**1. Introduction**

Concrete is one of the most commonly used materials in civil structures. From a scientific and technical point of view, it is a subject of interest both at the stage of designing a recipe, manufacturing various types of elements and during its operation. Ensuring safe and long-term use of concrete structures and elements requires, among other things, appropriate diagnostics. It can use destructive testing (e.g., by testing the strength of drilled cores [1,2]), semi-destructive testing (e.g., pull-out [1,3], pull-off [4,5] methods) and non-destructive testing (NDT, e.g., using sclerometer tests [1,3,6], thermal imaging techniques [7,8], analysis of natural frequencies [9,10], stereological investigations [11], acoustic emission [12,13], X-ray tomography [12], ground-penetrating radars [7,14,15], ultrasound [1,3,7,15–25] including ultrasound tomography [15,19–25]). The choice of method depends on the material characteristics that we want or are able to measure. All three types of tests are widely known, but especially NDT, thanks to the introduction of a number of modern measurement techniques in the building industry and intensive research, is becoming more and more popular and reliable. Particularly interesting in this area are ultrasound techniques which use at their basis typical phenomena associated

with wave motion physics—e.g., reflection, diffraction, attenuation, change of propagation velocity depending on changes in stiffness and density of the medium. The simplest method in the case of concrete structures is an assessment of the velocity of longitudinal waves between selected points of the tested element—the lower the velocity, the lower the stiffness of concrete and its quality [16]. However, it concerns the average speed measurement on the section between the ultrasonic transducers. An interesting, practical case of this type of analysis is article [18] where ultrasonic measurements carried out on a river dam were verified by means of visual inspection of cores taken from it. In the literature, there are also analyses concerning the change of time and intensity of ultrasound wave passing through the area of concrete where a single crack builds up [17]. The use of such research, however, requires in advance knowledge of where such a defect may develop in an element. On the other hand, tomographic methods are deprived of this type of inconvenience, where only a set of transceiver converters suitable for concrete is required and external access to the tested element on one side in a reflective mode (e.g., Reference [21]) or with access on two or more sides in a transmission mode (e.g., Reference [15,20,22–25]). In the latter view, the state of the material is most often shown indirectly by means of reconstructed maps of the propagation velocity of a selected type of ultrasound wave. For the sake of convenience, longitudinal waves are usually selected for this purpose in the unambiguous interpretation of measurements, as they move the fastest and are not dispersed. It should be emphasized that very accurate maps of cracks in the concrete structure can be made, on the other hand, using X-ray tomography [12]; however, currently, due to the cost of the equipment and the possibility of its use, it is practically impossible to use it directly on real building structures in field research. It is also worth mentioning at this point that, from the point of view of developing mathematical foundations for tomography, its beginning dates back to 1917, when Johann Radon proposed a solution to the problem of reconstruction of the shape of an object on the basis of its projections [26].

The tomographic imaging concrete elements available in the literature focus mainly on the identification of defects with much lower acoustic resistance than the surrounding concrete: e.g., artificially introduced defects, for research purposes, in the form of inclusions from foamed polystyrene [21–23], expanded polypropylene [25], prisms from cracked concrete [23] or air-filled pipes [21], cavities in defectively injected pipes for placing prestressing cables [15], areas strongly cracked as a result of excessive loads [20,22,24] or freeze-thaw cycles [19]. Therefore, the first goal that the authors set for themselves in this paper was to carry out an analysis of the extent to which it is possible to detect brittle defects in concrete beams starting from the early stage of their development, when microcracks do not yet form defects capable of effective reflection of waves or their significant slowing down. For this purpose, the methodology of damage mechanics was applied in terms of one of the most recognized concrete models in this field, formulated by Chaboche [27] and Mazars [28], and, in the non-local terms, developed by Pijaudier-Cabot [29–31]. Then, depending on the degree of brittle damage described in a "fuzzy" way by the damage parameter, it is possible to model the development of a localized decrease in material stiffness and the associated reduction in the speed of sound waves in concrete. This fact can also be used in the tomographic assessment of concrete [32,33]. For this reason, in order to reliably calculate the distributions of drop in stiffness and changes in the velocity of the longitudinal wave around the forming crack, the authors of the paper proposed an effective way of identifying the parameters of a non-local model of brittle damage evolution using experimental data from [34]. These data were then used in computer simulations of tomographic identification of this type of defect in various phases of its formation and were confronted with the results of our own experimental research.

Another important aspect of ultrasound tomography measurements is its accuracy which may be affected by the diffraction of waves when passing through and around areas of different acoustic resistance than the rest of the medium. It is commonly assumed in order to significantly simplify calculations that paths of the fastest wave propagation are rectilinear (e.g., Reference [15,19–24,35]) which is then called rays. This introduces disturbances in tomographic reconstructions when the actual

paths differ strongly from the geometry assumed so far. The studies available in the literature show that this assumption does not cause any disturbances in the location of the damaged areas [21–24]. However, the obtained values of wave propagation velocities on the reconstructed maps differ significantly from the actual values at high levels of concrete degradation [32,33] or inclusions with significantly different acoustic resistance from the matrix [25]. In this case, it should be taken into account that according to the Fermat principle, wave disturbance travels from one point to another such a path that needs minimum or maximum time, or the same in comparison to other, adjacent paths [36], which determines the course of its fastest propagation. As stated in Reference [25], the first attempt to consider the Fermat principle in ultrasound imaging was made by Johnson et al. [37]. They proposed the use of the ray-tracing technique which in the case of concrete structures was first adopted in works [38,39]. An alternative to this type of approach is the use of graph theory, in particular Dijkstra's algorithm [40]. If we apply it in the analyzed issue to a full graph the nodes of which will cover the studied area, and the weights of the edges will be equal to the time of the wave passing through them, then in this way we can approximately determine the shape of paths of the fastest sound propagation and the time needed for the wave to travel along them. The results will be the more accurate the denser the network of nodes. This very interesting concept, inspired by the works of various authors and applied to seismic waves, was developed in Moser's work [41] in 1991 (as in Reference [25]). Extensive studies in this field with examples of calculations using experimental data from concrete cubes with inclusions from expanded polypropylene have been presented in Reference [25]. A proposal to use this methodology in the ultrasonic testing of structural elements was also presented by the authors in Reference [33,42]. In the light of the quoted information, it can be noted that so far there is no analysis in the literature concerning this type of problem in the case of tomographic assessment of the damage evolution described according to the concept of damage mechanics. Therefore, the second aim of this article was to present in this area appropriate numerical analyses with the use of Dijkstra's algorithm in determining shape of the paths of the fastest ultrasound wave propagation. On this basis, the author's own method of improving the accuracy of tomographic calculations was formulated because of the inconsistency of the assumptions with reality in the case of using straight-line rays to approximate the geometry of the fastest propagation paths. For this purpose, it is proposed to not interfere with the assumption of straightness of ultrasound pulse pathways but, on the other hand, to properly scale the measured times of their propagation between the assumed transceiver points elongating them in the case of rays that pass through elastically degraded areas. The advantage of such an approach is that it does not complicate well established mathematical methods (e.g., Reference [35,43–46]) which have been implemented in tomographic imaging techniques.

The paper also raises the practical aspect of conducting such measurements by analyzing in the light of the presented arguments how the accuracy of results may be affected by the introduction of the so-called fictitious transceiver points for which the times of propagation of ultrasound waves are interpolated on the basis of measurements from the real points. In this respect, the authors were inspired by work [47] where this approach was presented in the study of moisture distribution in walls. It may significantly reduce the number of points used and the labor intensity in real measurement situations, which is particularly important in the case where fewer ultrasonic transducers are available.

Due to the scope of studies within the framework of the presented calculation examples and experiments, the authors limited themselves to the case of concrete beams with elastically degraded zones perpendicular to the beam axis. All the calculations made in the article were done with the use of the authors' computer programs written in the MATLAB software environment.
