Variable Coefficients Research Articles

Biquadratic element discrete duality finite volume method for solving elliptic equations on quadrilateral mesh

Nowadays numerical methods for handling partial differential equations is so vast that it is almost taken for granted. In this paper, we propose a discrete duality finite volume scheme with biquadratic elements and its application in solving two-dimensional elliptic equation on quadrilateral meshes. To test the robustness and efficiency of this method, we investigate several examples from physical and engineering fields with different parameters and coefficients. Numerical experiments, encompassing linear and nonlinear elliptic equations with constant or variable coefficients, reveal that the new method achieves optimal convergence in the continuous H1-norm, with a 2nd-order convergence, and in the L2-norm, with a 3rd-order convergence. Moreover, comparison with the existing biquadratic element finite volume scheme shows that the above method has an advantage of accuracy in the discrete L∞-norm on several distorted meshes.

Thermoelastic analysis of variable thickness truncated conical shell subjected to thermomechanical load with internal heat generation using perturbation technique

In this article, the behavior of the truncated conical shell subjected to thermomechanical loading in their inner and outer layers with internal heat generation source is investigated. The displacement field obeys kinematic of the first order shear deformation theory and the first-order temperature theory is used. Two-dimensional temperature analysis has been performed along the thickness and axis of the shell, which can be defined under various loading and thermomechanical boundary conditions. The set of governing equations is a system of differential equations with variable coefficients which are solved by using the analytical matched asymptotic expansion of the perturbations technique. The mentioned solution has little computational cost, therefore can be used well in parametric studies for optimization. It was shown, axial displacement is somehow independent of the radial axis and its maximum value occurs near the upper boundary. The conical shell has expanded in the radial direction. Also the maximum temperature happens approximately in the middle of the length of the conical shell. The parametric study showed, with the increase heat flux in the outer layer thereby expanding the area of heat application which would let more heat in than out. Also it was found, controlling the lifetime of the structure is the result of directing the cone angle. The results obtained from the analytical solution were compared with the Finite Element Method (FEM) analysis and the results of similar related articles, only to show a good agreement.

New Soliton Solutions of the Burgers Equation with Additional Time-dependent Variable Coefficient

In this article, we use the functional variational method to solve the Burgers equation with an additional time-dependent variable coefficient. The main advantage of the proposed method over other methods is that it allows to obtain more new solutions of the equation. Among the solutions obtained, new soliton solutions should be noted, which are of great importance for revealing the internal mechanism of physical phenomena. Three-dimensional graphs of solutions are constructed using the mathematical program Matlab. For a better understanding of the physical properties of some of the resulting solutions, their graphical representations are shown. This method is effective for finding exact solutions to many other similar wave equations.

Separation of solutions and the attractivity of fractional-order positive linear delay systems with variable coefficients

This article has two purposes. First, we study the separation of solutions to mixed-order positive linear delay systems with variable coefficients and the lower estimates of the separation of solutions to those systems. Second, we consider the global attractivity of fractional-order positive linear delay systems and describe precisely the rate of convergence of solutions to their equilibrium in some specific cases. To do this, the unified approach we use here is that the comparison principles have been modified to accommodate fractional-order delay systems. In addition, numerical simulations are introduced to illustrate the validity of the theoretical results.

Plasma kinetic simulation in Eulerian approach fully respecting f≥0 requirement and total particle number conservation

Abstract Plasma kinetic simulation in Eulerian approach (PKSiEA) can yield a dynamic description on plasmas and hence is a popular method in theoretical plasmas physics. Its mathematical model consists of 8 members: 5 partial differential equations (PDEs), a phase space boundary condition and an initial condition of the distribution function (PDF) f, and a mandatory non-negativity requirement on f. Because Vlasov equation (VE) is a PDE with variable coefficients and the non-negativity requirement is mandatory, when 5 PDEs (4 Maxwell equations and the VE) are treated as an initial-value problem, the temporal evolution of f is difficult to be attacked by existing popular computational mathematics techniques, such as Fourier analysis technique and finite-difference technique, and hence these techniques do not yield efficient and reliable schemes of the PKSiEA. In a universal scheme of the PKSiEA presented here, because the phase space boundary condition f∣ r=∞,∣υ∣=c = 0 causes the VE to yield a conservation of total particle number ∫fd 3 rd 3 υ, we propose an efficient and strict expression of the f-profile in terms of two classes of functions of phase space coordinates r , υ , and the VE displays a universal relation between two classes. By this new expression, we express the initial-value problem of 5 PDEs in terms of allowed deformation modes from an initial f-profile, and ensure the PKSiEA fully respecting the non-negativity requirement on f.

Transport of graphene oxide in the capillary fringe: Insights from sandbox experiments and numerical simulation

Studying contaminant transport in the capillary fringe (CF), a crucial part of the vadose zone, offers insights into the mechanisms controlling pollution in soils and groundwater aquifers. This paper investigated contaminant transport in the CF by continuously injecting a conservative tracer (NaCl) and graphene oxide nanoparticle (GONP), an adsorptive contaminant, into a sandbox. After entering the CF from the unsaturated zone, both NaCl and GONP underwent lateral transport. The breakthrough curves (BTCs) for NaCl and GONP were derived from water samples collected at predetermined sampling holes. Subsequently, contaminant transport in the CF was modeled using a one-dimensional–two-dimensional (1D-2D) coupled hydrodynamic model. This model incorporated lateral dispersivity (αL = 1.198 cm) and longitudinal dispersivity (αT = 0.286 cm), calculated using a point-by-point method. The hydrodynamic dispersion coefficients obtained were then applied to the Brooks and Corey (BC) and the van Genuchten (VG) parametric models. The BC model more accurately simulated the NaCl migration compared to the VG model, leading to its application in simulating GONP transport in the CF. However, the simulated BTCs for GONP showed a lag behind the measured data, especially at high ionic strengths. This discrepancy was attributed to the variable adsorption partition coefficient of GONP under different ionic conditions. During the experiment, GONP adsorption onto the porous media's surface altered the capillary dynamics, notably increasing capillary rise height, decreasing seepage velocity, and reducing GONP dispersion. Therefore, it is necessary to consider the adsorption capacity of the contaminants in order to accurately assess their transport within the vadose zone.

Global behavior of the solutions to nonlinear wave equations with combined power-type nonlinearities with variable coefficients

In this paper we study the initial boundary value problem for the nonlinear wave equation with combined power-type nonlinearities with variable coefficients. Existence and uniqueness of local weak solutions are proved. The global behavior of the solutions with non-positive and sub-critical energy is completely investigated. The threshold between global existence and finite time blow up is found. For super-critical energy, two new sufficient conditions guaranteeing blow up of the solutions for a finite time, are given. One of them is proved for an arbitrary sign of the scalar product of the initial data, while the other one is derived only for a positive sign.

Numerical simulation of the distributed-order time-space fractional Bloch-Torrey equation with variable coefficients

The purpose of this research is to establish the generalised fractional Bloch-Torrey equation for better simulating anomalous diffusion in heterogeneous biological tissues. The introduction of the distributed-order time fractional derivative allows for an improved interpretation of the complex diffusion behaviours with multi-scale effects. The use of variable coefficients in the model increases its applicability for describing the spatial heterogeneity evident in the cellular structures. The proposed distributed-order time and space fractional Bloch-Torrey equation is discretised in time and space by the L2-1σ formula and the finite element method, respectively. The stability and convergence analyses of these numerical methods are provided. To further improve the computational efficiency, a reduced-order extrapolation scheme is developed. We verify the effectiveness of the proposed methods by numerical examples. Moreover, the coupled fractional dynamic system solution behaviour is explored on a human brain-like domain divided into the white matter and grey matter regions. Compared with the model having constant coefficients, solution behaviours suggest that variable diffusion coefficients offer an effective way to differentiate the distinct diffusion phenomena evolving in different tissue micro-environments. Furthermore, to evaluate the impacts of the weight function in the distributed-order operator, we choose three types of beta distributions with the same mean but different values of the variance. The results indicate that the larger value of the variance leads to a more remarkable fluctuation and a slower decay of the transverse magnetisation. This generalised distributed-order fractional model may provide further insights into capturing anomalous diffusion in heterogeneous media.

Almost minimizers for the thin obstacle problem with variable coefficients

Almost minimizers for the thin obstacle problem with variable coefficients

Are input-output coefficients really fixed?

ABSTRACT It is standard practice to treat input-output (I-O) technical coefficients as fixed. The coefficients are fixed when σ , the elasticity of substitution, is zero, a restriction mostly not supported by 25 years of I-O tables for Australia. We offer a simple way of allowing for variable coefficients with a CES approach. Finally, using various values of σ in CGE simulations establishes that taking coefficients as fixed has misleading economic implications.

Some new lump molecules and hybrid molecular states of a (3 + 1)-dimensional generalized variable coefficient Kadomtsev–Petviashvili equation

In this paper, a (3 + 1)-dimensional generalized variable coefficient Kadomtsev–Petviashvili equation is investigated systematically, which can characterize evolution of the long water waves and small amplitude surface waves with the weak nonlinearity, weak dispersion, and weak perturbation in fluid mechanics. We investigate one lump and lump molecules obtained from one breather and breather molecules by a new degenerating breather method, respectively. In addition, the bound state of lump molecules and other localized waves is derived theoretically by velocity resonance. Considering the condition of variable coefficient, the several sets of interesting solutions having a complex structure are obtained, which include the type of parabolic, S-shaped, and periodic. The analysis method can also help us to study lump molecules existing in other integrable systems from a new perspective.

Research on Speed Cooperative Control Strategy for Rudderless Dual-PMSM Propulsion Ships

In the navigation of rudderless dual-motor propulsion ships, it is often necessary for the dual motor to run stably at different given speeds. However, the existing coupling strategies cannot achieve precise cooperative control of the dual motor under different given commands. Therefore, in this study, we proposed a variable error difference compensation-coupling control (VEDC-CC) strategy. Simultaneously, a coupling compensation coefficient determination method based on the self-error difference of the dual motor was introduced. The VEDC-CC utilizes the self-error difference of the dual motor as the input for the coupling compensation controller, which enables the mutual coupling of self-errors, and effectively improves the cooperative controlling performance under different given commands. Moreover, the employed variable compensation coefficient determination method enhances the disturbance coupling performance and the response speed of the system. In this study, we analyzed the different navigation conditions and dual-propeller load characteristics of a rudderless dual-motor propulsion ship. Taking a rudderless dual-motor propulsion ship designed based on the cruise ship “Feng Hua Xue Yue” as the object, the VEDC-CC strategy was validated through simulation tests, considering the propeller load characteristics and actual operating conditions. The experimental results demonstrated that the VEDC-CC strategy could meet the practical control needs of the target ship.

On the upscaling of the diffusion equation in a heterogeneous medium by the two different methods

Upscaling the flow equations appears in many studies related to diffusion, heat conductivity, and flows in porous media. Variable coefficients of the original fine-scale process description are substituted by averaged constant values. The different upscaling procedures have been suggested in the literature. The question arises, whether they result in the same or different upscaled models; moreover, whether the solutions of the coarse-scale equations provide a reasonably accurate description of the fine scale. In this work, we consider three sample 1D diffusion problems with periodic coefficients: diffusion with or without the external source, as well as diffusion with advection. These problems are upscaled by the two methods. The method of direct upscaling selects the averaged coefficients to provide the minimum difference between the solutions of the coarse-scale and fine-scale models. The method of continuous upscaling, developed previously, consists of the continuous averaging transformation between the distant scales. New expressions for the upscaled diffusion coefficients were derived for this case. It turns out that the direct upscaling results in multiple optimum parameters of the upscaled model. Meanwhile, continuous upscaling points at one of them. The coarse-scale approximation of a fine-scale solution may be unsatisfactory, even for the best choice of the upscaled coefficient. A numerical study demonstrates a nontrivial behavior of the diffusion coefficient under continuous upscaling, while it converges to an asymptotic value.

Integrated control strategy of BESS in primary frequency modulation considering SOC recovery

AbstractThis paper proposes a comprehensive control strategy for a battery energy storage system (BESS) participating in primary frequency modulation (FM) while considering the state of charge (SOC) recovery. On the one hand, when the frequency fluctuations are outside the dead zone of the FM, variable coefficient virtual sag control and variable coefficient virtual inertia control are coordinated for more accurate primary FM. Also, an assisted FM strategy is proposed, where the temporal alignment of assisted FM is determined by the regional segmentation of the frequency deviation and SOC, and it enables the BESS to recover SOC concurrently during its involvement in primary FM. On the other hand, when the frequency fluctuations are inside the dead zone of the FM, typical S‐type functions are deployed to dynamically establish the mutual constraints relationships of SOC recovery demand coefficient and frequency deviation. Subsequently, the recovery control coefficients are derived, and accordingly, a SOC recovery strategy is proposed. Finally, simulation results verify the efficacy of the proposed control strategy in the grid primary FM system and demonstrate that the strategy expedites the SOC recovery and concurrently diminishes the conventional unit output.

On the Dynamic Characteristics of Orthotropic Rectangular Plates under the Influence of Moving Distributed Masses and Resting on a Variable Elastic Pasternak Foundation

This work studies the dynamic characteristics of orthotropic rectangular plates under the influence of moving distributed masses and resting on a variable elastic Pasternak foundation. The governing equation is a fourth order partial differential equation with variable and singular coefficients. The solutions to the problem are obtained by transforming the fourth order partial differential equation for the problem to a set of coupled second order ordinary differential equations using the technique of Shadnam et al, then simplified using asymptotic method of Struble [11]. The closed form solution is analyzed, resonance conditions are obtained and the results are depicted graphically for both cases of moving distributed mass and moving distributed force

Boundary Value and Control Problems for Mass Transfer Equations with Variable Coefficients

Boundary Value and Control Problems for Mass Transfer Equations with Variable Coefficients

The impact of imputation methods on the performance of Phase I Hotelling’s T 2 control chart

The objective of this study was to evaluate the impact of three different methods of handling missing data on the performance of Phase I Hotelling’s T 2 multivariate control chart. Using a Monte Carlo simulation, we studied the average, median, and standard deviation of the run length performance of multivariate data imputed using mean substitution, regression imputation, and predictive mean matching at three different levels of missingness ( 1 % , 10 % , and 25 % ) and three levels of variable correlation coefficients (0.2, 0.4, and 0.8). We found that predictive mean matching has average run length performance results comparable to that of the complete in-control data set at all levels of missingness and variable correlation, while the performance of mean substitution was adversely affected by high levels of missingness and by strong variable correlation. Based on the simulation (multivariate normal data), we concluded that predictive mean matching is superior to both regression imputation and mean substitution as a method for imputing missing values for the analysis of Phase I Hotelling’s T 2 control chart. Two applications were presented using the Altenrhein wastewater treatment plant and Olive oil datasets.

Research on Variable Transmission Ratio Control Method to Improve Vehicle Handling Comfort Based on Steer-by-Wire System

The steer-by-wire system severs the mechanical link between the steering wheel and the steering gear. This configuration enhances the angular transmission characteristics. Entering the nonlinear region of the tires could result in a reduction in the vehicle’s steering gain. In order to improve the comfort of vehicle steering operation, we have developed a variable transmission ratio controller for the steer-by-wire (SBW) system. This controller utilizes information on the vehicle speed and steering wheel angle to generate a variable transmission ratio coefficient, thereby adjusting the steering ratio. We introduce a multi-objective comprehensive evaluation index that takes into account vehicle lateral deviation, driver steering burden, vehicle stability, and safety. To harmonize the transmission ratio weights of constant steering gain, we employ the coefficient of variation method. Ultimately, a fuzzy neural network is employed to craft a nonlinear controller. We conducted steady-state circular motion tests, double lane-change tests, and step input tests to validate the performance of the variable transmission ratio control. The results suggest that, in comparison to conventional fixed transmission ratio systems, the variable transmission ratio control within the steer-by-wire system significantly alleviates the driver’s operational burden while enhancing the vehicle’s handling stability and safety.

Coordinated control of path tracking and yaw stability for distributed drive electric vehicle based on AMPC and DYC

Maintaining both path-tracking accuracy and yaw stability of distributed drive electric vehicles (DDEVs) under various driving conditions presents a significant challenge in the field of vehicle control. To address this limitation, a coordinated control strategy that integrates adaptive model predictive control (AMPC) path-tracking control and direct yaw moment control (DYC) is proposed for DDEVs. The proposed strategy, inspired by a hierarchical framework, is coordinated by the upper layer of path-tracking control and the lower layer of direct yaw moment control. Based on the linear time-varying model predictive control (LTV MPC) algorithm, the effects of prediction horizon and weight coefficients on the path-tracking accuracy and yaw stability of the vehicle are compared and analyzed first. According to the aforementioned analysis, an AMPC path-tracking controller with variable prediction horizon and weight coefficients is designed considering the change of vehicle speed in the upper layer. The lower layer involves DYC based on the linear quadratic regulator (LQR) technique. Specifically, the intervention rule of DYC is determined by the threshold of the yaw rate error and the phase diagram of the sideslip angle. Extensive simulation experiments are conducted to evaluate the proposed coordinated control strategy under different driving conditions. The results show that, under variable speed and low adhesion conditions, the vehicle yaw stability and path-tracking accuracy have been improved by 21.58% and 14.43%, respectively, compared to AMPC. Similarly, under high speed and low adhesion conditions, the vehicle yaw stability and path-tracking accuracy have been improved by 44.30% and 14.25%, respectively, compared to the coordination of LTV MPC and DYC. The results indicate that the proposed adaptive path-tracking controller is effective across different speeds. Furthermore, the proposed coordinated control strategy successfully enhances the vehicle stability while maintaining good path-tracking accuracy even under extreme conditions.

Parallel Physics-Informed Neural Networks Method with Regularization Strategies for the Forward-Inverse Problems of the Variable Coefficient Modified KdV Equation

Parallel Physics-Informed Neural Networks Method with Regularization Strategies for the Forward-Inverse Problems of the Variable Coefficient Modified KdV Equation