Paper63 lmspolymaxcurvefitter

A POLY-REFERENCE IMPLEMENTATION OF THE LEAST-SQUARES COMPLEX FREQUENCY-DOMAIN ESTIMATOR (1)

(1)

(1)

Patrick Guillaume , Peter Verboven , Steve Vanlanduit , Herman Van der Auweraer(2) and Bart Peeters(2) (1)

Vrije Universiteit Brussel (VUB) Acoustics & Vibration Research Group (AVRG) Department of Mechanical Engineering Pleinlaan 2, B-1050 Brussel, BELGIUM http://www.avrg.vub.ac.be (2)

LMS International Interleuvenlaan 68, Researchpark Haasrode Z1 B-3001 Leuven, BELGIUM http://www.lmsintl.com

ABSTRACT The Least-Squares Complex Frequency-domain (LSCF) estimator can be viewed as a frequency-domain implementation of the well-known Least-Squares Complex Exponential (LSCE) estimator. An important advantage of the LSCF estimator is the fact that it produces “fast-stabilizing” stabilization charts. In this contribution, the LSCF estimator will be generalized to a “poly-reference” estimator.

1. INTRODUCTION Recently, the Least-Squares Complex Frequency-domain (LSCF) estimator has been introduced and applied in modal analysis. The LSCF estimator can be viewed as a frequency-domain implementation of the well-known LSCE estimator [1-2]. The LSCF estimator has several advantages: (1) the use of frequency-dependent weighting functions (the inclusion of weights in the LeastSquares cost function allows to improve accuracy of the estimates) [3-5]; (2) beside the Least Squares implementation, the LSCF estimator can easily be adapted to more sophisticated solvers such as the Generalized Total Least-Squares implementation [6]; (3) and maybe the most important advantage of the LSCF estimator is the fact that it produces “faststabilizing” stabilization charts. [7-8]

model is used. This implies that only poles are available during the construction of the stabilization diagram.

In this contribution, the LSCF estimator will be generalized to a “poly-reference” estimator. This can be realized by means of a so-called Right Matrix-Fraction Description (RMFD) [5, 9]. With this approach, the modal participation factor can be estimated directly together with the poles. In Section 2.1 the derivation and main numerical aspects of the LSCF estimator will be summarized. In Section 2.2 the generalization to the poly-reference version will be given. In Section 3 the advantages of a poly-reference implementation of the LSCF estimator will be illustrated and finally the conclusions will be drawn.

2. FREQUENCY-DOMAIN IDENTIFICATION 2.1 Least Squares Complex Frequency (LSCF) 2.1.1

Common-Denominator Transfer Function Model

The relationship between output o ( o = 1,!,No ) and input i ( i = 1,!, Ni ) can be modeled in the frequency domain by means of a common-denominator transfer function

N (ω ) Hˆ k (ω ) = k d (ω )

(1)

for k = 1,!,No Ni (where k = (o − 1)Ni + i ) and with n

In the existing implementation of the LSCF estimator, a multivariable common-denominator transfer function

Nk (ω ) = ∑ Ω j (ω )Bkj j =0

(2)

the numerator polynomial between output o and input i and n

d (ω ) = ∑ â„Ś j (ω ) Aj

(3)

j =0

the common-denominator polynomial. The coefficients Aj and Bkj are the parameters to be estimated. All these

This nonlinear least-squares problem can be approximated by a (sub-optimal) linear least-squares one.

Indeed, by multiplying ξ kNLS (ω f , ) with d (ω , ) , one obtains an equation error that is linear in the parameters

ξ kLS (ω f , ) = d (ω f , ) ⋅ ξ kNLS (ω f , ) = Wk (ω f ) (Nk (ω f , n

k

j =0

ℌ j (ω )

are

(

)

Because the equations (8), for f = 1,!,Nf , are “linear-in-

 A0    A  =  1 #   An 

(4)

the-parameters�, they can be reformulated in matrix notations as

LS k

Several choices are possible for the polynomial basis functions ℌ j (ω ) . For a discrete-time domain model, the functions

usually

given

by

 ξ kLS (ω1,  ( )= ξ LS (ω , Nf  k

)   =  ) 

k

  ⋅  

k

Linearity in the Parameters

Starting from measured FRFs, Hk (ω f ) (with k = 1,!,No Ni and

 = 

k

 Wk ( ω1 )[ ℌ0 (ω1 ), ℌ1 (ω1 ), , ℌn (ω1 )]  Xk =  W (ω )[ ℌ (ω ), ℌ (ω ), , ℌ (ω 0 Nf 1 Nf n Nf  k Nf

for a continuous-time domain model ℌ j (ω ) = (iω ) j . The bad numerical conditioning of the continuous-time domain approach can be improved by using for instance orthogonal Forsythe polynomials (at the expense of an increase of the computation time).

k

 ⋅ 

k

  

(9)

with

â„Ś j (ω ) = exp( −iωTs â‹… j ) (with Ts the sampling period) while

2.1.2

(8)

= Wk (ω f )∑ â„Śj (ωf )Bkj − â„Śj (ωf )Aj Hk (ωf )

coefficients are grouped together in one column vector = [ T1 ,!, TNo Ni , T ]T with Bk 0    B  =  k1  ,  #   Bkn 

) − d (ωf , )Hk (ω ))

k

   )]

 −Wk ( ω1 )[ â„Ś0( ω1 )Hk (ω1 ), , â„Śn (ω1 )Hk (ω1 )]  Yk =   −W (ω )[ â„Ś (ω )H (ω ), , â„Ś (ω )H (ω 0 Nf k Nf n Nf k Nf  k Nf

(10)

   (11) )]

f = 1,!,Nf ), estimates of the transfer-function

coefficients can be obtained by minimizing the following nonlinear least-squares (NLS) cost function with respect to the parameter vector

NLS

( )=

No Ni Nf

âˆ‘âˆ‘Îľ k =1 f =1

NLS k

(ω f , )

2

(5)

where the (weighted) NLS equation error, ξ kNLS (ω f , ) , is

2.1.3

Reduced Normal Equations

The (weighted) linear least-squares estimates are found by minimizing A LS ( ) = =

Îľ

âˆ‘âˆ‘Îľ k =1 f =1

LS k

∑ Re ((

 N (ω , )  (ω f , ) = Wk (ω f )  k f k − Hk (ω f )  d ( ω , ) f  

(6)

with Wk (ω f ) an arbitrary weighting functions. The quality of the estimate can often be further improved by using an adequate weighting function such as Wk (ω f ) =

1 var {Hk (ω f )}

(7)

which takes the quality of the measured FRFs into account: FRF measurements with a small variance, var{Hk (ω f )} , have an important contribution to the cost function while noisy FRF measurements are penalized.

=

(ω f , )

No Ni

k =1

defined as NLS k

No Ni Nf

No Ni



∑   k =1



LS k

T k

( ) T

2

)

H

â‹…

LS k

R  ⋅  Tk  S  k

( )

)

Sk   ⋅ Tk  

(12) k

   

with R k = Re(X Hk X k ) , Sk = Re( X Hk Yk ) and Tk = Re(YkH Yk ) .

Note that cost function (12) is equivalent to

LS

( )=

T

â‹… Re(

) â‹… H

(13)

with J the so-called Jacobian matrix, which is given by  X1 0  0 X2 J=    0 0

0 0 X No Ni

Y1   Y2    YNo Ni 

(14)

The Jacobian matrix J of this least-squares problem has Nf No Ni rows and (n+1)(No Ni +1) columns (with Nf >> n, where n is the order of the polynomials).

In the minimum of the cost function the derivatives of (12) with respect to the unknown coefficients k and have to be zero ∂ ∂

A LS ( ) = 2 (5 k

k

+ 6k

)= ,

k = 1,!,No Ni

(15)

k

∂ ∂

LS

 No Ni ( ) = 2 ∑  k =1

(

Substitution of (15),

T k

k

k

+

=−



) =

k

−1 k



k

 No Ni  T −1 2 ∑ Tk − Sk R k Sk  ⋅  k =1 

(

)

(

)

⋅

(16)

=

⋅

=

S1   1    S2   2   #  H # ⋅  = 2Re( J J) ⋅ =    No Ni N N Tk   o i  ∑    k =1  

(18)

Fast Implementation of the Reduced Normal Equations

entries of these matrices equal

(

 Nf 2 Tk (r , s ) = Re  ∑ Wk (ωf )Hk (ω f ) ⋅ ei2π ( r −s ) f f 1 = 

 Nf  2 Rk (r , s ) = Re  ∑ Wk (ωf ) ⋅ Ω rH−1(ω f )Ω s −1(ω f )   f =1 

N

)

(20)

)

One can readily verify that the above matrices have a Toeplitz structure, i.e. entry (r , s ) only depends on (r − s ) . For instance  sk (0) sk (−1) % sk ( −n )   s (1) sk (0) % %  Sk =  k  % % % sk (−1)    % sk (1) sk (0)  sk (n)

(

 Nf 2 with sk (r ) = − Re  ∑ Wk (ω f ) Hk (ω f ) ⋅ ei2π r f N f 1 =  For symmetric Toeplitz matrices (such as

(21)

) R k and Tk )

only the first column has to be computed. Moreover, these entries can be computed in a time-efficient way by means of the Fast Fourier Transform (FFT) algorithm.

Solving the Reduced Normal Equations

To remove the parameter redundancy of transfer function model (1) (and to avoid the trivial solution with all coefficient equal to zero), a constraint has to be imposed on the coefficients. This can be done, for instance, by imposing that one of the coefficients is equal to a nonzero constant value. Assume, for instance, that the last coefficient of is constrained to 1 (i.e. (n + 1) = 1 ). In that case, the “reduced” normal equations become A⋅x = b

(22)

with (23)

b = −M(1: n, n + 1)

)

(

 Nf  2 Sk (r , s ) = − Re  ∑ Wk (ω f ) Hk (ω f ) ⋅ Ω Hr −1(ω f )Ω s −1(ω f )  (19)  f =1 

(

N

A = M(1: n,1: n)

)

(

)

(

Examining the matrices R k = Re(X Hk X k ) , Sk = Re( X Hk Yk ) , and Tk = Re(YkH Yk ) in more details reveals that the

N

 Nf 2 Sk (r , s ) = − Re  ∑ Wk (ω f ) Hk (ω f ) ⋅ ei2π ( r −s )f  f =1

2.1.5

The size of the square matrix M in the “reduced” normal equations (17) is n+1, and thus much smaller than the size of Re(JH J) in (18).

2.1.4

(

 Nf 2 Rk (r , s ) = Re  ∑ Wk (ωf ) ⋅ ei2π ( r − s )f f 1 = 

(17)

Equations (15) and (16) are the so-called normal equations, which are usually formulated as  R1 0 " 0 R 2  2 # %  ST ST " 2  1

rewritten as

, in (16) yields

with M = 2∑ k =o 1 i Tk − STk R k−1Sk .

N N

∆ω = 2π NTs ), then, the above summations can be

)

 Nf  2 Tk (r , s ) = Re  ∑ Wk (ωf )Hk (ω f ) ⋅ ΩrH−1(ω f )Ωs −1(ω f )   f =1 

If a discrete time-domain model is used, i.e. Ω j (ω f ) = exp(−iω f Ts ⋅ j ) , and if the frequencies are uniformly distributed (i.e. ω f = f ⋅ ∆ω , f = 1,!,Nf , with

The least-squares estimate of

is given by

x  ˆ LS =    1 with x = A −1 ⋅ b . Once ˆ LS is known,

(24)

k

=−

−1 k

k

⋅

(see

(15)) can be used to derive all ˆ LS,k coefficients. This approach, which takes into account the structure of the normal equations, is much faster than solving (18) directly (approximately No2 Ni2 times faster).

2.1.6

Stabilization Chart

In modal analysis, a stabilization chart is an important tool that is often used to assist the user in separating the physical system poles from mathematical ones [1]. A stabilization chart is obtained by repeating the analysis for increasing model order n. For each model order, the poles are calculated from the estimated denominator coefficients. The stable poles (i.e. the poles with a negative real part in the Laplace domain) are then presented graphically for ascending model order in a socalled “stabilization chart” (see Figure 1). On the vertical axis the model order is given; the horizontal axis represents the damped natural frequency of the estimated stable poles. (a)

The symbols in Figure 6 are used to denote how well poles are stabilizing. E.g., the symbol ‘s’ means that the variation over consecutive model orders of the damped natural frequency is smaller than 1% while the damping ratio varies with less than 5%.

Estimated poles corresponding to physically relevant system modes tend to appear for each estimation order at nearly identical locations, while the so-called mathematical poles, i.e. poles resulting from the mathematical solution of the normal equations but meaningless with respect to the physical interpretation, tend to jump around. These mathematical poles are mainly due to the presence of disturbing noise on the measurements.

In Figure 1 the stabilization chart of the LSCE estimator is compared with the LSCF estimator. It turns out that in many applications, the frequency-domain estimator is able to generate quite clear stabilization diagrams compared to the LSCE approach.

In the common-denominator implementation of the LSCF estimator only poles are available during the construction of the stabilization diagram. In the next section, the LSCF estimator will be generalized to a “poly-reference” estimator. This can be realized by means of a so-called Right Matrix-Fraction Description (RMFD). With this approach, the modal participation factor can be estimated directly together with the poles.

(b) Figure 1. Stabilization chart obtained with (a) a timedomain estimator (LSCE) and (b) the frequency-domain least-squares estimator.

for output o = 1,!,No with n

No (ω ) = ∑ Ω j (ω )Boj

(26)

j =0

the numerator row-vector ( No (ω ) ∈ 1×Ni ) and

polynomial

of

output

o

2.2 Poly-reference LSCF n

2.2.1

D(ω ) = ∑ Ω j (ω )A j

Right Matrix-Fraction Description

(27)

j =0

The relationship between output o ( o = 1,!,No ) and all

the denominator matrix polynomial ( D(ω ) ∈ Ni ×Ni ). The

inputs can be modeled in the frequency domain by means of right matrix-fraction description (RMFD) [9]

matrix coefficients A j and Boj are the parameters to be

ˆ (ω ) = N (ω ) ⋅ D −1(ω ) H o o

(25)

estimated. All these coefficients are grouped together in one matrix = [ T1 ,!, TNo , T ]T with

o

2.2.2

Bo 0    B  =  o1  ,  #  Bon 

with ⊗ the Kronecker product and INi the identity matrix.

 A0    A  =  1  #   An 

(28)

2.2.3

Reduced Normal Equations

The (weighted) linear least-squares problem is found by minimizing No

Linearity in the Parameters

A LS ( ) = ∑ tr o =1

Estimates of the transfer-function matrix coefficients can be obtained by minimizing the following nonlinear leastsquares (NLS) cost function with respect to the parameter matrix No Nf

NLS

( ) = ∑∑ tr o =1 f =1

((

(ω f , )

NLS o

)

H

â‹…

NLS o

(ω f , )

)

with tr(â‹…) the trace operator and where the (weighted) NLS o

NLS equation error,

NLS o

(ω , ) = W (ω ) ( f

o

f

o

(ω f ,

o

)â‹…

−1

)−

(ω f ,

o

(ω f )

)

(30)

obtains an equation error that is linear in the parameters k ) − +o (ω f ) â‹… '(ω f , ) )

n

(31)

(

= Wo (ω f )∑ â„Śj (ωf )â‹… Boj − â„Śj (ωf )Ho (ω f ) â‹… A j j =0

)

the-parameters�, they can be reformulated in matrix notations as

(ω1, )    =  LS  o (ω Nf , )  LS o

o

with

(

 W (ω )  ℌ (ω ) o 1  0 1   X0 =   Wo (ω N )  ℌ0(ω N ) f f  

(

(

 − W (ω )  â„Ś (ω ) o 1  0 1   Y0 =   − Wo (ωN )  â„Ś 0(ω N ) f f  

(

o

  ⋅  

o

 = 

o

 ⋅ 

o

  

LS

( ) = tr

(

LS o

( )

)    

(35)

So = Re( X Ho Yo ) ,

and

R  ⋅  To  S  o

â‹… Re(

T

(32)

So   ⋅ To  

o

) â‹…

)

H

(36)

Y1   Y2    YNo 

0 0

X No

(37)

In the minimum of the cost function the derivatives of (35) with respect to the unknown matrix coefficients o and have to be zero ∂ ∂ ∂

Because the equations (31), for f = 1,!,Nf , are “linear-in-

  LS ( ) =  o  

T

 X1 0  0 X2 J=    0 0

∂

(ω f , ) ⋅ '(ω f , )

= Wo (ω f ) (No (ω f ,

T o

â‹…

with J the so-called Jacobian matrix, which is given by

This nonlinear least-squares problem can be approximated by a (sub-optimal) linear least-squares one.

Indeed, by right multiplying NLS o (ω f , ) with D(ω f , ) , one NLS o

H

To = Re( Y Yo ) . Note that cost function (35) is equivalent

to

the o-th row of the FRF matrix.

(ω f , ) =

)

H o

with Wo (ω f ) an arbitrary weighting function and Ho (ω f )

LS o

( )

R o = Re( XoH X o ) ,

(ω f , ) , is now a row-vector

defined as

LS o

No  = ∑ tr    o =1 

with (29)

((

A LS ( ) = 2 (5 o

o

+ 6o

)= ,

o = 1,!,No

(38)

o

LS

 No ( ) = 2 ďŁŻâˆ‘  o =1

(

Substitution of (38),

T o

o

o

+

=−



o

) =

o

 No  T −1 2∑ To − So R o So  â‹…  o =1 

)

(

)

(39)



−1 o

(

â‹…

, in (39) yields =

â‹…

=

(40)

with M = 2∑ o =o 1 To − STo R o−1So .

N

Equations (38) and (39) are the so-called normal equations, which can also be formulated as

)

ℌn (ω1 ) ⊗ INi     ℌ 0(ω Nf ) ⊗ INi  

(33)

)

)

ℌn (ω1 ) ⊗ Ho ( ω1 )    (34)  ℌ0(ω Nf ) ⊗ Ho (ω N ) 

)

f

 R1 0 " 0 R 2  2 # %  ST ST " 2  1

  1      2   â‹…  #  = 2Re( JH J) â‹… =    No N To   o  ∑    o =1  S1 S2 #

(41)

The size of the square matrix M in the “reduced� normal equations (40) is Ni (n + 1) , and thus much smaller than the size of Re(JH J) in (41).

2.2.4

Solving the Reduced Normal Equations

To remove the parameter redundancy of the RMFD transfer function model (25) (and to avoid the trivial solution with all matrix coefficient equal to zero), a constraint has to be imposed on the matrix coefficients.

This can be done, for instance, by imposing that one of the denominator matrix coefficients is equal to a regular constant matrix. Assume, for instance, that the last matrix coefficient of is constrained to the identity matrix (i.e. (n + 1) = Ni ).

In that case, the “reduced” normal equations become

A⋅X = B

(42)

with A = M(1: Ni n,1: Ni n )

m

⋅

T m

(46)

This is only possible when the residue matrix R m is of rank one. If this is not the case (e.g., pole with multiplicity 2 or more), then it is not possible to uniquely derive the corresponding mode shape and modal participation vector.

When poles and modal participation vectors are a-priori known (e.g., poly-reference LSCF and LSCE estimators), the well-known Least-Squares Frequency-Domain (LSFD) estimator can be used to directly estimate the mode shapes, m , occurring in [1]

Nm ∗  ⋅ T ⋅ H H(s ) = ∑  m m + m ∗m s − λm m =1  s − λm

 LR  + 2 + UR  s

(47)

is given by

X ˆ LS =   INi 

(44)

−1

with X = A ⋅ B . Once ˆ LS is known,

o

=−

−1 o

o

⋅

(see

(38)) can be used to derive all ˆ LS,o matrix coefficients.

This approach, which takes into account the structure of the normal equations, is much faster than solving (41) directly.

From the knowledge of the denominator matrix coefficients it is now possible to compute the poles and corresponding modal participation factors in a similar way as for the LSCE estimator, i.e., by computing the eigenvalue decomposition of the so-called companion matrix [1-2, 9].

2.3 Mode Shape Estimators Once the poles (LSCF) or the poles together with the modal participation factors (poly-reference LSCF) have been determined by means of a stabilization diagram, the mode shapes still have to be estimated.

2.3.1

Rm =

(43)

B = −M(1: Ni n,Ni n + 1: Ni (n + 1))

The least-squares estimate of

modal participation factors can be obtained by means of a Singular Value Decomposition (SVD)

3. ESTIMATION RESULTS In this section the LSCF estimator will be compared to the Poly-reference LSCF estimator. To do so, simulated data will be used. A system with 12 outputs and 2 inputs has been considered that contains 10 modes in the frequency band 0–500 Hz. Disturbing noise was added to the synthesized FRFs resulting in FRF data with a constant signal-to-noise ratio of 30 dB. In Table 1 the exact poles are given together with the estimated ones. Note that there are 2 pairs of close modes. Indeed, the relative difference between pole 3 and 4 is 4 % while the relative difference between pole 7 and 8 is 0.1 %.

In Figures 2 and 3 the stabilization charts of respectively the LSCF and the Poly-reference LSCF estimator are given. Both stabilization charts are well ‘stabilized’ such that it is quite straightforward to select the ‘physical’ modes. In Figures 4 and 5 a zoom around 0.1 kHz is given (containing poles 3 and 4). Both estimators are able to estimate the 2 close poles.

Least-Squares Frequency-Domain Residue (LSFDR) estimator

Table 1: Estimated poles (in k rad/s).

When the poles are available (LSCF), the residues occurring in Nm  Rm R ∗m H(s ) = ∑  + s − λm∗ m =1  s − λm

 LR  + 2 + UR  s

(45)

can be estimated with a Least-Squares FrequencyDomain estimator (LSFDR). From these estimated residues, R m , the corresponding mode shapes and

1 2 3 4 5 6 7 8 9 10

Exact Poles -0.0110+0.3028i -0.0118+0.3593i -0.0115+0.6283i -0.0106+0.6551i -0.0102+1.1474i -0.0110+1.5541i -0.0090+2.0311i -0.0101+2.0334i -0.0107+2.4863i -0.0092+2.7966i

LSCF -0.0110+0.3028i -0.0117+0.3594i -0.0114+0.6283i -0.0105+0.6551i -0.0102+1.1474i -0.0109+1.5541i -0.0090+2.0321i -0.0106+2.4860i -0.0090+2.7962i

Poly-ref. LSCF -0.0109+0.3028i -0.0116+0.3593i -0.0114+0.6284i -0.0104+0.6552i -0.0102+1.1474i -0.0109+1.5541i -0.0085+2.0312i -0.0100+2.0333i -0.0107+2.4861i -0.0092+2.7963i

Figure 2: Stabilization chart of the LSCF estimator.

Figure 4: Stabilization chart of the LSCF estimator (zoom around 0.1 kHz, poles 3 and 4).

Figure 3: Stabilization chart of the Poly-reference LSCF estimator.

Figure 5: Stabilization chart of the Poly-reference LSCF estimator (zoom around 0.1 kHz, poles 3 and 4).

In Figures 6 and 7 a zoom around 0.3234 kHz is given (i.e., poles 7 and 8). In this case, only the Poly-reference LSCF estimator finds both poles. The LSCF estimator finds only one stable pole in between the two exact poles.

LSCF-LSFD approach is to be preferred for very closely coupled modes.

The MAC values between the exact mode shapes and the mode shapes resulting from the LSCF poles (see Table I) followed by the LSFDR approach are given in Figure 8. Figure 8 shows that the estimated mode shape corresponding with pole -0.0090+2.0321i (see Table I, LSCF) does not correlate well with the exact mode shape of the 2 adjacent poles 7 and 8.

The MAC values between the exact mode shapes and the mode shapes resulting from the Poly-reference LSCF poles (see Table I) followed by the LSFD approach are given in Figure 9. The correlation is now very good for all modes. A comparison of the synthesized FRFs is given in Figure 10. One concludes that the Poly-reference

4. CONCLUSION A Poly-reference implementation of the Least-Squares Complex Frequency-domain (LSCF) estimator has been given. This poly-reference estimator produces quite clear stabilization charts (comparable to the original LSCF estimator and usually much easier to interpret than the stabilization chart produced by the LSCE estimator [7-8]).

The Poly-reference LSCF implementation is superior for close-coupled modes resulting in improved modal parameter estimates.

Figure 6: Stabilization chart of the LSCF estimator (zoom around 0.3234 kHz).

Figure 9: MAC values between exact mode shapes and the ones from the Poly-reference LSCF approach.

Figure 7: Stabilization chart of the Poly-reference LSCF estimator (zoom around 0.3234 kHz, poles 7 and 8).

Figure 10: Synthesized FRF between response 12 and input 1. Measured FRF (dots); LSCF-LSFDR approach (dashed line); Poly-reference LSCF-LSFD approach (solid line).

ACKNOWLEDGEMENTS

Figure 8: MAC values between the exact mode shapes and the ones resulting from the LSCF approach.

This research has been supported by the Fund for Scientific Research – Flanders (Belgium) (FWO); the Concerted Research Action “OPTIMech” of the Flemish Community; the Flemish Institute for the Improvement of the Scientific and Technological Research in Industry (IWT); and the Research Council (OZR) of the Vrije Universiteit Brussel (VUB).

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