Search Results (484)

Background: Rock–blast design is a canonical inverse problem that joins elastodynamic partial differential equations (PDEs), fracture mechanics, and stochastic heterogeneity. Objective: Guided by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) protocol, a systematic review of mathematical methods for geomechanically informed blast modelling and optimisation is provided. Methods: A Scopus–Web of Science search (2000–2025) retrieved 2415 records; semantic filtering and expert screening reduced the corpus to 97 studies. Topic modelling with Bidirectional Encoder Representations from Transformers Topic (BERTOPIC) and bibliometrics organised them into (i) finite-element and finite–discrete element simulations, including arbitrary Lagrangian–Eulerian (ALE) formulations; (ii) geomechanics-enhanced empirical laws; and (iii) machine-learning surrogates and multi-objective optimisers. Results: High-fidelity simulations delimit blast-induced damage with ≤0.2 m mean absolute error; extensions of the Kuznetsov–Ram equation cut median-size mean absolute percentage error (MAPE) from 27% to 15%; Gaussian-process and ensemble learners reach a coefficient of determination ($R^2>0.95$) while providing closed-form uncertainty; Pareto optimisers lower peak particle velocity (PPV) by up to 48% without productivity loss. Synthesis: Four themes emerge—surrogate-assisted PDE-constrained optimisation, probabilistic domain adaptation, Bayesian model fusion for digital-twin updating, and entropy-based energy metrics. Conclusions: Persisting challenges in scalable uncertainty quantification, coupled discrete–continuous fracture solvers, and rigorous fusion of physics-informed and data-driven models position blast design as a fertile test bed for advances in applied mathematics, numerical analysis, and machine-learning theory. Full article

Atmospheric climate science lacks the capacity to integrate thermodynamics with the gravitational potential of air in a classical quantum theory. To what extent can we identify Carnot’s ideal heat engine cycle in reversible isothermal and isentropic phases between dual temperatures partitioning heat flow with coupled work processes in the atmosphere? Using statistical action mechanics to describe Carnot’s cycle, the maximum rate of work possible can be integrated for the working gases as equal to variations in the absolute Gibbs energy, estimated as sustaining field quanta consistent with Carnot’s definition of heat as caloric. His treatise of 1824 even gave equations expressing work potential as a function of differences in temperature and the logarithm of the change in density and volume. Second, Carnot’s mechanical principle of cooling caused by gas dilation or warming by compression can be applied to tropospheric heat–work cycles in anticyclones and cyclones. Third, the virial theorem of Lagrange and Clausius based on least action predicts a more accurate temperature gradient with altitude near 6.5–6.9 °C per km, requiring that the Gibbs rotational quantum energies of gas molecules exchange reversibly with gravitational potential. This predicts a diminished role for the radiative transfer of energy from the atmosphere to the surface, in contrast to the Trenberth global radiative budget of ≈330 watts per square metre as downwelling radiation. The spectral absorptivity of greenhouse gas for surface radiation into the troposphere enables thermal recycling, sustaining air masses in Lagrangian action. This obviates the current paradigm of cooling with altitude by adiabatic expansion. The virial-action theorem must also control non-reversible heat–work Carnot cycles, with turbulent friction raising the surface temperature. Dissipative surface warming raises the surface pressure by heating, sustaining the weight of the atmosphere to varying altitudes according to latitude and seasonal angles of insolation. New predictions for experimental testing are now emerging from this virial-action hypothesis for climate, linking vortical energy potential with convective and turbulent exchanges of work and heat, proposed as the efficient cause setting the thermal temperature of surface materials. Full article

In this paper, we study a geometric framework for second-order differential systems arising in classical and relativistic mechanics. For this class of systems, we derive necessary and sufficient conditions for their Lagrangian description. The main objectives of this work are to construct efficient structure-preserving variational integrators in a variational framework. To achieve this, we develop new variational integrators through Lagrangian splitting and prove their equivalence to composition methods. We display the superiority of the newly derived numerical methods for the Kepler problem and provide rigorous error estimates by analysing the Laplace–Runge–Lenz vector. The framework provides tools applicable to geometric numerical integration of both ordinary and partial differential equations. Full article

Infectious diseases can spread through virus-laden respiratory droplets exhaled into the air. Ventilation systems are crucial in indoor settings as they can dilute or eliminate these droplets, underscoring the importance of understanding their efficacy in the management of indoor infections. Within the field of fluid dynamics methods, the dispersed droplets may be approached through either a Lagrangian framework or an Eulerian framework. In this study, various Eulerian methodologies are systematically compared against the Eulerian–Lagrangian (E-L) approach across three different scenarios: the pseudo-single-phase model (PSPM) for assessing the transport of gaseous pollutants in an office with displacement ventilation (DV), stratum ventilation (SV), and mixing ventilation (MV); the two-fluid model (TFM) for evaluating the transport of non-evaporating particles within an office with DV and MV; and the two-fluid model-population balance equation (TFM-PBE) approach for analyzing the transport of evaporating droplets in a ward with MV. The Eulerian and Lagrangian approaches present similar agreement with the experimental data, indicating that the two approaches are comparable in accuracy. The computational cost of the E-L approach is closely related to the number of tracked droplets; therefore, the Eulerian approach is recommended when the number of droplets required by the simulation is large. Finally, the performances of DV, SV, and MV are presented and discussed. DV creates a stratified environment due to buoyant flows, which transport respiratory droplets upward. MV provides a well-mixed environment, resulting in a uniform dispersion of droplets. SV supplies fresh air directly to the breathing zone, thereby effectively reducing infection risk. Consequently, DV and SV are preferred to reduce indoor infection. Full article

The dynamic control of vibrations in skyscrapers is a critical consideration in sustainable building design, particularly in response to environmental excitations such as wind impact or seismic activity. Effective vibration neutralisation plays a crucial role in providing the safety of high-rise buildings. This research introduces an innovative concept for an active vibration damper that operates based on fluid dynamic transport to adaptively alter a skyscraper’s natural frequency, thereby counteracting resonant vibrations. A distinctive feature of this system is an adjustable-length cable mechanism, allowing for the dynamic modification of the pendulum’s effective length in real time. The structure, based on cable length adjustment, enables the PTMD to precisely tune its natural frequency to variable excitation conditions, thereby improving damping during transient or resonance phenomena of the building’s dynamic behaviour. A comprehensive mathematical model based on Lagrangian mechanics outlines the governing equations for this system, capturing the interactions between pendulum motion, fluid flow, and the damping forces necessary to maintain stability. Simulation analyses examine the role of initial excitation frequency and variable damping coefficients, revealing critical insights into optimal damper performance under varied structural conditions. The findings indicate that the proposed pendulum damper effectively mitigates resonance risks, paving the way for sustainable skyscraper design through enhanced structural adaptability and resilience. This adaptive PTMD, featuring an adjustable-length cable, provides a solution for creating safe and energy-efficient skyscraper designs, aligning with sustainable architectural practices and advancing future trends in vibration management technology. The study presented in this article supports the development of modern skyscraper design, with a focus on dynamic vibration control for sustainability and structural safety. It combines advanced numerical modelling, data-driven control algorithms, and experimental validation. From a sustainability perspective, the proposed PTMD system reduces the need for oversized structural components by providing adaptive, efficient damping, thereby lowering material consumption and embedded carbon. Through dynamically retuning structural stiffness and mass, the proposed PTMD enhances resilience and energy efficiency in skyscrapers, lowers lifetime energy use associated with passive damping devices, and enhances occupant comfort. This aligns with global sustainability objectives and new-generation building standards. Full article

This work presents a one-dimensional (1D) model for simulating the behavior of an FCC riser reactor processing bio-oil. The FCC riser is modeled as a plug-flow reactor, where the bio-oil feed undergoes vaporization followed by catalytic cracking reactions. The bio-oil droplets are represented using a Lagrangian framework, which accounts for their movement and evaporation within the gas-solid flow field, enabling the assessment of droplet size impact on reactor performance. The cracking reactions are modeled using a four-lumped kinetic scheme, representing the conversion of bio-oil into gasoline, kerosene, gas, and coke. The resulting set of ordinary differential equations is solved using a stiff, second- to third-order solver. The simulation results are validated against experimental data from a full-scale FCC unit, demonstrating good agreement in terms of product yields. The findings indicate that heat exchange by radiation is negligible and that the Buchanan correlation best represents the heat transfer between the droplets and the catalyst particles/gas phase. Another significant observation is that droplet size, across a wide range, does not significantly affect conversion rates due to the bio-oil’s high vaporization heat. The proposed reduced-order model provides valuable insights into optimizing FCC riser reactors for bio-oil processing while avoiding the high computational costs of 3D CFD simulations. The model can be applied across multiple applications, provided the chemical reaction mechanism is known. Compared to full models such as CFD, this approach can reduce computational costs by thousands of computing hours. Full article

This study investigates the potential Kadomtsev–Petviashvili equation incorporating a power-type nonlinearity (PKPp), a model that features prominently in various nonlinear phenomena encountered in physics and applied mathematics. A complete Noether symmetry classification of the PKPp equation is conducted, revealing four distinct scenarios based on different values of the exponent $p$, namely, the general case where $p\ne 1,-1,-2,$ and three special cases where $p=1,$$p=-1,$ and $p=-2.$ Corresponding to each case, conservation laws are derived through a second-order Lagrangian framework. Furthermore, Lie group analysis is employed to reduce the nonlinear partial differential Equation (NLPDE) to ordinary differential Equations (ODEs), thereby enabling the effective application of the Kudryashov method and direct integration techniques to construct exact solutions. In particular, exact solutions of of the considered nonlinear partial differential equation are obtained for the cases $p=1$ and $p=2,$ illustrating the practical implementation of the proposed approach. The solutions obtained include solitary wave, periodic, and rational-type solutions. These results enhance the analytical understanding of the PKPp equation and contribute to the broader theory of nonlinear dispersive equations. Full article

Compliant mechanisms are often utilized in precise positioning systems but have not been thoroughly examined in vibration-aided fine CNC machining. This study aims to develop a new 02-DOF flexure stage for vibration-aided fine CNC milling. A hybrid displacement amplifier, featuring a two-lever mechanism, two Scott–Russell mechanisms, and a parallel leading mechanism, was integrated into a symmetric perpendicular series configuration to create an innovative design. The pseudo-rigid body model (PRBM), Lagrangian approach, finite element analysis (FEA), and Firefly optimization algorithm were employed to develop, verify, and optimize the quality response of the new positioner. The PRBM and Lagrangian methods were used to construct an analytical model, while finite element analysis was used to validate the theoretical solution. The primary natural frequency results from theoretical and FEM methods were 318.16 Hz and 308.79 Hz, respectively. The difference between these techniques was 3.04%, demonstrating a reliable modelling strategy. The Firefly optimization approach applied mathematical equations to enhance the key design factors of the mechanism. The prototype was then built, revealing an error of 7.23% between the experimental and simulated frequencies of 331.116 Hz and 308.79 Hz, respectively. The specimen was subsequently mounted on the fabricated optimization positioner, and vibration-assisted fine CNC milling was performed at 100–1000 Hz. At 400 Hz, the specimen achieved ideal surface roughness with a Ra value of 0.187 µm. The developed design is a potential structure that generates non-resonant frequency power for vibration-aided fine CNC milling. Full article

This study investigates three-dimensional two-phase flows in complex geometries found in industrial process equipment design using finite-element numerical simulations. The governing equations are formulated in three-dimensional Cartesian coordinates and solved on unstructured meshes employing the Taylor–Hood “Mini” element, selected for its numerical stability and convergence properties. The convective term in the momentum equation is discretized using a first-order semi-Lagrangian scheme. The two fluid phases are separated by an interface mesh composed of triangular surface elements, which is independent of the primary volumetric fluid mesh. Surface tension effects are incorporated as a source term using the continuum surface force (CSF) model, with the curvature computed via the Laplace–Beltrami operator. At each time step, the positions of the interface mesh nodes are updated according to the local fluid velocity field. The results show that the methodology is stable and can be used to accurately model two-phase flows in complex geometries found in several engineering solutions. Full article

Especially in real-world circumstances with uneven road surfaces and impulsive shocks, nonlinear dynamic effects in vehicle systems can greatly skew biometric data utilized to track passenger and driver physiological states. By creating a thorough multibody human–seat–chassis model, this work tackles the effect of vehicle-induced vibrations on the accuracy and dependability of biometric measures. The model includes external excitation from road-induced inputs, nonlinear damping between structural linkages, and vertical and angular degrees of freedom in the head–neck system. Motion equations are derived using a second-order Lagrangian method; simulations are run using representative values of a typical car and human body segments. Results show that higher vehicle speed generates more vibrational energy input, which especially in the head and torso enhances vertical and angular accelerations. Modal studies, on the other hand, show that while resonant frequencies stay constant, speed causes a considerable rise in amplitude and frequency dispersion. At speeds ≥ 50 km/h, RMS and VDV values exceed ISO 2631 comfort standards in the body and head. The results highlight the need to include vibration-optimized suspension systems and ergonomic design approaches to safeguard sensitive body areas and preserve biometric data integrity. This study helps to increase comfort and safety in both traditional and autonomous car uses. Full article

The presented study demonstrates that UAVs can be flown with a morphing wing to develop essential aerodynamic efficiency without a tail structure, which decides the operational cost and flight safety. The mechanical control for morphing is discussed, where the system design, simulation, and experimental realization of ±15° SDOF sweep motion for a 7 kg eVTOL wing are detailed. The methodology, developed through a mathematical modeling of the mechanism’s kinematics and dynamics, is explained using Denavit–Hartenberg (D-H) convention, Lagrangian mechanics, and Euler–Lagrangian equations. The simulation and MBD analyses were performed in MATLAB R2021 and by Altair Motion Solve, respectively. The experiment was conducted on a dedicated test rig with two wing variants fitted with IMUs and an autopilot. The results from various methods were analyzed and experimentally compared to provide an accurate insight into the system’s design, modeling, and performance of the sweep morphing wing. The theoretical calculations by the mathematical model were compared with the test results. The sweep requirement is essential for eVTOL to have long endurance and multi-mission capabilities. Therefore, the developed sweep morphing mechanism is very useful, meeting such a demand. However, the results for three-dimensional morphing, operating sweep, pitch, and roll together are also presented, for the sake of completeness. Full article

The Stewart-platform-based bumper plays a critical role in shipborne strap-down inertial navigation systems (SINSs), effectively mitigating shock-induced disturbances to ensure measurement accuracy. Dynamic modeling for the bumper under a huge impact is a key issue in predicting the shock isolation performance of the bumper. In this paper, the dynamic modeling of shock isolation performance for Stewart-platform-based bumpers under huge impacts is proposed and validated experimentally. Firstly, a model of a Stewart-platform-based bumper is established considering the geometric configuration and dynamic parameters of the bumper by calculating the Jacobian matrix, stiffness matrix, damping matrix and mass matrix. Secondly, an analytic simulation of the impact is presented based on the measured impact acceleration, and the impact force on the load is derived according to the non-displacement assumption in the impact stage. Then, the Lagrangian formulation was systematically applied to establish governing equations characterizing the six-degree-of-freedom (DOF) dynamics of the bumper, incorporating both inertial coupling effects and nonlinear shock energy dissipation mechanisms. Afterwards, dynamic equations were solved via the Runge–Kutta method to obtain the theoretical results. Finally, the proposed dynamic modeling and shock isolation performance analysis method was validated via impact experiments for the bumper. Full article

This study proposes an Index-Based Neural Network (IBNN) framework for the static analysis of truss structures, employing a Lagrangian dual optimization technique grounded in the force method. A truss is a discrete structural system composed of linear members connected to nodes. Despite their geometric simplicity, analysis of large-scale truss systems requires significant computational resources. The proposed model simplifies the input structure and enhances the scalability of the model using member and node indices as inputs instead of spatial coordinates. The IBNN framework approximates member forces and nodal displacements using separate neural networks and incorporates structural equations derived from the force method as mechanics-informed constraints within the loss function. Training was conducted using the Augmented Lagrangian Method (ALM), which improves the convergence stability and learning efficiency through a combination of penalty terms and Lagrange multipliers. The efficiency and accuracy of the framework were numerically validated using various examples, including spatial trusses, square grid-type space frames, lattice domes, and domes exhibiting radial flow characteristics. Multi-index mapping and domain decomposition techniques contribute to enhanced analysis performance, yielding superior prediction accuracy and numerical stability compared to conventional methods. Furthermore, by reflecting the structured and discrete nature of structural problems, the proposed framework demonstrates high potential for integration with next-generation neural network models such as Quantum Neural Networks (QNNs). Full article

In this paper, we construct the conformable actuated pendulum model in the conformable Lagrangian formalism. We solve the equations of motion in the absence of force and in the case of a specific force resulting from torques, which generalizes a well known mechanical model. Our study shows that the conformable model captures essential information about the physical system encoded in the parameters which depend on the conformability factor $\alpha$. This dependence can describe internal variations such as viscous friction, transmission, or environmental effects. We solve the equations of motion analytically for $\alpha =1/2$ and using Frobenius’ method for $\alpha \ne 1/2$. Full article

The simulation of human soft tissue deformation is a key issue in the research of surgical simulators. The most mathematically accurate model for soft tissue behavior is the finite element model (FEM), being the most widely adopted numerical approach for nonlinear continuum mechanics equations. The total Lagrangian explicit dynamics (TLED) model is a nonlinear FEM that could simulate the nonlinear deformation of soft tissues accurately and in real time. However, the main problems faced by this method are the high computational cost and the real-time performance of the simulation. Therefore, the linear FEM is used for ensuring computational efficiency and real-time performance of the simulation, though it is inadequate for capturing true biomechanical behavior. Consequently, we have to solve the problems of real-time performance and computational efficiency of nonlinear finite elements in simulating soft tissue deformation. To address this computational challenge, we propose a Dual-Mode Hybrid Dynamics Finite Element Algorithm (DHD-FEA). First, we divide the deformed soft tissues into the surgical area and the non-surgical area. Then, the TLED nonlinear FEM is applied to the simulation of soft tissue deformation in the surgical area, ensuring the accuracy of the simulation effect. Simultaneously, the simulation of soft tissue deformation in the non-surgical area using the linear FEM improves the real-time performance of the simulation and reduces the overall computational cost. Numerical results demonstrate that the error rate in the simulation of the DHD-FEA is lower than that of the complete linear FEM, and the computational efficiency is higher than that of the TLED. Therefore, the DHD-FEA not only ensures the accuracy of soft tissue simulation in the surgical area but also reduces the computational cost. Full article