Field Quantities Research Articles

On some energy‐based variational principles in non‐dissipative magneto‐mechanics using a vector potential approach

AbstractThis contribution covers the variational‐based modeling of non‐dissipative magneto‐mechanical systems using a vector potential approach and the thorough analysis and discussion of corresponding conforming finite element methods. Since the construction of divergence‐free finite element spaces explicitly enforcing the Coulomb gauge poses some major challenges, we propose some primal and mixed variational principles that ensure well posedness of the problem and allow to seek the vector potential in unconstrained function spaces. The performance of these methods is assessed in two comparative benchmark studies. The focus of both studies lies on the accurate approximation of field quantities in systems with material discontinuities and re‐entrant corners.

Weight Assignment Method and Application of Key Parameters in Shale Gas Resource Evaluation

The evaluation methods for shale gas resources and the key parameters involved are diverse. The main research object of this article is the key parameters in shale gas resource evaluation, and this study aims to quantitatively evaluate the required key parameters for calculation, thereby improving the credibility of shale gas resource evaluation results. This article mainly analyzes the weight analysis of the key parameters used in the resource evaluation process, focusing on the analysis and determination of key parameters controlling the generation, enrichment, and preservation of shale gas. The key parameters are graded and evaluated layer by layer, and the common key parameters that participate in the determination are extracted. Based on the above analysis, a weight conversion model is proposed to quantitatively evaluate the actual importance of key parameters in resource evaluation. By combining statistical research, the partial correlation coefficients in the correlation coefficients are applied to estimate the resource quantities in the exploration field in order to quantitatively evaluate the credibility of the resource evaluation results. This article focuses on the weight assignment of key parameters for calculating the resource quantity of the Jiaoshiba shale gas field in the Fuling area of Chongqing and obtains different weight results for different parameters. Using this method to assign key parameter weights to the target block can provide reliable parameter support for shale gas resource evaluation and provide strong support for shale gas resource estimation.

Two-dimensional analysis of electrical behavior in composite semiconductor shell structures with magnetic field tuning

To precisely investigate the electric properties and reveal the intrinsic interaction mechanisms between multiple fields in the composite magneto-electric-semiconductor shell structure, two-dimensional analyses were conducted within the framework of the third-order shear deformation theory and the coupled field theory. The governing equations for the composite piezoelectric semiconductor (PS) shell structures were derived using the principle of virtual work. Field distributions of the composite PS shell structures were obtained by expanding basic field quantities into Fourier series along the width direction and using the differential quadrature method. Prior to analysis, the convergence and correctness of the adopted method were evaluated. Several numerical examples were presented to demonstrate how magnetic manipulation can tune electrical behavior. It was found that inhomogeneous shear strain induced by applied magnetic fields is the key factor affecting potential variation in the composite PS shells. The appearance of potential barriers and wells in the composite PS shell structure with an opposite c axis is explained thoroughly. Additionally, the effect of magnetic field non-uniformity on the composite PS shell structure was investigated using an equivalent body force model, offering valuable insights for designing thin-film devices.

A modified quasi-3D theory and mixed beam element method for static behaviour analysis of functionally graded beams

This paper develops a modified quasi-3D theory and the mixed beam element model for static analysis of functionally graded beams. The modified quasi-3D theory encompasses two innovations. Firstly, the accurate distribution of transverse shear stress is derived from the differential equilibrium equation that describes the relationship between stresses, rather than the traditional one derived from geometric relations and constitutive equations. Secondly, the effect of distributed loads associated with transverse stretching deformation, a factor typically overlooked in the existing quasi-3D theory-based beam models, is involved. In the development of beam finite element model, the mixed variational principle is firstly utilized to formulate the model of a quasi-3D theory-based beam element, where generalized displacements and internal forces are treated as two types of independent field quantities. Two numerical examples are conducted to investigate the validity and accuracy of the proposed theory and beam element. Numerical results indicate that the proposed mixed beam element can accurately predict the stress distributions over the cross-section and produce accurate displacement solutions. Meanwhile, it is noted that the effect of distributed loads related to the transverse stretching deformation should be correctly considered in the analysis of functionally graded beams based on quasi-3D theory.

Analytical technique for hygrothermo-electroelastic field in piezoelectric bodies with D ∞ symmetry

A general solution technique for hygrothermo-electroelastic problems in bodies with D ∞ symmetry is developed. The displacement and electric field are expressed in terms of their respective potential functions, and the hygrothermo-electroelastic field quantities are expressed in terms of elastic and piezoelastic potential functions, each of which satisfies a Laplace equation with respect to the appropriately transformed spatial coordinates, combined with the two hygrothermoelastic displacement potential functions. As an application of this technique, a theoretical analysis of an infinite cylinder subjected to distributions of moisture content, temperature, and stress was performed, and the significance of the hygrothermo-electroelastic analyses was demonstrated.

Analysis of axisymmetric thermoelastic diffusive half-space with variable material properties using hyperbolic two-temperature model

The aim of the current study is to investigate the variable thermal conductivity and diffusivity impact on the transient response of an axisymmetric thermodiffusive elastic half-space under axisymmetric thermal, mechanical, and concentration loads in the light of hyperbolic two-temperature generalized fractional thermoelastic diffusion model. With the help of Kirchhoff’s transform, the governing equations are converted into linear form. The resulting equations are solved by imposing the combined Laplace–Hankel transform. In converted space, the solutions for different field quantities are presented in compact form. The copper material is considered for numerical computation. By employing a mathematical inversion procedure, the solutions are converted into the original space. Numerically computed solutions for various physical quantities are illustrated in the form of graphs to show the impact of variable thermal conductivity and diffusivity, hyperbolic two temperature, and fractional parameters. Validation of the obtained results is also presented.

Position-dependent Radiation Fields near Accretion Disks

In disk-wind models for active galactic nuclei outflows, high-energy radiation poses a significant problem wherein the gas can become overionized, effectively disabling what is often inferred to be the largest force acting on the gas: the radiation force due to spectral line opacity. Calculations of this radiation force depend on the magnitude of ionizing radiation, which can strongly depend on the position above a disk where the radiation is anisotropic. As our first step to quantify the position and direction dependence of the radiation field, we assumed free streaming of photons and computed energy distributions of the mean intensity and components of flux as well as energy-integrated quantities such as mean photon energy. We find a significant dependence of radiation-field properties on position, but this dependence is not necessarily the same for different field quantities. A key example is that the mean intensity is much softer than the radial flux at many points near the disk. Because the mean intensity largely controls ionization, this softening decreases the severity of the overionization problem. The position dependence of mean intensity implies the position dependence of gas opacity, which we illustrate by computing the radiation force a fluid element feels in an accelerating wind. We find that in a vertical accelerating flow, the force due to radiation is not parallel to the radiation flux. This misalignment is due to the force’s geometric weighting by both the velocity field’s directionality and the position dependence of the mean intensity.

Study of internal heat source, rotation, magnetic field, and initial stress influence on p-waves propagation in a photothermal semiconducting medium

The purpose of the current study is to establish a novel mathematical model in the p-waves in a photothermal semiconducting medium with an internal heat source. The fundamental equations in the context of isotropic and homogeneous medium have been presented. For the solution of the required problem, the normal mode analysis along with the displacement components, stress components and temperature has been utilized. For graphical representation of different physical quantities such as displacement components, stress components and carrier density as well as the temperature distribution. Using MATLAB R2023a software, a parametric analysis is performed, and the resulting data is represented graphically. A comparison is made to show the effect of the new parameters on the phenomenon. A graphic representation of the relationship between rotation, magnetic field, and initial stress in relation to the fluctuations in non-dimensional field quantities is provided, along with an analysis of the findings.

A modified Green-Naghdi fractional order model for analyzing thermoelectric MHD

PurposeWhereas, the classical Green-Naghdi Type II (GN-II) model struggles to accurately represent the thermo-mechanical behavior of thermoelectric MHD due to its inability to account for the memory effect. A new mathematical model of the GN-II theory incorporates a fractional order of heat transport to address this issue.Design/methodology/approachThe employment of the matrix exponential method, which forms the basis of the state-space approach in contemporary theory, is central to this strategy. The resulting formulation, together with the Laplace transform techniques, is applied to a variety of problems. Solutions to a thermal shock problem and to a problem of a layer media both without heat sources are obtained. Also, a problem with the distribution of heat sources is considered. The numerical technique is used to achieve the Laplace transform inversion.FindingsAccording to the numerical results and its graphs, the influences of the fractional order parameters, figure-of-merit factor, thermoelectric power and Peltier coefficient on the behavior of the field quantities are investigated in the new theory.Originality/valueThe new modeling of thermoelectric MHD has advanced significantly as a result of this work, providing a more thorough and precise tool for forecasting the behavior of these materials under a range of thermal and magnetic conditions.

Nonlinear Wave Interaction of Nanorods Embedded in a Viscoelastic Medium

Abstract Purpose Nonlinear interactions between two acoustic waves in nanorods traveling at various wave numbers, group velocities, and frequencies are examined in this study. Methods The nonlinear equation of the nanorod in a viscoelastic medium is obtained using the theory of nonlocal elasticity. Furthermore, the multiple-scale expansion method is applied to study strongly dispersive, weakly nonlinear waves in a nonlocal viscoelastic medium. Using this expansion technique, we can derive the coupled nonlinear Schrödinger equations as the governing equations, which we solve as differential equations of some parameters by expanding the field quantities into an asymptotic series of the smallness parameter. Results We give the nonlinear plane wave solutions to these equations in several special cases. The plane wave solutions show how the wave amplitude affects the frequencies of nonlinear plane waves. Additionally, we show numerically how the real and imaginary parts of the group velocities and natural frequency of the system for a carbon nanotube in a viscoelastic medium are affected by the nonlocal, damping, and stiffness parameters.

Analysis of functionally graded piezoelectric structures by Hermite interpolation element-free Galerkin method

Functionally graded piezoelectric structures (FGPSs) have excellent electromechanical properties and are promising for engineering applications. However, the inhomogeneous material properties and piezoelectric effects create great difficulties in the mechanical analysis of the FGPSs. In this paper, a Hermite interpolation element-free Galerkin method (HIEFGM) is proposed for the numerical analysis of the FGPSs. The HIEFGM utilizes a set of nodes to represent the problem domain, which can ideally reflect the inhomogeneous material properties of the FGPSs. Firstly, the governing equations of the FGPSs are derived through the constitutive equation, equilibrium equation, and boundary conditions. Secondly, the approximating function of the field quantities is obtained by the improved moving least-square method and Hermite interpolation. The HIEFGM formulation of the FGPSs is obtained using the variational principle. Furtherly, the influence of weight function, scaling parameter, and node densities on the HIEFGM of the FGPSs is discussed in detail through a parameter study. Finally, the availability of present method is estimated by several examples with different configurations and boundary conditions. The influence of gradation exponent on the electromechanical responses of the FGPSs is analyzed. The results show that the present method has excellent accuracy and stability in analyzing the FGPSs. As the gradation exponent increases, the distribution of the field quantities of the FGPSs exhibits nonlinear characteristics. This work may provide an effective methodology for the analysis of the FGPSs, and contribute to the theoretical research and engineering application of the FGPSs.

Nonlinear analysis on electro-elastic coupling properties in bended piezoelectric semiconductor beams with variable cross section

In this paper, the effects of nonlinear constitutive relation on the electro-elastic coupling behaviors in non-uniform piezoelectric semiconductor beams are investigated. On the basis of three-dimensional theory for piezoelectric semiconductor and double power series, one-dimensional bending model is developed. In order to deal with the nonlinear problem, a differential quadrature method based iteration procedure is proposed. Comparing with the results from finite element method, the proposed method has good convergence and high accuracy. In numerical analysis, the effects of nonlinearity and non-uniformity in piezoelectric semiconductor are discussed thoroughly. It is found the introduction of nonlinearity induces the loss of symmetric for all electrical field quantities but little changes the mechanical quantities. While the reduction of width along the length direction produces larger mechanical and electrical responses. Importantly, the effects of nonlinearity on the electrical field quantities are more evident in a piezoelectric semiconductor beam with rapidly varied cross section. Besides, we designed the piezoelectric semiconductor beams subject to double shear forces and a beam with locally varied cross section. In the designed structures, the potential barriers and wells are realized, which provide new approaches adjusting the electro-elastic behaviors in piezoelectric semiconductors. The study in this paper could be the guidance designing high performance electronic devices.

On the role of aortic valve architecture for physiological hemodynamics and valve replacement, Part II: Spectral analysis and anisotropy

Severe aortic valve stenosis can lead to heart failure and aortic valve replacement (AVR) is the primary treatment. However, increasing prevalence of aortic stenosis cases reveal limitations in current replacement options, necessitating improved prosthetic aortic valves. We investigate flow disturbances downstream of severe aortic stenosis and two bioprosthetic aortic valve (BioAV) designs using advanced energy-based analyses. Three-dimensional high-fidelity fluid–structure interaction simulations have been conducted and a dedicated and novel spectral analysis has been developed to characterise the kinetic energy (KE) carried by eddies in the wavenumber space. In addition, new field quantities, i.e. modal KE anisotropy intensity as well as normalised helicity intensity, are introduced. Spectral analysis shows kinetic energy (KE) decay variations, with the stenotic case aligning with Kolmogorov’s theory, while BioAV cases differing. We explore the impact of flow helicity on KE transfer and decay in BioAVs. Probability distributions of modal KE anisotropy unveil flow asymmetries in the stenotic and one BioAV cases. Moreover, an inverse correlation between temporally averaged modal KE anisotropy and normalised instantaneous helicity intensity is noted, with the coefficient of determination varying among the valve configurations. Leaflet dynamics analysis highlights a stronger correlation between flow and biomechanical KE anisotropy in one BioAV due to higher leaflet displacement magnitude. These findings emphasise the role of valve architecture in aortic turbulence as well as its importance for BioAV performance and energy-based design enhancement.

Design And Optimization Of Aeronautical Components And Digital Twins Development

In this work, the feasibility of using reduced models (ROMs) for optimizing a scoop air intake for aeronautical applications was evaluated. Since the air intake is exposed to aerodynamic loads, a two-way fluid-structure interaction workflow was used to characterize the component. The goal is to create an optimization dashboard that allows both scalar quantities (the parameters that are intended to be monitored during the design and optimization of the air intake) and field quantities to be evaluated in real time. In this way, the designer can have a full understanding of the physics of the problem and make more informed design choices. In addition, in this way it is possible to visualize results from different physics in a single dashboard, linking different components, interacting with models in real time. A mesh morphing technique based on Radial Basis Functions (RBFs) was used. The result was very interesting both from a structural point of view (mass reduction over 90% and maximum strain reduction of 36%) and from a fluid dynamic point of view (outlet pressure 86% higher and drag 32% lower) and the generated ROMs proved to be a very accurate (ROM relative error in the order of 7%) and flexible tool.

Axially Non-Uniform Properties Effected Electro-Elastic Fields in Bended Piezoelectric Semiconductor Beams

This paper studied the effects of axially non-uniform material and non-uniform doping on the electro-elastic fields in bent piezoelectric semiconductor beams. The mathematical model is established based on the phenomenological theory of piezoelectric semiconductor and Euler’s beam theory. To solve the governing equations with variable coefficients, the differential quadrature method is adopted. As two representative structures, the pure bending beam and cantilever beam are studied, respectively. From the calculated results, it can be found the varied material properties changed the effective stiffness, as a result, all the field quantities are altered. Meanwhile, the symmetry of all field quantities in a pure bending beam is broken. Considering different Gaussian doping parameters [Formula: see text] and [Formula: see text], it can be observed that the distribution rule of perturbation carrier density becomes complex. The increment of [Formula: see text] broadens the width of optimum range-producing carriers, however, the increment of [Formula: see text] describes the moving of the optimum range from left to right. Importantly, the introduction of non-uniform doping breaks the limitation of perturbation carrier density in a cantilever beam with uniform material properties, the maximum value of perturbation carrier density does not appear at the fixed end only. The obtained results could be the guidance in designing high-performance electric devices.

Analysis and modeling of two-dimensional piezoelectric semiconductor shell theory

In recent years, there has been a surge in research on piezoelectric semiconductors (PSs). However, studies specifically focused on two-dimensional PS structures remain scarce. In this paper, based on the principle of the virtual work method, the condition of charge continuity and the first-order thickness shear theory, we have established a two-dimensional model to investigate the electromechanical properties of two-dimensional PS shells. The basic field quantities, including the displacement, the electric potential and the carrier concentration perturbations were Fourier expanded along the width direction, and then the governing equations were successfully derived. The differential quadrature method was used to obtain numerical results for the PS shells with different loading modes and structural configurations. It was found that the inhomogeneous shear strain S13 is the key to the potential change in the PS shell structure. The emergence of the potential barriers and wells in the shell structure of PS is well explained. In addition, the position of the potential barriers can be modulated by varying the magnitude of the load, which provides a reference for the design of thin-film devices.

Data Assimilation of Ideally Expanded Supersonic Jet Using RANS Simulation for High-Resolution PIV Data

Data assimilation using particle image velocimetry (PIV) and Reynolds-averaged Navier–Stokes (RANS) simulation was performed for an ideally expanded supersonic jet flying at a Mach number of 2.0. The present study aims to efficiently reconstruct all the physical quantities in the aeroacoustic fields that match well with a realistic, experimentally obtained flow field. The two-dimensional, two-component PIV measurement was applied to the jet axis plane, and the time-averaged velocity field was obtained using single-pixel ensemble correlation. Two-dimensional axisymmetric RANS simulation using the Menter shear stress transport (SST) model was also performed, and the parameters of the SST model were optimized via data assimilation using the ensemble Kalman filter. The standard deviation of the observation noise σ, which is a parameter of the ensemble Kalman filter, is estimated by the previously proposed method (Nakamura et al., Low-Grid-Resolution-RANS-Based Data Assimilation of Time-Averaged Separated Flow Obtained by LES. Int. J. Comp. Fluid. Dyn., 2022), and its effectiveness was investigated for the first time. This method effectively estimated the magnitude of σ at each generation without tuning the hyperparameters. The assimilated flow fields exhibited similar flow structures observed in PIV such as the potential core length or shear layer. Therefore, the present framework can be used to estimate time-averaged full flow fields that match well with experimentally observed flow fields, and has the potential to construct a database for the Navier-Stokes-based stability analysis that requires a full flow field.

Multiphysics coupled sensing mechanism of Pd/Ni alloy thin-film coated SAW hydrogen sensor

Multiphysics coupled sensing mechanism of palladium/nickel (Pd/Ni) alloy thin-film coated surface acoustic wave (SAW) hydrogen (H2) sensor is demonstrated theoretically and experimentally to allow the optimization of the sensing device in this work. The resistor-capacitance circuit model is used to describe the interaction between Pd/Ni film and H2. Referring to the perturbation theory, the relationship between the changes in SAW velocity/phase and the multi-physical field quantities of the Pd/Ni film are analyzed. To verify the theoretical model, the Pd/Ni film is sputtered on the Y35°X quartz substrate to build the delay-line patterned SAW H2 sensor. Experimental results have well verified the theoretical predictions. That is, the main response mechanism is the mass loading effect, and the contribution of the acoustoelectric effect can be neglected. The expansion effect induced by hydrogen adsorption is completely different from the mass loading effect, which causes the sensing response failure, but it can be effectively improved by increasing the working temperature or decreasing the thickness of the Pd/Ni thin-film. Wide detection range (100 ppm ∼ 38 v/v %), rapid response (t 90 ∼ 7 s), and good humidity stability are achieved from the optimized SAW H2 sensor.

A canonical Hamiltonian formulation of the Navier–Stokes problem

This paper presents a novel Hamiltonian formulation of the isotropic Navier–Stokes problem based on a minimum-action principle derived from the principle of least squares. This formulation uses the velocities $u_{i}(x_{j},t)$ and pressure $p(x_{j},t)$ as the field quantities to be varied, along with canonically conjugate momenta deduced from the analysis. From these, a conserved Hamiltonian functional $H^{*}$ satisfying Hamilton's canonical equations is constructed, and the associated Hamilton–Jacobi equation is formulated for both compressible and incompressible flows. This Hamilton–Jacobi equation reduces the problem of finding four separate field quantities ( $u_{i}$ , $p$ ) to that of finding a single scalar functional in those fields – Hamilton's principal functional ${S}^{*}[u_{i},p,t]$ . Moreover, the transformation theory of Hamilton and Jacobi now provides a prescribed recipe for solving the Navier–Stokes problem: find ${S}^{*}$ . If an analytical expression for ${S}^{*}$ can be obtained, it will lead via canonical transformation to a new set of fields which are simply equal to their initial values, giving analytical expressions for the original velocity and pressure fields. Failing that, if one can only show that a complete solution to this Hamilton–Jacobi equation does or does not exist, that will also resolve the question of existence of solutions. The method employed here is not specific to the Navier–Stokes problem or even to classical mechanics, and can be applied to any traditionally non-Hamiltonian problem.

An Analytical Determination of the Magnetic Field in a System of Finite-Length Ribbon Busbars

Using the analytic method based on the Biot–Savart law for the electromagnetic field, the distribution of the magnetic field of a ribbon busbar of finite length was determined. The analytical formulas describing the magnetic field in all areas of the considered ribbon busbars were obtained. The Mathematica program was used to visualize the solutions obtained. The Mathematica programme is a good and convenient tool for analytical measurements using the integration function and conversion of the analytical solutions, for the determination of field quantities, and for the graphical visualisation of the obtained final solutions. This allowed for quick field analysis to be conducted after changes were made in the geometrical or electrical parameters of the systems under examination.