Presence Of Corners Research Articles

A numerical study of the generalized Steklov problem in planar domains

Abstract We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry.

Flow around a slender body with sharp edges

AbstractThe behavior of harmonic functions near corners of the domain boundary has a universal character that may be applied to the study of the flow around slender bodies with sharp edges. Depending of the energy of the potential function near the tip of the corner, this may yield a fast rotating flow or a vortex in its vicinity. For internal angles, such as occur in grooves of the body, a stagnation region or an eddy may occur. Lateral forces are also affected by the presence of corners, yielding configurations that resemble either a circulation lift or a vortex lift.

Cornering gravitational entropy

We present a new derivation of gravitational entropy functionals in higher-curvature theories of gravity using corner terms that are needed to ensure well-posedness of the variational principle in the presence of corners. This is accomplished by cutting open a manifold with a conical singularity into a wedge with boundaries intersecting at a corner. Notably, our observation provides a rigorous definition of the action of a conical singularity that does not require regularization. For Einstein gravity, we compute the Rényi entropy of gravitational states with either fixed-periodicity or fixed-area boundary conditions. The entropy functional for fixed-area states is equal to the corner term, whose extremization follows from the variation of the Einstein action of the wedge under transverse diffeomorphisms. For general Lovelock gravity the entropy functional of fixed-periodicity states is equal to the Jacobson-Myers (JM) functional, while fixed-area states generalize to fixed-JM-functional states, having a flat spectrum. Extremization of the JM functional is shown to coincide with the variation of the Lovelock action of the wedge. For arbitrary F(Riemann) gravity, under special periodic boundary conditions, we recover the Dong-Lewkowycz entropy for fixed-periodicity states. Since the variational problem in the presence of corners is not well-posed, we conjecture the generalization of fixed-area states does not exist for such theories without additional boundary conditions. Thus, our work suggests the existence of entropy functionals is tied to the existence of corner terms which make the Dirichlet variational problem well-posed.

Modeling Approach to Determine Static Rivulet Height in Regular Polygonal Capillary Tubes.

The rise of partially wetting liquids along the corners of noncircular capillary tubes is observed in many practical science and engineering applications such as wastewater treatment using membranes, remediation, oil recovery from petroleum reservoirs, and blood flow. In this paper, rivulet rise at the corners of polygonal capillary tubes is studied for partially wetting liquids with contact angles below the critical value. The presence of corners changes the distribution of a liquid in an incomplete wetting condition. In this study, geometrical models are proposed to better understand the capillary rise and flow behavior at the corners. A geometrical solution for the capillary rivulet height and profile is derived under gravity in triangular, square, and pentagonal capillary tubes. The effects of several factors including contact angle, number of polygon sides, and liquid properties on the capillary rivulet height are examined. It was found that the ratio of liquid surface tension to density directly affects the corner rise, while it has an inverse relationship with other factors. The maximum rivulet height of 91.6 mm is obtained in the triangular capillary tube with a side length of 1 mm and a contact angle of 30° for polydimethylsiloxane (PDMS-20)-air fluid pair. The minimum capillary rivulet height of 6.2 mm, on the other hand, is achieved in the pentagonal capillary tube, with a side length of 3 mm and a contact angle of 30°. To validate the developed analytical approach, comparisons are made between the model results, literature predictions, and experimental data. In addition, the geometrical model for a square capillary tube is compared with previous published studies, revealing a good agreement. This study provides quantitative results for the influence of capillary tube shape on the flow behavior of fluids in noncircular tubes that can be useful for control and optimization of transport phenomena in corresponding systems.

Almost flat angles in surface superconductivity

Type-II superconductivity is known to persist close to the sample surface in presence of a strong magnetic field. As a consequence, the ground state energy in the Ginzburg–Landau theory is approximated by an effective one-dimensional model. As shown by Correggi and Giacomelli (2021 Calc. Var. Partial Differential Equations in press), the presence of corners on the surface affects the energy of the sample with a non-trivial contribution. In (Correggi and Giacomelli 2021 Calc. Var. Partial Differential Equations in press), the two-dimensional model problem providing the corner energy is implicitly identified and, although no explicit dependence of the energy on the corner opening angle is derived, a conjecture about its form is proposed. We study here such a conjecture and confirm it, at least to leading order, for corners with almost flat opening angle.

Universal signatures of Dirac fermions in entanglement and charge fluctuations

We investigate the entanglement entropy (EE) and charge fluctuations in models where the low energy physics is governed by massless Dirac fermions. We focus on the response to flux insertion which, for the EE, is widely assumed to be universal, \emph{i.e.}, independent of the microscopic details. We provide an analytical derivation of the EE and charge fluctuations for the seminal example of graphene, using the dimensional reduction of its tight-binding model to the one-dimensional Su-Schrieffer-Heeger model. Our asymptotic expression for the EE matches the conformal field theory prediction. We show that the charge variance has the same asymptotic behavior, up to a constant prefactor. To check the validity of universality arguments, we numerically consider several models, with different geometries and number of Dirac cones, and either for strictly two-dimensional models or for gapless surface mode of three-dimensional topological insulators. We also show that the flux response does not depend on the entangling surface geometry as long as it encloses the flux. Finally we consider the universal corner contributions to the EE. We show that in the presence of corners, the Kitaev-Preskill subtraction scheme provides non-universal, geometry dependent results.

Influence of corners and road conditions on cycling individual time trial performance and ‘optimal’ pacing strategy: A simulation study

A mathematical model of a bike-rider’s longitudinal and lateral dynamics was used to study the influence of road conditions (tyre-road friction coefficient) on cycling individual time trial (ITT) performance and pacing strategy. A dynamic optimisation approach was used on different simulated 40-km-ITT courses, where environmental variables (i.e. slope and wind), the presence of corners and the tyre-road friction coefficient were varied. The objective of the optimisation was the performance time. Maximal velocity was constrained by road geometry and the tyre-road friction coefficient. The maximal deliverable power output was constrained accordingly to the critical power model. The simulation results suggest that when technical sections constitute 25% of the entire course, road conditions can meaningfully affect the final performance time and peak power required, but not the pacing strategy. In fact, the time lost in slow technical sections cannot be regained during fast straight sections, even if technical sections are used to restore anaerobic energy stores. However, more experimental research is needed to test the applicability of these predictions.

Surface effects in superconductors with corners

We review some recent results on the phenomenon of surface superconductivity in the framework of Ginzburg-Landau theory for extreme type-II materials. In particular, we focus on the response of the superconductor to a strong longitudinal magnetic field in the regime where superconductivity survives only along the boundary of the wire. We derive the energy and density asymptotics for samples with smooth cross section, up to curvature-dependent terms. Furthermore, we discuss the corrections in presence of corners at the boundary of the sample.

On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates

We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameterε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly inε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniquesà laGrisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence inεin all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework.

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

This paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal and polyhedral computational domains for the finite element solution of high-frequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two- and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle.

Energy-corrected FEM and explicit time-stepping for parabolic problems

The presence of corners in the computational domain, in general, reduces the regularity of solutions of parabolic problems and diminishes the convergence properties of the finite element approximation introducing a so-called “pollution effect”. Standard remedies based on mesh refinement around the singular corner result in very restrictive stability requirements on the time-step size when explicit time integration is applied. In this article, we introduce and analyse the energy-corrected finite element method for parabolic problems, which works on quasi-uniform meshes, and, based on it, create fast explicit time discretisation. We illustrate these results with extensive numerical investigations not only confirming the theoretical results but also showing the flexibility of the method, which can be applied in the presence of multiple singular corners and a three-dimensional setting. We also propose a fast explicit time-stepping scheme based on a piecewise cubic energy-corrected discretisation in space completed with mass-lumping techniques and numerically verify its efficiency.

Nucleation of crystal surfaces with corner energy regularization

The thermodynamics of strongly anisotropic crystalline surfaces is analogous to that of a binary mixture exhibiting phase separation. On a metastable planar surface, formation of stable orientations requires a nucleation process, in which the energy associated with the presence of corners must be considered. In this context, a nucleation event corresponds to the formation of a critical shape for the crystalline surface before the system enters the growth regime. We first derive the Euler-Lagrange equation for crystal surface nucleation, in two dimensions, and show that the saddle-point condition corresponds to a vanishing chemical potential along this critical surface. We then perform numerical simulation of the equation of motion for the crystal surface and show that, as compared with saddle point nucleation, ridge crossing is dynamically favoured.

Fracture of PBX notched specimens: Experimental research and numerical prediction

Polymer-bonded explosives (PBXs) are being increasingly applied for both military and civil applications, especially when high performances are required. From a material structure point of view, they can be considered as a kind of composite material since they are made of a polymer matrix filled with a high content of explosive granules. They are intended for applications in which common explosives are not easily melted into the required final shape or are difficult to machine. For this reason, their use is normally associated to complex shapes with presence of corners and many different stress concentrators. Since unexpected failure and crack propagation in these materials may lead to malfunctioning and even to safety issues, the study of their fracture behavior is of paramount importance.In this paper the results of an experimental campaign on PBX semi-circular notched specimens subjected to mode-I bending are presented. In the tests different notch lengths were considered while the notch tip geometry was preserved. Besides this, the critical loads experimentally obtained were also evaluated through different criteria, such as the Embedded Cohesive Crack Model or the Theory of Critical Distances, in order to study the best approach to the fracture assessment of notched components made PBX.

The square lattice Ising model on the rectangle II: finite-size scaling limit

Based on the results published recently (Hucht 2017 J. Phys. A: Math. Theor. 50 065201), the universal finite-size contributions to the free energy of the square lattice Ising model on the rectangle, with open boundary conditions in both directions, are calculated exactly in the finite-size scaling limit , , with fixed temperature scaling variable and fixed aspect ratio . We derive exponentially fast converging series for the related Casimir potential and Casimir force scaling functions. At the critical point we confirm predictions from conformal field theory (Cardy and Peschel 1988 Nucl. Phys. B 300 377, Kleban and Vassileva 1991 J. Phys. A: Math. Gen. 24 3407). The presence of corners and the related corner free energy has dramatic impact on the Casimir scaling functions and leads to a logarithmic divergence of the Casimir potential scaling function at criticality.

Surface superconductivity in presence of corners

We consider an extreme type-II superconducting wire with non-smooth cross section, i.e. with one or more corners at the boundary, in the framework of the Ginzburg–Landau theory. We prove the existence of an interval of values of the applied field, where superconductivity is spread uniformly along the boundary of the sample. More precisely, the energy is not affected to leading order by the presence of corners and the modulus of the Ginzburg–Landau minimizer is approximately constant along the transversal direction. The critical fields delimiting this surface superconductivity regime coincide with the ones in absence of boundary singularities.

A comprehensive analysis on the effect of shape on the microwave heating dynamics of food materials

Abstract A detailed theoretical analysis has been carried out to investigate the effect of the shape of the sample on the microwave heating dynamics of food materials. The food materials are divided into four groups based on the microwave propagation parameters and the heating dynamics of each group of materials have been simulated for one representative material involving semiinfinite cylindrical samples of circular and square shapes. The selection of the two shapes (circular and square) is to investigate the influence of the presence of sharp corners in the square samples on the microwave propagation dynamics within the food materials. The detailed study reveals that the presence of the corner can have a pronounced effect (either beneficial or detrimental) on the spatial heating dynamics for the intermediate and thick samples based on the propagation parameters and orientation of microwave sources with respect to the corner. Industrial relevance Microwave processing of food materials strongly depends on shapes of the materials as microwave propagation within food materials can be altered based on various incidence and reflected waves which are largely dependent on the structure of food materials. In general, spatial non-uniformities of heating patterns occur during microwave processing and there are no generalized guidelines available for industrial food processing. The entire spectrum of food materials can be categorized in four groups and generalized heating patterns can be deduced apriori for each group of food materials. Numerical simulation results predict that the shapes or cross section of food materials (squares, circular etc.) play important roles as the square shape exhibits interesting corner heating patterns and a trivial circular sample may not be a recommended food sample for efficient industrial microwave processing.

Stationary states of a pair of tangent identical vortex spots in a barotropic ocean

The method for constructing limiting forms of steady states of vortex patches characterized by the presence of corners on the boundary is presented. The method is based on a continuation of the solution (the streamline which must coincide with the vortex boundary) when passing through the singular point to those part of the common vortex border whose tangent is continuous at the critical point. Limiting steady states of a pair of identical touching vortex patches are constructed for the cases of unlimited and circular barotropic oceans. It is found that, for the case of a circular ocean, the solution of maximum area is the domain bounded by two diameters intersecting at right angles. This conclusion is also valid for an unlimited ocean when the vortex pair of infinite area takes even/odd quadrants whose boundaries are formed by the asymptotes of solutions of finite area. The results add new members to the set of known exact analytical solutions of the problem of steady states of vortex patches.

Analysis of Natural Convection in Horizontal Concentric Annuli of Varying Inner Shape

An investigation is presented of free convection in horizontal concentric annuli of varying with inner shape where the inner and outer surfaces are kept at a constant temperature. The simulation is categorized into four groups based on the shape of the inner entity, which is either cylindrical, elliptical, square, or triangular. Flow and thermal fields are exhibited by means of streamlines and isotherms. Overall heat transfer correlations incorporating thermal radiation are established and presented in terms of the Nusselt number. It is observed that the surface radiation and presence of corners and larger top space enhances the heat transfer rate. As the reference temperature increases, surface radiation plays a more prominent role in the overall heat transfer performance.

A unified approach for embedded boundary conditions for fourth‐order elliptic problems

SummaryAn efficient procedure for embedding kinematic boundary conditions in the biharmonic equation, for problems such as the pure streamfunction formulation of the Navier–Stokes equations and thin plate bending, is based on a stabilized variational formulation, obtained by Nitsche's approach for enforcing boundary constraints. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the higher continuity requirements typical of these problems. Variationally conjugate pairs weakly enforce kinematic boundary conditions. The use of a scaling factor leads to a formulation with a single stabilization parameter. For plates, the enforcement of tangential derivatives of deflections obviates the need for pointwise enforcement of corner values in the presence of corners. The single stabilization parameter is determined from a local generalized eigenvalue problem, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic B‐splines, providing guidance to the determination of the scaling and exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameter. Copyright © 2014 John Wiley & Sons, Ltd.

Embedded kinematic boundary conditions for thin plate bending by Nitsche's approach

SUMMARYA stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C1‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C2 B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.