fem validation

5/21/2018 FEM validation

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VERIFICATION OF FE MODELS FOR MODEL UPDATING

G. Chen and D. J. Ewins

Dynamics Section, Mechanical Engineering Department

Imperial College of Science, Technology and Medicine

London SW7 2AZ, United Kingdom

Email: [email protected]

SUMMARY:This paper describes a study of FE model verification. Before undertaking a

model updating procedure, it is important to determine whether the initial model can be

updated or not. Two methods for model verification - a convergence check and a

configuration check - are proposed here to address the problems of discretisation and

configuration errors in FE models, respectively. The differences between the predictions froma given FE model by using different mass matrix formulations are seen to be closely related to

the convergence range of the model prediction. The residuals between the experimental mode

shapes and those obtained by curve-fitting the experimental mode shapes with eigenvectors

predicted from the FE model are studied here. It is concluded that these residuals can reveal

the existence of configuration errors in the FE model. Case studies based on theoretical

models show that these two methods are efficient at distinguishing configuration errors from

parameter errors and estimating discretisation errors and the compensation for them.

KEYWORDS: model updating, model verification, model convergence, curve-fitting.

1 INTRODUCTION

Model updating is a step in model validation process that modifies the values of parameters in

an FE model in order to bring the FE model prediction into better agreement with

experimental data. In general, there are three kinds of error in an FE model which cause the

discrepancies between the model predictions and the experimental data. Discretisation errorscome from modelling a continuous structural system by a discrete numerical system.

Configuration errors arise when the structure is approximated with some unsuitable kinds of

element. When undertaking an updating step on an FE model with all these types of error, the

results from the procedure are usually not satisfactory either the parameters lose their

physical meaning, or the correlation with the experimental data does not achieve significant

improvement.

Model verification is another step in the process of model validation. In this step, the model is"debugged" to verify that it is modelled according to the initial requirement on the model. In

the viewpoint of model validation, model verification should provide a verified model which

can be updated to match the experimental data by only modifying the parameters of the model.

Thus, model verification addresses the discretisation and configuration errors.

In this paper, two methods of model verification one addressing the discretisation errors and

the other the configuration errors are proposed.

2 CONVERGENCE CHECK

In general, an FE model is a discrete numerical model of a continuous structural system. The

differences between the actual dynamic properties of the structural system and those predictedby the FE model, when there is no other error in the model, are discretisation errors of the

model. When this model is undertaken an updating procedure, the obtained updating

parameter values are the compensation for the discretisation errors [1]. If the discretisation
mailto:[email protected]:[email protected]:[email protected]


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errors are large, or the required dynamic property prediction from the model is beyond the

convergence range of the model, the compensation for the discretisation errors will distort the

physical meanings of the updating parameters. Thus, before undertaking an updating

procedure on an FE model, a convergence check on the model is critical.

Many people have written papers on the estimation of discretisation errors of FE models [2, 3,

4]. In this paper, the authors propose a simple and efficient way to estimate the discretisation

errors.

2.1 Lumped Mass and Consistent Mass Approaches

When performing a dynamic calculation with an FE model, a mass matrix will be formed for

each element. There are two different formulations for the mass matrix of most kinds of

element lumped mass matrix formulation and consistent mass matrix formulation [5].

For the lumped mass formulation, the mass of an element is simply divided and distributed

around all the grids of the element. For example, the mass matrix for a rod element is:

[ ]

=

10

01

2

lm

(1)

In this approach, the moment of inertia properties of the element will be greater than those of

the real structure. If there are no other errors in an FE model with only rod elements, the

predicted natural frequencies will be lower than those of the real structure. The estimated

eigenvalue error for the bar element is:

42

, 06

11

+

=

NNi

il

(2)

In the above equation, the subscript l in l,i is for lumped mass, and N is the number ofelements per wavelength. For a given model, the higher order of a mode, the smaller this

number will be. For a specific mode, the more elements a model has, the larger this number

will be. From the above equation, it can be seen that the lumped mass approach will produce

lower natural frequencies.

By the consistent mass formulation, the mass matrix of an element has off-diagonal

coefficients that couple the grids of the element. For a rod element, the mass matrix is:

[ ]

=

21

12

6

lm

(3)

In this way, not only the mass but also the higher mass moments of the element can berepresented accurately by the mass matrix construction. The estimated eigenvalue error for a

model with only rod elements is:

42

, 06

11

+

+=

NNi

ic

(4)

In the above equation, the subscript c in c,i is for consistent mass. This mass matrixapproach has an effect of stiffening the model.

If two normal mode solutions from the same model are obtained by choosing the two mass

matrix formulations separately, the natural frequency difference for the same mode is

approximately proportional to the discretisation error of the model for this mode. For a model

with only rod elements:


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42

,

, 03

11

+

+

NNil

ic

(5)

As the number of elements in a model approaches infinity, the above equation will approach

one, and the eigenvalues predicted for the same mode by the same model with the two mass

matrix approaches will be the same and equal to the "true" eigenvalue. When the number ofelements in a model is less than infinity, the predicted eigenvalues for a specific mode by the

same model with the two different mass matrix approaches are different from the "true"

eigenvalue. The eigenvalue obtained with the lumped mass approach is lower than the "true"

eigenvalue while that with the consistent mass approach is higher than the "true" eigenvalue.

The two-mass-matrix-approach can be applied to most types of element, such as beam and

plate elements. It is the same for a model with these types of element the two-mass-matrix-

approach for getting normal mode solution will result in two sets of eigenvalues and

eigenvectors, and the difference of the eigenvalues of the same mode from the different sets is

related to the discretisation error of the model for this mode. The eigenvectors obtained from

the two-mass-matrix-approach for the same mode are almost the same with a little differencein the amplitudes because of the mass normalisation process.

From the above analysis, it can be seen that the difference between the natural frequencies of

the same mode predicted by the same model with the two different mass matrix approaches

can be used as an indicator for model convergence. Once a threshold on natural frequency

differences is set for the convergence check, the frequency range over which the difference

between the natural frequencies of the same mode from the two predictions is smaller than the

threshold can be considered as the convergence range of the model.

2.2 Estimate of Compensation for Discretisation Errors

The above comparison can be also used to estimate the amplitude of the compensation of theupdating parameters for discretisation errors. As an example, a model with only rod elements

is considered here.

If all the physical parameters of the FE model are correct, and the model is to be updated in

order to reduce the discrepancies of the natural frequencies, the Youngs modulus (or the

mass density) of the model needs to be modified [1]. When the relative natural frequency

differences of the first mmodes are taken as comprising the residue in the updating equation,

the objective of the model updating procedure is to minimise:

=

=

m

i i

iR1

2

(6)

If only one parameter - the Youngs modulus (or the mass density) of all elements is

selected as the updating parameter, the final value of the parameter cannot make all modes

have zero natural frequency difference, but will shift the natural frequencies of all modes

down in the consistent mass case or up in the lumped mass case so that the residue R is

minimised. Thus, the modification of the updating parameter is:

=

m

i i

i

mE

E

1

2

(7)

Although the differences between the natural frequencies predicted by the model and those of

the structure are unknown, they are related to the differences between the natural frequencies

predicted by the model with the two different mass matrix formulations. Comparing equations

(2, 4) and (5), it can be shown that:


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i

il

i

ic

icl

icl

,,

,

,22 (8)

The bar over indicates as mean value. From equations (7) and (8), the compensation of theupdating parameter for the mesh size can be estimated as:

==

m

i icl

iclm

i i

ic

mmE

E

1 ,

,

1

, 12

(9)

In general, the relation between i/iand cl,i/ cl,iis not as clear as in the case of a modelof only rod elements and with the lumped mass and the consistent mass approaches. For

some kinds of element in NASTRAN, the mass matrix formulations for the coupled mass

approach are not the same as for the consistent approach. According to [5], the predicted

natural frequencies from the coupled mass approach have higher precision than those from

the lumped mass approach for most types of element. Thus, as a conservative estimation,

the compensation can be calculated by the following equation.

=

m

i icl

icl

mE

E

1 ,

,2

(10)

The subscript cin the above equation is for "coupled mass" which in NASTRAN takes the

place of "consistent mass" for most kinds of element. If several parameters are selected in the

updating procedure, the compensation for the discretisation errors will be different from the

above estimation. In this case, the estimation for the compensation can be obtained by an

updating procedure that takes the prediction from the model with one mass matrix approach

as the target and updates the model with the other mass matrix approach.

It is worth mentioning that the convergence check method proposed here is based on thediscretisation errors that are caused by the construction of mass matrices. The discretisation

errors that are caused by the construction of stiffness matrices will make the estimated

eigenvalues higher than the "true" eigenvalues. However, the discrepancies caused by these

errors are usually smaller than those caused by the construction of mass matrices, as shown in

equations (2, 4) in the case of rod elements. The difference between the natural frequencies of

the same mode predicted by a model with the two different mass matrix formulations will

cover the natural frequency discrepancy on the same mode caused by the construction of

stiffness matrices. Thus, the convergence check and estimation of the compensation proposed

here are conservative.

3 CONFIGURATION CHECK

When an FE model is constructed for predicting the dynamic properties of a structure, there

are usually some simplifications made when representing complicated parts in the structure by

standard elements in the model. Although in this construction process the simplifications are

made according to the experience of the modelling engineer, in general, the effects of the

simplifications on the dynamic properties of the model are unknown, or at least not clear for

some of them.

If a simplification made to the model has the capability to provide all the key features of the

corresponding part of the structure for predicting the required dynamic properties, even if the

predicted properties are not accurate, this simplification does not cause configuration errors.

However, if a simplification results in a loss of some key features and makes the model

unable to predict the required properties accurately, even by modifying parameters in the

model, this simplification results in the model having configuration errors. In general, if some

key features in the structure are missing in the FE model, and this makes the model unable to


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predict the dynamic properties in a specific frequency range with the required precision, the

discrepancies of the predicted properties from those of the structure cannot be reduced

significantly even by modifying the parameters in the model, and the model is said to have

configuration errors.

3.1 Eigenvector (Mode Shape) Curve-fitting Function

Any structure, no matter how simple or complicated, can be considered as a system that has

particular dynamic properties in a specific frequency range. Suppose that there is an FE model

that can produce the same dynamic properties in the range, this model is called the structural

modelin this paper.

An FE model for predicting the dynamic properties of that structure, called the analytical

model, can also be considered as a system that has its own properties. In general, this

analytical model would not be the same as the structural model that is defined above. The

differences between the dynamic properties of the structural model and those of the analytical

model are caused by uncertainties in the model parameters, the discretisation errors and

possibly the wrong configuration of the analytical model.

After a convergence check on an analytical model, the discretisation errors can be limited to a

range and reduced by the compensation of updating parameters. Thus, the object of the model

configuration check is to distinguish configuration errors from parameter errors of the

analytical model.

The aims of the method proposed here are (1) to determine whether there is any configuration

error, and (2) to try to find out the unsuitable simplifications in the FE model that are

considered to be configuration errors.

The dynamic properties of a structural model in a specific frequency range can be described

by the eigenvalues and eigenvectors in that range (plus the residual of the modes outside therange). Suppose that all the eigenvectors in the range can be expressed by a kind of functionwith the coordinate data of the DOFs as the variables. For the i

thelement in thej

theigenvector,

( ){ }

=

Nj

j

j

T

iii

X

ij

a

a

a

zyxf

2

1

,, (11)

( )iii

zyxf ,, called eigenvector (mode shape) curve-fitting function for the sake of

convenience, is the function with the coordinate data of the DOFs as the variables. akj (k

=1,2, , N) are constants used to curve-fit all elements in the jth

eigenvector. For the ith

element in the kth

eigenvector, the function has the same format but different constants.

( ){ }

=

Nk

k

k

T

iii

X

ik

a

a

a

zyxf

2

1

,, (12)

For all DOFs and all eigenvectors, it follows that:

[ ] { } { } { }( )[ ] [ ]AzyxFX

=,,

(13)


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where { } { } { }( )[ ]

( ){ }

( ){ }

( ){ }

=

T

nnn

T

T

zyxf

zyxf

zyxf

zyxF

,,

,,

,,

,, 222

111

and [ ]

=

Nm

m

m

NNa

a

a

a

a

a

a

a

a

A

2

1

2

22

12

1

21

11

.

For example, the eigenvectors for the out-of-plane displacement of a rectangular plate can be

expressed by polynomial functions with the variables as the two coordinates that determine

the position of a point in the plane of the plate. In another example, the eigenvectors for the

displacement of an axisymmetric ring in the radial direction can be expressed by

trigonometric functions with the spatial phase coordinate as the variable.

N in equation (11) is the number of the constants needed for the equation to express an

eigenvector. It is also the order of the eigenvector curve-fitting function, and it depends on the

complexity of the eigenvector and the format of the function used. Generally, an eigenvector

of a higher order needs a greater value ofNthan that of a lower order.

Actually, the eigenvector curve-fitting functions with constant vectors have the same basis asthe functions used in the Rayleigh-Ritz method, [6]. In the Rayleigh-Ritz method, the

functions are also functions of coordinates. When the constants in the functions are separated

in the form of equation (11),the eigenvector curve-fitting functions are obtained.

3.2 Configuration Check with Eigenvector Curve-fitting Functions

As one or more parameters of an FE model are changed, this will alter the eigenvectors

predicted by the model. The eigenvector curve-fitting functions will change as well. If the

change in the parameters is small, it will not change the basic features of the eigenvectors.

This means that the format and the order of the eigenvector curve-fitting functions may,

possibly, remain un-changed, and only the constants of the functions will be changed.However, if some key features of the model are changed or missing, it will affect the

eigenvectors significantly. Therefore, not only the constants but also the format or the order of

the eigenvector curve-fitting functions will change. This phenomenon is used to develop a

method for verifying FE models with configuration errors.

Consider a structure and an FE model ("the analytical model") that is constructed to predict

the dynamic properties of the structure. From a modal test on the structure, a set of mode

shapes can be obtained. From the analytical model, eigenvectors of some modes can be

predicted. Suppose that eigenvector curve-fitting functions can be found to express the mode

shapes from the experiment and the eigenvectors from the analytical model. Let the mode

shapes from the experiment be expressed as in equation (13),and the eigenvectors from theanalytical model be expressed as:

[ ] { } { } { }( )[ ] [ ]BzyxGA = ,, (14)

where [G] is a function matrix with the coordinates of DOFs as the variables and [B] is a

constant matrix with the same form as [A] in equation (13).

If the analytical model can predict the dynamic properties of the structure by having a small

change for the values of some of the parameters in the model, the eigenvector curve-fitting

functions for both the mode shapes from the experiment and the eigenvectors from the FE

model should have the same format and the same order. This is because a small change in

only the parameters would not change significantly the basic features of the eigenvectors.

Thus, we have:

{ } { } { }( )[ ] { } { } { }( )[ ]zyxGzyxF ,,,, (15)


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If there are enough modes included in equation (14),the matrix [G] can be obtained by:

{ } { } { }( )[ ] [ ] [ ]+= BzyxG A,, (16)

where, [ ]+ is the pseudo-inverse of the matrix. Combining the above two equations and

putting them into equation (13) gives the following:

[ ] [ ] [ ] [ ]

[ ] [ ]C

AB

A

AX

=

= +

(17)

The condition for achieving equality in the above equation is that the FE model can be made

to predict the dynamic properties of the structure just by modifying some of its parameters.

Thus, this equation gives a necessary condition for an FE model to be configuration error free.

If the above equation does not hold, the FE model has some configuration errors that make

the model unable to predict the dynamic properties of the structure described by the mode

shapes, even after modifying the parameter values in the model.

In this equation, the mode shapes from the experiment on the structure are expressed as linear

combinations of the eigenvectors predicted by the FE model. If the number of the

eigenvectors equals to the number of DOFs and the eigenvector matrix is of full rank, the

matrix [C] must exist to satisfy the equation. This will give false information for detecting

configuration error. In real cases, there must be some noise in the experimental mode shapes,

and this will make the equation not stand even when there is no configuration error.

Considering these two factors, the practical use of the method is described below.

For the experimental mode shapes of all modes in a given frequency range, the eigenvectors

predicted by an FE model in a little wider frequency range than the experiment frequency

range are selected. Putting the experimental mode shapes and analytical eigenvectors into

equation (17),the matrix [C] can be obtained as, by the least-squares method:

[ ] [ ] [ ]XAC = +

(18)

Then, a set of curve-fitted mode shapes can be calculated by the equation:

[ ] [ ] [ ]CAX = ~ (19)

The matrix [C] is called the "curve-fitting matrix" because it serves to curve-fit the mode

shapes. The correlation between the experimental mode shapes and the curve-fitted mode

shapes is obtained using the MAC, as follows:

[ ]

{ }{ }( ){ } { }( ){ } { }( )

X

i

TX

i

X

i

TX

i

X

i

TX

i

iiXXMAC

~~

~ 2

~

= (20)

The values of the diagonal elements of the MAC matrix give the curve-fit results. By

examining both the MAC matrix and the curve-fitting matrix, [C], it is possible to verify the

model with or without configuration errors.

First, consider the situation where there is no noise on the experimental mode shapes. If the

value of the ith

diagonal element of the MAC matrix is close to 1.0, and there are only one or a

few elements in the ith

column of [C] that have their absolute values much greater than other

elements in the same column, it can be said that the mode shape of the ith

mode can be curve-

fitted with the selected eigenvectors. Further, the analytical model, by which the eigenvectors

are predicted, has no configuration error or the configuration errors in the model do not affectthe ability of the model to predict this mode with a high accuracy.

If the value of the ith

diagonal element of the MAC matrix is close to 0.0, that means that the


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Figure 1. FE model for a plate

Figure 2. Convergence Chek

ith

mode shape cannot be curve-fitted at all by the selected eigenvectors. Thus, the model is

unable to predict this mode, even by modifying parameters in the model, and the model has

configuration errors.


diagonal element of the MAC matrix is somewhere between 0.0 and 1.0,

the value reflects the degree to which the ith

mode shape can be curve-fitted by the selected

eigenvectors.

Usually, a low MAC value, say lower than 0.8, means that even if many modes predicted by

the analytical model are included in curve-fitting the ith

experimental mode shape, the curve-

fitted shape is still not the same as the original one. Therefore, it is concluded that the model

has configuration errors that make the model unable to predict this mode.


diagonal element of the MAC matrix is between 0.8 and 1.0, and there is

only one element in the ith

column of [C] that has a larger absolute value than the other

elements in the same column, there is a possibility that not enough eigenvectors have been

selected to curve-fit this mode.

When there is noise on the experimental mode shapes, the values of the diagonal elements inthe MAC matrix will drop to some extent, and the number of elements in columns of the

matrix [C] that have larger absolute values than others will increase. Thus, when using this

method in practical cases, these effects should be taken into account.

4 CASE STUDIES

4.1 Case study for convergence check

An FE model for a flat plate is shown in Figure 1. There are

150 4-grid shell elements in the model. 20 flexible modes

were predicted by this model which are in the frequency

range up to 400Hz. Using the convergence check proposed in

this paper, the frequency differences of the modes by the two

mass matrix formulations are plotted in Figure 2.

From this figure, it can be easily seen that only the first

fourteen modes predicted by the model have the natural

frequency differences smaller than 10%, while the natural

frequency differences for the other modes are greater than

10%.

If we set the threshold to 6%, the model can only be

considered as a converged model to predict the first eight

modes. It cannot be used to predict other modes in the senseof model convergence.

Equation (10) was used to estimate the

compensation of updating parameters

for the discretisation errors. For the

first eight modes it resulted in the

following:

071.08

2 8

1 ,

, =

=i icl

icl

E

E

This value is the estimation for the casewith one updating parameter

representing the Youngs module of all


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elements in the model.

Another FE model for the same plate was formed with the same geometrical and physical

parameters as the model in Figure 1 but with 3750 4-grid shell elements, which is 5 times

finer in each planar direction than the model shown in Figure 1. Taking the first eight modes

predicted by this fine model as an "experimental" data set, two model-updating procedures

were undertaken on the model shown in Figure 1.

In the first procedure, there was one updating parameter, which represented the Youngs

modules of all elements. The obtained updating parameter value was 0.057. It is reasonably

close to the estimated value of 0.071.

In the second procedure, three updating parameters were selected. An updating procedure for

estimating the compensation of these parameters was undertaken as described in Section 2 of

this paper. This procedure resulted in an estimation of the updating parameter compensations

as (0.156, -0.022, 0.107). The updating procedure, which took the eigenvalues from the fine

model as the target, resulted in the updating parameter values (0.124, -0.0082, 0.0818).

Comparing with the estimation, it can be seen that the estimated values are relatively close to

the obtained updating parameter values. This is consistent with the analysis described in this

paper.

4.2 Case studies for configuration check

Each case study for the configuration check

uses two FE models with the same mesh size

for avoiding discretisation errors. Figure 3

shows the models. There are two flat plates

(modelled by 4-grid shell elements) joined

by some connecting elements. The structural

model provides a set of "experimental"mode shape data for checking the

configurations of the analytical models.

The analytical model in each case may have

different parameter values for elements or a different type of element for the joints in order to

introduce parameter errors or configuration errors into the model. The parameter values (and

the type for the connecting elements) in all models are listed in Table 1. Compared with the

structural model, Analytical models 1 and 2 have no configuration error but do have

parameter errors, while Analytical model 3 has a configuration different from the structure

model but no parameter errors.

Table 1. Parameters for case studies of configuration check

ParaStructure

modelAnalyticalmodel 1

Analyticalmodel 2

Analyticalmodel 3

E 2.06*1011

3.09*1011

2.06*1011

2.06*1011

Group A

7800 7800 7800 7800

E 2.06*1011

2.06*1011

2.06*1011

2.06*1011

Group B

7800 7800 7800 7800

kx 1.0*10

6

1.0*10

6

1.0*10

6

ky 1.0*106 1.0*106 1.0*106

Connecting

Elements

kz 2.0*104 2.0*10

4 8.0*10

4

MPC

Figure 3. FE model for configuration check


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The configuration check for the three analytical models described above was undertaken with

the results shown in Figures 4 to 9. Figures 4, 6, and 8 are the MAC matrices between the

eigenvectors predicted by the structural model and those predicted by the analytical models.

From these MAC matrices, it is difficult, if not impossible, to identify which analytical model

has configuration errors.

Figures 5, 7, and 9 are the MAC matrices between the initial eigenvectors from the structure

model and those curve-fitted by eigenvectors from the analytical models.

From Figure 5 it can be seen that the first 36 "experimental" mode shapes can be curve-fitted

by the eigenvectors predicted by Analytical model 1 with MAC values higher than 80%. Thus,

for these 36 modes, Analytical model 1 has no configuration error. The reason why the MAC

values for the last few modes are small is that only 40 analytical eigenvectors were used in the

curve-fitting process.

The significant change in the stiffness of the connecting springs in Analytical model 2 causes

poor correlation with very low MAC values of some modes when compared with those fromthe structural model as shown in Figure 6. However, the correlation between the initial

"experimental" mode shapes and the those curve-fitted by eigenvectors from this analytical

model, as shown in Figure 7, has high MAC values in the diagonal elements of the MAC

matrix. This indicates that the analytical model has no configuration errors when this model is

used to predict the first 40 modes of the Structure Model.

Figure 4. Mode shape correlation

(Structural model vs. Analytical model 1) Figure 5. Mode shape correlation

(curve-fitted by Analytical model 1)


(Structural model vs. Analytical model 2)Figure 7. Mode shape correlation



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From Table 1 it is known that the configuration of Analytical model 3 is different from that ofthe structural model. In the MAC matrix shown in Figure 9, there are some diagonal elements

with values lower than 80% and some of them are even lower than 40%. Figure 10 shows the

"experimental" mode shape of Mode 13 and the one curve-fitted by the eigenvectors from

Analytical model 3. It can be seen that the analytical model cannot predict the relative

displacements of the two plates around the joints. Thus, the MAC matrix shown in Figure 9

indicates that the analytical model has configuration errors. By comparing the "experimental"

mode shapes and the curve-fitted ones that have low MAC values, it is possible to point out

unsuitable elements in the analytical model.

Figure 10. The initial (left) and the curve-fitted (right) mode shapes for Mode 13

5 COMCLUSION

Estimating discretisation errors and distinguishing configuration errors from parameter errorsare two important requirements before an FE model can undertake the model-updating

procedure in a model validation process. Two methods are proposed in this paper to address

these requirements separately. Through case studies based on theoretical models, it can be

seen that the methods proposed here are efficient to determine whether an FE model fulfils

these requirements. Furthermore, by using the method for configuration check, it is possible

to identify unsuitable elements in the model that contribute to configuration errors of the

model.

6 ACKNOWLEDGEMENT

The authors of the paper gratefully acknowledge the financial and technical support from

Rolls-Royce plc. for this project.

7 REFERENCES

[1] G. Chen & D.J. Ewins A Perspective on Model Updating Performance, 18th

IMAC,


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February 2000

[2] J.E. Mottershead, M.I. Friswell & Y. Zhang On Discretisation Error Estimates for Finite

Element Model Updating, Modal Analysis: The International Journal of Analytical and

Experimental Modal Analysis, 11(3&4) December 1996, p 155-164

[3] H. Ahmadian, M.I. Friswell & J.E. Mottershead, Minimisation of the Discretisation Errorin Mass and Stiffness Formulations by an Inverse Method, International Journal for

Numerical Methods in Engineering, Vol. 41, 1998, p 371-387

[4] O.C. Zienkiewicz & J.Z. Zhu, A Simple Error Estimator and Adaptive Procedure for

Practical Engineering Analysis, International Journal for Numerical Methods in Engineering,

Vol. 24, 1987, p 337-357

[5] R. H. MacNeal, The NASTRAN Theoretical Manual,The MACNEAL-SCHWENDLER

Corporation, December 1972

[6] M. Geradin and D. Rixen, Mechanical Vibration Theory and Application to Structural

Dynamics, second Edition, John Wiley & Sons, NY, USA 1997

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