Convex Hull Research Articles

3D X-ray Tomography Analysis of Mg–Si–Zn Alloys for Biomedical Applications: Elucidating the Morphology of the MgZn Phase

Temporary metal implants, made from materials like titanium (Ti) or stainless steel, can cause metabolic issues, raise toxicity levels within the body, and negatively impact the patient’s long-term health. This necessitates a subsequent operation to extract these implants once the healing process is complete or when they are outgrown by the patient. In contrast, medical devices fabricated from absorbable alloys have the advantage of being biodegradable, allowing them to be naturally absorbed by the body once they have fulfilled their role in facilitating tissue healing. Among the various absorbable alloy systems studied, magnesium (Mg) alloys stand out due to their biocompatibility, mechanical properties, and corrosion behavior. The existing literature on absorbable Mg alloys highlights the effectiveness of silicon (Si) and zinc (Zn) additions in improving mechanical properties and controlling corrosion susceptibility; however, there is a lack of comprehensive quantitative morphological analysis of the intermetallic phases within these alloy systems. The quantification of the complex morphology of intermetallic particles is a challenging task and has significant implications for the micromechanical properties of the alloys. This study, therefore, aims to introduce a robust set of morphometric parameters for evaluating the morphology of intermetallic phases within two as-cast Mg alloys with Si and Zn additions. X-ray Computed Tomography (XCT) was used to capture the 3D tomographic data of the alloys, and a novel pair of morphological parameters (ratio of convex hull to particle volume and convex hull sphericity) was applied to the 3D tomographic data to assess the MgZn phase formed in the two alloys. In addition to the impact of composition, the effect of solidification rate on the morphological parameters was also studied. Furthermore, Scanning Electron Microscopy (SEM) and Energy-Dispersive Spectroscopy (EDS) were employed to gather detailed 2D microstructural and compositional information on the intermetallics. The comprehensive characterization reveals that the morphological complexity and size distribution of the MgZn phase are influenced by both compositional changes and the solidification rate. However, the change in MgZn intermetallic particle morphology with size was found to follow a predictable trend, which was relatively agnostic of the chosen casting conditions.

Bayesian optimization acquisition functions for accelerated search of cluster expansion convex hull of multi-component alloys

Atomistic simulations are crucial for predicting material properties and understanding phase stability, essential for materials selection and development. However, the high computational cost of density functional theory calculations challenges the design of materials with complex structures and composition. This study introduces new data acquisition strategies using Bayesian-Gaussian optimization that efficiently integrate the geometry of the convex hull to optimize the yield of batch experiments. We developed uncertainty-based acquisition functions to prioritize the computation tasks of configurations of multi-component alloys, enhancing our ability to identify the ground-state line. Our methods were validated across diverse materials systems including Co-Ni alloys, Zr-O compounds, Ni-Al-Cr ternary alloys, and a planar defect system in intermetallic (Ni1−x, Cox)3Al. Compared to traditional genetic algorithms, our strategies reduce training parameters and user interaction, cutting the number of experiments needed to accurately determine the ground-state line by over 30%. These approaches can be expanded to multi-component systems and integrated with cost functions to further optimize experimental designs.

Stability and Electronic Properties of K‐Sb and Na‐Sb Binary Crystals from High‐Throughput Ab Initio Calculations

AbstractThe study of the fundamental properties of alkali antimonide photocathodes for particle accelerators is currently hindered by the limited purity of the samples. First‐principles studies can effectively complement experiments to gain insight into the stability and the electronic structure of these compounds. In this high‐throughput analysis based on density‐functional theory (DFT), two families of binary crystals with K‐Sb and Na‐Sb compositions expected to form during evaporation of multi‐alkali antimonide photocathodes are investigated. Starting from an initial pool of structures mined from existing computational databases, automatized routines included in the in‐house developed library aim2dat are employed to determine the stability and the electronic properties of the aforementioned systems. By analyzing the formation energy, the structures are ranked in a convex hull retaining the information of their crystalline arrangement. Next, the band structure and the projected density of states of selected stable compounds are analyzed. Adopting the r2SCAN functional for the DFT calculations, reliable estimates of the character and size of the bandgaps are obtained and discussed in relation to the relative alkali content in the crystals. These results provide useful indications to predict and characterize binary phases forming during the growth of multi‐alkali antimonide photocathodes.

Vectorcardiography Predicts Heart Failure in Patients Following ST Elevation Myocardial Infarction.

Modeling outcomes, such as onset of heart failure (HF) or mortality, in patients following ST elevation myocardial infarction (STEMI) is challenging but clinically very useful. The acute insult following a myocardial infarction and chronic degeneration seen in HF involve a similar process where a loss of cardiomyocytes and abnormal remodeling lead to pump failure. This process may alter the strength and direction of the heart's net depolarization signal. We hypothesize that changes over time in unique parameters extracted using vectorcardiography (VCG) have the potential to predict outcomes in patients post-STEMI and could eventually be used as a noninvasive and cost-effective surveillance tool for characterizing the severity and progression of HF to guide evidence-based therapies. We identified 162 patients discharged from Michigan Medicine between 2016 and 2021 with a diagnosis of acute STEMI. For each patient, a single 12-lead ECG > 1 week pre-STEMI and > 1 week post-STEMI were collected. A set of unique VCG parameters were derived by analyzing features of the QRS complex. We used LASSO regression analysis incorporating clinical variables and VCG parameters to create a predictive model for HF, mortality, or the composite at 90, 180, and 365 days post-STEMI. The VCG model is most predictive for HF onset at 90 days with a robust AUC. Variables from the HF model mitigating or driving risk, at a p < 0.05, were primarily parameters that assess the area swept by the depolarization vector including the 3D integral and convex hull in select spatial octants and quadrants.

Cooperative learning‐based practical formation‐containment control with prescribed performance for heterogeneous clusters of UAV/USV

SummaryIn this paper, a new approach for formation‐containment control with prescribed performances is introduced for heterogeneous autonomous vehicles involving a cluster of leader unmanned aerial vehicles (UAVs) and follower unmanned surface vessels (USVs). We introduce a two‐layer distributed control system: The upper layer focuses on guiding the UAVs to form a scalable lattice while synchronizing their movement along a predefined path, and the second layer guides the USVs to enter the convex hull formed by the UAVs, ensuring collision‐free operation with static/dynamic objects. To prevent collisions and ensure lattice formation, a set of well‐defined bump functions are utilized in the design of the backstepping control algorithm. Managing virtual controls, we incorporate a nonlinear dynamic surface control (NDSC), while a universal barrier function enhances the convergence of formation tracking errors. Furthermore, each USV employs a cooperative adaptive learning neural network to handle uncertainties in heterogeneous vehicle models. Utilizing the Lyapunov theorem, the stability of the formation‐containment of UAV/USV is achieved, and all signals in the formation‐containment systems are semiglobal uniform ultimate bounded (SGUUB). A simulation example showcases the effectiveness of our proposed approach, highlighting contributions in collision avoidance, synchronization speed, and adaptive learning. Our work advances the heterogeneous formation‐containment literature towards more realistic scenarios with safety‐critical considerations amidst multiple layers of uncertainties and unknown parameters.

Hierarchical rank-one sequence convexification for the relaxation of variational problems with microstructures

This paper presents an efficient algorithm for the approximation of the rank-one convex hull in the context of nonlinear solid mechanics. It is based on hierarchical rank-one sequences and simultaneously provides first and second derivative information essential for the calculation of mechanical stresses and the computational minimisation of discretised energies. For materials, whose microstructure can be well approximated in terms of laminates and where each laminate stage achieves energetic optimality with respect to the current stage, the approximate envelope coincides with the rank-one convex envelope. Although the proposed method provides only an upper bound for the rank-one convex hull, a careful examination of the resulting constraints shows a decent applicability in mechanical problems. Various aspects of the algorithm are discussed, including the restoration of rotational invariance, microstructure reconstruction, comparisons with other semi-convexification algorithms, and mesh independency. Overall, this paper demonstrates the efficiency of the algorithm for both, well-established mathematical benchmark problems as well as nonconvex isotropic finite-strain continuum damage models in two and three dimensions. Thereby, for the first time, a feasible concurrent numerical relaxation is established for an incremental, dissipative large-strain model with relevant applications in engineering problems.

Concrete Crack Detection and Segregation: A Feature Fusion, Crack Isolation, and Explainable AI-Based Approach.

Scientific knowledge of image-based crack detection methods is limited in understanding their performance across diverse crack sizes, types, and environmental conditions. Builders and engineers often face difficulties with image resolution, detecting fine cracks, and differentiating between structural and non-structural issues. Enhanced algorithms and analysis techniques are needed for more accurate assessments. Hence, this research aims to generate an intelligent scheme that can recognize the presence of cracks and visualize the percentage of cracks from an image along with an explanation. The proposed method fuses features from concrete surface images through a ResNet-50 convolutional neural network (CNN) and curvelet transform handcrafted (HC) method, optimized by linear discriminant analysis (LDA), and the eXtreme gradient boosting (XGB) classifier then uses these features to recognize cracks. This study evaluates several CNN models, including VGG-16, VGG-19, Inception-V3, and ResNet-50, and various HC techniques, such as wavelet transform, counterlet transform, and curvelet transform for feature extraction. Principal component analysis (PCA) and LDA are assessed for feature optimization. For classification, XGB, random forest (RF), adaptive boosting (AdaBoost), and category boosting (CatBoost) are tested. To isolate and quantify the crack region, this research combines image thresholding, morphological operations, and contour detection with the convex hulls method and forms a novel algorithm. Two explainable AI (XAI) tools, local interpretable model-agnostic explanations (LIMEs) and gradient-weighted class activation mapping++ (Grad-CAM++) are integrated with the proposed method to enhance result clarity. This research introduces a novel feature fusion approach that enhances crack detection accuracy and interpretability. The method demonstrates superior performance by achieving 99.93% and 99.69% accuracy on two existing datasets, outperforming state-of-the-art methods. Additionally, the development of an algorithm for isolating and quantifying crack regions represents a significant advancement in image processing for structural analysis. The proposed approach provides a robust and reliable tool for real-time crack detection and assessment in concrete structures, facilitating timely maintenance and improving structural safety. By offering detailed explanations of the model's decisions, the research addresses the critical need for transparency in AI applications, thus increasing trust and adoption in engineering practice.

Fully distributed convex hull pricing based on alternating direction method of multipliers

In certain power markets, due to the non-convex operation characteristics, generators adhering to the decisions of Independent System Operators (ISO) may struggle to recover costs through local marginal energy sales. ISOs impose discriminatory additional payments as incentives for generator compliance. Convex hull pricing is a unified scheme that significantly reduces these supplementary payments. The Lagrangian dual problem of the Unit Commitment (UC) problem is solved within the dual space to determine convex hull prices. To navigate the computational challenges posed by the real large-scale power systems, we propose a methodology for a global convergence distributed solution, which addresses the Lagrangian dual problem. This methodology is based on the Alternating Direction Method of Multipliers (ADMM) algorithm and incorporates convex hull cut planes to enhance computational efficiency. Moreover, a tight and compact UC model is employed to reduce the number of iterations. Numerical results indicate that if the convex hull descriptions of units can be obtained, our algorithm is capable of providing precise convex hull prices and high-quality solutions within a feasible timeframe, while also maintaining the confidentiality of individual subset unit information.

Numerical prediction of beam capacity using the new constitutive model based on concrete biaxial behavior

The biaxial strength of concrete exhibits intriguing nonlinearity and scattering, posing challenges to the assessment of building structural safety. Experiments have confirmed that biaxial strength envelopes can manifest concave, quasi-linear, and convex shapes. Despite various empirical formulas being proposed to describe this behavior, they cannot be applied to finite element simulations due to the absence of a constitutive perspective. To study the mechanisms forming the shape of strength envelopes, a novel three-invariant model constitutive model is proposed, which reveals the characteristics of the envelopes resulting from the coupling effect of hydrostatic pressure and Lode angle. By adjusting the model parameters, it effectively describes concave, quasi-linear, and convex envelopes. The model's accuracy and compatibility were validated against experimental data spanning various concrete types, with compression strengths ranging from 19.29 to 96.5 MPa and tension-compression ratios form 0.056 to 0.095. To investigate how biaxial strength impacts the capacity of beams, numerical experiments were conducted on notched and intact beams based on the proposed model. The results underscored that beams prepared using concrete with convex envelopes exhibited higher capacity compared to those with quasi-linear envelopes. Therefore, conducting biaxial tension-compression experiments is recommended to accurately evaluate beam bearing capacity. The proposed model is expected to have further applications in the safety evaluation of building structures.

Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications

Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications

Expanding the reach of quantum optimization with fermionic embeddings

Quadratic programming over orthogonal matrices encompasses a broad class of hard optimization problems that do not have an efficient quantum representation. Such problems are instances of the little noncommutative Grothendieck problem (LNCG), a generalization of binary quadratic programs to continuous, noncommutative variables. In this work, we establish a natural embedding for this class of LNCG problems onto a fermionic Hamiltonian, thereby enabling the study of this classical problem with the tools of quantum information. This embedding is accomplished by a new representation of orthogonal matrices as fermionic quantum states, which we achieve through the well-known double covering of the orthogonal group. Correspondingly, the embedded LNCG Hamiltonian is a two-body fermion model. Determining extremal states of this Hamiltonian provides an outer approximation to the original problem, a quantum analogue to classical semidefinite relaxations. In particular, when optimizing over the special orthogonal group our quantum relaxation obeys additional, powerful constraints based on the convex hull of rotation matrices. The classical size of this convex-hull representation is exponential in matrix dimension, whereas our quantum representation requires only a linear number of qubits. Finally, to project the relaxed solution back into the feasible space, we propose rounding procedures which return orthogonal matrices from appropriate measurements of the quantum state. Through numerical experiments we provide evidence that this rounded quantum relaxation can produce high-quality approximations.

Investigation of thermodynamical stability and generation of large half-metallic gap along with optical properties in N-doped YHfO3 (YHO) (Y = Ca, Sr, and Ba) perovskite materials by first principle calculations

Considerable energy gap (Eg) in insulating channel, high spin-filtering, significant mean free path, and high magnetic transition temperature (TC) escalate half-metallic (HM) ferromagnetic (FM) materials very enticing for spintronic applications. Herein, ab-initio calculations have been executed comprehensively to inquest the structural, thermodynamical, electronic, magnetic, and optical properties of pure as well as nitrogen (N)-doped YHfO3 (YHO) (Y = Ca/Sr/Ba) materials with one N atom added at one of the oxygen (O) site. First, the thermodynamical stability of both bulk and doped YHO systems has been examined by estimating the formation enthalpies (ΔHf) by Convex Hull analysis, which confirms the practical realization of these materials on experimental grounds. Moreover, mechanical stability of these systems is affirmed by estimating the necessary elastic constants and respective parameters. Strikingly, all doped systems possess complete (100%) splitting in spin↑ and spin↓ channels at Fermi level and depict HM FM character with significant Eg of 3.31, 3.49, and 3.24 eV in spin↓ channels for CaHO (CHO), SrHO (SHO), and BaHO (BHO) systems, respectively. It is also found that N 2p orbitals are totally responsible for the emergence of metatallicity and magnetic character in all these doped structures. Moreover, the ground state long range FM stability between two added N atoms in doped systems is affirmed by computing the difference of total energies of FM and anti-ferromagnetic (AFM) spin ordering (ΔE) along with estimated TC at different distances. Lastly, it is found that N-doped systems have a lower optical energy loss than pristine YHO systems in the incident photon energy range of 0-50 eV. Hence, the present study proposes that an unconventional doping approach with intrinsically non-magnetic element in a variety of YHO perovskite systems could be a useful method to alter their various physical properties and forecast their potential applications in the development of novel spintronic and optoelectronic devices.

Observer-based time-varying formation-containment tracking of general linear multi-agent systems with input saturation

In this article, we study the time-varying formation-containment (TVFC) tracking problem for linear multi-agent systems (MASs) and take the input saturation problem of the systems into account. An observer-based TVFC tracking protocol is proposed. It has been proven that the real leaders can realize the expected formation and track the virtual leader’s trajectory, and followers can get into the convex envelope spanned by the leaders. In addition, to avoid unexpected large chattering caused by nonzero input, we propose a continuous protocol. Under this protocol, each error is uniformly ultimately bounded and is able to converge to any small zero neighborhood. At the end, the feasibility of the TVFC theory is demonstrated by simulation examples.

Description lattices of generalised convex hulls

In this article, we present a new approach to tackle lattice generation for complex and heterogeneous data using the concept of convexity. This is a work that we have already carried out, albeit intuitively [11] where we proposed the NextPriorityConcept algorithm for generating a meet-semilattice of concepts based on suitable descriptions and strategies. Now, we revisit the essential properties of our description spaces using a stronger formalism based on the properties of closure operators.

On the strength of Lagrangian duality in multiobjective integer programming

AbstractThis paper investigates the potential of Lagrangian relaxations to generate quality bounds on non-dominated images of multiobjective integer programs (MOIPs). Under some conditions on the relaxed constraints, we show that a set of Lagrangian relaxations can provide bounds that coincide with every bound generated by the convex hull relaxation. We also provide a guarantee of the relative quality of the Lagrangian bound at unsupported solutions. These results imply that, if the relaxed feasible region is bounded, some Lagrangian bounds will be strictly better than some convex hull bounds. We demonstrate that there exist Lagrangian multipliers which are sparse, satisfy a complementary slackness property, and generate tight relaxations at supported solutions. However, if all constraints are dualized, a relaxation can never be tight at an unsupported solution. These results characterize the strength of the Lagrangian dual at efficient solutions of an MOIP.

Containment-Based Distributed Secondary Control for AC Shipboard Microgrids under General Noise

This paper investigates the secondary control problem of shipboard microgrids (SMGs) with a high percentage of new energy sources under general noise. Firstly, a polymorphic SMG model is constructed, which enables the software-defined functionality of the control strategy and allows heterogeneous distributed generators (DGs) in AC SMGs to exchange packets of different types. Secondly, due to the presence of highly dynamic and high-power loads in the SMGs, a containment-based distributed secondary control strategy is proposed to improve the flexibility of the DG voltage regulation. Then, considering the complexity and diversity of disturbances during ship navigation, general noise is introduced instead of white noise to describe various disturbances. Furthermore, based on the random differential equations (RDEs), the NOS stability of the proposed strategy is proved using Lyapunov theory, which proves the effectiveness of the containment-based distributed secondary control strategy under general noise. And, the containment error is obtained to prove that the voltage and frequency of the system converge to the convex hull spanned by the virtual leaders, ensuring the high quality of the power supply. Finally, the validity of the proposed containment-based strategy is verified by an AC SMG model with four DGs in three cases.

Deep insights into the sodium transport mechanism and phase evolution in Na4VP2O9 cathode/anode material

For the entire intercalation/deintercalation range of Na4VP2O9, quantum mechanics is applied to investigate sodium transport mechanism and a machine learning force field is used instead of ab initio method to offers an applicable method for the full understanding of the complicated Na + movements dynamically. It shows chain of '8′ shaped units horizontally and repeated alternating round aggregate and '8′ shaped long chain vertically. For primitive Na4VP2O9, facile Na+ transportation along a-axis is observed with an energy barrier of 0.16 V. As a cathode material, Na3VP2O9 also shows fast Na + diffusion with a minimal diffusion energy barrier 0.27 eV along a-axis. However, for the fully sodiated state of Na4VP2O9, Na+ diffusion in the lattice is significantly restricted with an energy barrier of 0.92 eV along b-axis. Accordingly, the sodium ion diffusion coefficients at room temperature for Na3VP2O9, Na4VP2O9 and Na5VP2O9 are 1.2 × 10−9 cm2 s−1, 5.9 × 10−9 cm2 s−1 and 8.2 × 10−11 cm2 s−1, respectively. The formation energy convex hull originated from the removal of Na1 sites and the occupation of E8cO8 cavities reveals an intermediate phase, Na3.5VP2O9, for the NaxVP2O9 (3 ≤ x ≤ 4) and three intermediate phases, Na4.5VP2O9, Na4.625VP2O9 and Na4.75VP2O9, for the NaxVP2O9 (4 ≤ x ≤ 5).

Viscosity solutions of obstacle equations by inhomogeneous convex envelopes

We characterize the convex envelope of a continuous function as a viscosity solution of a second order differential equation of obstacle style. The notion of convexity that we investigate is an inhomogeneous generalization of the classical concept of convexity motivated by problems of singular control in finance.

Virtual skeleton and body mass for revealing the life strategies of Sinosaurus

ABSTRACT Sinosaurus from the Lufeng Basin, Yunnan Province, is the largest and most famous predatory dinosaur from the Early Jurassic in China. Although a number of specimens have been discovered in the past, no studies to date have focused exclusively on complete skeletal articulation, estimation of mass, or life strategies. Based on surface scans of a new well-preserved specimen, a digital skeleton of Sinosaurus was produced in two poses: osteologically neutral pose and extended pose. We estimated the possible living body mass by three volumetric-density methods, advocating the use of the expanding convex hulls method of 849.6 kg with the COM (0.41, 1.31, 0.00) as a more reliable result, which is greater than the mass of traditional regression formulas. The body shape and skull/body length suggested that Sinosaurus was a great runner, and this, combined with the weak crests and large forelimbs, further led to the conclusion that both the head and forelimbs were required to prey. This finding is also consistent with the finding that Sinosaurus has a moderate ability to metabolise body heat in the skin, as reflected by the surface/volume ratio.

Minimally fine exhausters

In this article, we study upper exhausters of positively homogenous functions defined on locally convex space. We prove the existence of minimally fine exhauster finer than the given one. Such exhauster has to be unique for two-element exhauster. We show that a positively homogenous function is upper semicontinuous if and only if it has an upper exhauster. We prove that Clarke subdifferential of a positively homogenous function is the closed convex hull of the union of its minimally fine exhauster. We also present a number of appropriate examples.