Multi-objective optimization to evaluate the compressive strength of concrete by combining non-destructive techniques

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Introduction
Rebound hammer (RH) and ultrasonic pulse velocity (UPV) techniques are often used to evaluate concrete characteristics and estimate compressive strength in the laboratory and in situ. The idea of correlating concrete strength (Fc) simultaneously with rebound hammer (RH) and ultrasonic pulse velocity (UPV), which is the origin of the SonReb method, dates back to the early 1980s, thanks to Facaoaru [1]. However, the assessment of strength variability has received very little attention. Recent work of Alwash [2] has proposed the use of a bi-objective method capable of assessing variability with better accuracy than conventional methods. The main limitation of this method is that it does not allow combining two NDT techniques to assess strength and its variability. This article proposes to answer this problem by introducing the multi-objective optimization and exposing the interest of combining two NDT techniques.

Synthetic data
Due to the limited number of experimental data, synthetic simulations were performed, allowing the control of some parameters such as the effect of humidity and measurement uncertainty. These data are generated by Monte-Carlo simulation using the procedure proposed by Breysse [3]. The advantage of such an approach is to build a synthetic database in order to simulate different configurations and to estimate the prediction error and therefore to evaluate the quality of the methodology. The main steps of the simulation are: (1) Definition of the simulation input variables and their domains, (2) Generation of random values for the input variables using the appropriate probability distribution, (3) Calculation of the output variables using the relationships between the inputs and outputs, (4) Repetition of the simulations in order to account for the sampling effect. We considered as input variables of the simulation, the in situ strength as well as the concrete humidity with the following characteristics: ̅ = 25 , ( ) = 2 , ̅ = 75%, ( ) = 2.25%

Experimental data
The experimental database used is from the work of K. Ali-Benyahia [4]. Non-destructive evaluations of rebound hammer and ultrasonic pulse velocity measurements were applied on the elements (columns and beams) of a reinforced concrete building. In parallel, compressive tests of sampled cores were also performed.

Simple linear regression model
Regression approach is the most popular statistical approach that is used to identify the conversion model between the core strengths and NDT measurements. Assuming a linear model between the strength and NDT measurement, where represents the NDT measurement and is the core strength, both corresponding to one test location: = + + Where and are the unknown parameters of the true regression model, and is a random error with mean value of zero and unknown variance [5]. Least squares method is used to provide the estimators and respectively.

The bi-objective approach
Another innovative approach, named "bi-objective", was proposed by Alwash [2]. which is devoted to capture the concrete strength variability in addition to the mean strength. Two conditions are required in order to derive the values of the model parameters and respectively: Where ̅ and ̅ are respectively the estimated and measured mean strength values, while, ( ) and ( ) are respectively the estimated strength variability and the variability calculated from the core strengths.

Multiple linear regression model (mono-objective approach)
Regression approach illustrated above for the case of single NDT technique as an independent variable can be extended to the case of more than one NDT technique as independent variables. Therefore, the fitted regression model that can be used to estimate the strength is: = + 1 1 + 2 2 + ⋯ + + The parameters (a, 1 , 2 , … , ) are obtained through least squares minimization [6]. The equation above can also be written in matrix form or more compactly: Y= X*A Where A is the model parameters matrix. To get to the parameters of the model we are looking for, we have to invert the equation. This inversion can be done by the method of least squares for example, but also by calculating the inverse matrix.

Multi-objective approach
Nowadays, many complex problems are treated with multi-objective optimization. The formalization of an optimization program includes the same steps whatever the techniques required later for the treatment: -The detection of the problem and the identification of the variables, in our case it is to identify the parameters of the combination model. -The formulation of the objective function. In the assessment methodology using the multi-objective approach, the optimization procedure is based on the minimization of three objective functions, which are respectively the mean square error, the relative error on the mean and the relative error on the standard deviation: is the estimated concrete strength and is the true in situ concrete strength.

Analysing the model identification approaches
In this section, the predictive ability of different approaches is investigated, namely regression and bi-objective approach for the case of single NDT techniques, as well as mono-objective and multi-objective optimization for the case of combined NDT techniques. For this purpose, two main sources of data are used in this study: experimental and synthetic data sets.

Case of single nondestructive technique
First, the ability to estimate the mean strength and concrete strength variability using the biobjective approach was compared to the regression method. From the results presented in Figures 1 and 2, the following observations can be highlighted: The two existing model identification approaches, namely regression and bi-objective approach, are generally able to estimate the mean strength with an improvement in estimation efficiency as number of cores NC increases. On the other hand, concerning the estimation of concrete strength variability, it is clear that the bi-objective approach is the only one that can efficiently obtain the true reference value of the concrete strength variability. While the regression approach remains limited.

Case of non-destructive techniques combination
Since the proposed bi-objective approach gives encouraging results for evaluating the mean and strength variability, our main prospect is to develop a similar approach applicable for the case of combined measurement techniques, in order to further improve the quality of the evaluation. The multi-objective optimization is able to estimate the mean strength and the concrete strength variability, providing a model with high predictive quality. On the other hand, using the monoobjective approach with a limited number of samples is less efficient and can lead to large errors. In addition, the quality of the results was checked in terms of the difference between the estimated and experimental strength (relative errors on the mean and standard deviation). The best results were considered the ones with small relative errors.

Effect of combining NDT techniques
The aim of combining NDT techniques is to improve the quality of assessment. The efficiency of combination studied herein is limited to the case of combining the rebound hammer and pulse velocity. Comparing the fitting and prediction capacity (fitting and prediction error) of the model identified for combination with that identified for each NDT technique separately is the only way to decide whether using combination of NDT techniques can improve the quality of assessment or not.
In what follows, we retain both approaches: bi-objective for single NDT and multi-objective for combined NDT. To study the efficiency of the combination, fitting and prediction errors corresponding to single and combined NDT techniques are plotted. Figure 7 highlights a comparative analysis between the values of identification and prediction errors for the single and combined techniques. It is shown here that the combination of NDT techniques is potentially effective in obtaining low identification and prediction errors.

Conclusion and perspectives
The advances of this project were mainly concerned with the analysis and comparison of the estimation capacity of the conversion model identification approaches, concerning the evaluation of the mean strength and variability of the concrete in the case of combining NDT methods, on a real case study and using synthetic data. Based on the results presented above, multi-objective optimization is particularly relevant for the evaluation of concrete mean strength and variability. In addition, the results of these analyses show the major effect of the number of test locations on the quality of the concrete strength evaluation. Moreover, the main goal of this work is to explore how to optimize the structural capacity of buildings by combining NDT techniques which has a positive effect on the quality of assessment A second contribution of this work is to study the effect of conditional selection of test locations, which improves the quality of the assessment without additional cost. Different sampling plans of core locations are being studied by generating risk curves as indicators of the estimation quality.