Model assessment

Chapter 5. Applicability of honeycomb as a sacrificial composite to resist low-

5.5 Probabilistic contact force model

5.5.4 Model assessment

However, it is evident that not all the candidate variables substantially upsurge the model accuracy. A probabilistic model needs to be parsimonious comprising minimum set of regressors. The insensitive regressors are deleted using step-wise element deletion technique to reduce the model complexity (Gardoni et al. 2002). Model parameters having coefficient of variation (c.o.v) significantly greater than standard deviation (



_D) of the model error does not contribute to its accuracy. Hence, these independent variables can be conveniently deleted for the model simplicity. For the first trial, the model is assessed considering all the assumed variables as listed in Table 5.4. The largest c.o.v (



₃) is found to be 0.69 while the standard deviation of the model error is 0.19. The model is stepwise reassessed by deleting the parameters have c.o.v greater than the standard deviation of model. The parameter deletion step is terminated when there is significant increase in



_D. Figure 5.9 tracks the changes in c.o.v and



_D during the stepwise deletion process. In step 5, the model parameter with largest c.o.v is 0.25, which is approximately equal to the



_D (0.21). The final regression model includesX₁,X₂,X₃, and,X₄ terms, as any further parameter deletion increases



Dconsiderably. The least square estimates of the model parameter

 

^ˆ and their standard deviation are summarized in Table 5.5.

.

characterisation

Table 5.4 Tabulation of explanatory regressor functions

Table 5.5 Estimates of Model Parameters

Variable Mean Standard error



1 0.0590 0.0075



2 -0.8017 0.1090



4 -0.0388 0.0099



D 0.2114 0.0216

Regressor Variable Expressions Nomenclature

Constant bias

Xo 1

Velocity ratio X₁

1 c s

X V

V

1

1 2

s o

V M V

M M

  ;

0 c

c

V q c

 ;

f o

alu

c E

 

Vs-Velocity of composite structure

Vc-Deformation speed of the core

c0- Speed of sound wave

Slenderness

ratio X² 2

c

X c

h c- cell size

hc- height of core

Strength

ratio X³ 3

c

X q



q- crushing strength

c- yield strength

Stiffness ratio X₄ 4 b m

X K

D K^b-bending stiffness

Dm-membrane rigidity

Mass ratio X₅ 5 ^{1 2}

2

M M

X M

 

M1- mass of the impactor

M2-mass of the panel

Resonance-

Time ratio ⁶

X

6 c

X T X₄

1 2

1 2

( )

s M M

 M M^ ;

2 1/5 1

*2 e

2.87 R

c

o

T M

E V

 

  

 

s P





 ; PK_sp_z³²

-contact resonance (s^1)

Tc-contact time(s) s- effective stiffness

Figure 5.9 Step-wise element deletion of regressors

The correlation coefficient matrix of the standardized explanatory functions is shown in Table 5.6. The diagonal terms of inverse of correlation coefficient matrix are defined as variance inflation factors (VIF). The VIF for the model lies between 1 and 1.8 indicating no significant multi- collinearity.

Table 5.6 Correlation coefficient matrix of the explanatory functions



1



₂



₄



_D



1 ^{1.000 - - -}



2 ^{-0.0863 1.000 - -}



4 ^{0.0018 -0.646 1.000 -}



D ^{0.000 0.000 0.000 1.000}

The probabilistic model for the prediction of the peak contact force on the honeycomb composite structure due to low-velocity impact is given as

ˆ_P ˆ exp 0.059 ^c 0 02.8 0.039 ^b

m T

s c

F V

V

c

F h K

D

 

    

  (5.17)

The residual of the model is normally distributed with mean zero. Several residual plots are examined to determine the usefulness of the individual candidate variables.

In all the cases, the standardize residue is randomly scattered with respect to explanatory function in the given bounds. Validation of the predicted results is indispensable for determining the accuracy of the probabilistic model. The comparison between deterministic responses and design of experiments data set is shown in Figure 5.10 (a). The analytical results are compared against the numerical model of design of experiments data set. The results depicted in Figure 5.5 is also a data point in the

characterisation

Figure 5.10 (a). As expected, due to the uncertainties involved, the response variables are highly dispersed in the vicinity of 1:1 line. This figure lays the strong justification to incorporate of the uncertainties in geometrical and material parameter in analytical models. For an idealistic well behaving system, the analytical and experimental results will lie exactly on an 1:1 line. To prove the strength of the proposed technique, the results predicted from the probabilistic model is plotted against design of experiments data set. Figure 5.10 (b) shows the prediction capability of the probabilistic model as evaluated from Eq. (5.17). In Figure 5.10 (b), few points are the validation data points which are obtained from the experiments. Here, the validation data points are established by comparing experimental results with finite element simulation as discussed in section 5.4 (Figure 5.5). Later they are compared with proposed probabilistic model which does incorporate the uncertainty. The dotted lines indicate the confidence bound interval of the probabilistic model in form of 1:1



_D. The proposed model enhanced the accuracy of the prediction in low force regime. However, it is observed that the model tends to deviate for extreme combinations of material, geometric, and high impact energies. In such extreme scenarios, the honeycomb composite is excessively damaged due to perforation and it no longer remains a case of low-velocity impact.

For the input parameters within the acceptable range as established in Table 5.3, the probabilistic model predicts accurate results. The proposed model was validated by conducting experiments for set of input variables outside the training data set. This step ensures that the proposed model is not restricted to be applied only for the training data set. Also, the upper bound of the applicability of the model is evaluated through the experimental validation in upper bound data set. These experimental data points are overlaid on the Figure 5.10 (b) and lies in vicinity of 1:1 line. As seen in Figure 5.10 (b) the validation points lie within the confidence bounds and ensure applicability of the model. The proposed model performs satisfactorily for the considered range of input parameters but need to be judiciously applied for other cases.

Figure 5.10 Comparison between normalized contact force: (a) Analytical and FE simulation, and (b) Probabilistic and FE simulation