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Data driven condition assessment of railway infrastructure
Published in Hiroshi Yokota, Dan M. Frangopol, Bridge Maintenance, Safety, Management, Life-Cycle Sustainability and Innovations, 2021
C. Hoelzl, V. Dertimanis, E. Chatzi, D. Winklehner, S. Züger, A. Oprandi
Fractal analysis is a method that was originally developed as a way to approximate the length of the coastline of Great Britain. Using Euclidian geometry, the coastline can be approximated using a polygonal chain. The term of fractal dimension first appeared in 1967 in a paper on self-similarity written by Benoit Mandelbrot (Mandelbrot B. 1967). The fractal dimension corresponds to the statistical indicator of how the ratio between the details in a pattern changes with the measurement scale.
Fundamentals of Molecular Dynamics (MD) Simulations and Tools for Examining Nanostructured Materials
Published in Junko Habasaki, Molecular Dynamics of Nanostructures and Nanoionics, 2020
Fractal dimension analyses are useful tools to examine the complexity of structures and dynamics. Then results of fractal dimension analyses of density profiles used for characterization of structures in lithium silicates and ionic liquid (IL), (1-ethyl 3-methyl imidazolium nitrate, EMIM- NO3 ) will be shown. Dynamics of these systems are explained by the multifractal nature of the profile and the existence of corresponding two-length scale regions of trajectories of walks. Namely, through multifractal analyses of density profiles and walks, structures can be connected to dynamics. This kind of analysis provides a common framework for the characterization of more complex systems.
Fractal Dimension
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
Pablo M. Blanco, Sergio Madurga, Adriana Isvoran, Laura Pitulice, Francesc Mas
Fractal dimension can be used to characterize the complex geometry of natural macroscopic systems such as coasts, clouds or snowflakes with direct applicability in geography, paleontology, geology and many other scientific areas.
Pattern Detection on Glioblastoma’s Waddington Landscape via Generative Adversarial Networks
Published in Cybernetics and Systems, 2022
Fractal Analysis: The fractal dimension is a non-integer dimension, FD. A fractal dimension suggests the existence of a scaling law describing the complexity and roughness (self-similarity) of the pattern (Mandelbrot 1982). A fractal dimension is also a characteristic signature of deterministic chaos in dynamical systems (Frederickson et al. 1983). Strange attractors, the causal patterns to which the trajectories of a chaotic system are bound to, occupy fractal dimensions in phase-space (Ruelle 1980). The ImageJ plugin FracLac was used to compute the fractal dimension (FD) of the GAN-generated patterns. The images of the GAN-reconstructed attractors were enhanced in saturation to 25% and converted to binarized images on the ImageJ Plugin. FracLac then performs FD calculation via the Box-counting algorithm. The description of the Box-counting algorithm is as follows: Let L be the line length, be the box size, and be the number of boxes which can divide the pattern/object into self-similar substructures. The slope of the log-log plot of N and if it exists, provides the box-count fractal dimension of the object/pattern, as given by:
Study on the optimal compaction effort of asphalt mixture based on the distribution of contact points of coarse aggregates
Published in Road Materials and Pavement Design, 2021
Xu Cai, Kuanghuai Wu, Wenke Huang
Table 3 also provides the fractal dimensions of the gradations of asphalt mixture. The fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is measured. Song (2010) suggested that the gradations of asphalt mixture can be characterised by fractal dimensions, that the values of the fractal dimensions are related to the complexity of the structural system of asphalt mixture, and that the gradation with a greater fractal dimension generally indicates a compact structure. To simplify a calculation, Song studied the correlation between the slope k of the logarithmic curve of a mineral aggregate gradation and the fractal dimension value D, and proposed a simplified fractal dimension calculation method, as shown in Equation (1). where k represents the slope of a mineral aggregate gradation in the logarithmic curve coordinate system and D represents the fractal dimension value of a gradation.
Mechanical behaviour and seepage characteristics of coal under the loading path of roadway excavation and coal mining
Published in Geomatics, Natural Hazards and Risk, 2021
Chenghang Fu, Heping Xie, Mingzhong Gao, Fei Wang, Jing Xie, Junjun Liu, Bengao Yang, Ruifeng Tang
Fractal dimension is a feature quantity that can effectively reflect the space occupation and complexity of complex shapes. It is often used to quantitatively describe the irregularity and complexity of cracks. In this study, the fractal dimension of cracks is calculated by box covering method. Firstly, the cracks depicted in Figures 10–12 are transformed into bitmaps and binarized, and then the binarized image is divided into several squares with sides of δ. Calculate the number of squares containing at least one pixel on the image, and record the number as Nδ. Change the size of δ and repeat the above process. Finally, the fractal dimension is obtained from Equation (13) (Gao, Zhang, et al. 2020):