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This is an example of a stochastic fractal, meaning that it is built out of probabilities and randomness. Unlike the deterministic tree-branching structure, it is statistically self-similar. As we go through the examples in this chapter, we will look at both deterministic and stochastic techniques for generating fractal patterns.

Writing a function that recursively calls itself is one technique for generating a fractal pattern on screen. However, what if you wanted the lines in the above Cantor set to exist as individual objects that could be moved independently? The recursive function is simple and elegant, but it does not allow you to do much besides simply generating the pattern itself. However, there is another way we can apply recursion in combination with an ArrayList that will allow us to not only generate a fractal pattern, but keep track of all its individual parts as objects.

Influences of surface roughness of a crack and the appearance of the stress field at its tip, shown by a series of electron micrographs depicting the interaction between a crack and a grain boundary in molybdenum. As the crack approaches the boundary, a small crack is nucleated and eventually joined the main crack. Figure adapted with permission from Ohr [26].

Identification of structural crack location has become an intensely investigated subject due to its practical importance. In this paper, a hybrid method is presented to detect crack locations using wavelet transform and fractal dimension (FD) for beam structures. Wavelet transform is employed to decompose the mode shape of the cracked beam. In many cases, small crack location cannot be identified from approximation signal and detailed signals. And FD estimation method is applied to calculate FD parameters of detailed signals. The crack locations will be detected accurately by FD singularity of the detailed signals. The effectiveness of the proposed method is validated by numerical simulations and experimental investigations for a cantilever beam. The results indicate that the proposed method is feasible and can been extended to more complex structures.

In the present work, a hybrid method is presented to detect crack locations based on wavelet transform and fractal dimension (FD) for beam structures, which can enhance the high sensitivity to the singularities induced by structural crack. Using Db4 wavelet decomposition, approximation signal and wavelet detailed signals of the mode shape are obtained. Then FD estimation method is used to calculate FD parameter of detailed signal. The crack location can be determined by the abrupt change of the FD along the beam length. Numerical simulation and experimental investigation are performed to testify the present method. In the numerical simulation, the mode shapes of the beam with crack are obtained based on wavelet-based Euler beam elements using B-spline wavelet on the interval (BSWI), whereas in the experimental investigation, the mode shapes are measured using Polytec Vibrometer PSV-400.

Figure 1(a) shows a model of cracked cantilever beam with dimensions of length and uniform cross section (where is the width and is the depth of beam), two open cracks of depth locates at away from the clamped end, where the subscript expresses the serial number of cracks. Since the linear rotational spring model can describe open crack effectively, present work is based on this model (as show in Figure 1(b)). The stiffness of rotational springs can be written aswhere is the modulus of elasticity of beams, is the depth of beams, and is a dimensionless local compliance function expressed as follows [32]:To express briefly, we denote two dimensionless parameters to describe cracks: relative crack size and normalized location .

As shown in Figure 1(b), the continuity conditions at crack positions 1 and 2 indicate that the left nodes , and the right nodes , have the same transverse displacement, namely, and , whereas the rotations , and , are connected through the cracked stiffness submatrix () as follows:

The global stiffness matrix and mass matrix of BSWI Euler beam can be obtained according to the literature [33], in which the specific formulas about wavelet-based modeling for beam using BSWI bases are shown. According to relative locations and of the crack, we can assemble cracked stuffiness submatrix () into the global stuffiness matrix in the corresponding place. The global mass matrix of the cracked beam is the same as the uncracked one.

This method can also be applied to calculate the FD of the mode shapes. Since it exhibits high-noise insusceptibility [37], FD has been applied to crack detection. Crack locations are determined by peaks on the FD curve, which indicates the local irregularity of the fundamental mode shape introduced by the crack. However, inflexions appearing in higher mode shapes would cause false peaks covering up the peaks induced by cracks. Qiao and Cao [38] verified that these false peaks can be inhibited well, using a specific bijective linear mapping from vector space to as shown in where and are the vectors in vector spaces and , respectively. In this paper, and denote beam length and and certain order mode shape data in vector spaces and , respectively. is within interval . And the selection of is usually done by trial and error.

The cracked beam is modeled using the B-spline wavelet on the interval (BSWI) finite element method. The beam crack is modeled according to linear elastic fracture mechanics theory. The first several mode shapes of the cracked beam is acquired from eigen formulation.

The mode shapes, approximation and detailed signals, and FD are plotted in the geometry space of the beam structure. The crack locations can be identified by the peak points of the signals along the beam length.

The BSWI43 Euler beam element is used as approximation bases to model the cracked beam, where 4 and subscript 3 denote the order and the level of the BSWI wavelet. In the simulation, we use 20 BSWI43 Euler beam element (184 DOFs). The left of the beam is fixed and its right is free. In this paper, we do not consider damping.

The first three mode shapes of the cracked beam are acquired from eigen formulation. For comparison purpose, mode curvature-based method and the proposed method are adopted to detected crack locations.

Mode curvature-based method is applied to the first and third mode shapes, respectively. The detection results are show in Figure 2. It can be seen that the crack locations are not identified clearly. By comparison, it is seen that the proposed method is more effective to detect crack locations than mode curvature-based method.

The proposed method is used to detect crack locations. The detection results of crack locations are showed in Figures 3 and 4 using the first mode shape and the third mode shape. The mode shape S, the approximation signal A, and the detailed signals and are showed in Figures 3(a) and 4(a), respectively. According to Figure 3(b), the relative locations of two cracks are clearly identified by the peak points at = 0.4 and = 0.6, respectively. However, from Figure 3(a), crack locations are not detected from approximation signals and detailed signals.

The proposed method is directly applied to third mode shape, crack locations are shadowed by two false peaks induced by two inflexions of the third mode shape, as showed in Figure 4(b). As mentioned above, a linear mapping as shown in (7) was applied to solve the inflexion problem. The parameter is selected as . After a linear mapping, the detection result is displayed in Figure 4(c), and the crack locations are accurately identified at = 0.4 and = 0.6. However, from Figure 4(a), singular locations induced by small beam crack cannot be detected from approximation signal and detailed signals.

In this section, an experiment is conducted to validate the proposed method on steel cantilever beam with two cracks. The experimental setup is shown in Figure 9. The test system consists of a Polytec Vibrometer PSV-400 and its control system, a shaker, a power amplifier, and a cantilever beam with two cracks.

The first several mode shapes can be acquired by the vibrometer-dedicated software. The first mode shape and the third mode shape are processed based on the proposed method. The experiment results are showed in the Figures 10 and 11, respectively. In Figures 10(a) and 11(a), solid line denotes the FE mode shapes, and black dot denotes the measured mode shapes. By comparison, the first FE mode shape and the first measured mode shape are more fit than the third FE mode shape and the third measured mode shape. The reason resulting in this phenomenon is that the third mode shape is more sensitive to environmental noise and measuring error than the first mode shape. From Figure 10, the crack locations are at = 0.41 and =0.58 using the first mode shape, whose relative error is = 2.5% and = 3.3%, respectively. From Figure 11, the crack locations are at = 0.38 and using the third mode shape, whose relative error is = 5% and = 10%, respectively. According to the experimental results, the relative errors of crack location estimations are within 10%. Hence, the proposed method is effective and can be used for real applications with reasonable accuracy. However, by comparison, it can conclude that the detection accuracy is higher using the first mode shape than using the third mode shape, since high mode shape is more sensitive to environmental noise and measuring error than low mode shape.

Deep learning is a machine learning technique that utilizes the deep neural network [6]. It allows computational models of multiple processing layers to learn representations of data with multiple levels of abstraction [7]. By harnessing this feature of machine learning, it is possible to use a well-trained neural network to detect, and classify, defects in concrete structures such as bridges thereby aiding engineering judgements of the conditions of bridges. The Objectives of this paper are to develop a framework that can be used for automation of bridge inspection, train a network that can be used for concrete crack classification, develop an algorithm to obtain info. of crack size and location in structure and build a 3D crack visualization model to assist maintenance engineer to determine whether the crack needs immediate attention. 2b1af7f3a8