal properties (mode shapes, eigenfrequencies, damping). In this work the feasibility of representing the modal properties for the calculation of FRFs by a simple meta-model is studied. Since the intended application is directed towards the variability of FRFs, the correlations between the modal properties play an essential role. Naturally, the complexity of the meta-model is a function of the number of independent random variables. Moreover, it is pointed out that the meta-model provides a computationally efficient approximation of the structural response which is sufficiently accurate to be adopted for structural design. Hence, the proposed meta-model provides a tool for exploring the input parameter space, which is of great interest in context with reliability analysis or optimization, respectively. In the example section, the presented meta-model is applied to assess the FRF variability of mass–spring structures and subsequently of an automotive windshield. 2. Frequency response analysis 2.1. FRF by modal superposition The FRFs depend non-linearly on the eigensolution in terms of eigenfrequencies, mode shapes, modal damping and on the specific dynamic excitation, respectively. The standard procedure in FE analysis is to evaluate the FRF adopting the concept of modal superposition. For a linear viscoelastic system, the frequency response functions are given by (see e.g. [25])

rðX; hÞ ¼

X j

cik ~ /kj ðhÞXp /ij ðhÞ~

x

2 j ðhÞ

 X2 þ i  2nj ðhÞxj ðhÞX

;

the jth eigenfrequency; ~ /ij denotes the jth mode at the ith degree of freedom; cik accounts for the relation between excitation acting at degree of freedom i and response at degree of freedom k and the integer p is 0 for displacement–, 1 for velocity– and 2 for acceleration-FRFs, respectively. The variability of frequency response functions, which is observed especially in the lower frequency range, arises mainly from the influence of random effects and uncertainties in the mechanical properties. A simple approximation of the highly non-linear frequency response function as a function of the uncertain structural input parameters is not feasible. 2.2. The concept of linearization Many physical processes or phenomena are weakly non-linear functions of one or more parameters. For such cases a linear relation is then generally a convenient approach to approximate the underlying process. A small variability in the uncertain input parameters evokes mode shapes which differ slightly from the nominal ones. This is valid for well-separated eigenfrequencies and has been investigated e.g. in [26]. These variations of the eigenvectors can be interpreted as small rotations w.r.t. a reference solution. Therefore, the rotated eigenvectors can be approximated by a linear function of the base vectors of the reference solution (cf. Fig. 2, Section 2.4). However, double or multiple eigenfrequencies occur e.g. for symmetric systems such as a quadratic plate or a multi-story building. For such cases the subspace spanned by the eigenvectors of the multiple eigenfrequencies can be represented by a linear approximation of the subspace of the reference system as shown in Fig. 1. Small variations of the structural parameters might cause a large rotation of the eigenvectors within the subspace related to the multiple eigenfrequencies. Generally, the first few eigenfrequencies of civil structures are well separated. These eigenfrequencies are important for the assessment of structures under dynamic excitation. They contribute most to the response, a fact which can easily be understood when using the concept of modal superposition. However, the largest contribution to FRFs within a certain frequency band stems mainly from the neighboring modes. 2.3. Linear relation for the modal properties Contrary to the frequency response function, which are a strongly non-linear function of the random parameters of the structure, the modal properties depend almost linearly on these parameters. It is assumed that the number of random variables, which influence the modal properties, is high, e.g. more than 30. Since all modal properties depend uniquely on the random realization of the parameters and are approximated by a function of it, all correlations among those modal solutions are automatically incorporated. Eventually, the present correlations are also propagated correctly to the FRFs. Different approaches could be applied to approximate the modal properties as a function of the uncertain structural parameters. It

ð1Þ

where h indicates that the quantity is uncertain and modeled as means the modal damping ratio; i denotes random variable; np j ffiffiffiffiffiffiffi the imaginary unit 1; X is the excitation frequency; xj denotes

333

Fig. 1. Rotation of a subspace w.r.t. the reference solution.

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is suggested here to describe all mode shapes as linear combinations of the modes of a reference solution. This approximation is valid for a small variability of the random structural parameters, which causes a rotation of the mode shapes when compared to a reference solution. Several procedures, such as the least square method, regression, optimization or soft computing methods provide solutions for the parameters of the linear combination.

and consequently cannot be approximated by a linear function of the uncertain input parameters. Therefore, ak;k is expressed by

2.4. Quantification of the variability for the modal properties

Each mode can then be represented by

Considering the fact that the modes of a system form a mathematically complete space and also from a physical point of view, it is easy to see that the variability of the modes shapes can be interpreted as functions of neighboring modes of a reference system. Here it is assumed, that the kth mode shape can be described by a linear combination of m mode shapes of the reference solution (see also e.g. [25,27]) as shown in Fig. 2. In many cases the nominal solution, i.e. the solution for the mean value of the vector of uncertain parameters, might be used for this purpose:

ðjÞ ~ /k ¼ ak

ðjÞ ~ /k ¼

kþm Xk

~ð0Þ aðjÞ ki /i ;

ð2Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X a^ 2k;i ;

ak;k ¼ ak 1 

ð5Þ

i–k

where

a^ k;i ¼ ak;i =ak : kþm Xk

ð6Þ

~ð0Þ a^ ðjÞ k;i /i :

ð7Þ

i¼kmk

The corresponding eigenfrequency is given by ðjÞ ð0Þ ðjÞ ð0Þ xðjÞ DxðjÞ k ¼ ð1 þ Dxk Þxk ; k ¼ ðxk =xk  1Þ:

ð8Þ

ðjÞ k

ðjÞ The modal properties of each mode, i.e. x and ~ /k are then specðjÞ ðjÞ ðjÞ ð0Þ ak and the nominal solution xk ified as a function of ak ; Dxk ; ^ ð0Þ and Uk .

2.5. Meta-model for the modal quantities

i¼kmk

where Uð0Þ denotes the modal matrix containing all modes of the ðjÞ reference system, ~ /k the kth mode of simulation j and ak contains the ð2mk þ 1Þ coefficients of the linear combination. The associated ð2mk þ 1Þ modes of the nominal solution Uð0Þ are denoted as:

h i ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ Uk ¼ ~ /kmk ; ~ /kmk þ1 ; . . . ; ~ /k ; . . . ; ~ /kþmk 1 ; ~ /kþmk :

ð3Þ

The number ð2mk þ 1Þ of modes involved in the linear combination, increases with frequency. Few, say two to four modes, generally suffice to describe the first group of modes. With increasing frequency, the density of eigenfrequencies increases and 10 or more modes should be employed for an accurate approximation. One way to determine the vector ak is to solve a least square problem of the associated linear relation in Eq. (2), which in its normal form is given by:

h i1 ð0Þ ð0Þ ð0Þ ðjÞ ak ¼ ðUk ÞT Uk ðUk ÞT ~ /k :

ð4Þ

ðjÞ As shown in Fig. 2, the mode ~ /k can be interpreted as a rotation ð0Þ ð0Þ of the mode ~ /k in the modal space of the reference solution Uk . ðjÞ ~ Moreover, the length of vector /k might also slightly differ from ð0Þ the length of ~ /k . kþmk ðjÞ is the projection of ~ /k onto the Since the vector ak ¼ fak;i gi¼km k ð0Þ base vectors of Uk , the Euclidean norm ak ¼ kak k specifies the raðjÞ ð0Þ ðjÞ /k k. In the simplest case, where small rotations of ~ /k tio k~ /k k=k~ can be assumed for small variations of the uncertain structural parameters, all factors ak;i–k will be small and hence can be approximated sufficiently well by a linear function of the uncertain parameters. The largest factor ak;k , however, will be close to unity

Fig. 2. Schematic sketch of the rotation in the space of the reference solution Uð0Þ .

An overview and comparison of different meta-modeling techniques can be found in Refs. [1,2,4,5]. In this work linear relations are used for the meta-model of the modal quantities, which are subsequently needed for the FRF calculations. Let Y be a N  M matrix where N is the number of independent P random samples j ¼ 1; 2; . . . ; N and M ¼ Kk¼1 ð2mk þ 3Þ is the number of factors describing the modal properties of all considered K modes. Hence, the jth row of Y contains the quantities

h i ðjÞ ðjÞ ðjÞT ðjÞ ðjÞ ðjÞT YðjÞ ¼ a1 ; Dx1 ; ^a1 ; . . . ; aK ; DxK ; ^aK :

ð9Þ

Let further X be a N  n matrix which comprises in each row the vector xðjÞ of realizations of the n uncertain parameters. The task of the meta-model can then be described as establishment of an approximate functional relation between the matrices X and Y, respectively,

YðXÞ ¼ f ðXÞ;

ð10Þ

which allows for any vector x to establish the associated vector yðxÞ:

yðxÞ ¼ ½a1 ðxÞ; Dx1 ðxÞ; ^aT1 ðxÞ; . . . ; aK ðxÞ; DxK ðxÞ; ^aTK ðxÞ:

ð11Þ

The simplest relation between x and y ¼ yðxÞ is a linear relation of the form

y ¼ Sx;

ð12Þ

to be used as its first approximation, in which S is a constant matrix of size M  n. The matrix S needs to be established by exploring the available information given by the two matrices X and Y. As already mentioned in the introduction, several possibilities may be chosen for determining the parameters of the meta-model. Neural networks are well suited in case of few input parameters, where an acceptable amount of independent data points suffices for training the network. However, the efficiency of neural networks decreases with increasing number of input parameters, resulting in an infeasible large training set. For the sought linear relation, a least square solution is the preferable choice. The required sample size N to establish this linear relation grows only linearly with the number n of uncertain input parameters. For many technical applications it is well known that only a small set of the random structural parameters influence mainly the structural response. In such cases the application of gradient estimation procedures [28,29] may allow a fast determination of these ‘most important’ parameters.

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3. Examples

3.2. Mass–spring structure – truss

3.1. General remarks

The meta-model for the modal quantities is used to investigate the FRF statistics for the mass–spring structure shown in Fig. 3. The FE model consists of 33 masses m and 131 springs k that provide the connections between the single masses and the connections to the supports. All masses and springs – in total 164 – are modeled as Gaussian distributed random variables with mean values lm ¼ 1 kg and lk ¼ 1000 kg=s2 , respectively. The coefficients of variation (cov.) are equal to 5% (case I) and 10% (case II). All masses and springs are assumed to be independent random variables to increase the complexity for testing purposes. The structure consists of 10 geometrically identical units which are composed of equilateral triangles of lateral length L ¼ 1 m and crossing elements between the triangles. The distance between two triangles measures 1 m. In the following, the response statistics will be analyzed more detailed for case II, i.e. assuming a cov. of 10% for all 164 independent random variables. Fig. 4 shows a comparison for 3 FRFs, when using the FE analysis combined with solution of the associated eigenvalue problem and the meta-model. 250 samples have been applied in order to calculate the parameters for the meta-model, i.e. for calibrating the model, using the least squares method. The variability of the FRFs increases with frequency. The first 40 modes have been used for the meta-model. The first eigenfrequency is x1 ¼ 0:21 Hz and x40 ¼ 6:76 Hz for the nominal model. A comparison is shown for the frequency range from 0 through 30 [rad/s]. The meta-model fits the FE solution reasonably well within the considered frequency range.

In the following examples the variability of the FRFs arising from the variability of the input modeled as random variables, is investigated. In the first example (Section 3.2), the FRF statistics is investigated for a mass–spring truss-like structure. The statistics is shown for a coefficient of variation of 5% and 10% of the structural parameters, respectively. In order to be useful for reliability analysis, the maximum responses predicted adopting the metamodel need to be compared to those obtained from the associated FE solution. Such a comparison is provided. The second example (Section 3.3) treats another mass–spring structure, which represents a rough approximation of a roof modeled as a space truss. Both examples indicate that the presented meta-model provides conservative statistics w.r.t. the FE solution, i.e. the response predicted with the meta-model is generally larger than the corresponding FE response. Conservatism is an important property of surrogate models, especially in context with estimation of failure probabilities for reliability analysis. The third example (Section 3.4) deals with a laminate car windscreen consisting of five layers. The structural uncertainties are described by a random field for modeling the thicknesses of the five layers of the windscreen, which are approximated by Karhunen– Loève (KL) expansions. In addition, a comparison of the highest FRFs provided by the meta-model and the associated FE-solution is shown.

10 8

z

0.866

6

0.433

4

0 2 0.5 0 −0.5

x

0

y

acceleration FRF

Fig. 3. Mass–spring structure – truss.

10

10

10

3

2

FE solution Meta−model

1

0

5

10

15 frequency [rad/sec]

20

Fig. 4. Comparison of the FRFs: FE vs. meta-model for case II.

25

30

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L. Pichler et al. / Computers and Structures 87 (2009) 332–341 4

3

10

4

10

Nominal Solution envelope mean μ± σ

acceleration FRF

acceleration FRF

10

2

10

1

10

0

3

10

Nominal Solution envelope mean μ±σ

2

10

1

5

10 15 20 frequency [rad/sec]

25

10

30

0

5

(a) FE analysis

10 15 20 frequency [rad/sec] (b) Meta-model

25

30

Fig. 5. Comparison of the FRF statistics based on 1000 samples – case II.

Figs. 5 and 6 show a comparison of the statistics between the FE analysis and the meta-model. Fig. 5 demonstrates the variability of the FRF by presenting the envelope based on 1000 samples and the l-FRF and l  r-FRFs. Quantiles for the FRFs, which are of interest in context with reliability assessment, are shown in Fig. 6. The quantiles are based on 100,000 samples. It can be seen from the figures that the meta-model reveals a conservative behavior. This property is valuable if a method is applied for reliability analysis.

3.2.1. Reliability assessment The reliability assessment is carried out by specifying the maximum response at 20 rad/s for both case I (cov. = 5%) and case II (cov. = 10%). The solution obtained with the meta-model is then compared with the FE-solution. The highest 10 solutions at 20 rad/s out of 100,000 simulations using the meta-model are compared with the results of the FE analysis using the same random input for the uncertain parameters (see Figs. 7 and 10).

4

4

10 FE−Analysis Meta−model

CoV=0.05

acceleration FRF

acceleration FRF

10

99.99%

3

10

50%

2

10

0.01%

1

10

0

FE−Analysis Meta−model

CoV=0.10 99.99%

3

10

50%

2

0.01%

10

1

5

10 15 20 frequency [rad/sec]

25

10

30

0

5

10 15 20 frequency [rad/sec]

25

30

25

30

Fig. 6. Quantiles of the FRFs based on 100,000 samples: FE analysis and meta-model for cases I and II.

4

4

10 acceleration FRF

acceleration FRF

10

3

10

2

10

1

10

0

3

10

2

10

1

5

10 15 20 frequency [rad/sec]

(a) Meta-model

25

30

10

0

5

10 15 20 frequency [rad/sec]

(b) FE analysis

Fig. 7. Ten highest FRFs at 20 rad/s among 100,000 simulations by the meta-model and associated FRFs by FEM for case I.

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L. Pichler et al. / Computers and Structures 87 (2009) 332–341 4

4

10 acceleration FRF

acceleration FRF

10

3

10

2

10

3

10

2

10

1

10

1

0

5

10 15 20 frequency [rad/sec]

25

10

30

0

5

(a) FE analysis

10 15 20 frequency [rad/sec]

25

30

(b) Meta-model

Fig. 8. Ten highest FRFs at 20 rad/s among 100,000 simulations by FE and associated FRFs by the meta-model for case I.

Subsequently, also 100,000 FE analyses are performed and the 10 highest FRFs at 20 rad/s are compared with the corresponding responses of the meta-model. Figs. 8 and 9 show the 10 highest FRFs obtained by the Finite Element Method (FEM) and the corresponding FRFs from application of the meta-model. This second comparison is shown purely in order to obtain an idea how well the meta-model fits the maximum FE results, which are in general computationally too expensive. For this simple, small model the

solution of the associated eigenvalue problem is quite inexpensive. Nonetheless, the computational time for a reliability assessment based on 100,000 samples from the meta-model is  10 times less than the associated FE assessment. For case II the results are also in very good agreement. However, for higher variation of the uncertain structural parameters, the accuracy of the meta-model for approximating the FE results somewhat decreases. Nonetheless, the associated FRFs by FE

4

4

10 acceleration FRF

acceleration FRF

10

3

10

2

10

1

10

0

3

10

2

10

1

5

10 15 20 frequency [rad/sec]

25

30

10

0

5

(a) Meta-model

10 15 20 frequency [rad/sec]

25

30

(b) FE analysis

Fig. 9. Ten highest FRFs 20 rad/s among 100,000 simulations by FE and associated FRFs by the meta-model for case II.

4

4

10

acceleration FRF

acceleration FRF

10

3

10

2

10

1

10

0

3

10

2

10

1

5

10 15 20 frequency [rad/sec]

(a) FE analysis

25

30

10

0

5

10 15 20 frequency [rad/sec]

25

(b) Meta-model

Fig. 10. Ten highest FRFs 20 rad/s among 100,000 simulations by the meta-model and associated FRFs by FEM for case II.

30

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z

338

0.707 0 4 2 y

(a) 3d model, http://de.wikipedia.org

0

0

4

2

6

x

(b) FE model

Fig. 11. Mass–spring structure for a roof.

3

3

10 acceleration FRF

acceleration FRF

10

2

10

1

10

Nominal Solution envelope mean μ±σ

0

10

0

10 20 frequency [rad/sec]

2

10

1

10

Nominal Solution envelope mean μ±σ

0

10 30

(a) FE analysis

0

10 20 frequency [rad/sec]

30

(b) Meta-model

Fig. 12. Comparison of the FRF statistics based on 1000 samples for FE analysis and meta-model.

analysis (see Figs. 10 and 9) are in the range of their global maximum. The meta-model shows to provide a computationally efficient approximation of the FRFs. The approximation suffices to determine the critical domains in the uncertain parameter space. This critical domain might then be approximated in a refined manner by using again full FE analyses to achieve the required accuracy for the reliability assessment.

3

acceleration FRF

10

FE−Analysis

calibration: 750 samples

Meta−model

CoV=0.1

99.9% 2

10

50%

1

10

0.1%

0

3.3. Mass–spring structure – roof In this second example, a simplified space truss model representing a roof as shown in Fig. 11b is analyzed. The model consists of 5  8 overturned pyramids (tetrahedrons) whose tips are connected through a rectangular grid. The tetrahedrons have a lateral length of 1 m and the distance of the their tips on the regular grid x; y is 1 m. The 94 masses m and the 328 springs k are all modeled as independent Gaussian distributed RVs with mean values lm ¼ 1 kg and lk ¼ 1000 kg=s2 , respectively, and coefficient of variation cov. = 10%. The first eigenfrequency results in x1 ¼ 0:29 Hz and x80 ¼ 5:19 Hz for the nominal model. The FRFs are plotted within the range from 0 through 30 [rad/s], based on modal superposition of the first 80 modes. The parameters for the meta-model are evaluated by solving the least square problem based on 750 solutions of the eigenvalue problem. The statistics of the FRFs, including the l and l  r-FRFs are shown in Fig. 12 for both the FE model and the meta-model, respectively. Again, the variability of the response is slightly higher for the meta-model. Fig. 13 shows a comparison for the quantiles of the FRF. Also for this case the meta-model shows to behave conservative. 3.4. Laminate windshield – unmounted model

10

0

5

10 15 20 frequency [rad/sec]

25

30

Fig. 13. Roof structure, comparison of FRF quantiles obtained by meta-model and FE-analysis.

3.4.1. Model description In this example the FRF variability of an automotive windshield is analyzed. More specifically point FRFs 1/1 (point 1, Fig. 14a) and transfer FRFs 1/15 for the acceleration transverse to the plane of the windshield are investigated. The spatial variability of

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(a) Finite element model of the windshield

(b) Layers of the laminate windshield

Fig. 14. FE model and schematic view of the layers.

the outer layers are made of glass with a thickness of lG ¼ 2:14 mm. An overview of the model data of the windshield is provided in Table 1. For additional investigations on the windshield, including results from experimental and further numerical studies, the reader is referred to Refs. [30,31].

geometric and material properties of the windshield can be represented by random fields. The FE mesh is depicted in Fig. 14a; the windshield measures approximately 800 mm in the vertical and 1500 mm in the horizontal direction. This laminated windshield consists of five layers with different thicknesses and materials as shown in Fig. 14b. The three central layers are polymers with mean values of the thickness of lPVB ¼ 0:29 mm and lAP ¼ 0:1 mm and

3.4.2. Random fields The random variables describing a homogeneous isotropic Gaussian field are assumed to be normally distributed. Therefore, the distance between two points suffices when describing the interdependency of the associated random variables within the random field. The correlation between two Gaussian random variables can be described by an exponential correlation function of the form

Table 1 Summary of the model data of the windshield. Parameter

Unit

Mean value, l Coefficient of variation, cov. Correlation length, L Young’s modulus, E Poisson ratio, m Mass density, q

mm % mm kPa – kg/mm3

Layer Glass (1, 5)

PVB (2, 4)

AP (3)

2.14 4 200 7  107 0.215 2:5  106

0.29 12 200 2:4  106 0.491 1:06  106

0.29 12 200 5  104 0.491 1:06  106

  kxi  xj k Rðxi ; xj ; L; rÞ ¼ r2 exp  L

ð13Þ

with standard deviation r, correlation length L and the spatial distance kxi  xj k. A random field can be conveniently represented by a Karhunen–Loève expansion [32]

900 800 700 0.36 0.34 0.32 0.3 0.28 0.26

width [mm]

600 500 400

800

300

600

200

400

100

200

0 −100 −800

−600

−400

−200

0

200

400

600

800

−400 0 −600 −800

−200

0

200

400

600

800

length [mm] Fig. 15. Realization of a homogeneous isotropic Gaussian random field of a PVB layer, 2D and 3D views; mean value 0.29 mm, coefficient of variation 12%, correlation length 200 mm.

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 ðxÞ þ uðx; nÞ ¼ u

r6n pffiffiffiffi X ki /i ðxÞni ;

ð14Þ

i 0

C/i ¼ ki /i ;

ð15Þ

C ij ¼ Rðkxi  xj k; L; rÞ i; j ¼ 1; . . . ; n:

ð16Þ

From Eq. (14) it can be seen that the randomness is only associated with the parameters ni . In general, the Karhunen–Loève expansion can be truncated after the first m  n terms for an acceptably accurate approximation of the random field (see Fig. 15).

acceleration FRF

3.4.3. Discussion of the FRF variability For the numerical investigations the KL expansion used for the representation of each of the random fields is truncated after 25 terms. It can be seen from Fig. 16 that using only 5 KL-terms would be insufficient to represent the FRFs sufficiently accurate at some maxima. 25 KL-terms however suffice for an accurate representation of the related FRFs. With regard to the sample size it can be seen from Fig. 17 that only N ¼ 125 samples for the calibration set are not sufficient for a

−1

10

−0.3

10 −1.1

10

−2

10

5 KL−terms 25 758

0

50

−0.6

10 60 70 80

150 160 170

100 150 frequency [rad/sec]

200

0.01% and 99.99% quantiles meta−model μ(10) associated FEM − μ(10)

0

50

0

10

100

200

stable solution. For this example however N ¼ 150 FE analyses do suffice for the calibration of the meta-model. For comparison, 25 FRFs are plotted for the meta-model together with the associated FRFs obtained with FE analysis. In Fig. 18 the mean of the 10 highest FRFs – out of 100,000 calculated by the meta-model – are compared to the mean of the associated 10 FRFs obtained with a full FE-analysis. It has to be noted that these FRFs are the highest at the observed peak. With growing distance to the examined peak the FRFs assume values in the upper quantile region. However, the examined associated FRFs by FE-analysis are close to the higher quantile curves (see Fig. 18) and are therefore meaningful when exploring the domain in the parameter space leading to extreme values of the searched response.

A meta-model for the modal properties to be used for the determination of FRFs of structures is developed. The parameters for this meta-model are obtained by solving a least square problem (see Eq. (4)). The suggested approach automatically provides correct correlations among the modal properties needed for a suitable representation of the uncertainties in the FRFs. The approximated variability of the modal properties, calculated with the meta-model is used in turn for investigating the variability of the FRFs.

0

10

FE solution meta−model

0

100 150 frequency [rad/sec]

Fig. 18. Quantiles of the FRFs by meta-model based on 100,000 simulations and 10 highest FRFs at 120 Hz by the meta-model and associated FRFs by FEM.

acceleration FRF

Fig. 16. Transfer FRF 1/15 – comparison: different number of Karhunen–Loève terms for random field representation.

acceleration FRF

−1

10

4. Summary and conclusions

−1.8

10

10

acceleration FRF

 is the mean, ni a standard normal variate and ki and /i are where u the eigenvalues and eigenvectors of the associated covariance matrix C, where n denotes the length of the vector u:

200 300 400 frequency [rad/sec] (a) calibrationset-125samples

500

FE solution meta−model

0

100

200 300 400 frequency [rad/sec] (b) calibrationset-150samples

500

Fig. 17. Comparison for two different sample sizes of the calibration set. Point FRF 1/1 – 25 independent samples of meta-model and associated FE solution.

L. Pichler et al. / Computers and Structures 87 (2009) 332–341

The accuracy of the results depends on: Linearity of the model, i.e. relation input–output. Number of samples used for the least square solution, i.e. the calibration data set. Number of random variables involved in the problem, which relates to the size of the calibration data set. The efficiency decreases with increasing numbers of RVs. The proposed meta-model is applicable if: The modes of a sample are representable as a rotation of the modes of a reference solution. The modes are well separated (the accuracy decreases with mode mixing). The least square solution can be applied for the evaluation of the parameters of the meta-model. It is shown that the presented meta-model for the modal properties can be effectively used for the assessment of the variability of the FRFs and in a further step for reliability analyses. The computationally far more efficient meta-model can also be applied effectively for (reliability based) optimization. In this context the availability of a computationally inexpensive meta-model is essential in order to avoid an exorbitant number of FE analyses of detailed FE models. The efficiency of the computational tools used for the uncertainty analysis is of critical importance, in order to extend their applicability to large-scale problems. As shown in this work, the capability to achieve high efficiency while preserving the accuracy is perfectly met by the meta-model. Acknowledgements This research was partially supported by the project ‘‘Modelling Product Variability and Data Uncertainty in Structural Dynamics Engineering (MADUSE)” of the European Union under contract number MRTN-CT-2003-505164 and the Austrian Research Council (FWF) under Project No. L269-N13 which is gratefully acknowledged by the authors. References [1] Giunta A, Watson L, Koehler J. A comparison of approximation modeling techniques: polynomial versus interpolating models. In: Proceedings of the seventh AIAA/USAF/ NASA/ISSMO symposium on multidisciplinary analysis and optimization, AIAA -98-4758; 1998. p. 1–13. [2] Jin R, Chen W, Simpson T. Comparative studies of metamodelling techniques under multiple modelling criteria. Struct Multidiscip Optim 2001;23(1): 1–13. [3] Sacks J, Welch WJ, Mitchell TJ, Wynn HP. Design and analysis of computer experiments. Stat Sci 1989;4(4):409–23. [4] Kleijnen JPC, Sargent RG. A methodology for fitting and validating metamodels in simulation. Eur J Oper Res 2000;120(1):14–29. [5] Reis dos Santos MI, Porta Nova AM. Statistical fitting and validation of nonlinear simulation metamodels: a case study. Eur J Oper Res 2006;171(1): 53–63.

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