Traditional Analytical Solutions Research Articles

Theoretical analysis of the mechanical response of a lined canal induced by soil frost heave behavior based on improved foundation beam models

Lined canals in cold regions often experience severe frost damage, posing a significant threat to the water supply safety. This paper proposes a modified analytical solution for the response of canal lining under soil frost heave, which is based on a Timoshenko beam on a Pasternak foundation considering tangential contact between the lining and frozen soil. This analytical solution can provide any solution using Timoshenko or Euler–Bernoulli beams on Pasternak or Winkler foundations with or without tangential contact. Then, the modified analytical solution, along with the traditional analytical solutions using Euler–Bernoulli beams on Winkle foundations, is compared against both model test and numerical simulation results. The modified analytical solution performs better than the traditional solution. Finally, the effects of foundation model, beam model, and tangential contact in simulating canal frost heave were discussed, and some measures to mitigate canal frost heave are proposed. The results show that solutions based on Winkler foundation overestimate frost heaves and tensile stresses of canal linings, and solutions without considering tangential contact only obtains overestimated frost heave. In addition, solutions based on Euler–Bernoulli beams underestimate frost heaves slightly and overestimate tensile stresses slightly. Therefore, the Pasternak foundation, Euler–Bernoulli beam, and tangential contact model can be used to simulate canal frost heave. The modified analytical solution directly uses field measurements of soil free frost heave to calculate canal frost heave, thereby enhancing result reliability. This analytical solution provides a simple method for canal frost heave design, and can be applied to frost heave analyses of flat and inclined structures.

Fractional-Order Extremum Seeking Method for Maximum Torque per Ampere Control of Permanent Magnet Synchronous Motor

Maximum torque per ampere (MTPA) control of internal permanent magnet synchronous motors (IPMSM) has become integral to high-efficiency motor drives. To minimize the influence of the traditional model-based analytical solution method on the MTPA control strategy due to the parameter variations during the motor operation, an online search MTPA method without model-based fractional-order extremum seeking control (FO-ESC) is proposed. Compared with the traditional integer-order ESC method, the Oustaloup approximation-based fractional-order calculus provides additional factors and possibilities for optimizing controller parameters to improve control performance. At the same time, the proposed approach does not require machine parameters and is thus not influenced by machine and drive nonlinearities. Simulation results show that the proposed method can ensure robust MTPA control under different loading conditions in real-time and improve the system’s dynamic response speed and steady-state accuracy.

Analysis of anchor force for pipeline walking induced by SCR tension

The deep-sea pipeline exposed on the seabed is prone to pipeline walking under the influence of thermal loading cycles. Anchors are commonly used to prevent the pipeline walking from causing the end connection of the pipeline system to break. The development of a simple and practical method for calculating anchor force is essential for the preliminary design of the project. In this paper, a finite element model for calculating anchor force is established, and the factors affecting the accumulated length of effective axial force are analyzed. A strain diagram method is proposed for calculating the anchor force under partial accumulated length, and an analytical solution for the rigid anchor force is derived based on the strain diagram method. To address the problem of large anchor force at the end anchor, a flexible anchor is proposed to reduce the anchor force, and a formula for predicting the flexible anchor force is fitted. The study shows that the accumulated length proportion of anchor force increases with the increase of loading temperature and SCR tension, and decreases with the increase of pipeline length and friction coefficient. Flexible anchors can significantly reduce the anchor force. The proposed analytical solution for anchor force can comprehensively consider the influence of factors such as pipeline length, SCR tension, loading temperature, and friction coefficient, and has certain advantages over traditional analytical solutions.

Analytical study on anchor force for pipeline walking laid on inclined seabed

Unburied subsea pipelines under cyclic thermal loading are often prone to axial walking. The axial movements accumulate over operational cycles, which can lead to potential failure at the mid-line or end connections. Anchoring is a commonly adopted method to -control pipeline walking, however, it requires accurate estimation of anchor force for efficient pipeline design. In view of this, the present study proposes a calculation method for anchor force of full mobilized lengths for different anchoring positions. Furthermore, the anchoring mechanism for partial mobilized lengths is analyzed. The anchor force is then determined using a strain diagram method for partial mobilized lengths. Finally, an analytical solution for anchor force based on the strain diagram method is derived and validated against finite element results. The results show that the mobilized length ratio of pipeline anchor force increases with the loading temperature and seabed slope but decreases with the pipeline length. The proposed analytical solution for anchor force which has advantages over traditional analytical solutions can comprehensively consider the effects of pipeline parameters, load parameters, and contact parameters.

Energy absorption analysis of origami structures based on small number of samples using conditional GAN

Origami-inspired metamaterials are increasingly being applied to the fields of science and engineering owing to their unique mechanical characteristics. Geometric and material arrangements of the panels and creases can lead to different mechanical properties of origami structures, such as energy absorption and multi-stability. The actual folding processes of the origami structures usually include crease rotation and panel deformation. However, the traditional analytical or empirical solutions based on rigid folding assumptions cannot sufficiently reflect the folding process precisely. Finite element analysis can provide detailed origami folding information, but it is a highly time-consuming cycle, including modeling and computing. Typically, conventional data-driven approaches necessitate a considerable amount of data samples to analyze the energy absorption of origami structures. The unavailability of extensive datasets poses a major impediment in employing data-driven methods to explore the authentic behavior of origami structures. Based on the situations above, a new data-driven framework with conditional generative adversarial networks with tabular data (CTGAN) is proposed in this study. To verify the feasibility of the framework, two study cases are presented herein: a Miura-ori structure in-plane quasi-static compression and the square-twist (Type 1) folding process. Energy absorption properties are predicted accurately and efficiently by the framework based on small number of samples, which are highly consistent with predicted results based on a large dataset. The framework not only provides a promising solution for energy absorption analysis of origami structures but also overcomes the bottlenecks (small number of sample datasets and a mixture of data types) associated with machine learning methods in origami structure area.

Analytical solution of longitudinal seismic response of circular tunnel across soil-rock stratum based on improved elastic foundation beam model

This study aims to analysis the longitudinal seismic responses of circular tunnels across soil-rock stratum longitudinally (“soil-rock” means two different stratums, not just the soil and rock ground). Based on an improved elastic foundation beam model, an analytical solution is derived to investigate the seismic response of long tunnels across soil-rock stratum, subjected to sinusoidal shear motions. The improved elastic foundation beam model adopts the Timoshenko beam and Winkler foundation, with considering tangential contact conditions at the ground-tunnel interface. The free field reaction considers the reflection and transmission by the soil-rock interface and the wave passage effect. Explicit formulations are obtained for tunnel displacement, rotation angles, bending moments and shearing forces. The proposed solution is verified by comparing with the dynamic numerical results, as well as traditional analytical solutions. Finally, a parametric study is conducted to investigate the influence of traveling wave effect, earthquake frequency, shear wave velocity ratio of two stratums, lining stiffness and contact condition on tunnel seismic responses. The proposed analytical solutions can be used to predict the longitudinal seismic response of deep circular tunnels crossing soil-rock stratum in engineering design.

Performance of Heat Transfer in Micropolar Fluid with Isothermal and Isoflux Boundary Conditions Using Supervised Neural Networks

The current study delivers a numerical investigation on the performance of heat transfer and flow of micropolar fluid in porous Darcy structures with isothermal and isoflux walls (boundary conditions) of a stretching sheet. The dynamics and mechanism of such fluid flows are modelled by nonlinear partial differential equations that are reduced to a system of nonlinear ordinary differential equations by utilizing the porosity of medium and similarity functions. Generally, the explicit or analytical solutions for such nonlinear problems are hard to calculate. Therefore, we have designed a computer or artificial intelligence-based numerical technique. The reliability of neural networks using the machine learning (ML) approach is used with a local optimization technique to investigate the behaviours of different material parameters such as the Prandtl number, micropolar parameters, Reynolds number, heat index parameter, injection/suction parameter on the temperature profile, fluid speed, and spin/rotational behaviour of the microstructures. The approximate solutions determined by the efficient machine learning approach are compared with the classical Runge–Kutta fourth-order method and generalized finite difference approximation on a quasi-uniform mesh. The accuracy of the errors lies around 10−8 to 10−10 between the traditional analytical solutions and machine learning strategy. ML-based techniques solve different problems without discretization or computational work, and are not subject to the continuity or differentiability of the governing model. Moreover, the results are illustrated briefly to help implement microfluids in drug administering, elegans immobilization, and pH controlling processes.

A semi-analytical solution technique for predicting circulating mud temperatures

Accurate prediction of circulating fluid temperature in wellbore is required for proper drilling operations and design of the closed-loop systems. This study involves the improvement of the traditional analytical solution based on the assumption of constant input values. The presented solutions are derived from the analytical solution of the transient radial heat conduction. Application of the superposition theorem and combination of the analytical solutions established for each segment constituting the wellbore to account for temporal and positional variations are the main features of the semi-analytical solution presented in this study. In this respect, it can be asserted that the solution is also valid for variable mud/water circulation in deviated and horizontal wells.The presented solution is validated using the pertinent results of the traditional analytical solution. The results of the semi-analytical solution for realistic wellbore design reveal the possible deviations from temperature profile corresponding to analytical solution. The temperature profile as a result of the circulation with variable rate in a designed horizontal well is examined to test the presented semi-analytical solution. The trials indicate that the semi-analytical solution which is able to be implemented with plausible computational effort can be an alternative improvement for the traditional analytical solution.

Machine Learning in Conventional Tunnel Deformation in High In Situ Stress Regions

Deformation prediction of extremely high in situ stress in soft-rock tunnels is a complex problem involving many parameters, and traditional analytical solutions and numerical simulations have difficulty achieving satisfactory results. This paper proposes the MIC-LSTM algorithm based on machine learning methods to predict the deformation of soft-rock tunnels under extremely high in situ stress conditions caused by construction. The study first analyzed the difficulties of engineering construction and the construction plan; then, numerical simulation was used to verify the modified construction plan. To prove that the construction plan was reasonable, machine learning was used to analyze the correlation of the various parameters that cause tunnel deformation; then, the future deformation of the tunnel was predicted. The study found that: (1) the new construction scheme contains symmetrical arrangement of bolts and two support structures along the tunnel vault can effectively control the deformation of the tunnel, and meet the requirements of the specification; (2) the rock uniaxial compressive strength had the greatest impact on tunnel deformation, and the rock humidity had the least influence on tunnel deformation; and (3) the prediction curve based on the deep learning model had a higher similarity to the monitoring curve compared with the traditional numerical analysis software. The MIC-LSTM machine algorithm provides a new approach to predicting the deformation of extremely high in situ stress soft-rock tunnels.

A CNN-based FBG demodulation method adopting the GAF-assisted ascending dimension of complicated signal

We introduce the convolutional neural network (CNN) to extract the effective information of some complex signals in the fiber sensing, and verify the feasibility in the specific process of fiber Bragg grating (FBG) demodulation. In the experiments, in order to eliminate the cross interference in the energized coil, classical double grating configuration is adopted to distinguish magnetic field and temperature. At the same time, the introduction of thermal induced chirp (TIC) phenomenon produces more wavelength components to achieve more precise demodulation. The signal received by oscilloscope presents complex waveform, and the traditional analytical solution cannot demodulate it precisely, while the deep learning capabilities of neural networks can be leveraged to solve this problem. The one-dimensional (1D) time-series signal is transformed into a two-dimensional (2D) image by using the Gramian angle field (GAF) method. On this basis, the data are augmented according to the characteristics of signal noise. Five classical neural network structures are used to process the data. The root mean square errors (RMSEs) of the demodulation results can achieve 0.3 °C and 17.1 Gs respectively, which are relatively high precision in terms of intensity demodulation. This method can be extended to demodulate 1D frequency domain signals and wavelength domain signals, rendering a demodulation scheme to process some complex signals in optical fiber sensing effectively.

Fourier-domain modeling of gravity effects caused by a vertical polyhedral prism, with application to a water reservoir storage process

We have developed a new Fourier-domain algorithm for the modeling of gravity effects caused by a vertical polyhedral prism. Simplified and more compact Fourier-domain expressions are derived for the vertical gravity anomaly on a 2D plane either above, in the middle of, or below the vertical polyhedral prism. A new Fourier-domain algorithm, which combines a low-order Gauss fast Fourier transform (FFT) algorithm and a low-order nonuniform FFT algorithm, to permit polar sampling near the zero wave vector, is introduced. To validate the numerical efficiency of the new algorithm, we carry out a series of synthetic tests. Numerical results show that the new algorithm is superior in numerical accuracy and efficiency compared to a pure Gauss-FFT algorithm, and both Fourier methods run much faster than traditional space-domain analytical solutions. We then apply the Fourier forward algorithms to the prediction of vertical component of gravity at many survey points, as a function of the water level of a reservoir located in Xianlin, Hangzhou, China. The reservoir geometry is approximated by a single vertical polyhedral prism. Water storage corresponds to increased heights of the polyhedral prism. Detailed maps of vertical gravity value changes corresponding to a water level rise of 5 and 10 m, with spatial resolution [Formula: see text], on the local topography covering a study area of approximately [Formula: see text] around the water reservoir, are calculated using the new algorithm. Then, the gravity values are classified according to the accuracy of the gravimeter, providing a reference of optimal site selection for a proposed cold-atom gravimetry survey to be carried out in the near future.

Dissection of Interaction Kinetics through Single-Molecule Interaction Simulation.

The ability to extract kinetic interaction parameters from single-molecule fluorescence resonance energy transfer trajectories without the need for solving complex single-molecule differential equations has the potential to address some of the critical biophysical questions. Here, we provide a noise-free single-molecule interaction simulation (SMIS) tool to give the expected dwell-time distributions and relative populations of each FRET level based on the assigned kinetic model and to dissect kinetic interaction parameters from single-molecule FRET trajectories. The method provides the expected dwell-time distributions, average transition rates, and relative populations of each FRET level based on the assigned kinetic model. By comparing with ground truth data and experimental data, we demonstrated that SMIS is useful to quantify the interaction kinetic rate constants without using the traditional single-molecule analytical solution approach.

Finite Difference Method for the One-Dimensional Non-linear Consolidation of Soft Ground Under Uniform Load

Initial stress and additional effective stress distributions in soil greatly influence the degree of ground consolidation when calculating one-dimensional soft clay ground consolidation in deep soil. The one-dimensional nonlinear consolidation governing equations of soft ground under uniform loads are derived and solved with the finite difference method. This method is based on the assumptions that the initial stress in soil varies with the ground depth and the additional effective stress caused by external loads changes with both the ground depth and consolidation time, and the hyperbolic model of the soil stress-strain relationship. Formulas for the degree of consolidation and the settlement of the ground are presented. A case study shows that the degree of consolidation of the ground calculated with the finite difference method agrees with the traditional analytical solution, and the computational efficiency of the finite difference method can be effectively improved when the segmental calculation method is used throughout the consolidation process. The results of another example show that the settlement of the ground calculated with the finite difference method agrees with the in situ data. The suggested method can greatly simplify the consolidation calculation and has a high application value in engineering.

Isoparametric B-Spline Elements for Immersed Boundary Explicit Dynamic Simulation

AbstractExplicit dynamic analysis has proven to be advantageous when simulating shock and impact loading, and very small time-scale events. In this article, the feasibility of using a background mesh of B-spline elements for immersed boundary explicit dynamic simulation is studied. In this approach, the geometry is immersed in a background mesh consisting of uniform regular shaped elements to avoid mesh generation difficulties. The boundary conditions are applied using the step boundary method, which uses the equations of the boundaries to construct trial functions that satisfy the essential boundary conditions. An isoparametric formulation is presented for quadratic and cubic B-spline elements and their shape functions are derived from the classical recursive definition of B-splines. The effectiveness of mass diagonalization for B-spline elements is also explored. This approach is validated using several examples by comparing with modal superposition solutions as well as past work using traditional finite element analysis (FEA) and analytical solutions when available.

Método rotacional para el análisis estructural de armaduras


 Objetivo: Proponer un método innovador y práctico para realizar el análisis estructural de armaduras con la intención de facilitar y simplificar el análisis estructural, y de esta forma constituirlo en una herramienta alterna de análisis de este tipo de estructuras. Metodología: Primero se presenta la fundamentación analítica y conceptual del tema. Seguidamente, se propone un procedimiento paso a paso, ejecutado sobre una nueva representación paramétrica, que simplifica el problema a resolver. Luego se desarrolla un ejemplo de análisis de una ménsula de concreto reforzado, la cual fue idealizada con el método de la biela. Finalmente, y con la intención de comparar la simplicidad y tiempo de solución del método propuesto, este es comparado con una solución analítica tradicional. Resultados: La metodología propuesta simplifica el análisis estructural de armaduras, además de ser eficiente, confiable y versátil. El método requiere, en concreto, un menor tiempo de solución y cálculos en comparación con los métodos analíticos tradicionales de solución de sistemas lineales de ecuaciones. Conclusiones: El método rotacional es apropiado para realizar el análisis estructural de armaduras, pues simplifica la solución analítica de este tipo de estructuras. El método rotacional es más versátil y eficiente, y puede usarse en el análisis estructural de armaduras estáticamente determinadas.

Optimization of Surface Mining Dig Limits with a Practical Heuristic Algorithm

This paper presents an algorithm for optimizing the classification of surface mine material subject to excavating constraints. High-resolution expected profit models are input and optimized to classification maps subject to site-specific rectangular excavating constraints. This optimization problem defies traditional closed-form analytical solutions; a practical heuristic algorithm has been developed to quickly determine the optimal final destination for the material subject to realistic constraints. The optimization is fast and generates results that achieve up to 98–99% of the total expected profit achieved with free selection. This algorithm provides a fast viable option for practical application in short-term grade control and in managing multiple realizations in long-term resource estimation.

Numerical simulation on modulational instability of ion-acoustic waves in plasma**Project supported by the National Natural Science Foundation of China (Grant No. 11675014).

In this paper, the one-dimensional (1D) particle-in-cell (PIC) simulation is used to study the modulation instability of ion acoustic waves in electron–ion plasmas. The ion acoustic wave is described by using a nonlinear Schrödinger equation (NLSE) derived from the reductive perturbation method. Form our numerical simulations, we are able to demonstrate that, after the modulation, the amplitude increases steadily over time. Furthermore, by comparing the numerical results with traditional analytical solutions, we acquire the application scope for the reductive perturbation method to obtain the NLSE. We also find this method can also be extended to other fields such as fluid dynamics, nonlinear optics, solid state physics, and the Bose–Einstein condensate to validate the application scope of the results from the traditional perturbation method.

Optimization of Compatible Meshfree Quadrature Rule for Nonlocal Problems with Applications to Peri Dynamics Study

The optimization of a compatible meshfree quadrature rule for nonlocal problems with applications to Peridynamics is investigated in this study. Peridynamics is a nonlocal continuum mechanics theory that models material behavior based on interactions between material points within a finite neighbourhood. In Peridynamics, the accurate evaluation of nonlocal integral equations is crucial for obtaining reliable and efficient solutions. Traditional numerical integration methods, such as Gaussian quadrature, are not directly applicable to nonlocal problems due to their local nature. Hence, the development of a compatible meshfree quadrature rule that can effectively handle nonlocal interactions is of great importance. The objective it to optimize a quadrature rule that accurately captures the nonlocal interactions in Peridynamics while maintaining computational efficiency. The optimization process involves the selection of appropriate quadrature points and weights that minimize the quadrature error and maximize the computational efficiency. Various optimization techniques, such as genetic algorithms, particle swarm optimization, or machine learning algorithms, are explored to search for an optimal quadrature rule. The optimized quadrature rule is then applied to several Peridynamics problems, including fracture mechanics, material failure, and dynamic response analysis. The performance of the optimized quadrature rule is evaluated by comparing the results with those obtained using traditional quadrature methods and analytical solutions, when available. The accuracy, stability, and computational efficiency of the optimized quadrature rule are analysed and discussed. The findings provide valuable insights into the development and optimization of compatible meshfree quadrature rules for nonlocal problems, particularly in the context of Peridynamics. The optimized quadrature rule offers improved accuracy in capturing nonlocal interactions and reduces the computational cost compared to traditional methods. The applications of the optimized quadrature rule to various Peridynamics problems demonstrate its effectiveness and reliability.
 The implications of it extend beyond Peridynamics and can be applicable to other nonlocal models in the field of computational mechanics. The optimized quadrature rule has the potential to enhance the accuracy and efficiency of numerical simulations involving nonlocal phenomena, enabling more reliable predictions of material behavior and structural response. The findings of this contribute to the advancement of numerical methods for nonlocal problems and provide a foundation for further research and developments in the field of compatible meshfree quadrature rules for nonlocal problems.

Impact of model complexity and multi-scale data integration on the estimation of hydrogeological parameters in a dual-porosity aquifer

This study investigates the impact of model complexity and multi-scale prior hydrogeological data on the interpretation of pumping test data in a dual-porosity aquifer (the Chalk aquifer in England, UK). In order to characterize the hydrogeological properties, different approaches ranging from a traditional analytical solution (Theis approach) to more sophisticated numerical models with automatically calibrated input parameters are applied. Comparisons of results from the different approaches show that neither traditional analytical solutions nor a numerical model assuming a homogenous and isotropic aquifer can adequately explain the observed drawdowns. A better reproduction of the observed drawdowns in all seven monitoring locations is instead achieved when medium and local-scale prior information about the vertical hydraulic conductivity (K) distribution is used to constrain the model calibration process. In particular, the integration of medium-scale vertical K variations based on flowmeter measurements lead to an improvement in the goodness-of-fit of the simulated drawdowns of about 30%. Further improvements (up to 70%) were observed when a simple upscaling approach was used to integrate small-scale K data to constrain the automatic calibration process of the numerical model. Although the analysis focuses on a specific case study, these results provide insights about the representativeness of the estimates of hydrogeological properties based on different interpretations of pumping test data, and promote the integration of multi-scale data for the characterization of heterogeneous aquifers in complex hydrogeological settings.

A mathematical model of a solar air thermosyphon integrated with building envelope

A mathematical model comprising of analytical and two dimensional network procedures is proposed for predicting steady heat flow and heat transport in a solar heated thermosyphon integrated with building envelope. The detail mathematical solution method is presented for solving partial differential heat flow and transport equations by performing two dimensional energy balances at surface and air nodes through conjugate heat exchange and heat transport analysis of a solar thermosyphon. The mathematical solution procedure is devised for conduction and radiation heat exchange between surface nodes to improve the accuracy of traditional analytical solution for predicting buoyancy-induced mass flow rate of air flowing through a solar thermosyphon. The matrix inversion solution of mathematical model is unconditionally stable, with accuracy dependent on magnitude of conductance terms and number of nodes in the grid. Only Δy is chosen as aspect ratios (Δx = L, L/H and W/H) are defined by the geometry of the thermosyphon. The conduction and convection conductance terms are based on discretisation height Δy, thermal capacity conductance (mcp) is based on air-gap length Δx, whilst integrated radiation conductance terms are based on both height Δy and width Δx of the grid. The proposed model has compared the results obtained from analytical method, two dimensional network method for a single set of environmental condition with given geometry of thermosyphon. The proposed model is validated with experimental results obtained from outdoor experimental setup comprising of thermosyphon based photovoltaic solar wall system.