PSI - Issue 64

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

146

2

1. Introduction It is common knowledge that many civil concrete structures grew old and their structural health condition needs to be investigated to predict the remaining lifetime or to plan reconstruction works carefully when necessary. Instead of unnecessary demolition and new construction of especially infrastructural constructions, it is wise to assess the current state. Here, particularly non-destructive methods are suited. For this reason, the experimental measurement design needs to be planned carefully, because useful measurement locations need to be chosen and redundant information needs to be avoided. These investigations also help to save costs and time as well as the reduction of data transport, storage and computational power for the data evaluation. The investigation by ultrasound is an established non-destructive tool for the condition assessment of concrete structures. Within this approach, the Full Waveform Inversion originating from geotechnical applications is used for the identification of structural defects. Additionally, Gaussian white noise on different levels is added to simulate measurement data. This procedure is followed by investigations on the optimal experimental design in the sense of sensor distance. 2. Methods Within this paper, the Full Waveform Inversion (FWI) is used to obtain information about defects inside concrete structures. Therefore, seismic waves are used to identify fault areas inside the structure. In general, seismic waves are divided into surface and body waves. Within this approach, we want to consider only body waves that can be distinguished in the primary wave (P-wave), which is a compression wave, and the secondary wave (S-wave) a shear wave. The latter only exists in solids. The FWI uses the information of these waves to minimize the residual energy between the experimentally obtained data and the simulated data of the numerical model as described in Tarantola (1984). The 2D numerical model is evaluated with the open-source program DENISE by Köhn (2011). Here, the elastic wave formulations are summarized as



i v     t

ij

( , ) x z

,

f



 

i

x

j









j i v v x x j     i      



(1)

ij

( , ) x z

( , ) x z

,





  

ij

t

t





, ; i x z j x z   , ;

+ boundary conditions and initial conditions,

where ( , ) x z  is the density, i v are the particle velocities, t stands for time, ij  denotes the stress tensor components, i f are the directed body forces, ( , ) x z  and ( , ) x z  are the Lamé parameters and ij  is known as Kronecker’ s delta. The wave equation is solved by the use of finite differences. On the boundaries of the discretized domain, perfectly matched layers as described in Berenger (1994) are used to prevent artificial wave reflections. The approach is here to minimize the difference between the experimental data exp u and the data obtained by the numerical model num u . Therefore, the cost function

(2)

num   m u m u ( ) ( )

exp

FWI

f C

is minimized in the sense of least squares and a model containing the values of the three describing model parameters ( , ) x z  , ( , ) x z  and ( , ) x z  is received. The wave equation is solved in each iteration step for each excitation source and frequency in order to obtain num u . The residuals of the experimental data are calculated and back propagated from the receiver positions to generate the adjoint wavefield. Simultaneously, the model parameters are updated by a Quasi Newton method. The P-wave and S-wave velocities can be derived from the density and Lamé parameters by