Geometric Assumptions Research Articles

Applying Fractal Theory: Solving the Geometric Challenge of Price Change and Scaling in Economics

<i>Background</i>: Price changes in economics present significant geometric challenges due to sharp discontinuities, which cannot be efficiently described by continuous processes like Brownian motion. Traditional models often rely on linear assumptions, yet financial data frequently exhibit irregular, complex patterns. Fractal theory, a mathematical framework, offers a more accurate way to describe these fluctuations by revealing the underlying self-similar structures in price changes and scaling phenomena. This study explores the use of fractal geometry to gain deeper insights into market behavior. <i>Objective</i>: The objective is to demonstrate that an alternative model, constructed based on geometric scaling assumptions, offers a more accurate description of price changes in competitive markets. <i>Method</i>: The study combined the scaling principle from fractal geometry with a stable Levy model to formulate an integrated model. The logarithmic transformation of the model was applied over successive price changes to observe the behavior of market prices. <i>Result</i>: The scaling principle asserts that no specific time interval (such as a day or a week) holds inherent significance in competitive markets. Instead, these time features are compensated or arbitrated away, supporting the idea that market behavior is self-similar across different time scales. <i>Conclusion</i>: The scaling principle provides a more reliable framework for modeling price changes and is recommended for consideration in economic analyses.

A 2D echocardiographic method for accurate right ventricular quantification using deep learning

Abstract Background/Introduction Transthoracic two-dimensional echocardiography (2DE) remains the backbone for non-invasive assessment of the right ventricle (RV) size and function. However, calculation of RV end-diastolic (RVEDV), end-systolic volume (RVESV) and ejection fraction (RVEF) cannot be performed using 2DE due to the complex RV shape (1), whereas currently used, respective 2DE indices lack diagnostic accuracy. Deep learning (DL) may be able to assist in developing new 2DE methods for RV quantification. Purpose To develop a novel DL-based 2DE method for quantitative evaluation of RV size and function without geometric assumptions. Methods 50 adult patients underwent 2DE and CMR within 30 days as part of routine cardiac care between 6/2020 – 9/2021 at a medical center. We excluded patients with prior cardiac surgery of the RV, tricuspid or pulmonic valve repair or replacement, complex congenital heart disease, more than small pericardial effusion, cardiac tamponade, atrial fibrillation at the time of either study, severe RV dysfunction or inadequate imaging quality. End-diastolic and end-systolic areas derived from planimetry of 8 2DE RV views, along with patient age and gender, were to used to train and test a DL algorithm, namely the Feature-Tokenizer and Transformer (Figure 1) in order to predict the corresponding, CMR-derived, RVEDV or RVESV, and subsequently calculate RVEF. We used 5-fold cross-validation for training and evaluation. We assessed the relative importance of each 2DE view using gain metric-powered explainability analysis and subsequently performed a sensitivity analysis to identify optimal combinations of views. Variability analysis was performed for a random subcohort of 10 patients. Results Median age was 51 (interquartile range 32-62) and 42% were women. Mean RVEDV, RVESV and RVEF) by CMR were 163±70ml, 82±42ml and 51±8% respectively. Six patients (12%) had RV dilatation and 9 (18%) had RV dysfunction when using published CMR criteria. Explainability analysis showed that the 3 most important views were parasternal long axis (PLAX) (relative feature importance [RFI] = 0.197), 4-chamber (RFI=0.145) and parasternal short axis at the base (PSAX-base) (RFI=0.112). The 3-view combination of PLAX, 4-chamber and PSAX-base achieved high accuracy when compared to CMR (R2=0.964, absolute percentage error [APE]=8.09±12.07% for RV volumes and APE=9.86±10.68% for RVEF). Good (intraclass correlation coefficient [ICC]>0.75) to excellent (ICC>0.90) intra- and interobserver variability was found for these 3 views. Conclusions We propose an innovative, clinically applicable DL-based 2DE method for quantification of RV size and function that requires manual RV planimetry of 3 views, namely 4-chamber, PLAX and PSAX-base (Figure 2). This method may revolutionize 2DE RV assessment. Further studies with largest number of patients and wider range of RV size and function are needed to validate this method.Deep learning architectureGraphical abstract

Discontinuous solutions for the Navier–Stokes equations with density-dependent viscosity

We prove the existence of a unique global-in-time weak solution of the Navier–Stokes equations that govern the motion of a compressible viscous fluid with density-dependent viscosity in two-dimensional space. The initial velocity belongs to the Sobolev space H^{1}(\mathbb{R}^{2}) , and the initial fluid density is \alpha -Hölder continuous on both sides of a \mathscr{C}^{1+\alpha} -regular interface with some geometrical assumption. We prove that this configuration persists over time: the initial interface is transported by the flow to an interface that maintains the same regularity as the initial one. Our result generalizes a previous known result of Hoff [Comm. Pure Appl. Math. 55 (2002), 1365–140] and Hoff and Santos [Arch. Ration. Mech. Anal. 188 (2008), 509–543] concerning the propagation of regularity for discontinuity surfaces by allowing a more general non-linear pressure law and density-dependent viscosity. Moreover, it supplements the work by Danchin, Fanelli, and Paicu [Anal. PDE 13 (2020), 275–316] with global-in-time well-posedness, even for density-dependent viscosity, and we achieve uniqueness in a large space.

The mitral to aortic/pulmonary velocity-time integral ratio is a simple, feasible and accurate discriminator for echocardiographic evaluation of severe isolated mitral regurgitation.

Echocardiographic quantification of mitral regurgitation (MR) remains challenging, requiring dedicated image acquisition, and is limited by potential error from geometric assumptions of annular dimensions. Volume is a product of area and flow and assuming proportional mitral/aortic areas, an increased mitral-inflow volume compared to LV/RV-outflow semi-quantitatively represents greater MR regurgitant volume. Therefore, we investigated the feasibility and diagnostic performance of the mitral-aortic velocity-time integral(VTI) ratio in isolated MR. We also investigated the use of the mitral-pulmonary VTI ratio as an alternative in clinical situations where the LV outflow tract(LVOT) VTI could not be used. We reviewed 166 consecutive patients (33%, n = 54 severe MR by multi-parameter integrated expert opinion). Pulsed wave Doppler VTI at the mitral leaflet tips and the left ventricular outflow and continuous-wave Doppler of the RV outflow tract were measured individually and independently by blinded readers(expert and trainee status) to derive the ratio. Receiver operator characteristic area under the curve(AUC) comparison was calculated and compared with effective regurgitant orifice area(EROA > 40mm), regurgitant volume(RVol > 60mL), vena contracta(VC > 0.7cm), E-velocity > 1.2cm, systolic flow reversal(SFR), left atrial and ventricular dilatation. Increasing ratio was associated with severe MR(AUC 0.94) with optimal threshold defined at 1.3. This provided significant discrimination for severe MR(AUC 0.81) compared to EROA(0.68), VC(0.52), LV dilatation(0.69), LA dilatation(0.70), SFR(0.73), E-velocity(0.68) all p < 0.05, with sensitivity 82% and specificity 94%. The mitral-pulmonary VTI ratio demonstrated similar discrimination(AUC 0.92) with optimal threshold defined at 1.14. Excellent inter-observer reproducibility(intra-class correlation 0.97) was seen between trainee and expert readers. There was no difference in AUC comparison by MR mechanism or patient rhythm. The mitral-aortic and mitral-pulmonary VTI ratios are simple, geometric-free parameters feasibly reproducible from routine echocardiographic datasets and are excellent discriminative tools for severe MR. Readers should consider integration of this parameter in routine reporting.

Multiscale molecular modeling of chromatin with MultiMM: From nucleosomes to the whole genome

Motivation: We present a user-friendly 3D chromatin simulation model for the human genome based on OpenMM, addressing the challenges posed by existing models with use-specific implementations. Our approach employs a multi-scale energy minimization strategy, capturing chromatin's hierarchical structure. Initiating with a Hilbert curve-based structure, users can input files specifying nucleosome positioning, loops, compartments, or subcompartments.Results: The model utilizes an energy minimization approach with a large choice of numerical integrators, providing the entire genome's structure within minutes. Output files include the generated structures for each chromosome, offering a versatile and accessible tool for chromatin simulation in bioinformatics studies. Furthermore, MultiMM is capable of producing nucleosome-resolution structures by making simplistic geometric assumptions about the structure and the density of nucleosomes on the DNA.Code availability: Open-source software and the manual are freely available on https://github.com/SFGLab/MultiMM or via pip https://pypi.org/project/MultiMM/.

$$A_p$$ weights on nonhomogeneous trees equipped with measures of exponential growth

AbstractThis paper aims to study $$A_p$$ A p weights in the context of a class of metric measure spaces with exponential volume growth, namely infinite trees with root at infinity equipped with the geodesic distance and flow measures. Our main result is a Muckenhoupt Theorem, which is a characterization of the weights for which a suitable Hardy–Littlewood maximal operator is bounded on the corresponding weighted $$L^p$$ L p spaces. We emphasise that this result does not require any geometric assumption on the tree or any condition on the flow measure. We also prove a reverse Hölder inequality in the case when the flow measure is locally doubling. We finally show that the logarithm of an $$A_p$$ A p weight is in BMO and discuss the connection between $$A_p$$ A p weights and quasisymmetric mappings.

Implementing Real Options Valuation under Macroeconomic Risk and Normally Distributed Cash Flows

The paper highlights the encountered problems in implementing real options under more realistic assumptions such as business cycle risk and normally distributed cash flows. The problems considered include (i) estimating empirical distribution of cash flows from real option investments; (ii) investment decisions across business cycles, and (iii) calculating the probability of investing with the above stated rich features. To this end, we estimate operating cash flows of US corporate firms using a Markov chain model under both geometric and arithmetic Brownian motions assumptions for cash flows and develop a valuation model of real option with normally distributed cash flows. Associated investment valuation models incorporating these estimates reveal that critical cash flow levels significantly differ across models and regimes.

Multi-cameras calibration System Based Deep Learning Approach and Beyond: A Survey

The process of determining camera settings to deduce geometric attributes from recorded sequences is known as camera calibration. This process is essential in the fields of robotics and computer vision, encompassing both two-dimensional and three-dimensional applications. Traditional calibration methods, however, are time-consuming and require specific expertise. Recent endeavors have demonstrated that learning-based systems can replace the monotonous tasks associated with manual calibrations. Responses have been examined through a range of learning techniques, networks, geometric assumptions, and datasets. A thorough examination of camera calibration systems that rely on learning algorithms is offered in this paper, assessing their advantages and disadvantages. The primary categories of calibration presented are the regular pinhole camera model, distortion camera model, cross-sensor model, and cross-view model. These categories align with current research trends and have diverse applications. As there is no existing standard in this field, a large dataset of calibration has been created, which can be used as a public platform to assess the effectiveness of current methods. This collection consists of both artificially generated and genuine data, including images and videos obtained from various cameras in different locations. The difficulties faced will be analyzed, and alternative avenues for further research will be suggested in the next stage of this project. This survey represents the initial attempt to perform camera calibration using learning-based methods spanning a period of eight years. Our findings indicate that learning-based methods significantly reduce the time and expertise required for calibration while maintaining or improving accuracy compared to traditional methods. Specifically, our research demonstrates a calibration error reduction of up to 20% and speed improvements by a factor of three compared to traditional methods, as well as better adaptability to different camera types and environments.

A geometric Elliott invariant and noncommutative rigidity of mapping tori

We prove a rigidity property for mapping tori associated to minimal topological dynamical systems using tools from noncommutative geometry. More precisely, we show that under mild geometric assumptions, a leafwise homotopy equivalence of two mapping tori associated to Zd-actions on a compact space can be lifted to an isomorphism of their foliation C⁎-algebras. This property is a noncommutative analogue of topological rigidity in the context of foliated spaces whose space of leaves is singular, where isomorphism type of the C⁎-algebra replaces homeomorphism type. Our technique is to develop a geometric approach to the Elliott invariant that relies on topological and index-theoretic data from the mapping torus. We also discuss how our construction can be extended to slightly more general homotopy quotients arising from actions of discrete cocompact subgroups of simply connected solvable Lie groups, as well as how the theory can be applied to the magnetic gap-labelling problem for certain Cantor minimal systems.

Elucidating the impact of non-spherical morphology on kinetic behavior of graphite using single-particle microelectrode

The precise assessment of the kinetic properties of graphite is crucial for the design of the material and the development of electrochemical models for lithium-ion batteries. However, conventional evaluation methods using composite electrodes fall short in excluding the influence of its porous structure and evaluating the influence of particle shape on kinetic behavior. In this work, we elucidate the impact of non-spherical morphology on kinetic behavior of single artificial graphite particle, employing the single-particle microelectrode technique. Three single artificial graphite particles with different sizes are pre-cycled and exhibit outstanding rate capability, maintaining a maximum capacity retention of 78.1 % at 200 C. Further, exchange current density and diffusion coefficients are determined through Tafel plots and the potentiostatic intermittent titration technique (PITT), respectively. It is found that the particle with a higher exposure of edge planes present elevated exchange current density and diffusion coefficients. Considering that the graphite particles typically feature an ellipsoidal rather than spherical shape, we propose an approach based on an ellipsoidal diffusion model for extracting diffusion coefficients. This model takes into account the non-spherical morphology of graphite, addressing the limitations inherent in the geometric assumptions of previous methods. The average relative error between the diffusion coefficients derived from the ellipsoidal diffusion model and the previous method using spherical assumptions is 215 %, indicating that accurately depicting the shape of ellipsoidal graphite particles in the model is crucial for obtaining correct estimates of kinetic parameters. This study offers direct experimental evidence of superior kinetic behavior of edge planes over basal planes. To leverage this characteristic, it is recommended to employ an out-of-plane aligned architecture for composite electrodes using novel techniques, e.g., 3D printing or magnetic alignment techniques.

Thermal vibration analysis of bi-directionally stepped porous functionally graded plates with segment-specific material property variation supported by Kerr foundation

This paper presents a comprehensive analysis of the nonlinear dynamic response of stepped rectangular plates made of functionally graded porous material (FGPM), supported by the Kerr foundation, in a thermal environment. A novel analytical framework is developed to explore geometric and material variations. The study investigates structural variations in plate thickness with abrupt changes in uni- or bi-directional orientations, examining both single and double stepped thickness profiles. Material properties vary with thickness, featuring distinct horizontal discontinuities across plate segments. Porosity distributions, both even and uneven, are addressed using modified mixture rules. Nonlinear kinematic relationships are established using Reddy’s third-order shear deformation plate theory and von Kármán’s nonlinear geometric assumptions, with equations of motion solved via Galerkin’s technique. This improved model effectively addresses non-continuous thickness variation through integral calculus, enhancing computational efficiency. Validation is achieved by comparing outcomes with published literature and Finite Element Analysis (FEA). The study investigates the influence of material properties, elastic foundation, boundary conditions, and geometric parameters on the free vibration and nonlinear behaviors of the plates. Some of the key findings include: increasing the thickness of the stepped segment significantly heightens the fundamental frequency while reducing vibrational amplitudes; optimizing the location of the stepped segment directly impacts the plate’s fundamental frequency and vibrational amplitudes; and as the load factor increases, the difference between linear and nonlinear deflection becomes evident. Therefore, accurate FGPM stepped plate design requires incorporating nonlinear terms in the strain–displacement relationships. Suggestions for future model modifications are also discussed, contributing to advancements in structural design and analysis.

Large strain consolidation of soft soils with partially penetrating PVDs: Analytical solutions incorporating variable well resistance

The large-strain geometric assumptions and nonlinear compressibility and permeability have significant effects on the consolidation of soft soils with high compressibility. However, analytical solutions for large-strain nonlinear consolidation of soft soils with partially penetrating PVDs have been rarely reported in the literature. A double logarithmic model is adopted to describe the nonlinear compressibility and permeability of soft soils with high compressibility, and a large-strain consolidation model for soft soils with partially penetrating PVDs under the condition that the excess pore water pressure at the interface between the improved and unimproved layers is equal is established based on Gibson’s large-strain consolidation theory. The analytical solution for the large-strain nonlinear consolidation model for soft soils with partially penetrating PVDs is obtained. The reliability of the analytical solution obtained in this study is verified by comparing it with the existing solutions under different conditions, and the maximum deviation between the two methods does not exceed 5 %. On this basis, consolidation behaviors of soft soils with partially penetrating PVDs under different conditions were analyzed by extensive calculations. Finally, the proposed analytical solution for the large strain consolidation model is applied to the settlement calculation of the Bachiem Highway Project, which further demonstrates the applicability of the consolidation model.

From Dawn to Dusk: High-Resolution Tree Shading Model Based on Terrestrial LiDAR Data

Light availability and distribution play an important role in every ecosystem as these affect a variety of ecosystem processes and functions. To estimate light availability and distribution, light simulations can be used. Many previous models were based on highly simplified tree models and geometrical assumptions about tree form, or were sophisticated and computationally demanding models based on 3D data which had to be acquired in every season to be simulated. The aim of this study was to model the shadow cast by individual trees at high spatial and temporal resolution without the need for repeated data collection during multiple seasons. For our approach, we captured trees under leaf-off conditions using terrestrial laser scanning and simulated leaf-on conditions for individual trees over the remainder of the year. The model was validated against light measurements (n=20,436) collected using 60 quantum sensors underneath an apple tree (Malus domestica Borkh.) on a sunny and cloudless summer day. On this day, the leaves and the shadow were simulated with a high spatial (1 cm) and temporal resolution (1 min). The simulated values were highly correlated with the measured radiation at r=0.84. Additionally, we simulated the radiation for a whole year for the sample apple tree (tree height: 6.6 m, crown width: 7.6 m) with a resolution of 10 cm and a temporal resolution of 10 min. Below the tree, an area of 49.55 m² is exposed to a radiation reduction of at least 10%, 17.74 m² to at least 20% and only 0.12 m² to at least 30%. The model could be further improved by incorporating branch growth, curved leaf surfaces, and gravity to take the weight of the foliage into account. The presented approach offers a high potential for modelling the light availability in the surroundings of trees with an unprecedented spatial and temporal resolution.

Common field measures and geometric assumptions of tree shape produce consistently biased estimates of tree and canopy structure in mixed Mediterranean forests

Crown shape and canopy structure are crucial determinants of tree and forest performance, and forest-climate interactions, but methods of quantifying these from field measurements are based on untested assumptions. Historically, measurements of crown size, shape and interactions have been limited to what can be quantified using simple tools such as a tape measure or clinometer, which are then combined with simple geometric assumptions to calculate metrics including crown area, crown volume and canopy cover and overlap. Terrestrial Laser Scanning (TLS) captures the three-dimensional structure of trees and can derive such metrics directly, bypassing the need for geometric assumptions. Here we use TLS data from c. 2500 trees in mixed Mediterranean forests to present a first test of the validity of common assumptions of tree shape. We simulate field measurements from TLS, use these to estimate crown and canopy properties based on geometric assumptions and compare with direct TLS estimates. We find that commonly used methods produce consistently biased estimates of individual tree crown area and crown volume, and whole-plot crown projected area and crown overlap. Although TLS is often not the practical choice for forest measurement, we show that it can guide best practice and here present recommendations of ground measurements that minimise errors.

Dimensionless index and outcome in patients with aortic valve disease.

Assessment of aortic stenosis (AS) is based on aortic valve (AV) gradients and calculation of aortic valve area (AVA). These parameters are influenced by flow and dependent on geometric assumptions. The dimensionless index (DI), the ratio of the LVOT time-velocity integral to that of the AV jet, is simple to perform, and is less susceptible to error but has only been examined in small selected groups of AS patients. The objective of this study was to assess the DI and prognosis in a large cohort. All subjects who underwent echocardiography with an assessment of the AV that included DI were included. Association between AV parameters including mean gradient, AVA, DI and AV resistance and mortality and cardiovascular hospitalizations was examined. A total of 9393 patients (mean age 71±16 years; 53% male) were included. 731 (7.7%) patients had DI less than.25. Increasing age and a diagnosis of heart failure were significantly associated with lower DI. Subjects with low DI had significantly lower ventricular function, a higher incidence of mitral and tricuspid regurgitation, worse diastolic function and more elevated pulmonary pressures. Decreasing DI was associated with significantly decreased survival and event-free survival which remained highly significant on multivariate analysis. In a large population of patients with AV disease, decreased DI, was associated with increased mortality and decreased event-free survival. The easily obtained DI identifies a broad range of AS subjects with worse prognosis and should be integrated into the assessment of these complex patients.

In vivo mechanical response of thigh soft tissues under compression: A two-layer model allows an improved representation of the local tissue kinematics

Biomechanical parameters have the potential to be used as physical markers for prevention and diagnosis. Finite Element Analysis (FEA) is a widely used tool to evaluate these parameters in vivo. However, the development of clinically relevant FEA requires personalisation of the geometry, boundary conditions, and constitutive parameters. This contribution focuses on the characterisation of mechanical properties in vivo which remains a significant challenge for the community. The aim of this retrospective study is to evaluate the sensitivity of the computed elastic parameters (shear modulus of fat and muscle tissues) derived by inverse analysis as a function of the geometrical modelling assumption (homogenised monolayer vs bilayer) and the formulation of the cost function. The methodology presented here proposes to extract the experimental force-displacement response for each tissue layer (muscle and fat) and construct the associated Finite Element Model for each volunteer, based on data previously collected in our group (N = 7 volunteers) as reported in (Fougeron et al., 2020). The sensitivity analysis indicates that the choice of the cost function has minimal impact on the topology of the response surface in the parametric space. Each surface displays a valley of parameters that minimises the cost function. The constitutive properties of the thigh (reported as median ± interquartile range) were determined to be (μ=198±322Pa,α=37) for the monolayer and (μmuscle=1675±1127Pa,αmuscle=22±14,μfat=537±1131Pa,αfat=32±7) for the bilayer. A comparison of the homogenised monolayer and bilayer models showed that adding a layer reduces the error on the local force displacement curves, increasing the accuracy of the local kinematics of soft tissues during indentation. This allows for an increased understanding of load transmission in soft tissue. The comparison of the two models in terms of strains indicates that the modelling choice significantly influences the localization of maximal compressive strains. These results support the idea that the biomechanical community should conduct further work to develop reliable methodologies for estimating in vivo strain in soft tissue.

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number \ell\geq 2 of components, for the Yamabe equation on a closed Riemannian manifold (M,g) . To this aim, we study a weakly coupled competitive elliptic system of \ell equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if \dim M\geq 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever \dim M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -\infty , giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For \ell=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.

Nonlinear dynamic analysis of aircraft CFRP sandwich wings under explosive blast loading: Introducing SVM-DNN algorithm to predict dynamical information

The aircraft wings are a crucial component, particularly during mechanical shock excitation. Thus, for the first time, an inventive adaptable model to simulate the nonlinear dynamics of the wings under explosive blast loading at the moment of the collision is described in this article. It is crucial to increase the structure's stability because of explosive blast loading. The mentioned structure is made of two composite face sheets and a carbon fiber-reinforced polymer (CFRP). Functionally graded (FG) graphene origami (GOri)-enabled auxetic metallic metamaterial (GOEAM) is introduced to improve the stability of this kind of applicable structure, especially in air maneuvers. The analysis is conducted utilizing Reddy's third-order shear deformation of Reddy theory (TSDRT), incorporating Hamilton's principle and Von Kármán nonlinear geometric assumptions. The governing and boundary equations are accurately computed at the domain and boundary edges of the doubly curved structure. The coupled mesh-free radial point interpolation method (MRPIM) and Newton-Raphson method are applied for numerically simulating the current doubly curved structure by employing Newmark's temporal integration method with a constant-average acceleration approach. The present findings are cross-checked against the machine learning method and open-source results from the literature. Based on physical data, machine learning integrates supervised machine learning to predict the nonlinear vibration of the system. In this case, nonlinear transient bending properties are found using hybrid machine learning methods and solutions. The findings demonstrate the significance of GOEAM for nonlinear vibrations of the composite system. The current study's conclusions could be used as a benchmark and useful recommendations for future structural design techniques with enhanced features.

이차계획법에 기반한 소파 이동 문제의 상한선 개선

The moving sofa problem is a problem in discrete geometry that involves finding a shape with the maximum area that can pass through a right-angled corner in a hallway with unit width, and has remained an open problem since it was posed by Moser in 1966. The largest lower bound on the maximum area known to date is 2.2195..., found by Gerver in 1992, and the smallest upper bound is 2.37, proved by Kallus and Romik in 2018. While the minimum upper bound of 2.37 appears to be somewhat loose, experimental evidence suggests that there is little room for improvement in the largest lower bound of 2.2195.... In this study, we aim to lower the upper bound to be closer to Gerver’s maximum lower bound. For this purpose, we construct an optimization algorithm based on geometric branch-and-bound and quadratic programming, and prove various geometric properties for efficient search of the algorithm. Under a reasonable geometric assumption, the proposed optimization algorithm obtains an improved upper bound of 2.3361 with much shorter computation time than Kallus and Romik.

Predictive modeling of 3D textile composites using realistic micromechanical representations

An innovative and efficient progressive damage model, informed by coupon tests, and using minimal, readily available material parameters, is proposed in this paper to comprehensively predict the tensile, compressive and shear stress–strain response and strengths of a 3D woven angle interlock composite within 5% of test data. Realistic, high-fidelity microgeometries, with process-induced defects, and devoid of geometric assumptions, are constructed using the digital element approach. Excellent correlations with test micrographs are observed for yarn cross-section areas, aspect ratios and crimp ratios, that are on average within 5.5%, 1.9% and 1% of the micrograph measurements respectively. The mesoscale predictions capture the detailed failure mechanisms under each loading condition. Analytical stiffness expressions are derived in terms of the constituent material elastic properties and architecture geometry, which also predict the stiffnesses within 10% error margin. A robust workflow is established that combines the high-fidelity microgeometry generation, automated conformal finite element model creation and comprehensive property prediction capabilities to provide a holistic solution.