Boundary Conditions Research Articles

Deep learning method for finding eigenpairs in Sturm-Liouville eigenvalue problems

Solving the eigenvalue problem for differential equations in inhomogeneous media poses a significant challenge across diverse scientific fields. While classical finite difference methods and finite element methods have produced numerous outcomes, they heavily rely on discretizing the computational domain, which can introduce complexities and limitations. In this study, we present an unsupervised neural network approach tailored for finding eigenpairs in Sturm-Liouville eigenvalue problems within inhomogeneous media. Our method introduces eigenvalues as trainable parameters, crafts a novel cost function, incorporates an adaptive hyper-parameter tuning strategy, and sequentially trains the eigenpairs. The simplicity, accuracy, and interpretability of our approach significantly expand its applicability across various domains. The method we present in this paper can easily tackle boundary value conditions with derivatives, resulting in orthogonal eigenfunctions. This is a very important advantage of deep learning methods that has not yet been noticed. Quantitative estimation of eigenpairs is given for the Sturm-Liouville eigenvalue problems. Furthermore, we extend the proposed methodology to tackle two-dimensional cases, periodic scenarios, demonstrating its versatility and broad potential. For more information see https://ejde.math.txstate.edu/Volumes/2024/53/abstr.html

Global well-posedness of strong solutions to the two-dimensional inhomogeneous biaxial nematic liquid crystal flow with vacuum

This paper considers the inhomogeneous biaxial nematic liquid crystal flow in a smooth bounded domain Ω⊂R2, where the velocity u and the orthogonal unit vector fields (m,n) admit the Dirichlet and Neumann boundary condition, respectively. By applying piecewise estimate and continuity method, we get the global existence of strong solutions, provided that the basic energy is suitably small. Our result may be regarded as an extension and improvement of Gong-Lin (2022) and Li-Liu-Zhong (2017) to the Neumann boundary condition, where the initial vacuum is allowed. Some new techniques are developed in order to deal with integral estimates caused by the boundary condition, and more complicated model.

Conditions for economic efficiency of latent heat thermal energy storage systems at nuclear power plants

In conditions of uneven schedules of electricity consumption and construction of power plants using renewable energy sources, nuclear power plants will participate in regulating the unevenness of daily energy consumption. At the same time, nuclear power plants have high economic feasibility of operating with a maximum installed capacity utilization factor due to significant capital investments at a relatively low fuel price. The use of nuclear power plants to cover uneven energy consumption will reduce the possibility of a return on investment, which will only be realized with greatly inflated capacity charges. At the same time, the annual operating costs will increase significantly. A possible decrease in the economic efficiency of a nuclear power plant is due to the fact that, at the request of the system operator of the energy system, the NPP will be to reduce electricity generation during off-peak hours or sell electricity on the market at significantly lower prices. In this regard, the problem of increasing economic efficiency of nuclear power plants under conditions of uneven daily energy consumption becomes particularly relevant. The solution to the problem could be the accumulation of excess energy and its subsequent use. Previously, the authors developed a method of additional redundancy of auxiliary needs of nuclear power plants based on the use of low-power steam turbines. In the present paper, schemes for increasing efficiency of using low-power steam turbines at nuclear power plants when regulating the load unevenness in the power system using thermal energy storage systems based on the latent heat thermal energy storage are developed. The design characteristics of the accumulator as part of a nuclear power plant are determined to ensure operation of the heat storage system throughout the entire period of energy consumption. In this research, the lithium nitrate was considered as a phase change material. The technical and system conditions under which the maximum economic effect is achieved have been determined. The boundary conditions under which a return on investment is achieved are shown.

Analytical solution of nano-concrete-epoxy interaction area considering static equilibrium

This research proposes an analytical solution of the nano-concrete-epoxy interaction area within nano crack region of the reinforced concrete beam by applying Newton’s third law in static equilibrium. For deriving the governing equation, the imaginary beam with free ends (no support) is considered within nano crack region. This imaginary beam is acted along the imaginary line of concrete-epoxy interface. Newton’s third law is applicable for deriving the governing equation because of assuming the absence of frictional and other external forces. The parametric study is performed for implementing the proposed formula of nano interactive area considering variable nano crack depths and thicknesses. The nano interactive area is increased gradually with the increment of depths and thicknesses based on the parametric study because of linear functionality of interactive area and geometry of nano crack region. The maximum interactive area is found to be 314 nm2 at 0.6 ratio of depths and thicknesses of the nano crack. The incremental differences in interactive area between the crack depth or thickness ratios of 0.1 and 0.6 are found to be 25.4% and 1.6% for variations of the crack depth and thickness ratios, respectively. So, the crack depth shows higher impact on the interaction area compared to the thickness of the crack. However, there is a scope for enhancing this research in future by deriving closed-formed analytical formulations to consider appropriate boundary conditions.

Exploring the significance of nanoparticle shapes and the impact of thermal radiation on MHD viscoelastic nanofluid flow in porous channels

ABSTRACT This study investigates analytically the influence of nanoparticle shapes on an unsteady mixed convective viscoelastic MHD nanofluid flow in a horizontal channel with oscillating walls in a saturated porous medium. The effect of radiation is also considered in the energy equation. The considered NPs are Cu, Ag, Al2O3, TiO2, and Fe3O4, with base fluids H2O and C2H6O2. The considered different shapes of NPs are cylinder, platelet, blade, and brick. The study examines three flow situations based on physical boundary conditions. The perturbation method is employed to solve the flow governing equations for three different flow situations based on physical boundary conditions: Case (i) flow in a horizontal channel with both lower and upper walls stationary; Case (ii) the upper wall of the channel is in oscillation while the lower one is constant; Case (iii) both walls of the channel are in oscillatory motion. The velocity and temperature distributions, as well as the skin friction coefficient and heat transfer rate, are analyzed through graphs and tables to assess the impact of various governing parameters. From the results, it is found that brick-shaped NPs exhibit more strength to the velocity and temperature distributions, followed by cylinder, platelet, and blade for both water-based and EG-based NF.

Short-time and long-time dynamics of the zeolite films in the light of alternative diffusion models

Analysis of the dynamics of diffusion of two mixtures of hydrogen and heavier components through a flat zeolite film was carried out. The concepts of short-time and long-time dynamics were introduced. They derive from the observation of temporal trajectories obtained by applying abrupt perturbations of concentrations on one side of the membrane.An alternative description of the diffusion dynamics was proposed compared to the commonly accepted model that employs a matrix of thermodynamic factors Γ. In this way, the difficulty of analytical determination of this matrix for more complex adsorption equilibria was eliminated.The model equations were expressed in a matrix form, which enables its easy extension to any number of components. The proposed research methodology was illustrated using an example of diffusion of two selected mixtures, i.e. {H2, n-C4H10} and {H2, CO2} in silicalite-1. These gaseous mixtures have applications in specific technological processes.The two models were used to perform quantitative benchmarking of short-time and long-time dynamics of flat zeolite membranes. The assumed boundary conditions correspond to a semi-batch Wicke-Kallenbach diffusion cell. It was demonstrated that the phenomena occurring in short-time and long-time diffusion represent two different dynamic features important from a scientific and practical point of view.A quantitative measure for degree of confinement was proposed based on dynamic adsorption-desorption hysteresis.

Corrective and new research frontier of dual-phase -lag non-fourier heat conduction in functionally graded cylindrical materials with bi-directional property variations

This study presents a novel and corrective analysis of Dual-Phase-Lag (DPL) non-fourier heat conduction in functionally graded cylindrical materials with bi-directional property variations. For general purposes, the 2-D DPL model was employed and solved in a polar coordinate system for an FGM cylinder whose material properties vary exponentially in the axial and radial directions. The proposed model's analytical and numerical solutions were obtained through the SOV method and implicit FDM with a non-uniform grid. Moreover, the effect of the inhomogeneity parameter in the Fourier, Cattaneo–Vernotte (C–V), and DPL models has been analyzed. The results indicate that the DPL model achieves temperature stability in less time when contrasted with the C–V model. In addition, the reduced inhomogeneity parameters result in quicker attainment of a steady temperature and the induction of higher temperatures. The thermal wave propagation in the DPL model is consistently greater than that in the C–V model. In addition, an increase in time lag for heat flux enhances thermal wave properties (amplitude, wavelength, propagation speed etc.); conversely, an increase in time lag for temperature gradient counteracts wave properties and augments heat release. Nevertheless, the present outcomes offer a straightforward multivariate analytical and numerical solution for a finite cylinder's non-Fourier heat conduction equation under diverse boundary conditions.

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Spline approximation methods for second order singularly perturbed convection-diffusion equation with integral boundary condition

Behavior in time of solutions to a degenerate chemotaxis system with flux limitation

We study a new class of Keller–Segel models, which presents a limited flux and an optimal transport of cells density according to chemical signal density. As a prototype of this class we study radially symmetric solutions to the parabolic–elliptic system ut=∇⋅(u∇uu2+|∇u|2)−χkf∇⋅(u∇v(1+|∇v|2)α),x∈Ω,t>0,0=Δv−μ+u,x∈Ω,t>0under no flux boundary conditions in a ball B=Ω⊂RN and initial condition u(x,0)=u0(x)>0,χ>0,α>0,kf>0 and μ=1|Ω|∫Ωu0dx. Under suitable conditions on α and u0 it is shown that the solution blows up in L∞-norm at a finite time Tmax and for some p>1 it blows up also in Lp-norm. The proofs are mainly based on an helpful change of variables, on comparison arguments and some suitable estimates.

ESG pay and corporate social irresponsibility: Does culture matter?

AbstractDespite the detrimental consequences of corporate social irresponsibility (CSiR), the role of monitoring and incentive‐based corporate governance (CG) mechanisms in mitigating stakeholder mismanagement has been largely neglected in the literature. At the same time, there has been growing interest in holding executives accountable for environmental, social, and governance (ESG) performance by linking their compensation to related targets. However, prior research provides scant and inconclusive evidence on the effectiveness of ESG pay in curbing CSiR. This study addresses these shortcomings and contributes to the emerging CSiR/CG literature by harnessing an institution‐based view on executive compensation. Specifically, it recognizes national culture as the “missing determinant” in comparative CG studies and investigates the culturally‐driven boundary conditions for the relationship between ESG pay and CSiR. The hypotheses were tested on a data set of 4066 firms from 56 countries during 2010–2020. The analysis supports a negative, though modest, effect of ESG pay on CSiR. Furthermore, the study shows that this relationship varies in statistical and economic significance across cultural contexts. Particularly, incentives prove effective for firms headquartered in nations characterized by low power distance, individualism, low uncertainty avoidance, and masculinity. Conversely, in other settings, compensation linked to ESG may either lack motivational impact or even yield counterproductive outcomes. These findings have implications for both the CSiR and CG literature, as well as for practice.

Research on multi-heat source arrangement optimization based on equivalent heat source method and reconstructed variational autoencoder

The variational autoencoder (VAE) architecture has significant advantages in predictive image generation. This study proposes a novel RFCNN-βVAE model, which combines residual-connected fully connected neural networks with VAE to handle multi-heat source arrangements. By integrating analytical solutions, polynomial fitting, and temperature field superposition, we accurately simulated the temperature rise distribution of a single heat source. We further explored the use of multiple equivalent heat sources to replace adiabatic boundary conditions. This enables the analytical method to effectively solve the two-dimensional conjugate convective heat transfer problem, providing a reliable alternative to traditional numerical solutions. Our results show that the model achieved high predictive accuracy. By adjusting the β parameter, we balanced reconstruction accuracy and latent space generalization. During the stable phase of the multi-heat source optimization iteration, 73.4% of the results outperformed the dense dataset benchmark, indicating that the model successfully optimized heat source coordinates and minimized peak temperature rise. This validates the feasibility and effectiveness of deep learning in thermal management system design. This research lays the groundwork for optimizing complex thermal environments and contributes valuable perspectives on the effective design of systems with multiple thermal sources or for applications like multi-beam lighting equalization.

General boundary conditions for a Boussinesq model with varying bathymetry

AbstractThis paper is devoted to the theoretical and numerical investigation of the initial boundary value problem for a system of equations used for the description of waves in coastal areas, namely, the Boussinesq–Abbott system in the presence of topography. We propose a procedure that allows one to handle very general linear or nonlinear boundary conditions. It consists in reducing the problem to a system of conservation laws with nonlocal fluxes and coupled to an ordinary differential equation. This reformulation is used to propose two hybrid finite volumes/finite differences schemes of first and second order, respectively. The possibility to use many kinds of boundary conditions is used to investigate numerically the asymptotic stability of the boundary conditions, which is an issue of practical relevance in coastal oceanography since asymptotically stable boundary conditions would allow one to reconstruct a wave field from the knowledge of the boundary data only, even if the initial data are not known.

Two Nitsche-based mixed finite element discretizations for the seepage problem in Richards’ equation

This paper proposes two algorithms to impose seepage boundary conditions in the context of Richards’ equation for groundwater flows in unsaturated media. Seepage conditions are non-linear boundary conditions, that can be formulated as a set of unilateral constraints on both the pressure head and the water flux at the ground surface, together with a complementarity condition: these conditions in practice require switching between Neumann and Dirichlet boundary conditions on unknown portions on the boundary. Upon realizing the similarities of these conditions with unilateral contact problems in mechanics, we take inspiration from that literature to propose two approaches: the first method relies on a strongly consistent penalization term, whereas the second one is obtained by an hybridization approach, in which the value of the pressure on the surface is treated as a separate set of unknowns. The flow problem is discretized in mixed form with div-conforming elements so that the water mass is preserved. Numerical experiments show the validity of the proposed strategy in handling the seepage boundary conditions on geometries with increasing complexity.

Design of nonlinear gradient sheet-based TPMS-lattice using artificial neural networks

Gradient triply periodic minimal surface (TPMS) structures are renowned for lightweight design and enhanced performance, but their complex and nonlinear configurations pose challenges in achieving targeted design goals. A new design methodology for the nonlinear gradient structure was proposed in this study, with the aim of achieving efficient and accurate modeling of complex and gradient sheet-based TPMS structures under specific performance objectives. This method utilized automated finite element (FE) simulations to obtain structure topology element densities under various boundary conditions. An artificial neural network (ANN) was then employed to efficiently predict the correspondence between these boundary conditions and topology element densities. A mapping was established between topology element densities and TPMS structural parameters, and the gradient structure was accurately constructed by using the voxel modeling technique. Taking a typical cantilever beam TPMS structure as an example of nonlinear gradient design, the results indicate that the error between the ANN-predicted and FE-simulated structure topology element densities is only 2.73 %, with prediction time being only 0.15 % of the simulation time. The thin regions of the gradient structure align with those geometrically removed in regular topology optimization scheme, achieving up to 65.45 % weight reduction, a 28.72 % improvement over the regular scheme, along with uniform structural stress transition and maximum stress reduction. TC4 alloy nonlinear gradient TPMS structures, printed by metal selective laser melting (SLM) technique, confirm the practical application value of this design method.

Existence of Positive Solutions for Hadamard-Type Fractional Boundary Value Problems at Resonance on an Infinite Interval

This paper investigates a class of resonance boundary value problems for Hadamard-type fractional differential equations on an infinite interval. Utilizing the Leggett-Williams norm-type theorem proposed by O’Regan and Zima, the existence of positive solutions is established. The main conclusions are illustrated with an example.

Oblique wave scattering by a pair of asymmetric inverse Π-shaped breakwater

Abstract The topic of wave scattering by different breakwaters has unfolded over the past few decades, largely driven by the profound impacts of global climate change. In this study, a completely new type of breakwater, inverse Π-shaped breakwater, has been taken into account for serving both purposes from analytical as well as an application point of view. During the formulation, the actual physical problem is transferred into a boundary value problem by employing the small amplitude water wave theory. With the aid of eigenfunction expansion, the problem is reduced to a set of integral equations having square-root singularities at the submerged edges of thin structures. To address such singularities, a multi-term Galerkin approximation technique is used along with Chebyshev polynomials (multiplied with suitable weights) as basis functions. The numerical methodology employed here validates with different previous literatures as special cases. Then, numerical solutions of the reflection and transmission coefficients, hydrodynamic forces are graphically illustrated across various dimensionless structural parameters. In summary, the present paper offers valuable insights into the dynamics of wave scattering by a pair of asymmetric inverse bottom-mounted Π-shaped breakwater, emphasizing the role of different non-dimensional parameters in this system.

Small scale pool fires: The case of toluene

Industrial applications adopting toluene as a solvent have been largely extended in recent years, including solutions within the framework of the energy transition and energy storage technologies. The potential use of this flammable compound in a different set of operative conditions and compositions requires a comprehensive and complete knowledge of its fire behaviour and combustion kinetic. To this scope, an innovative experimental procedure applicable to liquid reactive systems was developed for this scope and implemented at different boundary conditions. More specifically, the specimen was exposed to air and heat fluxes between 7 and 50 kW/m2, at a constant sample surface of 0.01 m2, an initial sample thickness of 0.01 m, and a distance between the sample and the horizontally oriented conical-shaped heater of 0.025 m. Measurements were compared with data from the current literature, when available, demonstrating the robustness and validity of the adopted procedure. Although an increase in the external flux leads to growing mass burning rates (i.e., from 0.47 g/s to 0.85 g/s), negligible effects on the ignitability were observed. Conversely, a peak in the heat release rate at 35 kW/m2 was measured. The observed reduction at higher external heat fluxes was attributed to less effective combustion, demonstrating that the maximum expected heat flux cannot be considered as an aprioristic worst-case scenario for the evaluation of pool fires. The collected data were, then, further utilized to obtain insights on the formation of the main products, including soot tendency. Based on the collected data a simplified kinetic model suitable for the computational fluid dynamics was proposed to reproduce the chemistry of the system.

A Symbolic Algorithm to Obtain Low or High Degree Splines from Discrete Fourier Transforms

Obtaining high-degree splines with the use of traditional spline interpolation methods is not an easy task; therefore, traditional spline interpolation is typically limited to cubic splines. In this paper, we present a symbolic and numeric algorithm to obtain splines of any degree, while providing detailed procedures and examples of how to use this algorithm so that it is immediately useful for an interested user. This method, which was initially developed by Beaudoin and Beauchemin [2, 3], works for splines of any degree and yields very accurate results when the boundary conditions are chosen wisely. It also provides approximations of higher order derivatives, something that is not available with the use of cubic splines. This paper presents formulas that can be used in a straightforward manner to obtain interpolation splines of first degree (linear splines), second degree (parabolic splines) and third degree (cubic splines). For splines of higher degree, a short but complete symbolic algorithm to compute the formulas is presented. The resulting formulas can be used in the same manner as those presented for splines of lower degree. A complete numerical example is included to show how the results are obtained and a link to the complete Maple source code is given.

Thickness-extensible higher order plate theory with enforced C1 continuity for the analysis of PEEK medical implants

Plate-like structures had been thoroughly studied in literature over years to reduce the computational space from 3D to 2D. Many of these theories suffer either from satisfying the free traction condition or thickness extensibility in addition to the consistency of transverse shear strain energy. This work presents a higher order shear deformation thickness-extensible plate theory (eHSDT) for the analysis of plates. The proposed eHSDT satisfies the condition of free traction as other theories do but it also satisfies the condition of consistency of transverse shear strain energy which is neglected by many theories in the area of plates and shells. The implementation of the proposed theory in displacement-based finite element procedure requires continuity of derivatives across elements. This necessary condition was achieved using the penalty enforcement method for derivative-based nodal degrees of freedom across the standard 9-nodes Lagrange element. The theory was tested for elastic bending deformation of Polyether-ether-ketone (PEEK) which is one of the basic materials for medical implants. The theory showed good accuracy compared to experimental data of the three-points bending test. The present eHSDT was also tested for different conditions with a wide range of aspects ratios (thin to thick plates) and different boundary conditions. The accuracy of the proposed eHSDT was verified against exact solutions for these conditions which showed the advantage over other approaches and commercial finite element packages.

The SU(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{SU}}(3)$$\\end{document} twisted gradient flow strong coupling without topological freezing

We investigate the role of topology on the lattice determination of the SU(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extrm{SU}}(3)$$\\end{document} strong coupling renormalized via gradient flow. To deal with the topological freezing of standard local algorithms, the definition of the coupling is usually projected onto the zero topological sector. However, it is not obvious that this definition is not biased by the loss of ergodicity. We instead avoid the topological freezing using a novel algorithm, the Parallel Tempering on Boundary Conditions. The comparison with a standard algorithm shows that, even in the case where the latter is severely frozen, one obtains the same projected coupling. Moreover, we show that the two definitions of the coupling, projected and non-projected, lead to the same flow of the renormalization scale. Our results imply that projecting the coupling does not affect the determination of the dynamically-generated scale of the theory Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varLambda $$\\end{document}, as obtained through the step-scaling method.