**2. Ultrasonic Transmission Tomography**

The considerations presented in the paper concern the case of concrete beams that contain elastically degraded zones (in the form of grouped micro-cracks or cracks) running across the entire cross-section—e.g., as a result of simple bending or tension. Therefore, tomographic analyses were narrowed down to the identification of brittle damage in a flat longitudinal section of beam which will also be the plane of symmetry of this element, focusing on the assessment of changes in the speed of ultrasonic waves due to a local change in the stiffness of the cracked material. For this reason, in all computational examples and experimental studies presented in the paper, a system of opposite transmitting/receiving points of ultrasound waves in the transmission mode was used (Figure 1). Longitudinal waves have been selected as non-dispersive and the fastest of all ultrasonic wave types for the study. This requires, however, that their length should be small enough in relation to the dimensions of the cross-section of the element to be effectively generated (practical recommendations in this respect can be found in, e.g., Reference [48]). On the other hand, it limits the distance between the intended transmitting/receiving points because of the attenuation of the longitudinal waves and the angle at which they may propagate from the transmitting point in an effective manner in reception because of the amplitude distribution. In the latter case, on the basis of individual experiences of the authors, the angle of inclination of diagonal rays to a bar axis was limited to range from 45◦ to 135◦; in Reference [33], a numerical simulation of the process of propagation of ultrasounds during its excitation by the transducer was performed where it was shown that the amplitudes of longitudinal waves are negligible in contrast to transverse waves outside this angular range, which may result in incorrect and increased propagation time reading.

**Figure 1.** Scheme of a cell system, transmitting/receiving points, and rays in a plane area examined tomographically.

From the mathematical point of view, by indirect visualization of the material structure by means of maps of distributions of quantities characterizing the propagation of ultrasound waves, it is necessary to build an appropriate system of equations (e.g., Reference [35]). The plane problem assumes that the reconstructed image consists of a finite number of plane cells which, in the examined area, is separated by an orthogonal grid of a step of δ<sup>1</sup> × δ<sup>2</sup> (Figure 1). The function is searched for in an approximate way:

$$f = c\_{\mathcal{L}}^{-1}(x, y),$$

where: *c*L—longitudinal wave velocity (m/s); *x*, *y*—spatial variables (m). For this purpose, it is assumed that *f* in each cell takes a constant value of *fk* where *k* = 1, 2, ... ,*K*.

In the problem, the longitudinal wave propagation times between the selected sending (*Sm*) and receiving (*Rn*) points must be given where *m* = 1, 2, ... , *M*; *n* = 1, 2, ... , *N*. It is assumed to simplify considerations that the fastest propagation path between these points can be modeled as a straight line, which in tomography is referred to as a ray. The propagation time of *t*ray,*<sup>i</sup>* (s) over the *i*-th ray, connecting points *Sm* and *Rn*, can be calculated from the integral:

$$t\_{\rm ray,i} = \int\_{S\_{\rm w}}^{\mathbb{R}\_{\pi}} f(l\_i) dl\_{i\prime} \tag{2}$$

where: *li*—variable describing the position on the *i*-th ray (m); *i* = 1, 2, ... , *I*. Considering that averaged values *f* are being sought in the cellular areas, we can write that:

$$t\_{\text{ray},i} = \sum\_{k=1}^{K} L\_{\text{ray},i,k} f\_k \to \mathbf{Ax} = \mathbf{b},\tag{3}$$

where: *L*ray,*i*,*k*—parts of the length of the *i*-th ray falling on the *k*-th cell (m) (if the ray does not pass through cell *k*, then *L*ray,*i*,*<sup>k</sup>* = 0), **A** = *L*ray,*i*,*<sup>k</sup> <sup>I</sup>*×*<sup>K</sup>* (m), **<sup>x</sup>** <sup>=</sup> [ *<sup>f</sup>*1, *<sup>f</sup>*2, ... , *fK*] <sup>T</sup> (s/m), **b** = *t*ray,1, *t*ray,2, ... , *t*ray,*<sup>I</sup>* <sup>T</sup> (s). The above is the definition of a system of equations in relation to the value of *fk*, which is used to determine the velocity distribution of ultrasound wave propagation in tomography. After its solution, we get that:

$$c\_{\text{L tom},k} = \frac{1}{f\_k} \tag{4}$$

where *c*L tom,*k*—mean velocity of the longitudinal wave in the *k*-th cell (m/s) (in the sense of the presented method). Please note that *I* (the number of rays used) must not be less than *K* (the number of the assumed cells) and the rays must evenly cover the test area. The system of equations formulated in such a way is an ill-conditioned one, which forces the use of iterative techniques of its solving. The basic method in this respect is an algorithm developed by the Polish mathematician Stefan Kaczmarz (1937). In 1970, Gordon and his collaborators, working on the application of this technique in medicine, rediscovered this method and named it Algebraic Reconstruction Technique (ART) [43]. It was this one that was used in the first in the world computed tomography scanner constructed by Hounsfield in 1972 [44]. On the basis of the Kaczmarz algorithm, many other methods were developed. Currently, the literature distinguishes three basic ways of imaging: the aforementioned ART, Simultaneous Iterative Reconstruction Technique (SIRT), and Simultaneous Algebraic Reconstruction Technique (SART) which is a combination linking the advantages of the ART and SIRT methods [35,46]. For this purpose, the Tikhonov regularization method can also be used in the method of least squares (e.g., Reference [24]).

In this article, all tomographic images presented below were solved with the use of a randomized Kaczmarz algorithm. The final result was taken as **x**mean,*q*, i.e., the mean of *q* independently obtained solutions of equation system (3). Hence:

$$\mathbf{x}\_{\text{mean},q} = \frac{1}{q} \sum\_{r=1}^{q} \mathbf{x}\_r. \tag{5}$$

In a given solution **x***r*, its subsequent approximations were made by projecting the previous approximation in a direction perpendicular to the randomly selected straight lines defined by the equations of system (3), but so that each of these lines would be used only once. The starting point for the iteration of each **x***r* was the vector:

$$\mathbf{x}\_0 = \begin{bmatrix} c\_{\text{L ref}}^{-1}, c\_{\text{L ref}}^{-1}, \dots, c\_{\text{L ref}}^{-1} \end{bmatrix}\_{\text{1} \times \text{K}'}^{\text{T}} \tag{6}$$

where *c*L ref—reference value of the longitudinal wave velocity (m/s). The number *q* of the averaged solutions of equation system (3) were selected so that the condition was met:

$$\left|\frac{\varepsilon\_{\mathfrak{E},\mathfrak{q}} - \varepsilon\_{\mathfrak{E},\mathfrak{q}-1}}{\varepsilon\_{\mathfrak{E},\mathfrak{q}}}\right| < 10^{-5} \text{ and } \varepsilon\_{\mathfrak{E},\mathfrak{q}} = \frac{\left\|\mathbf{Ax}\_{\text{mean},\mathfrak{q}} - \mathbf{b}\right\|}{\|\mathbf{b}\|}. \tag{7}$$

As mentioned in the introduction, another inconvenience of the presented method of tomographic imaging is the fact that ultrasonic waves are diffracted when avoiding areas with different acoustic resistance, so that the real paths of the fastest propagation are curvilinear. This is one of the basic sources of inaccuracy of the presented approach if there are sub-areas with significantly different acoustic resistances in the tested concrete element in relation to the rest of the element [25,32,33]. These issues will be discussed in the context of the following calculation examples and experimental studies concerning the detection of elastically degraded concrete zones.
