Boundary Curves Research Articles

Collective self-caging of active filaments in virtual confinement

Motility coupled to responsive behavior is essential for many microorganisms to seek and establish appropriate habitats. One of the simplest possible responses, reversing the direction of motion, is believed to enable filamentous cyanobacteria to form stable aggregates or accumulate in suitable light conditions. Here, we demonstrate that filamentous morphology in combination with responding to light gradients by reversals has consequences far beyond simple accumulation: Entangled aggregates form at the boundaries of illuminated regions, harnessing the boundary to establish local order. We explore how the light pattern, in particular its boundary curvature, impacts aggregation. A minimal mechanistic model of active flexible filaments resembles the experimental findings, thereby revealing the emergent and generic character of these structures. This phenomenon may enable elongated microorganisms to generate adaptive colony architectures in limited habitats or guide the assembly of biomimetic fibrous materials.

Innovative solid waste backfilling technology for top coal caving mining: analysing top coal movement and recovery optimisation

ABSTRACT The study introduces a novel solid backfilling technology, aiming to address gangue accumulation problem in top coal caving mining. To reveal its top coal drawing mechanism, discrete physical and numerical models were established. During the initial drawing stage, considerable amounts of low-position top coal are extracted, with yields fluctuating. As the working face advances, the drawing body shape shifts from knife-handle to semi-arch, and coal-gangue boundary curvature varies periodically. At low backfilling ratios, premature gangue exposure causes the loss of high-position top coal. Conversely, higher backfilling ratios facilitate the creation of a reconstructed floor, thereby enhancing top coal recovery.

The first Dirichlet eigenvalue and the width

For a geodesic ball with non-negative Ricci curvature and mean convex boundary, it is known that the first Dirichlet eigenvalue of this geodesic ball has a sharp lower bound in term of its radius. We show a quantitative explicit inequality, which bounds the width of geodesic ball in terms of the spectral gap between the first Dirichlet eigenvalue and the corresponding sharp lower bound.

Математическая модель в задаче идентификации формы летательного аппарата по диаграмме рассеяния звукового поля

This paper proposes a mathematical model to solve an inverse problem reconstructing the shape and size of possible models of an unmanned aerial vehicle based on its acoustic trail. To simplify the calculations, four two-dimensional simplest geometric shapes were chosen for numerical experiments - a circle, an ellipse, a square and a four-petal rose. The scattering diagrams of the sound field, as a solution to direct diffraction problem for each contour, were taken as the input data. A residual functional is constructed as a difference between the true data of the scattered sound field and the data obtained using the semi-analytical boundary integral equation method and the Fredholm integral equation of the second kind, in the direct diffraction problem. The minimization of this functional leads to the inverse diffraction problem. After discretizing the boundary curve, the problem is reduced to a linear algebraic equation system, from which a function is found that numerically describes the shape of the object.

Intelligent frequency control of AC microgrids with communication delay: An online tuning method subject to stabilizing parameters

Smart control techniques have been implemented to address fluctuating power levels within isolated microgrids, mitigating the risk of unstable frequencies and the potential degradation of power supply quality. However, a challenge lies in the fact that employing these computationally complex methods without stability preservation might not suffice to handle the rapid changes of this highly dynamic environment in real-world scenarios over communication delays. This study introduces a flexible real-time approach for the frequency control problem using an artificial neural network (ANN) constrained to stabilized regions. Our solution integrates stabilizing PID controllers, computed through small-signal analysis and tuned via an automated search for optimal ANN weights and reinforcement learning (RL)-based selected constraints. First, we design stabilizing PID controllers by applying the stability boundary locus method and the Mikhailov criterion, specifically addressing communication delays. Next, we refine the controller parameters online through an automated process that identifies optimal coefficient combinations, leveraging a constrained ANN to manage frequency deviations within a restricted parameter range. Our approach is further enhanced by employing the RL technique, which trains the tuning system using an interpolated stability boundary curve to ensure both stability and performance. This one-of-a-kind combination of ANN, RL, and advanced PID tuning methods is a big step forward in how we handle frequency control problems in isolated AC microgrids. The experiments show that our solution outperforms traditional methods due to its reduced parameter search space. In particular, the proposed method reduces transient and steady-state frequency deviations more than semi- and unconstrained methods. The improved metrics and stability analysis show that the method improves system performance and stability under changing conditions.

An improved moment distribution method for the analysis of concrete frames

The moment distribution method is considered one of the easiest and most reliable analysis methods. However, little attention has been given to modelling the stiffness of each member separately, as currently only one factor is being used to model all structural members without taking into account the loading conditions and curvature of the member. This can significantly influence the results when modelling columns, since unlike beams, which are usually bent in a single curvature configuration, columns can be bent in either a single or double curvature configuration. This paper presents a new set of stiffness factors to model each structural member separately depending on its boundary conditions and curvature. To validate this modification, an example concrete frame was modelled and analysed using the structural analysis software ETABS, and then the results were compared with that obtained from the standard moment distribution method and the modified moment distribution method. The results have revealed a significant enhancement in the accuracy of the obtained results when using the modified moment distribution method compared with the original moment distribution method, especially the values of the columns’ bending moments.

An Efficient Progressive Grid Generation Method Considering Internal Quality and Boundary Grid Adjustment

Due to its elegant appearance and high structural efficiency, free-form grid structures are increasingly adopted in architectural design. However, it is a challenging task for engineers to generate a uniform and well-shaped grid on a free-form surface while considering the processing of both interior and boundary of grid, especially for composite surfaces. To generate well-shaped grids with uniform rods, regular cells and smooth visual effects over free-form surfaces, this paper develops an innovative triangular grid generation method capable of automatically managing grid boundary with extending initial surface. In the method, nodes of grid structure are considered to be zero mass particles and are progressively added to the extended surface from geometric center to the boundary of surface. A boundary-processing algorithm is then established to evenly distribute nodes on initial boundary curve, while ensuring that grid cells at the boundary do not exhibit elongation. A mechanical simulation system based on spring-mass model is improved to optimize spatial grid structure, rods of grid are regarded as linear springs, a surface attraction force and an anchoring force are performed on grid nodes. Moreover, the proposed method allows architects change grid direction and rods length easily, which can greatly improve the efficiency of design. Several case studies show that the method can effectively avoid distortion of grids and generate well-shaped grids that can meet aesthetic requirements.

Competition between microstructural factors affecting growth of abnormally large grains in thin Cu foils

Grain boundary types and local boundary curvatures are generally considered to be important microstructural factors controlling grain boundary migration during grain growth. In this work, grain growth in thin copper foils is studied during annealing at a temperature of 1040 °C near the melting point by ex-situ experiments and Monte Carlo simulations. Few grains, stimulated by slight deformation at the sample edge, are observed to grow abnormally into the cube-oriented recrystallized microstructure with columnar grains spanning the foil thickness. The grain boundaries of these abnormally growing grains and the grain sizes in the adjacent polycrystalline recrystallized regions are analyzed. The experimental results suggest that spatial heterogeneities in the distribution of small recrystallized grains have a significant effect on the migrating boundaries. Potts model simulations confirm that grain boundary segments with small grains in front are more likely to migrate than segments facing coarser grains. The simulations also demonstrate the importance of grain morphology. Altogether, this work highlights the effects of a heterogeneous recrystallized microstructure on abnormal grain growth in thin foil samples.

Study on the abnormal growth of goss oriented grains in Fe-3.2%Si steel: Role of thermomechanical processing and inhibitors

Detailed characterization of Fe-3.2%Si steel after different stages of processing trails including hot-rolling, normalization annealing, cold-rolling with intermediate annealing, and finally, decarburization/primary annealing and secondary annealing has been carried out. The evolution of ferrite grain structure, grain boundary characteristics, and the micro-texture of the samples are characterized using electron backscatter diffraction (EBSD). The inhibiting precipitates such as MnS and CuS are studied under scanning and transmission electron microscopy (SEM and TEM). The evolution of ferrite grain structure and texture are explained considering the mechanisms (role of inhibitors, crystal orientation, special boundary, high energy medium angle boundary, grain boundary curvature) responsible for the abnormal growth of Goss oriented grains after final annealing treatment. Extremely coarse Goss oriented grains (as large as >20 mm) have been developed in this study. An analytical approach is presented at the end to explain the texture evolution during processing considering the Taylor factor maps (for different loading conditions) and the prediction of limiting ferrite grain size for abnormal grain growth, considering the size and fraction of the precipitates along with their coarsening and dissolution kinetics.

Riemann Problem for the Isentropic Euler Equations of Mixed Type in the Dark Energy Fluid

We are concerned with the Riemann problem for the isentropic Euler equations of mixed type in the dark energy fluid. This system is non-strictly hyperbolic on the boundary curve of elliptic and hyperbolic regions. We obtain the unique admissible shock waves by utilizing the viscosity criterion. Assuming fixed left states are in the elliptic and hyperbolic regions, respectively, we construct the unique Riemann solution for the mixed-type models with the initial right state in some feasible regions. Finally, we present numerical simulations which are consistent with our theoretical results.

Re-orientation of principal stresses during tunnel excavation: Model development and support strategy

The safety and stability during the excavation and production of the tunnel is a key issue that must be addressed in tunneling engineering, which is pronounced under principal stress re-orientation. A mechanical model is therefore developed to determine the boundary curve of the failure zone of a supported circular tunnel under rotating the principal stress direction. Meanwhile, the effects of various environmental factors on the pattern of the failure zone are quantitatively analyzed. Subsequently, two critical morphological indexes are determined to elucidate the uniformity of the failure zone and the functional relationship between these two morphological indices and the environmental factors, respectively. Furthermore, technical approaches and control measures for maintaining the surrounding rock are put forward. In particular, the “long and short” coordinated hierarchical support technology has been recommended and is validated through a comparison of the designed schemes and several engineering cases. The results indicate that the proposed support technology resulted in a gradual reduction of the “butterfly wings” pattern and a more consistent failure zone in the surrounding rock. These findings are useful to ensure the overall stability and anti-interference performance of the surrounding rock of a tunnel under the complicated stress field, especially with regards to the deflection of principal stress.

Robust self-assembly of nonconvex shapes in two dimensions.

We present fast simulation methods for the self-assembly of complex shapes in two dimensions. The shapes are modeled via a general boundary curve and interact via a standard volume term promoting overlap and an interpenetration penalty. To efficiently realize the Gibbs measure on the space of possible configurations we employ the hybrid Monte Carlo algorithm together with a careful use of signed distance functions for energy evaluation. Motivated by the self-assembly of identical coat proteins of the tobacco mosaic virus which assemble into a helical shell, we design a nonconvex two-dimensional model shape and demonstrate its robust self-assembly into a unique final state. Our numerical experiments reveal certain essential prerequisites for this self-assembly process: blocking and matching (i.e., local repulsion and attraction) of different parts of the boundary, and nonconvexity and handedness of the shape.

Boundary parameter matching for isogeometric analysis using Schwarz–Christoffel mapping

AbstractIsogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. Existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz–Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter matching method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.

Application of melt pool profiles for parameter calibration of Goldak’s heat source model

Goldak’s ellipsoidal heat source model is widely accepted in laser powder bed fusion (LPBF) process simulations. The model is dependent on three inherent flux distribution parameters and absorptivity which need to be calibrated before application. However, the uniqueness of the calibrated heat source parameters has not been established, which is crucial to accurately represent the energy density distribution in numerical studies. This paper proposed a novel parametric optimization method that can uniquely calibrate these four heat source parameters with high accuracy. Both the simulated cross-sectional (CS) profile and the mid-front (MF) profile of the melt pool were used during calibration. The differences between the experimental and simulated profiles formed the objective function of the parametric optimization problem, which was calculated by using the least power norm method on the deviations between melt pool boundary curves. A genetic algorithm (GA) with elitism selection approach was employed to optimize the heat source parameters by minimizing this single objective function value. To theoretically verify the applicability of the method, two theoretical case studies involving two distinct sets of heat source parameters were carried out. A unique solution of Goldak’s heat source parameters with less than 2 % error can be achieved within 50 generations of the optimization process. A single-track experiment was conducted to demonstrate the application of the proposed method in a real scenario. The CS and MF melt pool profiles were extracted and the calibration procedure was applied. The profiles from simulation with the calibrated heat source parameters exhibited a good agreement with experimental results. The calibrated parameters were used to simulate multi-track printing on a single layer for validation of the proposed method. The CS melt pool profiles obtained from the multi-track simulation and multi-track experiment agreed well, which validated the accuracy of the proposed calibration method and its utility in obtaining the key Goldak heat source parameters for LPBF simulation.

A novel Newton method for inverse elastic scattering problems

This work is concerned with an inverse elastic scattering problem of identifying the unknown rigid obstacle embedded in an open space filled with a homogeneous and isotropic elastic medium. A Newton-type iteration method relying on the boundary condition is designed to identify the boundary curve of the obstacle. Based on the Helmholtz decomposition and the Fourier–Bessel expansion, we explicitly derive the approximate scattered field and its derivative on each iterative curve. Rigorous mathematical justifications for the proposed method are provided. Numerical examples are presented to verify the effectiveness of the proposed method.

Plateau’s problem via the Allen–Cahn functional

Let Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} be a compact codimension-two submanifold of Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, and let L be a nontrivial real line bundle over X=Rn\\Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X = {\\mathbb {R}}^n {\\setminus } \\Gamma $$\\end{document}. We study the Allen–Cahn functional, Eε(u)=∫Xε|∇u|22+(1-|u|2)24εdx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}E_\\varepsilon (u) = \\int _X \\varepsilon \\frac{|\ abla u|^2}{2} + \\frac{(1-|u|^2)^2}{4\\varepsilon }\\,dx, \\\\\\end{aligned}$$\\end{document}on the space of sections u of L. Specifically, we are interested in critical sections for this functional and their relation to minimal hypersurfaces with boundary equal to Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. We first show that, for a family of critical sections with uniformly bounded energy, in the limit as ε→0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon \\rightarrow 0$$\\end{document}, the associated family of energy measures converges to an integer rectifiable (n-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-1)$$\\end{document}-varifold V. Moreover, V is stationary with respect to any variation which leaves Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} fixed. Away from Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}, this follows from work of Hutchinson–Tonegawa; our result extends their interior theory up to the boundary Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}. Under additional hypotheses, we can say more about V. When V arises as a limit of critical sections with uniformly bounded Morse index, Σ:=supp‖V‖\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma := {{\\,\ extrm{supp}\\,}}\\Vert V\\Vert $$\\end{document} is a minimal hypersurface, smooth away from Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} and a singular set of Hausdorff dimension at most n-8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-8$$\\end{document}. If the sections are globally energy minimizing and n=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n = 3$$\\end{document}, then Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} is a smooth surface with boundary, ∂Σ=Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial \\Sigma = \\Gamma $$\\end{document} (at least if L is chosen correctly), and Σ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma $$\\end{document} has least area among all surfaces with these properties. We thus obtain a new proof (originally suggested in a paper of Fröhlich and Struwe) that the smooth version of Plateau’s problem admits a solution for every boundary curve in R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^3$$\\end{document}. This also works if 4≤n≤7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$4 \\le n\\le 7$$\\end{document} and Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document} is assumed to lie in a strictly convex hypersurface.

Liquefaction resistance and small strain stiffness of silty sand: Effects of host sand gradation and fines content

In this paper, a well designed experimental study comprising undrained cyclic triaxial and bender element tests was carried out on five sand-silt mixtures with three grain size ratios (Rd) and three sand uniformity coefficients (Cus). The results reveal that at given fines content (FC) and void ratio, the liquefaction resistance (CRR) of the mixtures decreases as Rd increases from 9.7 to 27.8 or Cus from 1.50 to 4.46, and the CRR of the mixture at the highest Rd or Cus becomes as low as nearly 50% of that at the lowest Rd or Cus. The equivalent relative density (Dr*) based on the binary packing theory can be utilized to capture the coupled effects of host sand gradation and FC on the CRR of sand-silt mixtures. A novel finding is that a universal relationship exists between the liquefaction resistance and the shear modulus normalized by the minimum void ratio function of host sand (GN') for different FCs, initial effective confining pressures, sample preparation methods (SPMs), and host sand gradations. Based on the comprehensive laboratory test data, the empirical CRR15-GN' correlation is proposed and further adjusted for the field liquefaction triggering analysis based on the shear wave velocity (Vs). Soil can be considered non-liquefiable when its equivalent shear wave velocity exceeds 225 m/s or the applied cyclic stress ratio (CSR) is below 0.088. Comparisons with the previously established liquefaction triggering boundary curves validate the applicability of the proposed curve for initial evaluation of the liquefaction resistance of sandy soils, indicating laboratory studies can complement in-situ investigations for Vs-based liquefaction assessment.

The extended persistent homology transform of manifolds with boundary

A shortcoming of persistent homology is that when two domains have different numbers of components or holes the persistence diagrams of any filtration will have an infinite distance between them. We address this issue by revisiting the theory of extended persistence, initially developed by Cohen-Steiner, Edelsbrunner and Harer in 2009 to quantify the support of the essential homology classes for Morse functions on manifolds. We simplify the mathematical treatment of extended persistence by formulating it as a persistence module derived from a sequence of relative homology groups for pairs of spaces. Then, for n-manifolds with boundary embedded in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^n$$\\end{document}, we use Morse theory to show that the extended persistent homology of a height function over M can be deduced from the extended persistent homology of the same height function restricted to ∂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial M$$\\end{document}. As an application, we describe the extended persistent homology transform (XPHT); a topological transform which takes as input a shape embedded in Euclidean space, and to each unit vector assigns the extended persistence module of the height function over that shape with respect to that direction. We define a distance between two shapes by integrating over the sphere the distance between the respective extended persistence modules. By using extended persistence we get finite distances between shapes even when they have different numbers of essential classes. We study the application of the XPHT to binary images; outlining an algorithm for efficient calculation of the XPHT exploiting relationships between the PHT of the boundary curves to the extended persistence of the foreground.

The architectural application of shells whose boundaries subtend a constant solid angle

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements.In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners.Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown.We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions.

An efficient unstructured quadrilateral-dominated surface mesh generation method for arbitrary geometry with tiny features or crack defects in DiBFM

An efficient unstructured quadrilateral-dominated surface mesh generation algorithm is presented to generate high-quality surface elements for arbitrary geometry with tiny features or crack defects. For complex entities with proximity features, a novel type of meshing template has been proposed to solve the problem of entity boundary fitting of the outside-in grid-based method. The algorithm also works for regions with proximity features, as well as for arbitrary geometry with one or multiple cracks. The procedure incorporates aspects of well-known meshing techniques, but includes some original and efficient processes. The refinement field is constructed according to mesh size, boundary curvature, surface curvature and other geometry features. In order to effectively generate the core mesh, a fast intersection algorithm between topological elements and the solid boundary is established, which improves the efficiency of intersection calculation. The geometry-based techniques are employed for boundary matching of the core mesh. The method has been successfully employed in conjunction with the dual interpolation boundary face method (DiBFM). Several realistic geometries with cracks or proximity features are applied to demonstrate the accuracy and validity of the proposed approach.