Search Results (319)

This paper presents an analytical model for one-dimensional consolidation analysis of multi-layered unsaturated soils under depth-dependent initial conditions. The general solutions are derived explicitly using the Laplace transform. By combining these general solutions with interfacial continuity conditions between layers and the boundary conditions, the reduced-order system is solved via the Euler method to obtain analytical solutions in the Laplace domain. Numerical inversion of the Laplace transform is then performed using Crump’s method to yield the final analytical solutions in the time domain. The model incorporates initial conditions that account for both uniform and linear distributions of initial excess pore pressure within the soil stratum. The proposed solution is verified by reducing it to degenerated cases (e.g., uniform initial pressure) and comparing it with existing analytical solutions, showing excellent agreement. This confirms the model’s correctness and demonstrates its generalization to multi-layered systems with depth-dependent initial conditions. Focusing on a double-layered unsaturated soil system, the one-dimensional consolidation characteristics under depth-dependent initial conditions are investigated by varying the physical parameters of individual layers. The proposed solution can serve as a theoretical reference for the consolidation analysis of multi-layered unsaturated soils with depth-dependent initial conditions. Full article

Steel pipes play a vital role in energy and industrial transportation systems, where undetected defects such as cracks and wall thinning may lead to severe safety hazards. Although ultrasonic guided waves enable long-range inspection, their defect imaging performance is often limited by dispersion, multimode interference, and strong noise. In this work, a multi-frequency fusion imaging method integrating time-reversal, topological derivative, and sparse Bayesian learning is proposed for guided wave-based defect detection in steel pipes. Multi-frequency guided waves are employed to enhance defect sensitivity and suppress frequency-dependent ambiguity. Time-reversal focusing is used to concentrate scattered energy at defect locations, while the topological derivative provides a global sensitivity map as physics-guided prior information. These results are further fused within a sparse Bayesian learning framework to achieve probabilistic defect imaging and uncertainty quantification. Dispersion compensation based on the semi-analytical finite element method is introduced to ensure accurate wavefield reconstruction at different frequencies. Domain randomization is also incorporated to improve robustness against uncertainties in material properties, temperature, and measurement noise. Numerical simulation results verify that the proposed method achieves high localization accuracy and significantly outperforms conventional TR-based imaging in terms of resolution, false alarm suppression, and stability. The proposed approach provides a reliable and robust solution for guided wave inspection of steel pipelines and offers strong potential for engineering applications in nondestructive evaluation and structural health monitoring. Full article

In the wave theory of saturated soils, pore tortuosity is an important physical property for quantifying the added mass force caused by the relative acceleration between solid and liquid phases. However, this inertial force is often ignored for simplicity in practical applications. To investigate the influence of pore tortuosity on the propagation of compressional waves in saturated soils, a system of generalized governing equations for one-dimensional infinitesimal strain elastic waves is solved using the Laplace transform method. Semi-analytical solutions are obtained for the spatiotemporal distributions of the excess pore water pressure, the pore water velocity, and the soil particle velocity caused by a step load perturbation under undrained conditions. These solutions are used to evaluate the effects of pore tortuosity on the velocities and amplitudes of fast and slow compressional waves. The results show that pore tortuosity has an insignificant effect on the propagation of fast compressional waves, but for slow compressional waves, the larger the pore tortuosity is, the lower the wave velocity and the larger the wave amplitude. Ignoring the influence of pore tortuosity can lead to an underestimation of the arrival time of slow compressional wave. The propagation of this wave is limited to a distance of approximately 1 m away from the loading boundary. This research finding is purely theoretical. For further experimental validation, it is suggested to detect the slow compressional wave by placing miniature acoustic receiving transducers as close as possible to the loading or transmitting surface. The proposed solutions are also useful for calibrating sophisticated numerical codes for dynamic consolidation of saturated soils and wave transmission in porous media. Full article

Soil–structure interaction (SSI) under torsional loading plays a critical role in the dynamic performance of foundations supporting offshore structures and machine foundations. However, existing simplified or semi-analytical approaches often idealize the pile tip boundary and may not adequately capture the frequency-dependent torsional impedance induced by finite soil thickness beneath the pile tip in layered deposits. This study develops a Hamilton-based variational solution for dynamic torsional soil–pile interaction in layered viscoelastic soils by explicitly incorporating a fictitious soil pile (FSP) beneath the pile tip within an energy-consistent framework. Admissible torsional displacement fields for the pile, layered soil, and FSP are adopted to establish a frequency-domain variational functional, and an iterative scheme is used to obtain the convergent frequency-dependent torsional impedance at the pile head. The formulation is verified against an existing semi-analytical solution for piles in layered soils and shows excellent agreement. Parametric results indicate that introducing a finite FSP reduces torsional stiffness and increases damping compared with a rigid base condition, while the thickness and stiffness of the bearing stratum govern the variation in impedance, providing physical insight into torsional SSI in layered ground. Full article

Low-thrust propulsion systems have become mainstream for Low Earth Orbit (LEO) satellites due to their superior propellant efficiency, yet conventional low-thrust transfer strategies suffer from high computational costs and failure to achieve full orbital element convergence. To address these drawbacks, this paper proposes a novel semi-analytical three-phase low-thrust transfer strategy that leverages $J_2$ gravitational precession to realize convergence of all orbital elements for circular orbits. The core of the method lies in the design of two symmetric thrust arcs and an intermediate coasting period that utilizes $J_2$ precession. By solving the resulting polynomial equation, the strategy achieves simultaneous controlled convergence of the Right Ascension of the Ascending Node (RAAN) and the argument of latitude (AOL). Simulation results demonstrate that the proposed method achieves significant fuel savings compared to direct transfer strategies, while simultaneously achieving superior computational speed. Extensive validation via 100,000 Monte Carlo simulations confirms the method’s scope of applicability, and the sufficient conditions for the existence of a solution are provided. It is further found that the proposed method is particularly well-suited for missions involving medium-to-high inclination orbits and large RAAN gaps, such as constellation deployment. In conclusion, this strategy provides a fuel-efficient and computationally fast solution for low-thrust transfer, establishing the basis for the operational management of future large-scale space systems equipped with low-thrust propulsion. Full article

The vibrations of the dynamical system play an important role in biological processes, electrical engineering, and mechanical structures. In this work, we focus on the behaviors of dynamical systems, such as periodical or damped oscillations and asymptotic behaviors. Theorems for explicitly integrability of the dynamical system are established. The effect of the physical parameters $0<a\le 1$, $d\ge 0$ is semi-analytically analyzed by means of the Optimal Parametric Iteration Method (OPIM). We pointed out some cases when the investigated system admits only one first integral or two first integrals. These cases are reduced to a second-order nonlinear differential equations, which are solved by OPIM. The OPIM solutions are highlighted qualitatively by figures and quantitatively by tables, respectively, and are in good agreement with corresponding numerical ones. The accuracy of the obtained results are emphasized by comparison with the iterative solutions, via the classical iterative method and new optimal iterative method, respectively. Other advantages of the applied method are pointed out. Full article

Water depth in shallow marine environments is a fundamental parameter for oceanographic research and coastal engineering applications. High-resolution satellite imagery and long-term medium-resolution imagery offer significant potential for detailed bathymetric mapping and monitoring spatiotemporal variations in bathymetry. However, most of these images contain only three visible bands (blue, green, and red), making bathymetric mapping from such images challenging in practical applications. For the empirical approach, high-quality in situ depth calibration data, which are essential for establishing a reliable empirical bathymetric model, are either unavailable or excessively expensive. For the physics-based approach, images containing only three visible bands can be problematic in accurately deriving depths. To address this limitation, this study proposes a novel semi-analytical-empirical hybrid model for water depth retrieval. The core of the proposed method is the integration of a semi-analytical model with a physics-based dual-band model. This integration quantifies the relative depth relationships among pixels and uses them as a physical constraint. Through this constraint, the method identifies physically reliable depth estimates from the multiple numerical solutions of the semi-analytical model for a subset of shallow-water pixels, which then serve as an in situ–free calibration dataset. This dataset is subsequently used to determine the empirically based optimal retrieval model, which is finally applied to generate the complete bathymetric map. The results from four typical coral reef regions—Buck Island, Yongxing Island, Kaneohe Bay, and Yongle Atoll—demonstrated that the proposed model achieved root-mean-square errors (RMSE) of 0.98–1.62 m, mean absolute errors (MAE) of 0.73–1.13 m, and coefficients of determination (R²) of 0.91–0.95 in comparison to in situ measurements. Compared to both the physics-based dual-band model and the L-S model (i.e., the bathymetry mapping approach combining Log-ratio and Semi-analytical models), the proposed model reduced the RMSE by 9–55%, reduced the MAE by 4–56%, and improved the R² by 0.01–0.29. Additionally, the accuracy of the proposed model surpasses that of both the physics-based dual-band model and the L-S model across all depth intervals, particularly in deeper depth waters (>15 m). This study offers a robust solution for bathymetric mapping in areas lacking in situ depth data and contributes significantly to advancing optical remote sensing techniques for underwater topography detection. Full article

The mechanical integrity of advanced porous materials and perforated structures at the nanoscale is critically governed by the interaction of surface effects and stress concentration around pore architectures. This paper investigates the residual stress field induced by surface tension around two arbitrarily shaped nano-perforations within an infinite elastic matrix, a configuration highly relevant to nanoporous metals and functional composites. By leveraging the complex variable method and conformal mapping techniques, the physical domains of the perforations (approximated as triangular and square shapes, paired with an elliptical perforation) are transformed into unit circles. This transformation allows for the derivation of semi-analytical solutions for the complex potentials and the subsequent stress field. Systematic numerical case studies reveal that a reduced inter-perforation distance dramatically intensifies the hoop stress concentration at the adjacent vertices, identifying these sites as potential initiation points for mechanical failure. Conversely, an increase in the size of one perforation can effectively shield its neighbor and reduce the overall stress level. These findings provide quantitative, physics-based guidelines for the microstructural design of nanoporous materials. By consciously tailoring the spatial distribution, size, and shape of perforations, the mechanical reliability of nanomaterials can be rationally optimized for applications in nanoscale systems. Full article

This extensive study introduces the Rosenbrock method (RosM) for numerically integrating the chaotic Lorenz system, with a focus on its ability to preserve the system’s intrinsic dynamical and structural symmetries. The Lorenz system exhibits significant symmetry, most notably an inversion symmetry $(x,y,z)⟶(-x,-y,z)$, which is a fundamental feature of its chaotic attractor. We lay forth the algorithm and, after systematic comparisons to explicit Runge–Kutta higher-order schemes and semi-analytically obtained solutions, show that the second-order Rosenbrock method performs with excellent accuracy and stability. Crucially, we demonstrate that RosM reliably preserves the system’s symmetry over long-term integration, a property where some explicit methods can exhibit subtle drift. We give a formal error characterization, assess the computational efficiency, and verify the method via bifurcation analysis to support that RosM is a robust and symmetry-aware tool for simulating chaotic systems. Full article

This study proposes a semi-analytic harmonic modeling method that significantly improves the accuracy and efficiency of complex magnetic field modeling by integrating numerical and analytical approaches. Compared to traditional methods such as the equivalent charge method and finite element method, this approach optimizes the distribution of surface and body charges in the magnetic dipole model and introduces a finite and variable permeability model to accommodate material non-uniformity. Through harmonic expansion and analytical optimization, the method more accurately reflects the characteristics of real magnets, providing an efficient and precise solution for complex magnetic field problems, particularly in the design of high-performance magnets such as Halbach arrays. In this study, the effectiveness of the new modeling method is verified through the combination of simulation and experiment: the magnetic field distribution of the new Halbach array is accurately simulated, and the applicability of the model in the description of complex magnetic fields is analyzed. The dynamic response ability of the optimized model is verified by modeling and simulating the variation of the permeability under actual conditions. The distribution of scalar potential energy with permeability was simulated to evaluate the adaptability of the model to the real physical field. Through the comparative analysis of simulation and experimental results, the advantages of the new method in modeling accuracy and efficiency are clearly pointed out, and the effectiveness of the semi-analytic harmonic modeling method and its wide application potential in the design of new magnetic fields are proved. In this study, a semi-analytic harmonic modeling method is proposed by combining numerical and analytical methods, which breaks through the efficiency bottleneck of traditional modeling methods, and achieves the unity of high precision and high efficiency in the magnetic field modeling of the new Halbach array, providing a new solution for the study of complex magnetic field problems. Full article

This paper proposes a 2D semi-analytical model based on the subdomain method for the performance analysis of a brushless doubly fed reluctance machine (BDFRM) with a salient pole rotor. In particular, assuming an infinite magnetic permeability of the iron core and assuming a smooth stator, the field calculation region is divided into two solution subdomains, i.e., the rotor slot and air-gap. The magnetic vector potential in each subdomain is obtained by solving the governing PDE by the separation of variables method and employing the boundary conditions between adjacent interfaces. Moreover, based on the stored magnetic energy in the air-gap, the calculation of the three-phase windings’ self and mutual inductances is presented. Through a case study involving a 6/2 pole BDFRM, the accuracy of the developed subdomain model is confirmed by comparing its analytically predicted results with those obtained from two-dimensional finite element method (FEM) simulations. Full article

Mathematical optimization, in both continuous and discrete forms, is well established and widely applied. This work addresses a gap in the literature by focusing on large-number optimization, where integers or fractions with hundreds of digits occur in decision variables, objective functions, or constraints. Such problems challenge standard optimization tools, particularly when exact solutions are required. The suitability of computer algebra systems and high-precision arithmetic software for large-number optimization problems is discussed. Our first contribution is the development of Python implementations of an exact Simplex algorithm and a Branch-and-Bound algorithm for integer linear programming, capable of handling arbitrarily large integers. To test these implementations for correctness, analytic optimal solutions for nine specifically constructed linear, integer linear, and quadratic mixed-integer programming problems are derived. These examples are used to test and verify the developed software and can also serve as benchmarks for future research in large-number optimization. The second contribution concerns constructing partially increasing subsequences of the Collatz sequence. Motivated by this example, we quickly encountered the limits of commercial mixed-integer solvers and instead solved Diophantine equations or applied modular arithmetic techniques to obtain partial Collatz sequences. For any given number J, we obtain a sequence that begins at $2^J-1$ and repeats J times the pattern ud: multiply by $3x_j+1$ and then divide by 2. Further partially decreasing sequences are designed, which follow the pattern of multiplying by $3x_j+1$ and then dividing by $2^m$. The most general J-times increasing patterns (ududd, udududd, …, ududududddd) are constructed using analytic and semi-analytic methods that exploit modular arithmetic in combination with optimization techniques. Full article

Most existing multispecies transport analytical models primarily focus on inlet boundary sources, limiting their applicability in real-world contaminated sites where contaminants often arise from multiple internal sources. This study presents a novel semi-analytical model for simulating multispecies contaminant transport driven by multiple time-dependent internal sources. The model incorporates key transport mechanisms, including advection, dispersion, rate-limited sorption, and first-order degradation. In particular, the inclusion of rate-limited sorption addresses limitations in traditional equilibrium-based models, which often underestimate pollutant concentrations for degradable species. The derivation of this semi-analytical model utilizes the Laplace transform, finite cosine Fourier transform, generalized integral transform, and a sequence of inverse transformations. Results indicate that the concentrations of contaminants and their degradation products are highly sensitive to the variations in time-dependent sources. The model’s most significant contribution lies in its capability to simulate the contaminant transport from multiple internal pollution sources at a contaminated site under the influence of rate-limited sorption. By enabling the representation of multiple time-varying sources, this model fills a critical gap in analytical approaches and provides a necessary tool for accurately assessing contaminant transport in complex, realistic pollution scenarios. Full article

This study aims to evaluate the effectiveness of five analytical and semi-analytical methods for estimating Lamb wave dispersion in viscoelastic plates—the Rayleigh–Lamb solution, the Global Matrix Method (GMM), the Semi-Analytical Finite Element (SAFE) method, the Scaled Boundary Finite Element Method (SBFEM), and the Legendre Polynomial Method (LPM). The Rayleigh–Lamb equations are solved using an optimized Newton–Raphson algorithm, enhancing computational efficiency while maintaining comparable accuracy. The SAFE method exhibited a remarkable balance between computational efficiency and physical accuracy, outperforming SBFEM at high frequencies. For epoxy and high-performance polyethylene (HPPE) plates, the SAFE method and the LPM significantly outperform the GMM in relation to computational efficiency, with errors below $1\%$ for fundamental symmetric and antisymmetric modes across the tested frequency range of 0 to 100 kHz. In addition, the ability of the SAFE method to accurately predict both phase velocity and attenuation in viscous media supports their use in guided-wave-based structural health monitoring applications. Among the investigated approaches, the SAFE method emerges as the most robust and accurate for viscoelastic plates, while the SBFEM and LPM show limitations at higher frequencies. This study provides a quantitative and methodological foundation for selecting and implementing numerical methods for guided wave analysis, emphasizing the dual necessity of physical fidelity and numerical stability. Full article

This study presents the Fourier–Gegenbauer integral Galerkin (FGIG) method, a new numerical framework that uniquely integrates Fourier series and Gegenbauer polynomials to solve the one-dimensional advection–diffusion (AD) equation with spatially symmetric periodic boundary conditions, achieving exponential convergence and reduced computational cost compared to traditional methods. The FGIG method uniquely combines Fourier series for spatial periodicity and Gegenbauer polynomials for temporal integration within a Galerkin framework, resulting in highly accurate numerical and semi-analytical solutions. Unlike traditional approaches, this method eliminates the need for time-stepping procedures by reformulating the problem as a system of integral equations, reducing error accumulation over long-time simulations and improving computational efficiency. Key contributions include exponential convergence rates for smooth solutions, robustness under oscillatory conditions, and an inherently parallelizable structure, enabling scalable computation for large-scale problems. Additionally, the method introduces a barycentric formulation of the shifted Gegenbauer–Gauss (SGG) quadrature to ensure high accuracy and stability for relatively low Péclet numbers. This approach simplifies calculations of integrals, making the method faster and more reliable for diverse problems. Numerical experiments presented validate the method’s superior performance over traditional techniques, such as finite difference, finite element, and spline-based methods, achieving near-machine precision with significantly fewer mesh points. These results demonstrate its potential for extending to higher-dimensional problems and diverse applications in computational mathematics and engineering. The method’s fusion of spectral precision and integral reformulation marks a significant advancement in numerical PDE solvers, offering a scalable, high-fidelity alternative to conventional time-stepping techniques. Full article