Discrete 2D Research Articles

Architected Design and Fabrication of Soft Mechanical Metamaterials

This study proposes a systematic design approach to 3D print soft mechanical metamaterials by tuning material flow behavior and using optimal toolpaths and print parameters. Planar printing tool paths, utilized in layer‐by‐layer printing such as fused deposition modeling (FDM) and prevalently used in most slicing algorithms, severely limit the realization of complex topologies required in metamaterials. Utilizing parametric design principles, the proposed approach employs discrete and continuous 3D freeform tool paths for supported and unsupported direct ink writing (DIW) of elastomeric structures. The resulting textured, soft topologies significantly enhance the performance of various mechanisms in soft robotics and impact energy‐absorbing wearable devices. Various bioinspired structures such as cilia, webs, leaf‐like structures, and lattices are explored using extrusion‐based silicone 3D printing, achieving features with high aspect ratios (L/D ≤ 12) without support. These complex structures are challenging to 3D print using conventional planar toolpaths. To demonstrate the enhanced functionalities enabled by these new topologies, cilia arrays added to suction cups increased the pull‐off forces by 18%. Additionally, 3D‐printed elastomeric lattice slabs used as energy‐absorbing structures reduced the maximum impact peak forces by 85%. Further functionalities, including magnetic, thermochromic, and signal transmission properties, are also showcased using functional soft mechanical metamaterials.

An Effective Alternative to Current Mathematics

If you don't understand mathematics, ask yourself if I'm right, because others don't understand mathematics either. By effective alternative to current mathematics, we mean working in a more complete mathematical space than the classical 3D+t variety which is inadequate for generating well-defined definitions and hypotheses as well as its limited ability to solve time-dependent partial differential equations. The current classical discrete 3D+t space PDE, in which time is an external controller and not integrated into the 3D geometric space, cannot be integrated digitally. This space is logically incomplete and misleading in the production of definitions and hypotheses as well as in the resolution itself of time- dependent PDEs. It is no wonder that these definitions/assumptions are confusing and result in weak or intractable mathematics, leading to all kinds of misunderstandings, from horrible notations to undisciplined length of theorems containing a considerable amount of black magic and ending with a gray nature of the mathematical result obtained. In this article, we present some of the most inaccurate assumptions and definitions in current classical mathematics that arise from using the 3D+t manifold space to specify initial conditions, boundary conditions, and the source/sink term. Fortunately, these inaccurate assumptions that start with inadequate space for boundary conditions, initial conditions, and source/sink term can be spotted and analyzed via 4D unitary numerical statistical theory called Cairo techniques in the format of transition chains of matrix B to complete what is missing. In other words, we present how to spot some of the worst mathematical conclusions of classical 3D geometry plus t as an external control numerical space, and then show how to correct them via the 4D unit space which is the subject of this article.

STABILIZATION OF DISCRETE 2D LINEAR SYSTEMS IN FORNASINI-MARCHESINI MODEL

This paper is concerned with the stabilization problem of discrete 2D linear systems described by the second Fornasini-Marchesini model. A necessary and sufficient condition involving the characteristic polynomial is first quoted by which the unforced system is structurally or exponentially stable. On the basis of the derived stability condition, a tractable condition is formulated in the form of linear matrix inequality for obtaining the controller gain of a desired stabilizing state-feedback controller.

Psychometric evaluation of high-resolution electrotactile interface for conveying 3D spatial information

This study presents a detailed psychometric evaluation of a novel high-resolution electrotactile interface, which is developed to provide users with 3D spatial information and facilitate enhanced interaction with a Supernumerary Robotic Limb (SRL). The research introduces a novel electrotactile system that employs a multi-pad electrode configuration on the thigh, aimed at delivering intuitive feedback to users about the position of the SRL in a three-dimensional space. The interface's effectiveness was assessed through a series of psychometric tests, including static spatial discrimination, target-reaching with spatial feedback, frequency discrimination, and combined spatial/frequency modulation. The key findings demonstrate that participants could differentiate between 30 electrode pads with an average success rate of 62.7% when they were activated statically, while in the dynamic target-reaching task, the success rate increased to 88.1%. Frequency discrimination tests further revealed that four frequency levels could be distinguished with 86.0% success rate in single-pad feedback while the performance decreased to 74.3% in multi-pad distributed feedback. Finaly, in the closed-loop test with mixed spatial and frequency modulation, participants achieved an overall success rate of 78.8% in target reaching across 10 × 4 discrete 2D space. These results highlight the interface’s capability to transmit high-resolution spatial information through electrotactile feedback, offering a foundation for future applications in tactile-based navigation and control systems.

2D Discrete Element Simulation of Electrode Structural Evolutions in Li‐Ion Battery During Drying and Calendering

Drying and calendering are critical steps in the manufacture of electrodes for lithium‐ion battery that affect their mechanical and electrochemical properties. A 2D representative volume element (RVE) model, including active material and carbon binder domain particles of different shapes and sizes, is developed. The evolution of the RVE structure is simulated using the discrete element method, providing insight into changes in velocity, coordination number, porosity, pore size distribution, tortuosity, and stress. Based on this analysis, a three‐step drying scheme is proposed in accordance with the experimental drying results. In addition, the calendering process significantly improves the mechanical integrity and electronic conductivity of the electrode. Through simulations and experimental observations of changes in surface morphology and porosity, an optimal compression ratio of about 20% is determined for the electrode.

A Wavelet Shrinkage Mixed with a Single-level 2D Discrete Wavelet Transform for Image Denoising

The single-level 2D discrete wavelet transform method is a powerful technique for effectively removing Gaussian noise from natural images. Its effectiveness is attributed to its ability to capture a signal's energy at low energy conversion values, allowing for efficient noise reduction while preserving essential image details. The wavelet noise reduction method mitigates the noise present in the waveform coefficients produced by the discrete wavelet transform. In this study, three different wavelet families—Daubechies (db7), Coiflets (coif5), and Fejér-Korovkin (fk4)—were evaluated for their noise removal capabilities using the Bayes shrink method. This approach was applied to a set of images, and the performance was analyzed using the Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR) metrics. Our results demonstrated that among the wavelet families tested, the Fejér-Korovkin (fk4) wavelet consistently outperformed the others. The fk4 wavelet family yielded the lowest MSE values, indicating minimal reconstruction error, and the highest PSNR values, reflecting superior noise suppression and better image quality across all tested images. These findings suggest that the fk4 wavelet family, when combined with the Bayes shrink method, provides a robust framework for Gaussian noise reduction in natural images. The comparative analysis highlights the importance of selecting appropriate wavelet families to optimize noise reduction performance, paving the way for further research and potential improvements in image denoising techniques.

2D Discrete Yang–Mills Equations on the Torus

In this paper, we introduce a discretization scheme for the Yang–Mills equations in the two-dimensional case using a framework based on discrete exterior calculus. Within this framework, we define discrete versions of the exterior covariant derivative operator and its adjoint, which capture essential geometric features similar to their continuous counterparts. Our focus is on discrete models defined on a combinatorial torus, where the discrete Yang–Mills equations are presented in the form of both a system of difference equations and a matrix form.

Contribution to micromechanical modeling of the shear wave propagation in a sand deposit

The object of study is the vertical wave propagation in a sand deposit. This paper is aimed at analyzing the vertical wave propagation in a sand deposit through micromechanical modeling that inherently takes account of intergranular slips during deformation. Such a problem, which is part of the general framework of wave propagation in the soil, has long been analyzed using continuum models based on approximate behavior laws. For this purpose, a 2D Discrete Element Method (DEM) model is developed. The DEM model is based on molecular dynamics with the use of circular shaped elements. The intergranular normal forces at contacts are calculated through a linear viscoelastic law while the tangential forces are calculated through a perfectly plastic viscoelastic model. A model of rolling friction is incorporated in order to account for the damping of the grains rolling motion. Different boundary conditions of the profile have been implemented; a bedrock at the base, a free surface at the top and periodic boundaries in the horizontal direction. The sand deposit is subjected to a harmonic excitation at the base. Using this model, the fundamental and resonance frequencies of the deposit are first determined. The former is determined from the low-amplitude free vibration and the latter by performing a variable-frequency excitation test. It is noted that there is a significant gap between the two frequencies, this gap could be attributed to the degradation of the soil shear modulus in the vicinity of the resonance. Such degradation is well proven in classical soil dynamics. The effects of deposit height and confinement on resonance frequency and free-surface dynamic amplification factor are then investigated. The obtained results highlighted that the resonance frequency is inversely proportional to the deposit’s thickness whereas the dynamic amplification factor Rd increases with the deposit’s thickness. In the other hand, when the confinement increases the deposit becomes stiffer, which results in reducing the amplification. Such result is in accordance with theoretical knowledge which states that the most rigid profiles such as rocks do not amplify seismic movement.

Quasinormal modes and coupled mode theory of 1D metal‐dielectric photonic crystals

AbstractMetal‐dielectric photonic crystals (MDPCs) are a class of photonic structure formed through the coupling of metallic microcavities. Interaction of the cavities results in hybridization of the underlying microcavity states to form photonic bands defined by the crystal periodicity, metal‐to‐dielectric ratio, and number of unit cells. In this study, we provide analytical solutions for the resonant states of discrete 1D MDPCs with an arbitrary number of cavities using quasinormal mode (QNM) theory. Our results show that the QNM solutions closely match the predictions of coupled mode theory (CMT) in the tight‐binding regime and diverge as the metal thickness vanishes. We apply the QNM results to the CMT model to find analytical expressions for the inter‐cavity coupling coefficients. Additionally, we analyze the quality factor, transmission, and absorption of both ideal and lossy MDPCs, revealing important trends in bulk optical properties.

Metaheuristic Optimization of Functionally Graded 2D and 3D Discrete Structures Using the Red Fox Algorithm

Metaheuristic Optimization of Functionally Graded 2D and 3D Discrete Structures Using the Red Fox Algorithm

Image Segmentation Using 2D Discrete Wavelet Transform for Medical Image

Image Segmentation Using 2D Discrete Wavelet Transform for Medical Image

Partial differential equation modelling of the solid–liquid phase process in iron sintering and interface control with backstepping method

Due to the highly continuous temporal and spatial distribution characteristics of the sintering process, the solid–liquid phase interface during sintering is uneven and discontinuous, which seriously affects the quality of sintered ore. In order to achieve real-time and effective control of the solid–liquid interface in the sintering system, this study proposes a spatiotemporally distributed partial differential equation (PDE) thermal system with Stefan interface boundaries. Slice the 3D sintering mixing space and establish a discrete 1D space multi-distributed system to achieve distributed control of the sintering thermal system. Based on this system, a distributed boundary controller is designed using the backstepping method to drive the solid–liquid phase interface to the desired position. This method fully adapts to the nonlinear problems of the sintering thermal system and can quickly respond to system changes. Based on real data from the sintering site, a multi-physics simulation model was designed to verify the correctness of the theoretical research. Comparing the simulation results with the actual sintering state shows that this control method can effectively control the position of the sintering solid–liquid interface, which plays a vital role in improving the quality of sinter and sintering efficiency.

Automatic analysis of pottery sherds based on structure from motion scanning: The case of the Phoenician carinated-shoulder amphorae from Tell el-Burak (Lebanon)

Over the last few years, significant interest has been addressed in developing computer-based methods to document and analyze fragments of ceramics sherds in archaeology. This is because traditional manual processes do not allow for an objective, repeatable, and reproducible analysis of the large quantities of material needed to fully understand and explain human practices in various cultural contexts, such as the economy, daily life, and the material expression of religious beliefs.In that context, this paper proposes a fully digital methodology resulting from the constitution of an international research group coming from different scientific backgrounds: archaeologists with specific skills and experience in fast 3D geometry acquisition methods and researchers who developed and published the only available computer-based process for recognizing the geometric and morphological sherds features analyzed by archaeologists. The proposed methodology consists of two main parts: 1. 3D acquisition of sherds with the construction of the discrete 3D manifold model based on the Structure for Motion technologies; 2. recognition, segmentation, and dimensional characterization of morphological and geometrical features based on the codification and algorithmic implementation of the knowledge used by the archeologists in the traditional method. The method was applied to analyze a set of 133 sherds excavated at Tell el-Burak (Lebanon) to obtain, through the analysis of the namely Phoenician carinated-shoulder amphorae, new insights into the economic organization of the Phoenician homeland. The method demonstrated the potential for objectively, repeatedly, and reproducibly analyzing large quantities of sherds. Furthermore, it allowed studying sherds by generating new high-level knowledge from those acquired from 3D models; in particular, this paper introduces new morphological features that help the archaeologist classify fragments from an analysis of the rim's shape.

New Physical Insight into High-Spin Physics: Double Helix Level Scheme A nuclear structure data approach

To present a new nuclear structure data concept of a level scheme, we redesign the high-spin level scheme of 171Yb nucleus. We start with the description of the property of repeatability of the Eγ data based on which Eγ can be parametrized by the 2c(2I + k − 1) formula with I the nuclear spin and k an integer band offset different from band to band. This describes a beam of parallel and equidistant lines whose 2c average slope and k numbers can be determined by a least-squares fit to the experimental bands, which characterize the average rotational behavior of the experimental bands. Based on repeatability the level scheme of 171Yb nucleus can be shaped as a discrete 3D double helix structure, with the experimental bands superimposed as decay paths. Double helix suggests that semiclassically nuclear rotation can be seen as a vortex motion.

A new convolutional neural network based on combination of circlets and wavelets for macular OCT classification

Artificial intelligence (AI) algorithms, encompassing machine learning and deep learning, can assist ophthalmologists in early detection of various ocular abnormalities through the analysis of retinal optical coherence tomography (OCT) images. Despite considerable progress in these algorithms, several limitations persist in medical imaging fields, where a lack of data is a common issue. Accordingly, specific image processing techniques, such as time–frequency transforms, can be employed in conjunction with AI algorithms to enhance diagnostic accuracy. This research investigates the influence of non-data-adaptive time–frequency transforms, specifically X-lets, on the classification of OCT B-scans. For this purpose, each B-scan was transformed using every considered X-let individually, and all the sub-bands were utilized as the input for a designed 2D Convolutional Neural Network (CNN) to extract optimal features, which were subsequently fed to the classifiers. Evaluating per-class accuracy shows that the use of the 2D Discrete Wavelet Transform (2D-DWT) yields superior outcomes for normal cases, whereas the circlet transform outperforms other X-lets for abnormal cases characterized by circles in their retinal structure (due to the accumulation of fluid). As a result, we propose a novel transform named CircWave by concatenating all sub-bands from the 2D-DWT and the circlet transform. The objective is to enhance the per-class accuracy of both normal and abnormal cases simultaneously. Our findings show that classification results based on the CircWave transform outperform those derived from original images or any individual transform. Furthermore, Grad-CAM class activation visualization for B-scans reconstructed from CircWave sub-bands highlights a greater emphasis on circular formations in abnormal cases and straight lines in normal cases, in contrast to the focus on irrelevant regions in original B-scans. To assess the generalizability of our method, we applied it to another dataset obtained from a different imaging system. We achieved promising accuracies of 94.5% and 90% for the first and second datasets, respectively, which are comparable with results from previous studies. The proposed CNN based on CircWave sub-bands (i.e. CircWaveNet) not only produces superior outcomes but also offers more interpretable results with a heightened focus on features crucial for ophthalmologists.

Exploiting Newly Designed Fractional-Order 3D Lorenz Chaotic System and 2D Discrete Polynomial Hyper-Chaotic Map for High-Performance Multi-Image Encryption

Chaos-based image encryption has become a prominent area of research in recent years. In comparison to ordinary chaotic systems, fractional-order chaotic systems tend to have a greater number of control parameters and more complex dynamical characteristics. Thus, an increasing number of researchers are introducing fractional-order chaotic systems to enhance the security of chaos-based image encryption. However, their suggested algorithms still suffer from some security, practicality, and efficiency problems. To address these problems, we first constructed a new fractional-order 3D Lorenz chaotic system and a 2D sinusoidally constrained polynomial hyper-chaotic map (2D-SCPM). Then, we elaborately developed a multi-image encryption algorithm based on the new fractional-order 3D Lorenz chaotic system and 2D-SCPM (MIEA-FCSM). The introduction of the fractional-order 3D Lorenz chaotic system with the fourth parameter not only enables MIEA-FCSM to have a significantly large key space but also enhances its overall security. Compared with recent alternatives, the structure of 2D-SCPM is simpler and more conducive to application implementation. In our proposed MIEA-FCSM, multi-channel fusion initially reduces the number of pixels to one-sixth of the original. Next, after two rounds of plaintext-related chaotic random substitution, dynamic diffusion, and fast scrambling, the fused 2D pixel matrix is eventually encrypted into the ciphertext one. According to numerous experiments and analyses, MIEA-FCSM obtained excellent scores for key space (2541), correlation coefficients (<0.004), information entropy (7.9994), NPCR (99.6098%), and UACI (33.4659%). Significantly, MIEA-FCSM also attained an average encryption rate as high as 168.5608 Mbps. Due to the superiority of the new fractional-order chaotic system, 2D-SCPM, and targeted designs, MIEA-FCSM outperforms many recently reported leading image encryption algorithms.

Skin Imaging: A Digital Twin for Geometric Deviations on Manufactured Surfaces

Closed-loop manufacturing is crucial in Industry 4.0, since it provides an online detection–correction cycle to optimize the production line by using the live data provided from the product being manufactured. By integrating the inspection system and manufacturing processes, the production line achieves a new level of accuracy and savings on costs. This is far more crucial than only inspecting the finished product as an accepted or rejected part. Modeling the actual surface of the workpiece in production, including the manufacturing errors, enables the potential to process the provided live data and give feedback to production planning. Recently introduced “skin imaging” methodology can generate 2D images as a comprehensive digital twin for geometric deviations on any scanned 3D surface including analytical geometries and sculptured surfaces. Skin-Image has been addressed as a novel methodology for continuous representation of unorganized discrete 3D points, by which the geometric deviation on the surface is shown using image intensity. Skin-Image can be readily used in online surface inspection for automatic and precise 3D defect segmentation and characterization. It also facilitates search-guided sampling strategies. This paper presents the implementation of skin imaging for primary engineering surfaces. The results, supported by several industrial case studies, show high efficiency of skin imaging in providing models of the real manufactured surfaces.

Discrete 3D modeling of porous-cracked ceramic at the microstructure scale

The porous-cracked microstructure of plasma sprayed ceramics coatings directly influences their macroscopic mechanical response.A 3D micromechanical model based on the discrete element method (DEM) is developed to reproduce their behavior. 3D observations are conducted with FIB-SEM nano-tomography and image analysis is used to extract and discretize the microstructure. A modeling strategy is developed to incorporate both pores and cracks into the model while preserving the computation performance. The lattice nature of DEM is used for the modeling of crack thickness that are smaller than the discrete element size. A method to compute the crack thickness from gray level images is proposed.Virtual quasi-static tests are performed on a sample of yttria-stabilized-zirconia. The results are in accordance with the literature data, such as the anisotropy and the non-linear behavior. The model is helpful to precisely investigate the role of the microscopic components, and micro-cracking on the macroscopic failure.

Bearing capacity of lunar soil ground under extraterrestrial environmental effects

Lunar environment features a low gravity field and ultra-high vacuum which shows significant effects on the bearing behavior of lunar soil ground for future lunar habitation. Therefore, the 2D Discrete Element Method was employed to simulate the plate load tests under different gravity fields in non-vacuum (i.e., excluding the van der Waals force) and high-vacuum (i.e., including the van der Waals force) environments, analyzing the bearing behavior and soil response under extraterrestrial environmental effects. The results show that a bilinear line bounded by 1g can be employed to describe the gravity effect on the gravity factors for ultimate bearing capacity and coefficient of subgrade reaction. The slopes are different in low and high-gravity fields, indicating that the high-gravity effect in a centrifuge cannot be directly applied to predict the bearing behavior of lunar soil ground in a low-gravity field. The affected area becomes deeper in depth and narrower in width with the decreasing gravity level in non-vacuum environment, while the affected area becomes deeper in depth and wider in width with the decreasing gravity level in high-vacuum environment. Such observations can provide advice for developing analytical models for the bearing capacity of lunar soil ground considering the extraterrestrial environmental effects.

Comparison of lateral resistance between I-shaped and X-shaped sleepers using finite element and discrete element coupling method

The lateral resistance of ballasted track is a key factors to maintain the stability of continuously welded rails. In this study, a new type of X-shaped sleeper was developed to increase the lateral resistance compared to conventional I-shaped sleepers. Multi PD3D (Particle discrete 3D) elements were bonded by rigid Beam-type multi-point constraints (MPCs) to implement irregular shaped ballast. The performance of the X-shaped sleeper was examined using track panel push-out tests simulations. The results showed that the X-shaped sleeper could provide 27% greater lateral resistance compared to double I-shaped sleepers. This is due to the V-fork arms which distribute the central and lateral crib ballasts more effectively, enhancing the stability of the sleeper. Crib ballasts play a crucial role in the lateral resistance of X-shaped sleeper, providing nearly double the lateral resistance compared to double I-shaped sleepers. Overall, the proposed X-shaped sleeper has stronger lateral resistance and stability, making it a promising option for future applications.