Planar Geometry Research Articles

A reduced kinetic method for investigating non-local ion heat transport in ideal multi-species plasmas

A reduced kinetic method (RKM) with a first-principles collision operator is introduced in a 1D2V planar geometry and implemented in a computationally inexpensive code to investigate non-local ion heat transport in multi-species plasmas. The RKM successfully reproduces local results for multi-species ion systems and the important features expected to arise due to non-local effects on the heat flux are captured. In addition to this, novel features associated with multi-species, as opposed to single species, cases are found. Effects of non-locality on the heat flux are investigated in mass and charge symmetric and asymmetric ion mixtures with temperature, pressure, and concentration gradients. In particular, the enthalpy flux associated with diffusion is found to be insensitive to sharp pressure and concentration gradients, increasing its significance in comparison to the conductive heat flux driven by temperature gradients in non-local scenarios. The RKM code can be used for investigating other kinetic and non-local effects in a broader plasma physics context. Due to its relatively low computational cost it can also serve as a practical non-local ion heat flux closure in hydrodynamic simulations or as a training tool for machine learning surrogates.

Droplet asymmetry bouncing on structured surfaces: A simulation based on SPH method

The impact of a liquid droplet onto a macro-structured super-hydrophobic surface, results in an asymmetric spreading-retraction process and a significant decrease in contact time. The process involves large deformation of droplets and evolution of surface morphology, so it is difficult to efficiently simulate using traditional grid-based methods. In this study, a numerical model is established using the mesh-free smoothed particle hydrodynamics (SPH) method to simulate the interaction of liquid droplets with solid surfaces exhibiting complex macro-textures. Leveraging the Lagrange nature of SPH, liquid droplets can be model using free surface boundary condition. The weakly compressible SPH formulation is employed to discretize the governing equations of liquid. Surface tension is captured by introducing an additional negative pressure term, inducing attractive forces among fluid particles. Kernel functions are carefully chosen to mitigate stress instability resulting from droplet spreading and retraction. The model is then applied to simulate the dynamic processes of droplets impact on flat and non-flat super-hydrophobic surfaces. The bouncing dynamics of droplets are investigated under varying conditions, including Weber number, size, and morphology of macro-scale surface structures. Our simulation results for contact time, maximum diameter, and droplet morphology align closely with experimental observations. The contact time of the droplet exhibits a dependency on its size and surface geometry, with a transition towards planar geometry diminishing asymmetric spreading effects. The findings highlight the significance of Weber number and cylinder diameter in determining contact time, providing insights for the design of optimized surface structures for specific droplet impact parameters. In addition, the use of a free-surface condition for modeling proved computationally efficient for 3D simulations. The model effectively reproduces nonlinear behaviors such as droplet spreading, splashing, and bouncing, highlighting the significant advantage of SPH in simulating such problems.

Lattice-Based Generation of Euclidean Geometry Figures

We present a user-guided method to generate geometry figures appropriate for high school Euclidean geometry courses: a useful starting point for an intelligent tutoring system to provide meaningful, realistic figures for study. We first establish that a two-dimensional geometry figure can be represented abstractly using a complete, lattice we call a geometry figure lattice (GFL). As input, we take a user-defined vector of primitive geometry shapes and convert each into a GFL. We then exhaustively combine each these ‘primitive’ GFLs into a set of complex GFLs using a process we call gluing. We mitigate redundancy in GFLs by introducing a polynomial-time algorithm for determining if two GFLs are isomorphic. These lattices act as a template for the second step: instantiating GFLs into a sequence of concrete geometry figures. To identify figures that are structurally similar to textbook problems, we use a discriminator model trained on a corpus of textbook geometry figures.

Designing a Phenalenyl-Based Dinuclear Ni(II) Complex: An Electrocatalyst with Two Single Ni Sites for the Oxygen Evolution Reaction (OER).

A new dinuclear Ni(II) complex 1, [Ni2II(dtbh-PLY)2], is synthesized from 9-(2-(3,6-di-tert-butyl-2-hydroxybenzylidene)hydrazineyl)-1H-phenalen-1-one, dtbh-PLYH2 ligand, and structurally characterized by various analytical tools including the single-crystal X-ray diffraction (SCXRD) technique. In the solid state, both Ni(II) metal centers in complex 1 exist in a distorted square planar geometry and display the presence of rare Ni···H-C anagostic interactions to form a one-dimensional (1-D) linear motif in the supramolecular array. Complex 1 is further stabilized in the solid state by π-π-stacking interactions between the highly delocalized phenalenyl rings. The redox features of complex 1 have been analyzed by the cyclic voltammetry (CV) technique in solution as well as in the solid state, revealing the crucial involvement of both the Ni(II) metal centers for undergoing quasi-reversible oxidation reactions on the application of an anodic sweep. A complex 1-modified glassy carbon electrode, GC-1, is employed as an electrocatalyst for oxygen evolution reaction (OER) in 1.0 M KOH, giving an OER onset at 1.45 V, and very low OER overpotential, 300 mV vs the reversible hydrogen electrode (RHE) to reach 10 mA cm-2 current density. Furthermore, GC-1 displayed fast OER kinetics with a Tafel slope of 40 mV dec-1, a significantly lower Tafel slope value than those of previously reported molecular Ni(II) catalysts. In situ electrochemical experiments and postoperational UV-vis, Fourier transform infrared (FT-IR), scanning electron microscopy-energy-dispersive X-ray spectroscopy (SEM-EDS), and X-ray photoelectron spectroscopy (XPS) studies were performed to analyze the stability of the molecular nature of complex 1 and to gain reasonable insights into the true OER catalyst.

Monolithically grown CSPbBr3 by chemical vapor deposition for Self-Powered photodetector

The high-quality monolithic growth of perovskite films with larger grains on any substrate is a challenging task due to the instability of morphology, carrier mobility and chemical stability. In this work, we demonstrated the strategy to grow CsPbBr3 films with micron-sized grains along the (002) surface by chemical vapor deposition. The structural and optical characterizations have shown that the as-grown CsPbBr3 thin films are compact and packed without pin holes and have excellent optical quality for use in high-performance optoelectronic devices. The planar photodetector with the device configuration Ag/CsPbBr3/Ag shows a high photoresponsivity of 506.32 A/W and a specific detectivity of 8.29 × 1012 Jones at an incident light illumination of 22.3 µW/cm2. On the other hand, the vertically stacked device with ITO/SnO2/CsPbBr3/CuSCN/Ag configuration exhibits a photoresponsivity of 10.13 A/W at 0.05 V bias and 96 mA/W under 0 V bias. Specifically, the response time of the device in planar geometry is 241/205 ms, while the vertically stacked device has a response time of 20.51/21 ms at 0.05 V bias and 20.51/30 ms at 0 V bias. The results provide a new strategy for the growth of high-quality perovskite thin films and their effective application for high-performance optoelectronic devices.

A rare example of monoanionic coordination of thiourea ligands to chromium: synthesis and structure

First example of thioureato ligands with chromium(II) has been isolated and is reported here. The reaction of two equivalents of the in situ generated lithium salt of the bulky 1,3-bis(2,6-diethylphenyl)thiourea ligand (L) with [CrCl2] in tetrahydrofuran and subsequent crystallization from hexane led to the formation of the corresponding CrL2 complex. Single crystal X-ray analysis shows a centrosymmetric Cr complex with a square planar geometry in which the crystal packing of the complex is stabilized by C–H⋅⋅⋅H and C–H⋅⋅⋅S interactions. The strongest contributions (6.2%) due to S⋯H/H⋯S interactions have been further confirmed by Hirshfeld analysis that appear as dark red spots in the finger plots. The title compound, [C42H54CrN4S2], crystallized in the triclinic space group, P-1 with cell parameters: a = 9.3310(19) b = 10.987(2) c = 11.061(2) Å, α = 97.36(3) β = 107.26(3) γ = 108.88(3)°, V = 992.6(4) A3, Z = 1.

Antiferromagnetic resorcinol bridged dinuclear square-planar copper (II) complexes: Syntheses, structural, spectroscopic and magnetic properties

A new C2-symmetric Schiff base ligand (H2L) featuring a core resorcinol bridging unit has been created, utilizing the Duff formylation reaction followed by the Schiff base condensation reaction. The ligand was characterized by various spectroscopic techniques. Cu(II) salts with varying coordinating anions (Cl− and SCN−) in methanol were allowed to react with this ligand in a 2:1 M ratio to create dinuclear Cu(II) complexes. The complexes were isolated in pure form and characterized by, elemental analysis,single crystal X-ray crystallography and various spectroscopic techniques. X-ray analysis discloses that both complexes are dinuclear, where the ligand (H2L) being tridentate in nature, coordinates to Cu(II) through its two N donor atoms and one O donor atom in the equatorial positions. The respective anions (−Cl− for 1 and –SCN− for 2) coordinate to Cu(II) to complete its square planar geometry. Temperature dependent magnetic susceptibility measurements indicate weak antiferromagnetic exchange interactions with a coupling constant of J = −5.9(1) cm−1 and g = 2.148(1) for 1 and J = −10.08(1) cm−1 and g = 2.106(1) for 2.

An Exploration of Learners’ Understanding of Euclidean Geometric Concepts: A Case Study of Secondary Schools in the OR Tambo Inland District of the Eastern Cape

A major concern in South Africa is the poor performance of learners in mathematics, particularly in geometry. This paper therefore sought to explore the learners’ understanding of Euclidean geometric concepts. The Van Hiele model was a useful framework for understanding the reasoning and challenges that students encounter with geometry. This study focused on 15 participants from rural South African schools, including 5 grade 10 mathematics learners in each of the three secondary schools and three teachers who are teaching mathematics in grade 10, in three secondary schools. Through face-to-face interviews, learners’ comprehension of geometry was qualitatively assessed. Pedagogical and methodological difficulties, lack of learners’ interest and comprehension of numerous geometrical concepts, as well as the absence of technology use, were found to contribute to the challenges in learning and teaching Euclidean geometry. The recommendations suggest that teachers should plan and prepare their geometry classes with the students’ understanding in mind that Euclidean geometry has been a cornerstone of mathematical education for centuries, teaching students critical thinking, problem-solving, and logical reasoning skills. This study adds to the existing literature on introducing new concepts in mathematics into the educational system of South Africa. Keywords: Geometry, Students’ Interest, Self-efficacy, Understanding, Three-Dimensional Shape

Assembly-inspired multiferroicity with nontrivial Chern insulating phase from exohedral metallofullerenes.

Fullerene-assembled low-dimensional materials have been experimentally realized in polymorphic forms and have attracted significant interest very recently. Here, we predict a two-dimensional (2D) honeycomb lattice material TM2(C60)3 (TM = Cr, Mo, and W) assembled from exohedral metallofullerene clusters TM(C60)3 that could exhibit planar triangular geometries. According to first-principles calculations combined with Monte Carlo simulations, we suggest that these 2D assembled materials exhibit various exotic physical properties, including ferromagnetism, ferroelectricity, and quantum anomalous Hall effect. Interestingly, mechanical strains could effectively tune their magnetic moments and switch the conducting spin channel of the Dirac bands at the Fermi level. Our work provides a new cluster-assembly design strategy toward cluster-assembled 2D materials based on fullerene characters.

Improving student learning outcomes through the inquiry learning model with powtoon animation on plane geometry subject

Improving student learning outcomes through the inquiry learning model with powtoon animation on plane geometry subject

Elastocapillary interaction for particles trapped at the isotropic-nematic liquid crystal interface.

We present numerical simulations on pairwise interactions between particles trapped at an isotropic-nematic liquid crystal (Iso-N) interface. The particles are subject to elastocapillary interactions arising from interfacial deformations and elastic distortions of the nematic phase. We use a recent model based on a phase-field approach [see Qiu etal., Phys. Rev. E 103, 022706 (2021)2470-004510.1103/PhysRevE.103.022706] to take into account the coupling between elastic and capillary phenomena. The pair potential is computed in a two-dimensional geometry for a range of particle separations and two anchoring configurations. The first configuration leads to the presence of an anchoring conflict at the three-phase contact line, whereas such a conflict does not exist for the second one. In the first case, the results show that significant interfacial deformations and downward particle displacements occur, resulting in sizable attractive capillary interactions able to overcome repulsive elastic forces at intermediate separations. The pair potential exhibits an equilibrium distance since elastic repulsions prevail at short range and prevent the clustering of particles. However, in the absence of any anchoring conflict, the interfacial deformations are very small and the capillary forces have a negligible contribution to the pair potential, which is fully repulsive and overwhelmed by elastic forces. These results suggest that the self-assembly properties of particles floating at Iso-N interfaces might be controlled by tuning anchoring conflicts.

Accuracy of hygro-expansive curl predictions for paper sheets based on homogenised 2D and 3D network representations

Paper, a porous-fibrous network, is made up of hydrophilic fibres, which are notably susceptible to deformations due to variations in moisture content. The response of paper sheets to full or partial wetting, based on the water-induced fibre swelling, has mostly been predicted in the past using highly idealised two-dimensional network geometries. The purpose of the present paper is to establish how precisely the two-stage process of determining effective hygro-expansivity under uniform wetting by a two-dimensional model followed by bending analysis using a continuum model predicts the curl of a fully three-dimensional network. In the present multi-scale modelling work, the moisture-induced homogenised sheet scale curl, derived from hygro-elastic deformations using non-uniform moisture profiles, is analysed based on an idealised non-woven three-dimensional fibrous network finite element model. First, the effective paper sheet swelling properties, derived computationally for the uniform wetting condition, are compared with analytically homogenised hygro-elastic properties. Whereas the analytically predicted elastic modulus has an inaccuracy of only ∼5%, the hygro-expansivity coefficient is over-predicted by roughly a third for a network with straight fibres. The difference is smaller if waviness of the fibres and wrap-around in the fibre bonds are taken into account. Fibre alignment turns out to have little influence. Subsequently, paper sheet curl predictions made by the analytical model are compared with that computed from the three-dimensional network model. The inaccuracy of the fully analytical model, based on the two-dimensional network description, relative to the fully detailed three-dimensional computational model with straight fibres is on the order of 40%–50%. The fidelity of the simple model is primarily constrained by the inaccuracy of the effective hygro-expansion coefficient. If the expansivity extracted from the three-dimensional simulations under uniform wetting is inserted, the inaccuracy drops to less than 10%. The continuum bending description hence performs adequately, provided it is based on a reliable estimation of the uniform expansion behaviour of the material.

Highly efficient organic light‐emitting diodes and light‐emitting electrochemical cells employing multiresonant thermally activated delayed fluorescent emitters with bulky donor or acceptor peripheral groups

AbstractMultiresonant thermally activated delayed fluorescence (MR‐TADF) emitters have been the focus of extensive design efforts as they are recognized to show bright, narrowband emission, which makes them very appealing for display applications. However, the planar geometry and relatively large singlet–triplet energy gap lead to, respectively, severe aggregation‐caused quenching (ACQ) and slow reverse intersystem crossing (RISC). Here, a design strategy is proposed to address both issues. Two MR‐TADF emitters triphenylphosphine oxide (TPPO)‐tBu‐DiKTa and triphenylamine (TPA)‐tBu‐DiKTa have been synthesized. Twisted ortho‐substituted groups help increase the intermolecular distance and largely suppress the ACQ. In addition, the contributions from intermolecular charge transfer states in the case of TPA‐tBu‐DiKTa help to accelerate RISC. The organic light‐emitting diodes (OLEDs) with TPPO‐tBu‐DiKTa and TPA‐tBu‐DiKTa exhibit high maximum external quantum efficiencies (EQEmax) of 24.4% and 31.0%, respectively. Notably, the device with 25 wt% TPA‐tBu‐DiKTa showed both high EQEmax of 28.0% and reduced efficiency roll‐off (19.9% EQE at 1000 cd m−2) compared to the device with 5 wt% emitter (31.0% EQEmax and 11.0% EQE at 1000 cd m−2). The new emitters were also introduced into single‐layer light‐emitting electrochemical cells (LECs), equipped with air‐stable electrodes. The LEC containing TPA‐tBu‐DiKTa dispersed at 0.5 wt% in a matrix comprising a mobility‐balanced blend‐host and an ionic liquid electrolyte delivered blue luminance with an EQEmax of 2.6% at 425 cd m−2. The high efficiencies of the OLEDs and LECs with TPA‐tBu‐DiKTa illustrate the potential for improving device performance when the DiKTa core is decorated with twisted bulky donors.

Geometrically parametrised reduced order models for studying the hysteresis of the Coanda effect in finite element-based incompressible fluid dynamics

This article presents a general reduced order model (ROM) framework for addressing fluid dynamics problems involving time-dependent geometric parametrisations. The framework integrates Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) hyper-reduction techniques to effectively approximate incompressible computational fluid dynamics simulations. To demonstrate the applicability of this framework, we investigate the behaviour of a planar contraction-expansion channel geometry exhibiting bifurcating solutions known as the Coanda effect. By introducing time-dependent deformations to the channel geometry, we observe hysteresis phenomena in the solution.The paper provides a detailed formulation of the framework, including the stabilised finite elements full order model (FOM) and ROM, with a particular focus on the considerations related to geometric parametrisation. Subsequently, we present the results obtained from the simulations, analysing the solution behaviour in a phase space for the fluid velocity at a probe point, considered as the Quantity of Interest (QoI). Through qualitative and quantitative evaluations of the ROMs and hyper-reduced order models (HROMs), we demonstrate their ability to accurately reproduce the complete solution field and the QoI.While HROMs offer significant computational speedup, enabling efficient simulations, they do exhibit some errors, particularly for testing trajectories. However, their value lies in applications where the detection of the Coanda effect holds paramount importance, even if the selected bifurcation branch is incorrect. Alternatively, for more precise results, HROMs with lower speedups can be employed.

Mixing is easy: New insights for cosmochemical evolution from pre-stellar core collapse

Signposts of early planet formation are ubiquitous in substructured young discs. Dense, hot, and high-pressure regions that formed during the gravitational collapse process, integral to star formation, facilitate dynamical mixing of dust within the protostellar disc. This provides an incentive to constrain the role of gas and dust interaction and resolve potential zones of dust concentration during star and disc formation stages. We explore whether the thermal and dynamical conditions that developed during protostellar disc formation can generate gas flows that efficiently mix and transport the well-coupled gas and dust components. We simulated the collapse of dusty molecular cloud cores with the hydrodynamics code PLUTO augmented with radiation transport and self-gravity. We used a two-dimensional axisymmetric geometry and followed the azimuthal component of the velocity. The dust was treated as Lagrangian particles that are subject to drag from the gas, whose motion is computed on a Eulerian grid. We considered 1, 10, and 100 micron-sized neutral, spherical dust grains. Importantly, the equation of state accurately includes molecular hydrogen dissociation. We focus on molecular cloud core masses of 1 and 3 $M_ odot $ and explore the effects of different initial rotation rates and cloud core sizes. Our study underlines mechanisms for the early transport of dust from the inner hot disc regions via the occurrence of two transient gas motions, namely meridional flow and outflow. The vortical flow fosters dynamical mixing and retention of dust, while the thermal pressure driven outflow replenishes dust in the outer disc. Notably, these phenomena occur regardless of the initial cloud core mass, size, and rotation rate. Young dynamical precursors to planet-forming discs exhibit regions with complex hydrodynamical gas features and high-temperature structures. These can play a crucial role in concentrating dust for subsequent growth into protoplanets. Dust transport, especially, from sub-au scales surrounding the protostar to the outer relatively cooler parts, offers an efficient pathway for thermal reprocessing during pre-stellar core collapse.

Circling the Square with Straightedge & Compass in Euclidean Geometry

There are three classical problems remaining from ancient Greek mathematics which are extremely influential in the development of Geometry. They are Trisecting an Angle, Squaring the Circle, and Doubling the Cube problems. I solve the Squaring the Circle problem, of which paper is published in the International Journal of Mathematics Trends and Technology (Volume 69, June 2023). Upstream from this method of exact “Squaring the Circle”, we can deduce, conversely/inversely, to get a new Mathematical challenge "CIRCLING THE SQUARE" with a straightedge & a compass in Euclidean Geometry. This study idea came from my exact solution “Squaring the Circle by Straightedge & compass in Euclidean Geometry”, published by IJMTT in June 2023 at https://ijmttjournal.org/Volume-69/Issue-6/IJMTT-V69I6P506.pdf for this ancient Greek Geometry problem. In this research, I adopt the ANALYSIS method to prove the process of solving this new challenge problem, which has not existed in the Mathematics field till today. The process is an inverse/converse solution solving the ancient Greek Geometry challenge problem of “Squaring the Circle”, using a straightedge & a compass. I hereby commit that this is my own personal research project.

Highly integrable planar-structured printed circularly polarized antenna for emerging wideband internet of things applications in the millimeter-wave band

This paper proposes a numerically and experimentally validated printed wideband antenna with a planar geometry for Internet of Things (IoT) applications. This design tackles the challenges associated with deploying IoT sensors in remote areas or across extensive geographical regions. The proposed design exploits a coplanar-waveguide-fed modified microstrip line monopole for excitation of circularly polarized waves radiating in the broadside direction. The primary design is based on perturbations of the microstrip line protracted from a grounded coplanar waveguide. The capacitively coupled short rectangular stubs are periodically inserted alternately and excited asymmetrically on each side of the microstrip line parallel to the direction of the electric field vector. The sequential phase excitation of the periodic stubs generates a rectangular-cascaded electric field, which suppresses the stop band at the open end. As a result, the antenna radiates in the broadside direction. The impedance bandwidth of the antenna exceeds 8 GHz in the 28 GHz mm-wave band, i.e., it ranged from 25 to 33.5 GHz. Additionally, an axial ratio below 3 dB is achieved within the operating band from 26 to 33.5 GHz with the alterations of the surface current using straightforward topological adjustments of the physical parameters. The average in-band realized gain of the antenna is 10 dBic when measured in the broadside direction. These results indicate that the proposed design has the potential to improve the connectivity between IoT devices and the constantly varying orientation of satellites by mitigating the polarization mismatch.

Гален как представитель греческой paideia эпохи Второй софистики

Thanks to the extensive corpus of Galen's writings, which includes numerous autobiographical works and individual notes, we possess a wealth of information regarding his upbringing and education. According to a fami-ly tradition, he received a classical Greek education (known as paideia), which allowed him to transition from a provincial intellectual to one of the influential figures in Roman society and achieve the status of a court physi-cian. His brilliant command of language, extensive erudition, and active participation in the intellectual life of the Roman elite justify considering Galen as one of the authors of the so-called "Second Sophistic." This article scrutinizes Galen's various accounts of his mentors, early education, and scholastic experiences, with their veracity corroborated by contemporaneous testimonials. In addition to reading and scrutinizing the classical texts that formed the cornerstone of Greek paideia, Galen's education placed great emphasis on the study of exact sciences such as Euclidean geometry and arithmetic, which, together with Aristotle's logic, enabled Galen to adhere to the evidence-based method in medicine. An analysis of Galen's citation fre-quency of classical authors revealed that his foundational education was similar to other authors of his time, whereas the citation of medical authors is unparalleled in ancient literature. Unfortunately, we can only form a vague notion of his library since much of it was lost in a fire during Galen's lifetime in 192 AD. However, his collection of texts is a mix of various knowledge, rare writings, and quotes. Galen's interest in language and words, which confirmed his status as an intellectual, suggests his association with the "Second Sophistic" group. He wrote several works on ancient grammar, word usage, and rhetoric, criticiz-ing both extreme linguistic purism and loose interpretations of old-fashioned Attic language. This paints Galen not just as a doctor but also as a knowledgeable figure of his time.

Chen-Chvátal’s conjecture for graphs with restricted girth

A classic result in Euclidean geometry asserts that every non-collinear set of n points in the Euclidean plane determines at least n distinct lines. Chen and Chvátal conjectured that this holds for an arbitrary finite metric space, with an appropriate definition of line. This conjecture remains open even for graph metrics. In this paper, sufficient conditions on the girth to guarantee that a graph satisfies the conjecture are stated. We first study the existence of a universal line generated by an edge. Then, we focus on graphs of order n and maximum girth g with respect to their radius r. We prove that graphs with minimum degree δ ≥ 4 and girth g = 2r+1 or g = 2r have at least n distinct lines and we also partially solve the case when δ = 3. Graphs with cut-vertices are also studied, providing several upper bounds on the diameter, in terms of the girth, in order to assure that the graph satisfies the Chen-Chvátal conjecture. Finally, we prove that any triangle-free graph of order n, diameter 3 and minimum degree δ ≥ 4 has more than n distinct lines and we also partially solve the case when δ = 3.

Mathematical model for footwear upper material consumption estimation

Appropriate material consumption estimation since the design phase for footwear fabrication is a vital issue as material costs account for a sizeable portion of the overall production cost of a pair of shoes. This paper presents a mathematical model to predict the calculation of footwear upper material consumption to improve the utilization ratio of materials through a suitable nesting map onto leather. The proposed model concentrates on the two-dimensional geometry of footwear components and the application of rich mathematical concepts. The model reflects the outlines of individual footwear components while determining the area using definite integral calculus. Five distinct rotational kinds are applied for component arrangements that correlate to the physiognomy of leather because nesting onto the leather is intractable. The simple concept of a minimal-area polygonal enclosure is applied to maximize material utilization with minimum waste. Finally, the model was verified for four consecutive Oxford footwear sizes by comparing actual upper material consumption with predicted upper material consumption. The noble contribution of this analysis is to use ImageJ software to compute upper material consumption in real case analysis through image processing techniques and separate estimation of wastes, especially the fourth waste. The results of the comparison study show that the proposed model can reduce average material requirements by 2.06 %. This minimization of waste could be beneficial in terms of economic and environmental sustainability. Therefore, this study can be applied to estimate more accurately the amount of upper material required for footwear fabrication and support better utilization of material in the footwear industry.