PSI - Issue 64

Ina Reichert et al. / Procedia Structural Integrity 64 (2024) 145–152 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

147

3

2    

and

V



P

(3)

 

.

V



S

The received numerical models are subsequently compared to the real models using different distances between sources s  and receivers r  under variation of fault position and size x . That way, the optimal distance of sources and receivers are gained under consideration of various faults. The main goal is here to increase the quality of the results for damage detection by simultaneously reducing the number of sensors to a meaningful minimum. This helps to save time during the experiment’s installation, the data analysis, reduces the necessary data storage and last but not least minimizes the overall costs. There exist well-known approaches for the optimal design of experiments as the application of the Fisher Information M atrix (FIM) as described in Uciń ski (1999), where either the parameter sensitivities are maximized or the parameter uncertainties are minimized. Here, only random errors can be accounted for. Novel approaches often make use of the mean-squared error for finding optimal experimental designs (cf. Bardow (2008), Lahmer (2011) and Schenkendorf et al. (2009)). Within this research, the cost function is calculated by

(4)

( , ) w V s w s         x x x ( , ) ( , ), C s

1

2

f

S

where 1 2 1 w w   are weighting factors, the differences in the shear wave velocity S V and the density  between the real model and the model received by the FWI are computed as

S   

true V V

num

true

V

S

, 0.9

1.1

S  

and

S      V

num

n n

V

x z

S

0,

else

(5)

  

true n n   x z

num

true



, 0.9

1.1

 

.

     

num



0,

else

The values of the cost function are used for the comparison of different experimental designs and fault sizes and positions to achieve a trade-off in sensor positioning.