Two-dimensional Discretization Research Articles

Chaotic Dynamics of a Two-Dimensional Spatiotemporal Discrete System with Delayed Perturbations

This paper is mainly concerned with the existence of chaos in a two-dimensional spatiotemporal discrete system with delayed perturbations. By applying the snap-back repeller theory, two criteria of chaos in the sense of both Li–Yorke and Devaney are established. Moreover, the lower bounds of the parameters that make the system chaotic are obtained. Finally, an example and its computer simulations of the dynamical behavior, spatiotemporal graphs, and the trends of the largest Lyapunov exponents with respect to the initial values and parameters are given to demonstrate the correctness of the theoretical results.

The Extended Diamond Difference - Constant Nodal Method with Decoupled Cell Iteration Scheme in Two-Dimensional Discrete Ordinate Transport Problems

Analytical solutions for the Boltzmann transport equation are typically limited to specific scenarios, prompting the research transport community to extensively explore numerical approaches with special attention to the discrete ordinates (SN) formulation. The spectral nodal methods are widely recognized for their capability to preserve the analytical solution of the SN mathematical model. However, these formulations exhibit notable complexity with high-order polynomial approximations for the transverse leakage terms, motivating researchers to pursue simpler hybrid approaches by partially preserving the analytical solution. Both spectral nodal and hybrid spectral nodal methods introduce nonstandard auxiliary equations not amenable to standard iterative schemes due to a highly coupling in angular directions. Following these ideas, we present a finite element – spectral nodal hybrid method referred to as Extended Diamond Difference-Constant Nodal (EDD-CN) method for two-dimensional fixed-source discrete ordinate transport problems with constant nodal approximation for the leakage terms. The proposed method seeks to preserve eigenfunctions associated with the dominant pair of eigenvalues in the transverse integrated SN equations. In addition, we propose a novel iterative scheme called Decoupled Cell Iteration (DCI) scheme which decouples the angular directions in the nonstandard auxiliary equations and follows conventional iteration transport sweeps for solving SN problems.

Improve the thermal performance of precooler by enhancing the utilization of chemical heat sink: Refined design method and characteristic analysis

The chemical heat sink of the coolant plays a crucial role in enhancing the performance of the precooler, which affects the overall performance of the air-breathing precooled engines. The conventional thermal design method for precoolers cannot accurately evaluate the chemical heat sink of coolants. To address this crucial issue, this paper proposes a refined model that considers the chemical reaction processes of n-Decane, based on a two-dimensional discretization unit model and the plug flow reactor model incorporating molecular-level chemical reaction mechanisms. The validation of this model is conducted by comparing it with publicly accessible data and computational fluid dynamics simulations, thereby demonstrating its excellent accuracy. Subsequently, an analysis is conducted on the factors influencing the chemical heat sink release of the coolant within the precooler, and enhanced design criteria for optimizing chemical heat sink utilization are proposed. Building upon these considerations, the performance of a precooler with a design point of Mach number 5 under off-design conditions was analyzed. The findings indicate that employing rational design methods can enhance the chemical heat sink efficiency of endothermic fuels by 8.0%, along with a concurrent 10.3% increase in the total heat sink. This underscores the significance of considering the impact of the chemical heat sink of endothermic fuels in the design of precoolers. In the case presented in this paper, the contribution of the chemical heat sink to the total heat sink within the n-Decane precooler could reach 35.3%.

Handwritten Character Recognition using Deep Learning Algorithm with Machine Learning Classifier

Handwritten character recognition is a problem that has been worked on for many mainstream languages. Handwritten letter recognition has been proven to achieve promising results. Several studies using deep learning models have been conducted to achieve better accuracies. In this paper, the authors conducted two experiments on the EMNIST Letters dataset: Wavemix-Lite and CoAtNet. The Wavemix-Lite model uses Two-Dimensional Discrete Wavelet Transform Level 1 to reduce the parameters and speed up the runtime. The CoAtNet is a combined model of CNN and Visual Transformer where the image is broken down into fixed-size patches. The feature extraction part of the model is used to embed the input image into a feature vector. From those two models, the authors hooked the value of the features of the Global Average Pool layer using EMNIST Letters data. The features hooked from the training results of the two models, such as SVM, Random Forest, and XGBoost models, were used to train the machine learning classifier. The experiments conducted by the authors show that the best machine-learning model is the Random Forest, with 96.03% accuracy using the Wavemix-Lite model and 97.90% accuracy using the CoAtNet model. These results showcased the benefit of using a machine learning model for classifying image features that are extracted using a deep learning model.

Harmonizing Cross View Image Transformation Through Local and Global Insights- A Review

Hyperspectral and multispectral information processing systems and technologies have demonstrated their value in enhancing agricultural productivity and practices by providing farmers and crop managers with valuable information on factors influencing crop condition and growth. These technologies play a crucial role in various agricultural applications, such as crop management, crop yield forecasting, crop disease detection, and monitoring soil, water, and land usage. To enhance the process of agriculture through effective crop management and assistance to farmer, using an advanced image transformation techniques the study delves into the exploration of techniques for harmonizing cross-view image transformation, with a focus on integrating both local and global insights. Further the study proposes the application of the 2D-DWT (Two-Dimensional Discrete Wavelet Transform) technique for image data preprocessing and the Extreme Learning Machine (ELM) algorithm for image classification in the context of hyperspectral images. Hyperspectral information sensing allows for the coverage of the electromagnetic spectrum in a single acquisition with several hundred spectral bands, resulting in a data cube containing significant spectral and spatial information ELM is particularly effective for classification problems because of its quick training time and capacity to operate with hidden nodes whose parameters are randomly assigned rather than modified iteratively. This comprehensive review aims to provide valuable insights and a critical analysis of the suggested method, shedding light on its potential contributions to the advancement of image transformation techniques for agricultural development.

Modeling the Anisotropic Transport Behavior for Axisymmetric 2D Electrode Particles Using a Modified Pseudo-2D Model Framework

This presentation extends widely used pseudo-2D (P2D) Li-ion battery models to include electrode particles that have strong crystal-scale anisotropies and possibly non-spherical shapes. Typical P2D models predict Li transport with representative spherical electrode particles by solving a one-dimensional diffusion equation in spherical coordinates. However, battery electrodes (graphite, NMC, etc.) may have strong crystalline anisotropies, leading to transport and mechanical properties that vary by orders of magnitude depending on crystallographic orientations. The anisotropic transport can cause large concentration gradients, possibly leading to particle fracture associated with concentration-induced stress. The remainder of the P2D model, such as Li-ion transport within the electrolyte solvent, remains essentially unchanged. The charge-transfer boundary conditions at the particle surfaces do need to accommodate the crystallographic anisotropy.The present model discretizes representative electrode particles in a two-dimensional finite-volume axisymmetric geometry, accounting for the anisotropic behaviors. The computational burden is greater than it is for the one-dimensional spherical solution. However, the P2D model with two-dimensional axisymmetric particle discretization still runs in only a few minutes on a typical personal computer.The model shows how electrode-particle anisotropy affects the battery performance, such as charge-discharge characteristics and electrochemical impedance spectroscopy. The model also predicts the concentration gradients and correlated stress for various axisymmetric, non-spherical particles.

Techniques to embed channels in finite element shallow water equation models

Two new and novel techniques to efficiently represent channels in large-scale finite element shallow water equation models are introduced. The first enables the discontinuous representation of bathymetric depth by allowing nodes along a discontinuity to have two land surface elevations, one representing the bottom of the discontinuity and the other the top of the discontinuity. Elemental integration proceeds using the nodal depth corresponding to whether the element is on the upper or lower side of the discontinuity and enables efficient treatment of steep sided features in a two-dimensional horizontal discretization. The second technique consolidates nodal equations at paired nodes across a channel. Doing this eliminates cross-channel variability and consequently eliminates the across-channel Courant-Friedrichs-Lewy stability constraint on the model time step. Together the two techniques allow channels of various sizes as well as other instances of steep topography to be embedded seamlessly, efficiently and in a fully coupled manner within an otherwise two-dimensional horizontal spatial discretization. This new capability is especially useful in the interface zone subject to both coastal and hydrological influences as it effectively captures bi-directional channelized flow, bi-directional flow between the channel and the floodplain, and coupled flow when the channel and floodplain are fully submerged. This paper describes the methodological development and presents a series of tests that verify the performance of this novel solution approach.

An Alzheimer’s disease classification method using fusion of features from brain Magnetic Resonance Image transforms and deep convolutional networks

Alzheimer’s is a progressive and irreversible brain degenerative disorder, and presenting an accurate early-stage diagnosis tool is vital for preventing disease progression. The previous research considered the features only from the brain Magnetic Resonance Image (MRI) or its transforms. This study presents two efficient fusion schemes to combine the deep features obtained from the brain MR image and its transforms. We consider the Orthogonal Ripplet II Transform (ORT-II), the Two-Dimensional Discrete Orthonormal Stockwell Transform (2D DOST), and the original brain MR image. We present the decision- and feature-level fusion schemes to combine the information of different inputs. In decision-level fusion, Convolutional Neural Networks (CNNs) separately classify each matrix, where all CNNs have the same structure, and the majority role makes the final decision. In feature-level fusion, the same CNNs extract each matrix’s features and then concatenate them to construct the feature vector. Since this vector contains many features, we employ Neighborhood Component Analysis (NCA) to select the more relevant features for classification to determine the level of dementia. We consider the different CNN models and classifiers to classify the brain MR images in three two-class scenarios. The results demonstrate that EfficientNet-B7 with an Artificial Neural Network (ANN) provides the highest accuracy. Also, feature-level fusion reaches a higher accuracy compared to the decision-level one.

Numerical approaches in simulating Trichel pulse characteristics in point-plane configuration

In this work, a detailed comparison is made of a few different approaches to numerical modeling of non-equilibrium gas discharge plasmas in dry ambient air at atmospheric conditions, leading to Trichel pulse discharge. Simulation models are based on a two-dimensional axisymmetric finite element discretization of point-plane geometry. The negative corona discharge and the hydrodynamic approximation for generic ionic species (electrons, positive and negative ions) are used. The models account for the drift, diffusion, and reactions of the species. They comprise continuity equations coupled to Poisson’s equation for the electric field. Three different formulations were used to specify the ionic reaction rate coefficients. In the first one, the reaction coefficients are approximated by the analytical expressions as a function of the electric field intensity. Two others extract the reaction coefficients from the solution of the Boltzmann equation as a function of the reduced electric field or the electron energy. The effect of gas flow and heating on the pulse characteristics is also investigated. The accuracy of the models has been validated by comparing them with the experimental data.

A corrected finite-difference scheme for the flexure equation with abrupt changes in coefficient

Abstract. The fourth-order differential equation describing elastic flexure of the lithosphere is one of the cornerstones of geodynamics that is key to understanding topography, gravity, glacial isostatic rebound, foreland basin evolution, and a host of other phenomena. Despite being fully formulated in the 1940s, a number of significant issues concerning the basic equation have remained overlooked to this day. We first explain the different fundamental forms the equation can take and their difference in meaning and solution procedures. We then show how numerical solutions to flexure problems as they are currently formulated are in general potentially unreliable in an unpredictable manner for cases in which the coefficient of rigidity varies in space due to variations of the elastic thickness parameter. This is due to fundamental issues related to the numerical discretisation scheme employed. We demonstrate an alternative discretisation that is stable and accurate across the broadest conceivable range of conditions and variations of elastic thickness, and we show how such a scheme can simulate conditions up to and including a completely broken lithosphere more usually modelled as an end-loaded, single, continuous plate. Importantly, our scheme will allow breaks in plate interiors, allowing, for instance, the creation of separate blocks of lithosphere which can also share the support of loads. The scheme we use has been known for many years but remains rarely applied or discussed. We show that it is generally the most suitable finite-difference discretisation of fourth-order, elliptic equations of the kind describing many phenomena in elasticity, including the problem of bending of elastic beams. We compare the earlier discretisation scheme to the new one in one-dimensional form and also give the two-dimensional discretisation based on the new scheme. We also describe a general issue concerning the numerical stability of any second-order finite-difference discretisation of a fourth-order differential equation like that describing flexure wherein contrasting magnitudes of coefficients of different summed terms lead to round-off problems, which in turn destroy matrix positivity. We explain the use of 128 bit floating-point storage for variables to mitigate this issue.

Development of a cross-sectional finite element for the analysis of thin-walled composite beams like wind turbine blades

A method for structural analysis of thin-walled composite beams like wind turbine blades is presented. This method is based on the Nonhomogeneous Anisotropic Beam Section Analysis (NABSA) which consists in discretizing the beam cross section using finite elements. The proposed implementation uses 3-node line cross-sectional finite elements with nodes having rotational degrees of freedom to describe the cross-sectional warping displacements. Solutions obtained using this approach were verified against the corresponding analytical or numerical solutions. Agreement was very good to excellent for the computation of cross-sectional properties and distribution of stresses, strains and warping displacements for a broad range of possible composite beam behaviors including geometric and material couplings, open sections, multicell sections, and arbitrary laminates. For thin-walled layered structures, the proposed method provides models with fewer degrees of freedom than equivalent models based on a two-dimensional discretization of cross sections using triangular or quadrilateral elements such as conventional NABSA or VABS which suggests that computation time could be reduced.

Ballot Theorems for the Two-Dimensional Discrete Gaussian Free Field

We provide uniform bounds and asymptotics for the probability that a two-dimensional discrete Gaussian free field on an annulus-like domain and with Dirichlet boundary conditions, stays negative as the ratio of the radii of the outer and the inner boundary tends to infinity. Such estimates are often needed in the study of extreme values of the discrete Gaussian free field on planar domains.

Variation Inequality for the Two-Dimensional Discrete Hardy–Littlewood Maximal Function

Variation Inequality for the Two-Dimensional Discrete Hardy–Littlewood Maximal Function

A Single-Layer Dual-Mesh Boundary Element Method for Multiscale Electromagnetic Modeling of Penetrable Objects in Layered Media

A surface integral representation of Maxwell's equations allows the efficient electromagnetic (EM) modeling of three-dimensional structures with a two-dimensional discretization, via the boundary element method (BEM). However, existing BEM formulations either lead to a poorly conditioned system matrix for multiscale problems, or are computationally expensive for objects embedded in layered substrates. This article presents a new BEM formulation which leverages the surface equivalence principle and Buffa-Christiansen basis functions defined on a dual mesh, to obtain a well-conditioned system matrix suitable for multiscale EM modeling. Unlike existing methods involving dual meshes, the proposed formulation avoids the double-layer potential operator for the surrounding medium, which may be a stratified substrate requiring the use of an advanced Green's function. This feature greatly alleviates the computational expense associated with the use of Buffa-Christiansen functions. Numerical examples drawn from several applications, including remote sensing, chip-level EM analysis, and metasurface modeling, demonstrate speed-ups ranging from 3x to 7x compared to state-of-the-art formulations.

Different dimensional fractional-order discrete chaotic systems based on the Caputo h-difference discrete operator: dynamics, control, and synchronization

The paper investigates control and synchronization of fractional-order maps described by the Caputo h-difference operator. At first, two new fractional maps are introduced, i.e., the Two-Dimensional Fractional-order Lorenz Discrete System (2D-FoLDS) and Three-Dimensional Fractional-order Wang Discrete System (3D-FoWDS). Then, some novel theorems based on the Lyapunov approach are proved, with the aim of controlling and synchronizing the map dynamics. In particular, a new hybrid scheme is proposed, which enables synchronization to be achieved between a master system based on a 2D-FoLDS and a slave system based on a 3D-FoWDS. Simulation results are reported to highlight the effectiveness of the conceived approach.

A novel GBT-formulation for thin-walled FGM-beam-structures based on a reference beam problem

This paper proposes an efficient generalized beam theory (GBT) formulation, which accounts for cross-sectional deformations in slender prismatic structures. It was shown by the authors in a recent publication [1] that in-plane distortional deformations and accompanied out of plane warping deformations of the cross-section influence the accuracy of results in beam dynamics especially if thin-walled cross-sections are applied. The GBT formulation proposed in [1] overcomes the inaccuracies of classical beam mechanics, however, requires a two-dimensional plane discretization of the cross-section. The computational complexity can be reduced vastly, if thin-walled cross-sections can be discretized with one-dimensional elements. Consequently, this paper discusses an approach with a line mesh discretizing the cross-section and having six degrees of freedom at each node. The membrane part consists of massless micro-polar rotations (drilling rotations) and can be derived independently from the bending part, where a shear elastic formulation is selected.

Prediction of temperature and crystal growth evolution during 3D printing of polymeric materials via extrusion

Material Extrusion (ME), which is a type of Additive Manufacturing (AM), has become widely popular in the manufacturing world. However, the evolution of a material's temperature in this relatively new manufacturing method, which plays an important role in the polymer crystallinity, is not yet well researched. Next to voids, interlayer adhesion and surface roughness, the degree of crystallinity strongly determines the quality of printed products. Hence, a thorough and deep understanding of crystallinity in ME is essential for the improvement of the parts printed. In this paper, a primary generic model is proposed to predict the temperature evolution and crystal growth of printed polymer materials. The temperature evolution was developed based on the two-dimensional domain discretisation method and the crystal growth was simulated via the Hoffman-Lauritzen theory. In the simulation, key parameters have been taken into consideration, such as the printing speed, thermal convection coefficient, thermal contact conductance with the platform, nozzle diameter and latent heat in crystallisation. Then, a single-line printing scenario was tested to verify the accuracy of the model. A comparison of the model predictions with the experimental results showed only of up to 3.8 °C deviations which is a 2% of maximum percentage mismatch from the full scale reading.

Quantifying Stability and Chaoticity of One-dimensional and Two-dimensional Discrete Dynamical Systems using Stability Analysis and Lyapunov Exponents

Discrete dynamical system is a system that evolve dynamically with discrete time. In this paper, we consider two discrete systems which exhibit chaotic behaviour. We show that the chaoticity of a system is depend on the values of parameter in the system. The objective of this paper is to investigate both stability and chaoticity of the systems using stability analysis and Lyapunov exponent, respectively. The results show that there is only one fixed point for the one-dimensional system, while for the two-dimensional system, there exist four possible fixed points. We have proved the stability conditions for each fixed point obtained. The Lyapunov results show that the one-dimensional system is stable when r<π/2 and chaotic when r>π/2. Whereas for two-dimensional systems considered, the system is chaotic for the whole range of parameter α∈(0,1]. This study shows that it is significant to consider various values of parameter to study the stability of a dynamical systems in particular to control the chaos in a system.

Two-dimensional discrete feature based spatial attention CapsNet For sEMG signal recognition

Deep learning frameworks(such as deep convolutional networks) require data to have a regular shape. However, discrete features extracted from heterogeneous data cannot be collected in a regular shape to convolute. In this article, a Two-Dimensional Discrete Feature Based Spatial Attention CapsNet(TDACAPS) is proposed to convert one-dimensional discrete features into two-dimensional structured data through Cartesian Product for surface electromyogram(sEMG) signal recognition. sEMG signal varies from person to person is the main signal source of prosthetic control. Our model transforms multi-angle discrete features into structured data to find the inherent law of sEMG signal. Due to uneven information distribution of structured data, this model combines capsule network with attention mechanism to place emphasis on abundant information regions and reduce ancillary information loss. Extensive experiments show our model yields an improvement for sEMG signal recognition of almost 3% than capsule network and other neural networks under different conditions. Our attention mechanism that employs overlapping pooling to search feature map weight is preferable to the squeeze-and-excitation module, convolutional block attention module and others. Moreover, we validate that our model has great expansibility on Wine Quality Dataset and Breast Cancer Wisconsin.

Fault Diagnosability Analysis of Two-Dimensional Linear Discrete Systems

In this article, a systematic fault diagnosability evaluation, including fault detectability and isolability, is established in a quantitative manner for two-dimensional systems. With ingenious data formulation, a parity relation of two-dimensional systems is first established, then the Kullback-Leibler divergence is employed as the key measure for the diagnosability analysis based on the established parity relation. The basic idea is to quantify the distribution differences among each fault scenario-related system dynamic behavior. Explicit necessary and sufficient condition for fault diagnosability is further derived based on the appropriately introduced definitions corresponding to the two directions evolving system properties. Finally, the effectiveness of the proposed method is verified by two examples.