Boundary Conditions Research Articles

On crack simulation by mixed‐dimensional coupling in GFEM with global‐local enrichments

AbstractA strategy that combines the global‐local version of the generalized finite element method (GFEM) with a mixed‐dimensional coupling iterative method is proposed to simulate two‐dimensional crack propagation in structures globally represented by Timoshenko‐frame models. The region of interest called the local problem, where the crack propagates, is represented by a 2D elasticity model, where a fine mesh of plane stress/strain elements and special enrichment functions are used to describe this phenomenon accurately. A model of Timoshenko‐frame elements simulates the overall behavior of the structure. A coarse mesh of plane stress/strain elements provides a bridge between these two representation scales. The mixed‐dimensional coupling method imposes displacement compatibility and stress equilibrium at the interface between the two different element types by an iterative procedure based on the principle of virtual work. After establishing the constraint equations for the interface, the 2D elasticity model is related to the small‐scale model by the global‐local enrichment strategy of GFEM. In such a strategy, the numerical solutions of the local problem subjected to boundary conditions derived from the global‐scale problem enrich the approximation of this same global problem in an iterative procedure. Each step of the crack propagation requires a new sequence of this iterative global‐local procedure. On the other hand, the constraint equation for the interface is defined only once. The crack representation in a confined region by the global‐local strategy avoids a remeshing that would require new constraint equations. Two numerical problems illustrate the proposed strategy and assess the influence of the analysis parameters.

Minimum Principles for Sturm–Liouville Inequalities and Applications

A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then applied to prove that any nonconstant solutions of S-L inequalities subject to separated inequality boundary conditions (IBCs) must be strictly positive in the interiors of their domains and are increasing or decreasing for some of these IBCs. These positivity results are used to prove the uniqueness of the solutions of linear S-L equations with separated BCs. All of these results hold for the corresponding second-order differential inequalities (or equations), which are special cases of S-L inequalities (or equations). These results are applied to two models arising from the source distribution of the human head and chemical reactor theory. The first model is governed by a nonlinear S-L equation, while the second one is governed by a nonlinear second-order differential equation. For the first model, the explicit solutions are not available, and there are no results on the existence of solutions of the first model. Our results show that all the nonconstant solutions are increasing and are strictly positive solutions. For the second model, many results on the uniqueness of the solutions and the existence of multiple solutions have been obtained before. Our results are applied to prove that all the nonconstant solutions are decreasing and strictly positive.

Longitudinal Shear of a Composite Material in Which the Binder and Inclusions are Weakened by Cohesive Cracks

Background: An elastic medium is considered that is weakened by a doubly periodic system of round holes filled with washers made of a homogeneous elastic material and has cohesive cracks, the surface of which is uniformly covered with a cylindrical non-metallic film. The binder (medium) is weakened by two doubly periodic systems of rectilinear cohesive cracks, directed parallel to the abscissa and ordinate axes, and the length of the crack and their dimensions are not the same (different). General concepts are constructed that describe a class of problems with doubly periodic stress distribution outside circular holes and cohesive cracks under longitudinal shear. The analysis of the limit equilibrium of cracks within the framework of the end zone model is performed on the basis of a nonlocal failure criterion with a force condition for the advancement of the crack tip and a deformation condition for determining the advance of the edge of the end zone of the crack. In solving the problem, the main resolving equations are obtained in the form of infinite algebraic systems and three nonlinear singular integro-differential equations. The equations obtained in each approximation were solved by the Gaussian method with the choice of the main element for different order values depending on the radius of the holes. Calculations were carried out to determine the forces in the connections of the end zones and the ultimate loads causing crack growth. Objective: In the considered problem, the problem of longitudinal slip weakened by cracks parallel to the x axis in the filling medium with two pairs of two periodic cracks weakened by 2 periodic circular holes is considered. In the problem, the boundary problem along the contour of the circular holes is determined, and at the same time, the surface problem is established for the cracks parallel to the x and y axes. The solution of the problem is sought analytically, and taking into account the boundary condition, a system of linear algebraic equations for circular holes and 3 singular integral equations for cracks is obtained. Theoretical Framework: The problem is solved analytically, as the system of linear algebraic equations obtained for circular holes and the system of linear integral equations are jointly solved, and the stress intensity factor at the end of the cracks is determined. Method: The problem was solved using "Kolosova Muskhalashvili" functions and "Collondia's method of solving singular integral equations". Results and Discussion: The main essence of the considered issue is that the stress intensity factor at the end of the cracks is determined according to Fig. 1. Research Implications: For the first time, the stress intensity factor was calculated in composite materials with two pairs of biperiodic cracks of different lengths weakened by biperiodic round holes. Originality/Value: For the first time, the stress intensity coefficient was determined in composite materials with two periodic two pairs of cohesive cracks.

Root patterns and exact surface energy of the spin-1 Heisenberg model with generic open boundaries

Abstract We investigate the thermodynamic limit and exact surface energy of the isotropic spin-1 Heisenberg chain with integrable generic open boundary conditions by a novel Bethe ansatz method. We obtain the homogeneous (or two-term) Bethe ansatz like equations for the zero roots of the transfer matrix. Based on the patterns of the zero roots, we analytical calculate the densities of zero roots and the surface energies of the model in all regimes of the boundary parameters.

Abusive supervision and helping behavior among nursing staff: a moderated mediation model

PurposeThis study aims to explore the adverse impacts of abusive supervision on helping behaviors among employees, as mediating by intention to leave and moderating by Islamic work ethics (IWE).Design/methodology/approachA quantitative approach was employed, and the sample consisted of 283 nurses working in various public sector hospitals in Pakistan. The data analysis was conducted using SPSS and AMOS with the PROCESS macro.FindingsThe results suggest that abusive supervision diminishes helping behavior among nurses. Additionally, the study reveals that intention to leave mediates the relationship of abusive supervision and nurses' helping behavior. Moreover, the introduction of IWE as a boundary condition reveals that the mediated link is weaker when IWE is higher, and vice versa.Practical implicationsThis study provides valuable insights for hospital authorities to develop intervention strategies and policies aimed at reducing abusive supervision in hospitals. Hospital management should also be aware of the detrimental effects of abusive supervision on nurses' helping behaviors, which can be mitigated by promoting ethical values aligned with IWE.Originality/valueThis study makes a valuable contribution to the limited research on the link between abusive supervision and helping behaviors in hospital settings. It offers new perspectives by incorporating the Conservation of Resources theory, particularly within the healthcare sector. Furthermore, this research expands the current knowledge by investigating the mediating influence of intention to leave and the moderating effect of IWE in mitigating the adverse impact of abusive supervision on nurses' helping behavior in Pakistan's public sector hospitals.

Remarks on analytical solutions to compressible Navier–Stokes equations with free boundaries

Abstract In this paper, we consider the free boundary problem of the radially symmetric compressible Navier–Stokes equations with viscosity coefficients of the form μ(ρ) = ρ θ , λ(ρ) = (θ − 1)ρ θ in R N ${\mathbb{R}}^{N}$ . Under the continuous density boundary condition, we correct some errors in (Z. H. Guo and Z. P. Xin, “Analytical solutions to the compressible Navier–Stokes equations with density-dependent viscosity coefficients and free boundaries,” J. Differ. Equ., vol. 253, no. 1, pp. 1–19, 2012) for N = 3, θ = γ > 1 and improve the spreading rate of the free boundary, where γ is the adiabatic exponent. Moreover, we construct an analytical solution for θ = 2 3 $\theta =\frac{2}{3}$ , N = 3 and γ > 1, and we prove that the free boundary grows linearly in time by using some new techniques. When θ = 1, under the stress free boundary condition, we construct some analytical solutions for N = 2, γ = 2 and N = 3, γ = 5 3 $\gamma =\frac{5}{3}$ , respectively.

Thermomechanical bending of functionally graded carbon nanotubes reinforced composite plate by meshless method

AbstractFunctionally graded (FG) carbon nanotubes (CNTs) reinforced composite possessing continuously varied material properties have received extensive concern in the area of aviation, automotive, military, and spaceflight. The nonlinear bending of FG CNT composite is performed by developing a novel meshless model on the base of Smoothed Particle Hydrodynamics (SPH) method. The equivalent temperature relative material properties of composite is established in the light of the extended rule of mixture model. Different gradient dispersions of CNTs are taken into account and modeled. The governing equations are explored using the Mindlin theory and discretized though a symmetric SPH method with high prediction ability. The developed model is verified by analyzed several examples and compared to the solution obtained in literature. Influences of CNT dispersion, content, plate aspect ratio, boundary conditions and lamination angle on the bending response are elaborated. The present study provides an alternative potential method to solve the mechanical performances of FG CNT composites based on the meshless SPH formulation.Highlights Thermo‐mechanics of FG‐CNTRC is firstly solved by SPH method. The model is verified by solving the benchmarks in literature. Nonlinear bending behaviors of FG‐CNTRC are demonstrated. The minimum deflection take place in FG‐X type composite. For laminate plate, the smallest deformation occurs in 0° CNTRC.

Exploration of the effects of anisotropy and rotation on Rayleigh–Bénard convection of nanoliquid-saturated porous medium using general boundary conditions

This paper presents an analysis of Rayleigh–Bénard convection (RBC) of a Newtonian-nanoliquid-saturated anisotropic porous medium in the presence of rotation (Rayleigh–Bénard–Taylor convection). The investigation is performed using non-classical boundary conditions. The effect of various parameters on the onset of convection is presented graphically. The system sees stabilisation due to an increase in the rotation rate and thermal anisotropy parameter whereas the system destabilises due to an increase in the mechanical anisotropy parameter. The results of 82 limiting cases can be extracted from the current work. The results of free-free, rigid-free and rigid-rigid isothermal/adiabatic boundaries are obtained from the present study by considering appropriate limits. The results of the limiting cases of the present study are in excellent agreement with those observed in earlier investigations.

SENSITIVITY OF A 1D LAKE MODEL TO THE INPUT METEOROLOGICAL DATA

The paper evaluates how the choice of a source of meteorological data used as boundary conditions and the changes in model calibration parameters affect the adequate performance of a model of lake thermo- and hydrodynamics. The quality of simulation of water temperature dynamics in a small lake by a onedimensional GLM model using data from an automatic meteorological station on the lake shore, a state weather station and two global reanalyses was quantitatively assessed. It is shown that the best modeling result (RMSE of water temperature calculation 0,8-2,0°C, Nash-Sutcliffe efficiency >0,7) can be achieved using data sets that explicitly provide incoming shortwave and longwave radiation fluxes. At the same time, good representation of the bottom water temperature requires correct setting of wind regime, which is only possible if data from instrumental ground-based observations at a stationary weather station are applied. Combining ground-based observations of wind speed, air temperature and humidity with radiation fluxes from global reanalyses makes it possible to achieve a calculation error ≤1°C at all depths.The analysis of model sensitivity to calibration parameters in case when the reanalysis data are applied showed that parameters related to wind mixing, i.e. the scaling factor for wind speed and constants related to the in-model description of wind impact energy, have the greatest influence on the final calculation error. If we change the intensity coefficients of other types of mixing during calibration it does not lead to an obvious trend in the quantitative estimates of modeling quality. It could also be reasonable to introduce correction multipliers to the values of air temperature and solar radiation fluxes, thus neutralizing the effect of unrepresentativeness of the meteorological data applied for a particular lake, as well as to correct the value of solar radiation extinction coefficient relative to the value obtained from the Secchi depth. The results of the study could be used to justify the choice of meteorological data source and to optimize the methods of calibration of lake thermal stratification models

Legendre-Galerkin spectral algorithm for fractional-order BVPs: Application to the Bagley-Torvik equation

<p>Herein, we provide an efficient spectral Galerkin algorithm, according to a special type of shifted Legendre basis for finding a semi-analytic solution to the Liouville-Caputo fractional boundary value problem. The algorithm’s main goal is to transform the fractional differential problem into a linear system with efficiently invertible, well-structured matrices. The convergence rates of the algorithm are carefully obtained as well as the error bound.</p>

Hypercomplex operator calculus for the fractional Helmholtz equation

In this paper, we develop a hypercomplex operator calculus to treat fully analytically boundary value problems for the homogeneous and inhomogeneous fractional Helmholtz equation where fractional derivatives in the sense of Caputo and Riemann–Liouville are applied. Our method extends the recently proposed fractional reduced differential transform method (FRDTM) by using fractional derivatives in all directions. For the special separable case in three dimensions, we obtain completely explicit representations for the fundamental solution. This allows us to interpret and to understand the appearance of spatial steady‐state solutions or spatial blow‐ups of the fractional Helmholtz equation in a better way. More precisely, we were able to present explicit conditions for the parameters in the representation formulas of the fundamental solutions under which we obtain bounded or spatial decreasing steady‐solutions and when spatial blow‐ups occur. We also illustrate this with some representative numerical examples. Furthermore, we show that it is possible to recover the recently studied cases as well as the classical cases as particular limit cases within our more general setting. Using the hypercomplex operator approach also allows us to factorize the fractional Helmholtz operator and obtain some interesting duality relations between left and right derivatives, Caputo and Riemann–Liouville derivatives, and eigensolutions of antipodal eigenvalues in terms of a generalized Borel–Pompeiu formula. This factorization, in turn, allows us to tackle inhomogeneous fractional Helmholtz problems.

General Theory for Damped Beams with Elastic Supports Subjected to a Moving Damped Sprung Mass

Previously, classical boundary conditions (i.e. simple, fixed and cantilevered) have been adopted for the beams under the moving loads in deriving the dynamic response. In reality, most beams cannot be simply classified as the ones with classical supports, but are elastically restrained. For theoretical completeness, a general theory will be developed in this note for the damped beams with elastic restraints modeled by vertical and rotational springs subjected to a moving damped sprung mass. Essential to the present theory is the solution of the transcendental equation for the frequency by the bi-section method. The solution obtained can also be applied to the classical boundary conditions (i.e. simple, fixed and cantilevered) under a moving sprung mass. The reliability of the present theory, along with the bi-section method for solving the frequency and mode shape, is validated by comparison with the solution obtained by the finite element method (FEM) for various damping ratios, mass ratios and running speeds of the sprung mass.

Identification of ECA rules forming MACA in periodic boundary condition

Cellular automaton (CA) is a computing model which is emerging rapidly. It is largely used in different types of scientific applications and simulations due to its ability to solve complex problems using simple rule(s). Cellular automata (CAs) are used in different types of applications like cryptography, VLSI systems, fault detection, etc. Typically, most of these applications utilize one-dimensional, 2-state, 3-neighborhood CAs. This paper explores the concept of Next State RMT Transition Diagram (NSRTD) for characterization of all the Elementary Cellular Automata (ECA) rules in periodic boundary condition leading to the identification of all ECA rules forming more than two fixed points (referred to as Single Length Cycle Multi-Attractor CA (MACA)) for an arbitrary CA length (n). For this, the 88 Wolfram classification rules and their equivalent rules have been utilized to reduce the search complexity by avoiding exhaustive searching on all the 256 ECA rules.

Numerical investigation of discharge evolution and breakdown characteristics of ArF excimer lasers

Abstract The corona bar induced pre-ionization is a crucial preliminary process in the operation of ArF excimer lasers, directly impacting the uniformity and stability of output laser. However, the ultraviolet corona pre-ionization, as the mainstream method, is tightly coupled with the main discharge process, which complicates analysis. Here, we establish a numerical model of a single pulse discharge incorporating an external circuit to analyze the pre-ionization process and its influence on the breakdown characteristics. (1) By adopting detailed input parameters of photoionization model, we observe uniform and dispersed plasma propagation from the corona bar to the main gap. (2) An artificial boundary condition is proposed to investigate the phenomenological effect of high-energy electrons emission, emphasizing the influence of surface discharge along the cathode. (3) The differences between the two pre-ionization setup methods, photoionization and background electron density, are compared numerically. This study enhances the understanding of the pre-ionization process in ArF excimer lasers and provides theoretical insights for their optimization and design.

On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions

AbstractWe provide (upper and lower) scaling bounds for a singular perturbation model for the cubic-to-tetragonal phase transformation with (partial) displacement boundary data. We illustrate that the order of lamination of the affine displacement data determines the complexity of the microstructure. As in (Rüland and Tribuzio in ESAIM Control Optim. Calc. Var. 29:68, 2023) we heavily exploit careful Fourier space localization methods in distinguishing between the different lamination orders in the data.

The moral dark side of entrepreneurial leadership: How it leads to employee creative unethicality

AbstractDrawing on the social cognitive theory of morality, we propose a moderated mediation model to explain why and how employees under the supervision of entrepreneurial leadership engage in unethical behaviors to achieve creative performance. Specifically, we anticipated that entrepreneurial leadership would cause employees to engage in moral justification, which in turn would lead to creative unethicality. Furthermore, we argue that politics perception serves as a critical boundary condition, strengthening the indirect effect of entrepreneurial leadership on creative unethicality. We conducted two multiwave and multisource surveys and constructed a structural equation model to test the proposed hypotheses. Study 1, which involved 201 coworker–employee dyads in a single company, provided full support for all hypotheses. Study 2, conducted in 12 companies with 400 employees and 69 supervisors, replicated the findings of Study 1 and further confirmed the moderated mediation model. The theoretical and practical implications are discussed.

FAMILY FIRM NARCISSISM: CONCEPTUALISATION, MEASUREMENT, AND RELATIONSHIP TO INNOVATION

Family firm behaviour can be a potential root of innovation in family firms. Building on the theoretical perspectives of socioemotional wealth and narcissistic organisational identification, we introduce a novel concept and new measure for family firm narcissism (FFN). We suggest that FFN influences family firms’ innovation orientation (i.e., strategic innovativeness) and realised innovation outcome (i.e., digital inventiveness). Additionally, we assume that industry IT intensity, as an indicator of environmental pressure regarding digital transformation, moderates these relationships. To test the hypotheses, we investigated a longitudinal cross-industry sample, comprising 1,302 firm-year observations of S&P 500 family firms and analysed it with regression models. The results indicate that FFN is positively related to strategic innovativeness and negatively related to digital inventiveness. We further find that industry IT spending intensity amplifies both linkages. Our study contributes to the discussion on the family innovation agenda by (1) introducing the new concept and measure for FFN and (2) underlining its controversial impact and effectiveness on two central manifestations of innovation under the boundary conditions of digital transformation.

Existence and convergence of the length-preserving elastic flow of clamped curves

We study the evolution of curves with fixed length and clamped boundary conditions moving by the negative L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-gradient flow of the elastic energy. For any initial curve lying merely in the energy space we show existence and parabolic smoothing of the solution. Applying previous results on long-time existence and proving a constrained Łojasiewicz–Simon gradient inequality we furthermore show convergence to a critical point as time tends to infinity.

Variational approaches to the elasticity of deformable strings with and without mass redistribution

AbstractThe catenary is a curve that has played a significant role in the history of mathematics, finding applications in various disciplines such as mechanics, technology, architecture, the arts, and biology. In this paper, we introduce some generalizations by applying the variational method to deformable strings. We explore two specific cases: (i) in the first case, we investigate the nonlinear behavior of an elastic string with variable length, dependent on the applied boundary conditions; specifically, this analysis serves to introduce the variational method and demonstrate the process of finding analytical solutions; (ii) in the second case, we examine a deformable string with a constant length; however, we introduce mass redistribution within the string through nonlinear elastic interactions. In the first scenario, the deformation state of the string always describes elongation, as compression states prove to be unstable for fully flexible strings. In contrast, in the second scenario, the finite length constraint induces compressive states in specific configurations and regions of the string. However, it is worth noting that the solution to this problem exists only for values of the elastic constant that are not too low, a phenomenon that is studied in detail. We conduct here both analytical and graphical analyses of various geometries, comparing the elastic behavior of the two aforementioned types of strings. Understanding the elastic behavior of deformable strings, especially the second type involving mass redistribution, is crucial for enhancing comprehension in the study of biological filaments or fibers and soft matter. For instance, these investigations can contribute to understanding the mechanisms employed by cells to sense gravity or other mechanical conditions.

Domination of nonlinear semigroups generated by regular, local Dirichlet forms

AbstractIn this article we study perturbations of local, nonlinear Dirichlet forms on arbitrary topological measure spaces. As a main result, we show that the semigroup generated by a local, regular, nonlinear Dirichlet form $${\mathcal {E}}$$ E dominates the semigroup generated by another local functional $${\mathcal {F}}$$ F if, and only if, $${\mathcal {F}}$$ F is a specific zero order perturbation of $${\mathcal {E}}$$ E . On the way, we prove a nonlinear version of the Riesz–Markov representation theorem, we define an abstract boundary of a topological measure space, and apply the notion of nonlinear capacity. The main result helps to classify the perturbations that lie between Neumann and Dirichlet boundary conditions.