Analysis o f error influence

It has already been mentioned at the beginning o f chapter 4 that so-called inverse crime was committed in inverse calculations o f the parameters o f the curing kinetics model, done within the study referred to as the virtual experiment o f curing process. This means that the same numerical model was used to prepare data describing the cause o f curing reaction in the form o f temperature field as the one employed during inverse calculations aimed at finding the coefficients o f the curing kinetics model. More precisely, the numerical mesh, physical properties o f materials, initial condition and boundary conditions were the same.

The only difference between the model applied to simulate measurements and the

82 4.5. Validity o f the results o f inverse calculations

one utilized in the inverse procedure was the set o f the parameters o f the curing kinetics model. This idealistic approach allowed proving that the mathematical mechanism used in the inverse methodology works correctly and is free o f the computer programming errors.

Measured Initial Levenberg-Marquardt - PSO

/ '.

A-Profile *5 Monitoring point T5

2. 50 T im «, h

Figure 4.12: Temperature agreement observed for the modified boundary conditions with the curing kinetics data fitted for the original boundaiy

conditions.

The main focus o f the next step o f investigations was the influence o f error in the measured data on effectiveness o f the developed inverse methodology. For this purpose perturbations, simulating measurements errors, were introduced into the information about the measured temperatures and two different series o f inverse analyses were performed. In the first one the inverse crime was again committed, while in the second series the inverse formulation was deprived o f this problem. This was done by applying much finer mesh in the

4.5. Validity o f the results o f inverse calculations 83

model used to generate the measured data, whereas the numerical mesh o f the model applied during the inverse calculations was exactly the same as discussed in subchapter 4.3.2 and shown in Figure 4.3. The refined numerical mesh is presented in Figure 4.13. It consisted o f 56528 cells (36256 in the epoxy resin region and 20272 in the aluminium mould region) and was prepared by double adaption o f the original mesh. It means that the fine mesh had the same topology as in case o f the based mesh (Quad Elements Scheme with Pave and Submap Type Scheme in the casting system region and aluminium mould region respectively).

Figure 4.13: Fine numerical mesh used to generate the measured data avoiding inverse crime.

The main goal o f the error influence analysis was to check how different levels o f perturbations in the measured data affect the inverse problem solution.

The errors were introduced into the measurement data by using built-in M icrosoft Excel functions that compute a random num ber RN from the Gaussian distribution for a given standard deviation SD and mean M :

N O R M S IN V (R A N D ( ) • SD) + M (4.9)

where N O R M SIN V ( ) is a function that returns the inverse o f the standard normal cumulative, R A N D ( ) is a function that returns a random number between 0 and

1 (evenly distributed).

84 4.5. Validity o f the results o f inverse calculations

Basically, the function above generated random number from the Gaussian distribution with different levels o f standard deviation (1%, 2%, 5%) and with the mean value corresponding to the original measured temperature for a given monitoring point and time instance. In this way, the mentioned perturbations, simulating measurements errors, were introduced to the information about the measured temperatures, as shown in Figure 4.14. This, in turn, allowed judging more reliably the efficiency and accuracy o f the developed methodology. However, it is worth stressing that the error influence study was performed only by using the Levenberg-M arquardt approach, with the algorithm parameters tuned in comparison to the inverse analysis described in subchapter 4.4. The stochastic character o f the PSO algorithm excluded the sense o f performing the similar error influence analysis for this method, since the error influence on the solution could be diminished or enlarged by the mentioned randomness in the working principle o f the algorithm.

1 70 Figure 4.14: The level o f perturbations in the measurement data depending on

the value o f standard deviation.

The influence o f errors, introduced to the measured data, on the course of inverse analyses is illustrated in Figure 4.15. The left-hand side graph presents the objective function value in the consecutive iterations for the inverse calculations performed with different levels o f perturbations in the measurement data and committing the mentioned inverse crime (further referred to as ‘inverse crim e’ case). The right-hand side graph depicts analogous results obtained during inverse calculations without the inverse crime comm itm ent (further referred to as

‘no crim e’ case). Additionally, the included tables present the initial and final values o f the objective function (Gini and Gfin respectively) as well as the value o f improvement factor IF, calculated according to the following equation:

4.5. Validity o f the results o f inverse calculations 85

(Gim Gf ‘» ) . 100

Gi-ni (4-10)

Two main conclusions can be drawn based on the presented results.

Firstly, the developed inverse methodology improved the correlation between the measured and the estimated temperatures in all analysed cases. Secondly, the methodology is sensitive to the error in the measurement data, however, this dependence is strongly affected by the error level and, as expected, increases with its value. One can also notice that the efficiency o f the inverse methodology lowers more rapidly with the error growth in the ‘no crim e’ case. The probable reason for that is the summation o f the influence o f two uncertainties, the one connected with different numerical meshes used during the measurement data generation and during the inverse calculations, and the second one resulting from the introduced error. Furthermore, it can be seen in the results obtained for the

‘no crim e’ case that the final objective function value was relatively high even in the inverse analysis with the error-free measured temperatures. The analogous analysis performed for the ‘inverse crim e’ case finished successfully reaching the target value o f the objective function ju st after 4 iterations. Keeping in mind the conclusions above, it can be stated that much care should be taken both to prepare accurate numerical model o f a considered problem and to minimize the measurement errors when collecting the measurement data for inverse calculations done with the use o f the proposed inverse methodology. In this way the conditioning o f the inverse problem under investigation can be better, increasing simultaneously the probability o f the inverse calculations success.

SD = 0% ■ SD = 1 % — SD = 2%

Figure 4.15: The convergence o f the solution with (left) and without (right) commitment o f inverse crime depending on the error in the measurement data.

86 4.5. Validity o f the results o f inverse calculations fitting o f temperatures. This relationship is quite obvious when analysing the way the objective function is calculated (refer to Equation (3.14)). One can also see that, in the ‘inverse crim e’ case (refer to Figure 4.16), the matching between the measured and the estimated temperatures was significantly improved in all runs o f inverse analysis, although the best and the worst fitting was obtained, respectively, for the calculations without errors and with the highest errors in the input data. The disagreement between the measured and the estimated temperatures was also clearly lowered in the ‘no crim e’ case (see Figure 4.17), but the final fitting o f temperatures was generally worse for each error level when comparing to the ‘inverse crim e’ case. However, it seems that bigger discrepancy in the study based on the approach without inverse crime commitment is highly influenced by the difference in computational meshes applied in the numerical simulation acting as simulated measurement and in the computations performed within the inverse calculations.

Figure 4.16: The influence o f the error in the measurement data on temperatures agreement at selected monitoring points (with inverse crime committed).

2.1 2.4