Fractal Fract., Volume 8, Issue 6 (June 2024) – 30 articles

Over the last decade, dual active bridge (DAB) converters have become critical components in high-frequency power conversion systems. Recently, intensive efforts have been directed at optimizing DAB converter design and control. In particular, several strategies have been proposed to improve the performance of DAB control systems. For example, fractional-order (FO) control methods have proven potential in several applications since they offer improved controllability, flexibility, and robustness. However, the FO controller design process is critical for industrializing their use. Conventional FO control design methods use frequency domain-based design schemes, which result in complex and impractical designs. In addition, several nonlinear equations need to be solved to determine the optimum parameters. Currently, metaheuristic algorithms are used to design FO controllers due to their effectiveness in improving system performance and their ability to simultaneously tune possible design parameters. Moreover, metaheuristic algorithms do not require precise and detailed knowledge of the controlled system model. In this paper, a hybrid algorithm based on the chaotic artificial ecosystem-based optimization (AEO) and manta-ray foraging optimization (MRFO) algorithms is proposed with the aim of combining the best features of each. Unlike the conventional MRFO method, the newly proposed hybrid AEO-CMRFO algorithm enables the use of chaotic maps and weighting factors. Moreover, the AEO and CMRFO hybridization process enables better convergence performance and the avoidance of local optima. Therefore, superior FO controller performance was achieved compared to traditional control design methods and other studied metaheuristic algorithms. An exhaustive study is provided, and the proposed control method was compared with traditional control methods to verify its advantages and superiority. Full article

The balance between accuracy and computational complexity is currently a focal point of research in dynamical system modeling. From the perspective of model reduction, this paper addresses the mode selection strategy in Dynamic Mode Decomposition (DMD) by integrating an embedded fractal theory based on fractal dimension (FD). The existing model selection methods lack interpretability and exhibit arbitrariness in choosing mode dimension truncation levels. To address these issues, this paper analyzes the geometric features of modes for the dimensional characteristics of dynamical systems. By calculating the box counting dimension (BCD) of modes and the correlation dimension (CD) and embedding dimension (ED) of the original dynamical system, it achieves guidance on the importance ranking of modes and the truncation order of modes in DMD. To validate the practicality of this method, it is applied to the reduction applications on the reconstruction of the velocity field of cylinder wake flow and the force field of compressor blades. Theoretical results demonstrate that the proposed selection technique can effectively characterize the primary dynamic features of the original dynamical systems. By employing a loss function to measure the accuracy of the reconstruction models, the computed results show that the overall errors of the reconstruction models are below 5%. These results indicate that this method, based on fractal theory, ensures the model’s accuracy and significantly reduces the complexity of subsequent computations, exhibiting strong interpretability and practicality. Full article

This paper introduces the fractional Heston-type (fHt) model as a stochastic system comprising the stock price process modeled by a geometric Brownian motion. In this model, the infinitesimal return volatility is characterized by the square of a singular stochastic equation driven by a fractional Brownian motion with a Hurst parameter $H\in (0,1)$. We establish the Malliavin differentiability of the fHt model and derive an expression for the expected payoff function, revealing potential discontinuities. Simulation experiments are conducted to illustrate the dynamics of the stock price process and option prices. Full article

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a $\psi$-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results. Full article

In order to achieve time–frequency superresolution in comparison to the conventional Wigner distribution (WD), this study generalizes the well-known $\tau$-Wigner distribution ($\tau$-WD) with only one parameter $\tau$ to the multiple-parameter matrix-Wigner distribution (M-WD) with the parameter matrix $\mathcal{M}$. According to operator theory, we construct Heisenberg’s inequalities on the uncertainty product in M-WD domains and formulate two kinds of attainable lower bounds dependent on $\mathcal{M}$. We solve the problem of lower bound minimization and obtain the optimality condition of $\mathcal{M}$, under which the M-WD achieves superior time–frequency resolution. It turns out that the M-WD breaks through the limitation of the $\tau$-WD and gives birth to some novel distributions other than the WD that could generate the highest time–frequency resolution. As an example, the two-dimensional linear frequency-modulated signal is carried out to demonstrate the time–frequency concentration superiority of the M-WD over the short-time Fourier transform and wavelet transform. Full article

An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination. Full article

This work explores the existence and uniqueness criteria for the solution of hybrid Caputo–Hadamard fractional sequential differential equations (HCHFSDEs) by employing Darbo’s fixed-point theorem. Fractional differential equations play a pivotal role in modeling complex phenomena in various areas of science and engineering. The hybrid approach considered in this work combines the advantages of both the Caputo and Hadamard fractional derivatives, leading to a more comprehensive and versatile model for describing sequential processes. To address the problem of the existence and uniqueness of solutions for such hybrid fractional sequential differential equations, we turn to Darbo’s fixed-point theorem, a powerful mathematical tool that establishes the existence of fixed points for certain types of mappings. By appropriately transforming the differential equation into an equivalent fixed-point formulation, we can exploit the properties of Darbo’s theorem to analyze the solutions’ existence and uniqueness. The outcomes of this research expand the understanding of HCHFSDEs and contribute to the growing body of knowledge in fractional calculus and fixed-point theory. These findings are expected to have significant implications in various scientific and engineering applications, where sequential processes are prevalent, such as in physics, biology, finance, and control theory. Full article

In this study, we investigate the application of fractional calculus to the mathematical modeling of biological systems, focusing on fractional-order-in-time partial differential equations (FTPDEs). Fractional derivatives, especially those defined in the Caputo sense, provide a useful tool for modeling memory and hereditary characteristics, which are problems that are frequently faced with integer-order models. We use the Chebyshev spectral approach for spatial derivatives, which is known for its faster convergence rate, in conjunction with the $L1$ scheme for time-fractional derivatives because of its high accuracy and robustness in handling nonlocal effects. A detailed theoretical analysis, followed by a number of numerical experiments, is performed to confirmed the theoretical justification. Our simulation results show that our numerical technique significantly improves the convergence rates, effectively tackles computing difficulties, and provides a realistic simulation of biological population dynamics. Full article

In this article, we examine variational inequalities of the form $\left\{\begin{array}{c}⟨\mathcal{A}\left(u\right),v-u⟩+⟨\mathcal{F}\left(u\right),v-u⟩\ge 0,\forall v\in K\hfill \\ u\in K,\hfill \end{array}\right.$, where $\mathcal{A}$ is a generalized fractional Φ-Laplace operator, K is a closed convex set in a fractional Musielak–Orlicz–Sobolev space, and $\mathcal{F}$ is a multivalued integral operator. We consider a functional analytic framework for the above problem, including conditions on the multivalued lower order term $\mathcal{F}$ such that the problem can be properly formulated in a fractional Musielak–Orlicz–Sobolev space, and the involved mappings have certain useful monotonicity–continuity properties. Furthermore, we investigate the existence of solutions contingent upon certain coercivity conditions. Full article

This study proposes an enhanced Kepler Optimization (EKO) algorithm, incorporating fractional-order components to develop a Proportional-Integral-First-Order Double Derivative (PI–(1+DD) controller for frequency stability control in multi-area power systems with wind power integration. The fractional-order element facilitates efficient information and past experience sharing among participants, hence increasing the search efficiency of the EKO algorithm. Furthermore, a local escaping approach is included to improve the search process for avoiding local optimization. Applications were performed through comparisons with the 2020 IEEE Congress on Evolutionary Computation (CEC 2020) benchmark tests and applications in a two-area system, including thermal and wind power. In this regard, comparisons were implemented considering three different controllers of PI, PID, and PI–(1+DD) designs. The simulations show that the EKO algorithm demonstrates superior performance in optimizing load frequency control (LFC), significantly improving the stability of power systems with renewable energy systems (RES) integration. Full article

This paper introduces and explores the dynamics of a novel three-dimensional (3D) fractional map with hidden dynamics. The map is constructed through the integration of a discrete sinusoidal memristive into a discrete Duffing map. Moreover, a mathematical operator, namely, a fractional variable-order Caputo-like difference operator, is employed to establish the fractional form of the map with short memory. The numerical simulation results highlight its excellent dynamical behavior, revealing that the addition of the piecewise fractional order makes the memristive-based Duffing map even more chaotic. It is characterized by distinct features, including the absence of an equilibrium point and the presence of multiple hidden chaotic attractors. Full article

This paper investigates the dynamics of the SIR infectious disease model, with a specific emphasis on utilizing a harmonic mean-type incidence rate. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. A sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V-L) matrices, is developed to examine the overall trajectory of the model globally. In addition to that, we describe the transmission of infectious disease through a mathematical model using fractal-fractional differential operators. We prove the existence and uniqueness of solutions in the SIR model framework with a harmonic mean-type incidence rate by using the Banach contraction approach. Functional analysis is used together with the Ulam–Hyers (UH) stability approach to perform stability analysis. We simulate the numerical results by using a computational scheme with the help of MATLAB. This study advances our knowledge of the dynamics of epidemic dissemination and facilitates the development of disease prevention and mitigation tactics. Full article

This paper proposes a multifractal least squares support vector machine detrended fluctuation analysis (MF-LSSVM-DFA) model. The system is an extension of the traditional MF-DFA model. To address potential overfitting or underfitting caused by the fixed-order polynomial fitting in MF-DFA, LSSVM is employed as a superior alternative for fitting. This approach enhances model accuracy and adaptability, ensuring more reliable analysis results. We utilize the p model to construct a multiplicative cascade time series to evaluate the performance of MF-LSSVM-DFA, MF-DFA, and two other models that improve upon MF-DFA from recent studies. The results demonstrate that our proposed modified model yields generalized Hurst exponents $h\left(q\right)$ and scaling exponents $\tau \left(q\right)$ that align more closely with the analytical solutions, indicating superior correction effectiveness. In addition, we explore the sensitivity of MF-LSSVM-DFA to the overlapping window size s. We find that the sensitivity of our proposed model is less than that of MF-DFA. We find that when s exceeds the limited range of the traditional MF-DFA, $h\left(q\right)$ and $\tau \left(q\right)$ are closer than those obtained in MF-DFA when s is in a limited range. Meanwhile, we analyze the performances of the fitting of the two models and the results imply that MF-LSSVM-DFA achieves a better outstanding performance. In addition, we put the proposed MF-LSSVM-DFA into practice for applications in the medical field, and we found that MF-LSSVM-DFA improves the accuracy of ECG signal classification and the stability and robustness of the algorithm compared with MF-DFA. Finally, numerous image segmentation experiments are adopted to verify the effectiveness and robustness of our proposed method. Full article

The two-parameter Mittag–Leffler function $E_{\alpha ,\beta }$ is of fundamental importance in fractional calculus, and it appears frequently in the solutions of fractional differential and integral equations. However, the expense of calculating this function often prompts efforts to devise accurate approximations that are more cost-effective. When $\alpha >1$, the monotonicity property is largely lost, resulting in the emergence of roots and oscillations. As a result, current rational approximants constructed mainly for $\alpha \in (0,1)$ often fail to capture this oscillatory behavior. In this paper, we develop computationally efficient rational approximants for $E_{\alpha ,\beta }(-t)$, $t\ge 0$, with $\alpha \in (1,2)$. This process involves decomposing the Mittag–Leffler function with real roots into a weighted root-free Mittag–Leffler function and a polynomial. This provides approximants valid over extended intervals. These approximants are then extended to the matrix Mittag–Leffler function, and different implementation strategies are discussed, including using partial fraction decomposition. Numerical experiments are conducted to illustrate the performance of the proposed approximants. Full article

The main objective of this manuscript is to define the concepts of F-(⋏,h)-contraction and ($\alpha$,$\eta )$-Reich type interpolative contraction in the framework of orthogonal $\mathfrak{F}$-metric space and prove some fixed point results. Our primary result serves as a cornerstone, from which established findings in the literature emerge as natural consequences. To enhance the clarity of our novel contributions, we furnish a significant example that not only strengthens the innovative findings but also facilitates a deeper understanding of the established theory. The concluding section of our work is dedicated to the application of these results in establishing the existence and uniqueness of a solution for a fractional differential equation of anomalous diffusion. Full article

This paper discusses the time-fractional nonlinear Schrödinger model with optical soliton solutions. We employ the $\left(f+\left(\frac{G^{\prime }}{G}\right)\right)$-expansion method to attain the optical solution solutions. An important tool for explaining the particular explosion of brief pulses in optical fibers is the nonlinear Schrödinger model. It can also be utilized in a telecommunications system. The suggested method yields trigonometric solutions such as dark, bright, kink, and anti-kink-type optical soliton solutions. Mathematica 11 software creates 2D and 3D graphs for many physically important parameters. The computational method is effective and generally appropriate for solving analytical problems related to complicated nonlinear issues that have emerged in the recent history of nonlinear optics and mathematical physics. Furthermore, we venture into uncharted territory by subjecting our model to chaotic and sensitivity analysis, shedding light on its robustness and responsiveness to perturbations. The proposed technique is being applied to this model for the first time. Full article

To better simulate the prices of underlying assets and improve the accuracy of pricing financial derivatives, an increasing number of new models are being proposed. Among them, the Lévy process with jumps has received increasing attention because of its capacity to model sudden movements in asset prices. This paper explores the Hamilton–Jacobi–Bellman (HJB) equation with a fractional derivative and an integro-differential operator, which arise in the valuation of American options and stock loans based on the Lévy-$\alpha$-stable process with jumps model. We design a fast solution strategy that includes the policy iteration method, Krylov subspace method, and banded preconditioner, aiming to solve this equation rapidly. To solve the resulting HJB equation, a finite difference method including an upwind scheme, shifted Grünwald approximation, and trapezoidal method is developed with stability and convergence analysis. Then, an algorithmic framework involving the policy iteration method and the Krylov subspace method is employed. To improve the performance of the above solver, a banded preconditioner is proposed with condition number analysis. Finally, two examples, sugar option pricing and stock loan valuation, are provided to illustrate the effectiveness of the considered model and the efficiency of the proposed preconditioned policy–Krylov subspace method. Full article

In this paper, we consider a direct problem and an inverse problem involving a nonlinear fractional diffusion equation, which can be applied to many physical situations. The equation contains a Caputo fractional derivative, a symmetric uniformly elliptic operator and a source term consisting of the sum of two terms, one of which is linear and the other is nonlinear. The well-posedness of the direct problem is examined and the results are used to investigate the stability of an inverse problem of determining a function in the linear part of the source. The main tools in our study are the generalized eigenfunction expansions theory for nonlinear fractional diffusion equations, contraction mapping, Young’s convolution and generalized Grönwall’s inequalities. We present a stability estimate for the solution of the inverse source problem by means of observation data at a given point in the domain. Full article

The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with SrTiO₃, MnO₂ and SrTiO₃/MnO₂ were used for fractal reconstructions of particle agglomerates in the polymer matrix. Fractal analysis represents a valuable mathematical tool for the characterization of the microstructure and finding correlation between microstructural features and mechanical properties. Utilizing the mathematical affine fractal regression model, the Fractal Real Finder software was employed to reconstruct agglomerate shapes and estimate the Hausdorff dimensions (HD). Controlled energy impact and tensile tests were used to evaluate the mechanical performance of PMMA-MnO₂, PMMA-SrTiO₃ and PMMA-SrTiO₃/MnO₂ composites. It was determined that PMMA-SrTiO₃/MnO₂ had the highest total absorbed energy value (E_tot), corresponding to the lowest HD value of 1.03637 calculated for SrTiO₃/MnO₂ agglomerates. On the other hand, the highest HD value of 1.21521 was calculated for MnO₂ agglomerates, while the PMMA-MnO₂ showed the lowest E_tot. The linear correlation between the total absorbed impact energy of composites and the HD of the corresponding agglomerates was determined, with an R² value of 0.99486, showing the potential use of this approach in the optimization of composite materials’ microstructure–property relationship. Full article

Ordinary differential equations have recently been extended to fractional equations that are transformed using fractional differential equations. These fractional equations are believed to have high accuracy and low computational cost compared to ordinary differential equations. For the first time, this paper focuses on extending the nonlinear heat equations to a fractional order in a Caputo order. A new perturbation iteration algorithm (PIA) of the fractional order is applied to solve the nonlinear heat equations. Solving numerical problems that involve fractional differential equations can be challenging due to their inherent complexity and high computational cost. To overcome these challenges, there is a need to develop numerical schemes such as the PIA method. This method can provide approximate solutions to problems that involve classical fractional derivatives. The results obtained from this algorithm are compared with those obtained from the perturbation iteration method (PIM), the variational iteration method (VIM), and the Bezier curve method (BCM). All solutions are tested with numerical simulations. The study found that the new PIA algorithm performs better than the PIM, VIM, and BCM, achieving high accuracy and low computational cost. One significant advantage of this algorithm is that the solutions obtained have established that the fractional values of alpha, specifically $\alpha$, significantly influencing the accuracy of the outcome and the associated computational cost. Full article

The intricacy and fractal properties of human DNA sequences are examined in this work. The core of this study is to discern whether complete DNA sequences present distinct complexity and fractal attributes compared with sequences containing exclusively exon regions. In this regard, the entire base pair sequences of DNA are extracted from the NCBI (National Center for Biotechnology Information) database. In order to create a time series representation for the base pair sequence $\{G,C,T,A\}$, we use the Chaos Game Representation (CGR) approach and a mapping rule f, which enables us to apply the metric known as the Complexity–Entropy Plane (CEP) and multifractal detrended fluctuation analysis (MF-DFA). To carry out our investigation, we divided human DNA into two groups: the first is composed of the 24 chromosomes, which comprises all the base pairs that form the DNA sequence, and another group that also includes the 24 chromosomes, but the DNA sequences rely only on the exons’ presence. The results show that both sets provide fractal patterns in their structure, as obtained by the CGR approach. Complete DNA sequences show a sharper visual fractal pattern than sequences composed only of exons. Moreover, the sequences occupy distinct areas of the complexity–entropy plane, and the complete DNA sequences lead to greater statistical complexity and lower entropy than the exon sequences. Also, we observed that different fractal parameters between chromosomes indicate diversity in genomic sequences. All these results occur in different scales for all chromosomes. Full article

In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results. Full article

In this paper, we study the existence of multiple normalized solutions for a Choquard equation involving fractional p-Laplacian in ${\mathbb{R}}^N$. With the help of variational methods, minimization techniques, and the Lusternik–Schnirelmann category, the existence of multiple normalized solutions is obtained for the above problem. Full article

The modified second-grade fluid flow across a plate of semi-infinite extent, which is initiated by the plate’s movement, is considered herein. The relaxation parameters and fractional parameters are introduced to express the generalized constitutive relation. A convolution-based absorbing boundary condition (ABC) is developed based on the artificial boundary method (ABM), addressing issues related to the semi-infinite boundary. We adopt the finite difference method (FDM) for deriving the numerical solution by employing the L1 scheme to approximate the fractional derivative. To confirm the precision of this method, a source term is added to establish an exact solution for verification purposes. A comparative evaluation of the ABC versus the direct truncated boundary condition (DTBC) is conducted, with their effectiveness and soundness being visually scrutinized and assessed. This study investigates the impact of the motion of plates at different fluid flow velocities, focusing on the effects of dynamic elements influencing flow mechanisms and velocity. This research’s primary conclusion is that a higher fractional parameter correlates with the fluid flow. As relaxation parameters decrease, the delay effect intensifies and the fluid velocity decreases. Full article

The concept of subordination is expanded in this study from the fuzzy sets theory to the geometry theory of analytic functions with a single complex variable. This work aims to clarify fuzzy subordination as a notion and demonstrate its primary attributes. With this work’s assistance, new fuzzy differential subordinations will be presented. The first theorems lead to intriguing corollaries for specific aspects chosen to exhibit fuzzy best dominance. The work introduces a new integral operator for meromorphic functions and uses the newly developed integral operator, which is starlike and convex, respectively, to obtain conclusions on fuzzy differential subordination. Full article

In this paper, the characteristics of absolute value memristors are verified through the circuit implementation and construction of a chaotic system with a conditional symmetric fractional-order memristor. The dynamic behavior of fractional-order memristor systems is explored using fractional-order calculus theory and the Adomian Decomposition Method (ADM). Concurrently, the investigation probes into the existence of coexisting symmetric attractors, multiple coexisting bifurcation diagrams, and Lyapunov exponent spectra (LEs) utilizing system parameters as variables. Additionally, the system demonstrates an intriguing phenomenon known as offset boosting, where the embedding of an offset can adjust the position and size of the system’s attractors. To ensure the practical applicability of these findings, a fractional-order sliding mode synchronization control scheme, inspired by integer-order sliding mode theory, is designed. The rationality and feasibility of this scheme are validated through a theoretical analysis and numerical simulation. Full article

Mexico is a well-known seismically active country, which is primarily affected by several tectonic plate interactions along the southern Pacific coastline and by active structures in the Gulf of California. In this paper, we investigate this seismicity using the classical Gutenberg–Richter (GR) law and a non-extensive statistical approach based on Tsallis entropy. The analysis is performed using data from the corrected Mexican seismic catalog provided by the National Seismic Service, spanning the period from January 2000 to October 2023, and unlike previous work, it includes six different regions along the entire west coastline of Mexico. The Gutenberg–Richter law fitting to the earthquake sub-catalogs for all six regions studied indicates magnitudes of completeness between 3.30 and 3.76, implying that the majority of seismic movements occur for magnitudes less than 4. The cumulative distribution of earthquakes as derived from the Tsallis entropy was fitted to the corrected catalog data to estimate the q-entropic index for all six regions, which for values greater than one is a measure of the non-extensivity (i.e., non-equilibrium) of the system. All regions display values of the entropic index in the range $1.52\lesssim q\lesssim 1.61$, slightly lower than previously estimated ( $1.54\lesssim q\lesssim 1.70$) using catalog data from 1988 to 2010. The reason for this difference is related to the use of modern recording devices, which are sensitive to the detection of a larger number of low-magnitude events compared to older instrumentation. Full article

This paper considers a nonlinear impulsive fractional boundary value problem, which involves a $\psi$-Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results of infinitely many solutions are established depending on some simple algebraic conditions. Finally, examples are also presented, which show that Caputo-type fractional models can be more accurate by selecting different kernels for the fractional integral and derivative. Full article

Cement-based composites’, as the most widely used building material, macroscopic performance significantly influences the safety of engineering structures. Meanwhile, the macroscopic properties of cement-based composites are tightly related to their microscopic structure. The complexity of cement-based composites’ microscopic structure is challenging to describe geometrically, so fractal theory is extensively applied to quantify the microscopic structure of cement-based composites. However, existing studies have not clearly defined the quantification methods for various microscopic structures in CCs, nor have they provided a comprehensive evaluation of the correlation between the fractal dimensions of different microscopic structures and macroscopic performance. So, this study categorizes the commonly used testing methods in fractal theory into three categories: particle distribution (laser granulometry, etc.), pore structure (mercury intrusion porosity, etc.), and fracture (computed tomography, etc.). It systematically establishes a detailed process for the application of testing methods, the processing of test results, model building, and fractal dimension calculation. The applicability of different fractal dimension calculation models and the range of the same fractal dimension established by different models are compared and discussed, and the advantages and disadvantages of different models are analyzed. Finally, the research delves into an in-depth analysis of the relationship between the fractal dimension of cement-based composites’ microscopic structure and its macroscopic properties, such as compressive strength, corrosion resistance, impermeability, and high-temperature resistance. The principle that affects the positive and negative correlation between fractal dimension and macroscopic performance is discussed and revealed in this study. The comprehensive review in this paper provides scholars with methods and models for quantitative research on the microscopic structural parameters of cement-based composites and offers a pathway for the non-destructive assessment of the macroscopic performance of cement-based composites. Full article

As a geological barrier for high-level radioactive waste (HLW) disposal in China, granite is crucial for blocking nuclide migration into the biosphere. However, the high uncertainty associated with the 3D geological system, such as the stochastic discrete fracture networks in granite, significantly impedes practical safety assessments of HLW disposal. This study proposes a Monte Carlo simulation (MCS)-based simulation framework for evaluating the long-term barrier performance of nuclide migration in fractured rocks. Statistical data on fracture geometric parameters, on-site hydrogeological conditions, and relevant migration parameters are obtained from a research site in Northwestern China. The simulation models consider the migration of three key nuclides, Cs-135, Se-79, and Zr-93, in fractured granite, with mechanisms including adsorption, advection, diffusion, dispersion, and decay considered as factors. Subsequently, sixty MCS realizations are performed to conduct a sensitivity analysis using the open-source software OpenGeoSys-5 (OGS-5). The results reveal the maximum and minimum values of the nuclide breakthrough time T_t (12,000 and 3600 years, respectively) and the maximum and minimum values of the nuclide breakthrough concentration C_max (4.26 × 10⁻⁴ mSv/a and 2.64 × 10⁻⁵ mSv/a, respectively). These significant differences underscore the significant effect of the uncertainty in the discrete fracture network model on long-term barrier performance. After the failure of the waste tank (1000 years), nuclides are estimated to reach the outlet boundary 6480 years later. The individual effective dose in the biosphere initially increases and then decreases, reaching a peak value of C_max = 4.26 × 10⁻⁴ mSv/a around 350,000 years, which is below the critical dose of 0.01 mSv/a. These sensitivity analysis results concerning nuclide migration in discrete fractured granite can enhance the simulation and prediction accuracy for risk evaluation of HLW disposal. Full article