Two-dimensional Analytical Model Research Articles

Hybrid Analytical Modeling of Force Dense Segmented Magnetic Linear Actuator with Non-Dimensional Parametric Modeling of the Magnetic Flux Effects

A new advanced two-dimensional hybrid analytical model of a segmented magnet linear actuator (MLA) comprised of surface permanent magnets (PM) is developed in this paper. This model is used to predict and evaluate the performance of the segmented MLA with proper correction on magnetic Flux Effects, validated by computational modeling. An MLA design with non-uniform PM segmentation was applied in this research to improve its performance compared with conventional radially magnetized MLA and uniform segmented Halbach Array based MLA. For MLA thrust force prediction, the previous published analytical model does not consider losses due to two observed magnetic Flux Effects: (1) the magnetic edge effect—the diminishing nature of the magnetic flux at the edge of the MLA, and (2) the observed magnetic interaction effect—the inconsistent peaks of individual magnetic flux lines, lower than the overall peak flux. In the proposed hybrid model for the segmented MLA, the shaft magnetic field distribution is based on a scalar potential theory subdomain method and the ring magnetic field is based on equivalent surface distributed currents. Collectively, these models are combined with three-dimensional finite element analysis (FEA), to estimate the magnetic thrust force. A data driven pole correction factor is introduced, based on the FEA computational database of three-dimensional MLA, to capture the losses associated with the magnetic flux, which is not considered in the analytical subdomain method. Finally, a normalized pole correction is proposed to generalize the model to different magnetic grades, different dimensional constraints, and varying magnet ratios of the segmented magnets. The developed model provides the design basis for manufacturing optimized force dense segmented MLAs for rotary to linear actuation, based on the force required for the application without the need for running FEA analysis after each design iteration, reducing costs and time required for the optimal design.

Approximate analytical model of the refracted field of the Gaussian light beam on a plane dielectric interface

An approximate two-dimensional analytical model is proposed for description of a stationary monochromatic field arising at the Gaussian light beam refraction into a transparent or weakly absorbing homogeneous medium. Its formulae describe the spatial structure of a refracted field in a wide range of variation of incidence angles, when this field is characterized by free propagation, total internal reflection, or exponential decay. The model is based on the asymptotic calculation of the Fourier integral, taking into account possible diffraction effects of the first and the second order.

Use of the repeated integral transformation method to describe the transport of solute in soil

Predicting the fate and transport of contaminants in soil or groundwater systems using analytical or numerical models is crucial for environmental researchers. While the analytical models are a flexible approach to quantifying the subsurface contamination and remediation because they are non-susceptible to numerical dispersions, economical and handier; two-dimensional analytical models that describe a bilateral flow coupled with both sink and decay factors are rarely reported. Motivated by the case of a non-bare soil ridge with constant point-solute source lying internally but parallel to the longitudinal flow direction, a (2 + 1) dimensional Advection-Diffusion-Reaction Equation (ADRE) of bilateral flow coupled with the linear sorption, decay, and sink is formulated to model the transport of dissolved solute in a homogenous and isotropic non-fractured porous medium. Then, a brief review of exact and analytical methods for solving the ADRE is conducted to establish the right solution methods. The Repeated Integral Transformation Method (RITM) is employed to derive the approximate analytical solutions for the formulated model, maintaining the model's original terms for the efficient sensitivity analysis. The RITM uses Laplace and Fourier transforms with a wide range of computed results. We compare the approximate solutions with numerical simulations in COMSOL to verify the accuracy of the approximate analytical models. Then, the application of the solutions is demonstrated through a systematic analysis of the effect on solute transport of advection-diffusion, reaction, sorption, retardation, sink, and pore water velocity. Results show that the presence of sink, mimicked by plant-root uptake activity, and decay tend to reduce the solute concentration in the medium. While both the retardation and sorption factors affect the movement of the dissolved solutes, water content and pore-water velocity promote the spreading of dissolved solutes. Solute concentration in the medium increases at low Peclet numbers, signifying the influence of diffusive coefficients. The current proposed RITM-based solutions can characterize contamination in the soil, and should be useful to environmental researchers.

Two-Dimensional Eddy-Current Losses Model and Experimental Validation Through Thermal Measurements

Eddy-currents, generated by a spatio-temporal magnetic field variation, involve volumic losses, and consequently overheating. A two-dimensional (2-D) analytical model is developed to determine precisely the eddy-current distribution as well as losses. An experimental validation by thermal evolution measurements and an overheating of conductive massive parts is performed. Consequently, the eddy-current losses could be determined experimentally. A comparison of the results considering the segmentation influence of conductive massive parts is carried out. The difference between the theoretical and experimental results is less than 5%.

Two-dimensional model for composite beams and its state-space based mixed finite element solution by DQM

Most one-dimensional composite beam theories are based on simplified shear deformation assumptions, or even ignore shear deformation, which have limitations in analyzing multilayer beams with significant material differences. Therefore, this paper firstly proposed a two-dimensional analytical model for composite beams through the equivalent transformation of cross section. Based on the mixed variational principle, the state equations are then derived by finite element meshing and interpolation along the length of the beam, with nodal displacements and their energy-conjugated stresses as state variables. Subsequently, the differential quadrature method (DQM) is introduced to solve the equations and a two-dimensional analysis method for composite beams is established. Due to the use of displacements and stresses as fundamental variables in the state equations, various transfer characteristics of displacements and stresses at the interlayer interfaces of composite beams can be conveniently handled. This method is finally verified by numerical examples of steel-concrete composite beams and concrete beams with corrugated steel webs. Since no assumptions of displacement and stress distributions along the thickness of beams are required, the present method can provide benchmarks for various one-dimensional beam theories.

Validating a double Gaussian source model for small proton fields in a commercial Monte-Carlo dose calculation engine

PurposeThe primary fluence of a proton pencil beam exiting the accelerator is enveloped by a region of secondaries, commonly called “spray”. Although small in magnitude, this spray may affect dose distributions in pencil beam scanning mode e.g., in the calculation of the small field output, if not modelled properly in a treatment planning system (TPS). The purpose of this study was to dosimetrically benchmark the Monte Carlo (MC) dose engine of the RayStation TPS (v.10A) in small proton fields and systematically compare single Gaussian (SG) and double Gaussian (DG) modeling of initial proton fluence providing a more accurate representation of the nozzle spray. MethodsThe initial proton fluence distribution for SG/DG beam modeling was deduced from two-dimensional measurements in air with a scintillation screen with electronic readout. The DG model was either based on direct fits of the two Gaussians to the measured profiles, or by an iterative optimization procedure, which uses the measured profiles to mimic in-air scan-field factor (SF) measurements. To validate the DG beam models SFs, i.e. relative doses to a 10 × 10 cm2 field, were measured in water for three different initial proton energies (100MeV, 160MeV, 226.7MeV) and square field sizes from 1×1cm2 to 10×10cm2 using a small field ionization chamber (IBA CC01) and an IBA ProteusPlus system (universal nozzle). Furthermore, the dose to the center of spherical target volumes (diameters: 1cm to 10cm) was determined using the same small volume ionization chamber (IC). A comprehensive uncertainty analysis was performed, including estimates of influence factors typical for small field dosimetry deduced from a simple two-dimensional analytical model of the relative fluence distribution. Measurements were compared to the predictions of the RayStation TPS. ResultsSFs deviated by more than 2% from TPS predictions in all fields <4×4cm2 with a maximum deviation of 5.8% for SG modeling. In contrast, deviations were smaller than 2% for all field-sizes and proton energies when using the directly fitted DG model. The optimized DG model performed similarly except for slightly larger deviations in the 1×1cm2 scan-fields. The uncertainty estimates showed a significant impact of pencil beam size variations (±5%) resulting in up to 5.0% standard uncertainty. The point doses within spherical irradiation volumes deviated from calculations by up to 3.3% for the SG model and 2.0% for the DG model. ConclusionProperly representing nozzle spray in RayStation’s MC-based dose engine using a DG beam model was found to reduce the deviation to measurements in small spherical targets to below 2%. A thorough uncertainty analysis shows a similar magnitude for the combined standard uncertainty of such measurements.

Tube wall deformation behaviour of tensile-loaded blind-bolted connections in octagonal hollow section tubes

The structural behaviour of octagonal hollow section (OctHS) tubes under lateral tensile loads applied through blind-bolted T-stubs were experimentally and numerically investigated in this study. Two series of T-stub connections – single-side and double-side bolted T-stub connections, were tested to investigate the influence of the tube geometry and boundary conditions on the failure modes, deformation, and strength. Finite element (FE) models were developed and validated against the test results. Based on the validated FE models, parametric studies were conducted to further assess the influence of width-to-thickness ratios, tube length-to-width ratios and boundary conditions on the tube deformation and strength. Furthermore, the applicability of the tube deformation of 3% of a tube width to be the ultimate state criterion for OctHS tubes was evaluated using tube-face component shell FE models. A simplified two-dimensional analytical model was developed, and the calculation equations were proposed based on either small deformation assumption or large deformation assumption. The comparison of the strengths obtained from the proposed equations, numerical simulations and tests demonstrated that the proposed equations are able to give reasonable predictions on the yield and ultimate strengths of blind-bolted OctHS tube connections under tensile loading.

A two-dimensional analytical model for heavy metal contaminants transport in permeable reactive barrier.

Permeable reactive barrier (PRB) remediation technology has been widely used in the remediation of groundwater contamination. In numerical simulations, neglecting the non-uniform distribution of heavy metal contamination along the depth may lead to deviations between simulation results and reality. The distribution of heavy metals in the soil layer around a non-ferrous mining area in Hezhou, Guangxi, southern China was investigated, and it was found that the standard Gaussian function could well describe the non-uniform distribution of heavy metals in the soil layer. A two-dimensional analytical model solved by the finite element method was used to simulate the migration process of heavy metal contamination in the aquifer and PRB. The results show that the uniform distribution of contaminants along the depth ignores the dilution effect, which may underestimate the service life of the PRB and lead to an overly conservative design of the PRB. The breakthrough time of the PRB decreases with the increase of the maximum initial concentration (Cin,max) and the high concentration range (σ), and increases almost linearly with the barrier thickness (Lw). An optimal design method for PRB location and thickness is proposed, which can provide a reference for the engineering application of PRB.

A new model for predicting hydraulic fracture penetration or termination at an orthogonal interface between dissimilar formations

Vertical height growth of hydraulic fractures (HFs) can unexpectedly penetrate a stratigraphic interface and propagate into neighboring layers, thereby resulting in low gas-production efficiency and high risk of groundwater contamination or fault reactivation. Understanding of hydraulic fracture behavior at the interface is of pivotal importance for the successful development of layered reservoirs. In this paper, a two-dimensional analytical model was developed to examine HF penetration and termination behavior at an orthogonal interface between two dissimilar materials. This model involves changes in the stress singularity ahead of the HF tip, which may alter at the formation interface due to material heterogeneity. Three critical stress conditions were considered to assess possible fracture behavior (i.e., crossing, slippage, and opening) at the interface. Then, this model was verified by comparing its theoretical predictions to numerical simulations and three independent experiments. Good agreement with the simulation results and experimental data was observed, which shows the validity and reliability of this model. Finally, a parametric study was conducted to investigate the effects of key formation parameters (elastic modulus, Poisson's ratio, and fracture toughness) between adjacent layers. These results indicate that the variation in the introduced parameters can limit or promote vertical HF growth by redistributing the induced normal and shear stresses at the interface. Among the three studied parameters, Poisson's ratio has the least influence on the formation interface. When the fracture toughness and elastic modulus of the bounding layer are larger than those of the pay zone layer, the influence of fracture toughness will dominate the HF behavior at the interface; otherwise, the HF behavior will more likely be influenced by elastic modulus.

Face stability analysis of circular tunnels in layered rock masses using the upper bound theorem

An analysis of tunnel face stability generally assumes a single homogeneous rock mass. However, most rock tunnel projects are excavated in stratified rock masses. This paper presents a two-dimensional (2D) analytical model for estimating the face stability of a rock tunnel in the presence of rock mass stratification. The model uses the kinematical limit analysis approach combined with the block calculation technique. A virtual support force is applied to the tunnel face, and then solved using an optimization method based on the upper limit theorem of limit analysis and the nonlinear Hoek–Brown yield criterion. Several design charts are provided to analyze the effects of rock layer thickness on tunnel face stability, tunnel diameter, the arrangement sequence of weak and strong rock layers, and the variation in rock layer parameters at different positions. The results indicate that the thickness of the rock layer, tunnel diameter, and arrangement sequence of weak and strong rock layers significantly affect the tunnel face stability. Variations in the parameters of the lower layer of the tunnel face have a greater effect on tunnel stability than those of the upper layer.

The acoustic impedance of three-dimensional turbines

Thermoacoustic stability assessment of gas turbine combustors requires an acoustic impedance boundary condition for the downstream turbomachinery. Actuator disk type analytical methods offer rapid predictions of acoustic impedance in an iterative design process; previous studies show that this class of models works well for cascades consistent with a two-dimensional assumption. Real turbines, however, are three-dimensional with multiple stages, coolant flows, and leakage flows. The first part of this paper validates a cambered semi-actuator disk model for the acoustic impedance of a realistic multi-stage turbine using non-linear time-marching computations: the two methods agree to within 9% for incident pressure waves and 14% for incident entropy waves. Simulations of the multi-stage turbine with different inlet conditions confirm that, to a close approximation, inlet corrected flow and hence Mach numbers and acoustic impedance are constant during off-design operation. The second part of the paper then applies both analytical and computational approaches to families of parametrically generated turbine stages to quantify three-dimensional design effects. The results show that the influence of hub-to-tip ratio on acoustic impedance is weak, and the two-dimensional analytical model is accurate even for high aspect ratio stages. Front-loaded camber lines increase axial Mach number within blade passages, raising acoustic impedance by up to 51% compared to a datum quadratic camber line. Varying the stator–rotor axial gap changes the relative phase of reflections from vanes and blades, causing the total impedance to either increase or decrease, at different frequencies, by up to 11%. The cambered semi-actuator disk method consistently captures the correct trends, showing that the physical basis of the model is sufficient to produce a broadly applicable design tool for rapid assessment of turbine impedance boundary conditions.

Open-Circuit Electromagnetic Analysis of Interior Permanent Magnet Machines With Arbitrary Rotor Frame Using a 2-D Analytical Model

Two-dimensional (2-D) analytical modeling of interior permanent magnet (IPM) machines is quite challenging due to the nonuniform permeability of the rotor core. The choice of a non-cylindrical rotor frame (e.g., rotor frame with cosine shape) further increases the complexity of the model. This article presents a methodology for the 2-D modeling of IPM machines with cylindrical and non-cylindrical rotor frames. In this model, interior magnets are first mapped to the surface of the rotor body using the equivalent magnetic circuit model. In other words, the IPM machine is transformed into an equivalent surface-mounted permanent magnet (SPM) machine. The rotor shape is modeled by the proper adjustment of the permanent magnet (PM) magnetization pattern of the equivalent SPM machine. Next, the radial and tangential flux components of the equivalent (SPM) machine are computed by solving Maxwell’s relations and applying 2-D boundary conditions. The effect of the stator slotting on the distribution of magnetic flux is accounted using the theory of virtual current. The accuracy of the presented model is verified against the finite element (FE) and experimental measurements.

Two-Dimensional Winding Loss Analytical Model for High-Frequency Multilayer Air-Core Planar Inductor

Air-cored planar inductors play an important part in enabling high power-density, low weight, and low-profile design of high-frequency switched-mode power applications. However, the absence of ferromagnetic core in these inductors makes the magnetic field distribution in them different from that in the cored inductors and the transformers. Due to this difference, the conventional winding loss calculation models like the 1-D Dowell's model are not suitable for these inductors. Therefore, in this article, an improved two-dimensional analytical winding loss model for multilayered air-cored inductors with a single-turn per layer is proposed based on the fundamental electromagnetic field solution incorporating the 2-D boundary values and the edge effect in its formulation. The finite element method (FEM) simulation and experimental results validate the proposed model's high-accuracy and engineering precision over a wide frequency range (up to 10 MHz). Furthermore, the results ensure the proposed analytical model to be an efficient, easier, and better substitute for the FEM based calculation approach.

Enhanced delivery of amendments in fractured clay sites based on multi-point injection: An analytical study

Fracturing technology that can enhance the delivery of amendments has attracted attention in the remediation of low-permeability contaminated sites. However, there are few works on the enhanced delivery of amendments based on multi-point injection in a fracture-matrix system. This study develops a two-dimensional analytical model for enhanced delivery of amendments in a finite-domain low-permeability matrix through multi-point injection in a natural, hydraulic or pneumatic fracture. The mechanisms of advection, diffusion, dispersion, sorption and degradation are considered in the model and any injection form (e.g., pulse injection, periodic injection or slow-release injection) can be embedded to obtain a specified solution. Then, a new linear factor R*, which is the ratio of the peak concentration to the trough concentration on the same plane, is introduced to evaluate the concentration fluctuation in the fracture and matrix. Results show that with a stronger line source formed in the fracture right after injection (corresponding to a small R*), the concentration distribution of amendments in the matrix is more uniform at each depth resulting in a smaller residual rate, i.e., (R*-1) × 100%. If the injection wells have been installed unreasonably (e.g., a too large spacing), the continuous injection time is an effective controllable parameter to compensate for this defect. Moreover, a controlled slow-release system can maintain a more stable concentration distribution in the fracture than continuous injection and periodic injection systems, giving a longer residence time. Overall, this work is expected to provide some interesting guidelines for the design of multi-point injection in the fracturing low-permeability sites to enhance the remediation of contaminated soil.

In-plane flexible ring modeling and a nonlinear stiffness solution for heavy-load radial tires

This paper presents a complete tire model for heavy-load radial tires with a large section ratio that is based on a flexible carcass ring model, combined with a sidewall element with two-dimensional stiffness. It also establishes a two-dimensional analytical model of sidewall stiffness for heavy-duty tires based on preloaded beam deformation theory. The flexible ring model is used to examine the ground-contact modeling of heavy load tires and their stiffness characteristics are analyzed.Experimental and theoretical results show that flexible ring tire models based on nonlinear elastic foundations can accurately characterize the grounding mark of tires and their stiffness characteristics, with there being a good fit between the theoretical and experimental results. The analytic two-dimensional function for curved sidewalls developed in the paper can also be used to enrich other tire models, such as FTire and CDTire.

Characterizing THz Scattering Loss in Nano-Scale SOI Waveguides Exhibiting Stochastic Surface Roughness with Exponential Autocorrelation

Electromagnetic (EM) scattering may be a significant source of degradation in signal and power integrity of high-contrast silicon-on-insulator (SOI) nano-scale interconnects, such as opto-electronic or optical interconnects operating at 100 s of THz where two-dimensional (2D) analytical models of dielectric slab waveguides are often used to approximate scattering loss. In this work, a formulation is presented to relate the scattering (propagation) loss to the scattering parameters (S-parameters) for the smooth waveguide; the results are correlated with results from the finite-difference time-domain (FDTD) method in 2D space. We propose a normalization factor to the previous 2D analytical formulation for the stochastic scattering loss based on physical parameters of waveguides exhibiting random surface roughness under the exponential autocorrelation function (ACF), and validate the results by comparing against numerical experiments via the 2D FDTD method, through simulation of hundreds of rough waveguides; additionally, results are compared to other 2D analytical and previous 3D experimental results. The FDTD environment is described and validated by comparing results of the smooth waveguide against analytical solutions for wave impedance, propagation constant, and S-parameters. Results show that the FDTD model is in agreement with the analytical solution for the smooth waveguide and is a reasonable approximation of the stochastic scattering loss for the rough waveguide.

Analytical model for degradable contaminant transport through a cutoff wall-aquifer system under time-dependent point source pollution

The sources of pollution, in general, are divided into nonpoint sources and point sources. Cutoff wall as a low-permeability vertical barrier has been widely employed for control and containment of these two types of pollution sources. Previous cutoff wall models mainly were limited to the nonpoint source pollution with constant concentrations. Hence, this study developed a novel two-dimensional analytical model for rapidly assessing the service performance of cutoff walls under time-dependent point source pollution. The present model considers a transient migration process coupling advection–dispersion-sorption-biodegradation in a multi-layer system, and its general solutions are derived using a combined approach of Laplace transform, finite Fourier cosine transform, variable substitution, and global matrix method. Results reveal that the service life of a cutoff wall increases exponentially with the decrease of release time for finite pulse point sources, and it increases linearly with the increase of decay coefficient for decay point sources. The previous hypothesis about the Gaussian distribution of contaminant concentration at the entrance to a cutoff wall is invalid when the point sources are separated by more than 1 m. Moreover, a comprehensive design method of a cutoff wall system under point sources pollution is proposed.

Feasibility investigation of rail monitoring by wheel-mounted acoustic emission technology using two-dimensional analytical contact acoustic nonlinearity model

In an attempt to analyze the feasibility of a rail monitoring scheme with a wheel-mounted acoustic emission measurement method, a two-dimensional analytical spring model was proposed to interpret the interactions between acoustic waves and stationary wheel–rail contact interfaces. The spring model represents the coupling strength of the interfaces with their stiffness. It accommodates the environmental information of contact stiffness and the acoustic source information, such as the feature frequency and the incident angle in the analysis of interface transmissibility. The fractal dimension of the interface is introduced into the spring model to interpret the effect of the axle load on the interface stiffness and the variation in transmissibility. Discrete Rayleigh integration is further combined to obtain the directivity of the interface and estimate the acoustic field intensity in the entire wheel. A wheel–rail contact rig was designed to simulate the actual contact conditions in a railway. Experimental data acquired from this test rig were utilized to validate the model, with regard to the interfacial stiffness in normal incidence and amplitudes of the transmitted waves in oblique incidence. After verifying the reliability of the model, the acoustic field intensity in the wheel was visualized under the assumed environmental conditions. Finally, a discussion is presented to determine a proper angular separation for the wheel-mounted sensor system, and substantiate the feasibility of the system.

Two-dimensional steady-state thermal analytical model of permanent-magnet synchronous machines operating in generator mode

PurposeThermal analysis of electrical machines is usually performed by using numerical methods or lumped parameter thermal networks depending on the desired accuracy. The analytical prediction of temperature distribution based on the formal resolution of thermal partial differential equations (PDEs) by the harmonic modeling technique (or the Fourier method) is uncommon in electrical machines. Therefore, this paper aims to present a two-dimensional (2D) analytical model of steady-state temperature distribution for permanent-magnet (PM) synchronous machines (PMSM) operating in generator mode.Design/methodology/approachThe proposed model is based on the multi-layer models with the convolution theorem (i.e. Cauchy’s product theorem) by using complex Fourier’s series and the separation of variables method. This technique takes into the different thermal conductivities of the machine parts. The heat sources are determined by calculating the different power losses in the PMSM with the finite-element method (FEM).FindingsTo validate the proposed analytical model, the analytical results are compared with those obtained by thermal FEM. The comparisons show good results of the proposed model.Originality/valueA new 2D analytical model based on the PDE in steady-state for full prediction of temperature distribution in the PMSM takes into account the heat transfer by conduction, convection and radiation.

New refined analytical models for various bonding conditions of an adhesively bonded smart PZT transducer using the EMI technique

The electro-mechanical impedance (EMI) technique has emerged as a cost-effective and non-destructive technique to detect the possible damages in the structure using a piezoelectric transducer, especially, lead zirconate titanate (PZT). The adhesive bond layer plays an important role in the PZT patch-host structure interaction for monitoring structural damage. Two bonding conditions are investigated in this research paper. Primarily, the debonding phenomenon of the adhesive bond layer may misinterpret the EMI response on the damage caused in structure. Subsequently, the investigation included the protective layer at the top of the PZT transducer to avoid sensor degradation. However, the analytical models developed so far have not considered a protective layer at the top of the PZT transducer. This paper presents the novel two-dimensional (2D) analytical model for incorporating debonding concepts and the new refined 2D analytical model to include a protective layer in the study of surface-bonded PZT transducers. The proposed analytical models are verified with the experimental studies. The experimental and analytical results show good agreement, which confirms the effectiveness of the new models. This paper also incorporated the effect of each bonding condition for monitoring structural damage by implementing the EMI technique. For the simulation, the numerical investigations on the PZT transducer bonded on the metallic (aluminum and steel) and concrete blocks are performed using coupled field analysis through finite element (FE) modeling. It is found that each bonding condition has influenced the resulting signatures. The signatures obtained from developed theoretical models and numerical simulations using three-dimensional FE models for each bonding condition are compared to highlight the influence on structural damage detection. The trend of signatures is found to be matching satisfactory. Several parametric studies have been conducted to show the efficacy of the new refined model with a protective layer. It considers the different input properties of an adhesive layer, host structure, and temperature conditions. The influence of debonding of the protective layer is also studied, and the obtained results support the need for a protective layer in the models.