One-dimensional Boundary Research Articles

Using the Boundary Element Method for airfoil trailing edge noise prediction

This study investigates two aspects of the prediction of trailing edge noise from an airfoil using the Boundary Element Method to solve the scattering phenomenon. The first one consists in defining the surface pressure spectrum near the trailing edge as a source for the turbulent boundary layer-trailing edge interaction, using either an empirical model or one-dimensional turbulent boundary layer theory. The second aspect focuses on the capability of the Boundary Element Method to handle general boundary conditions. It is applied to predict the noise emitted by a porous trailing edge. This work is conducted in the context of aerodynamic noise from wind turbine blades, but has a wider range of engineering applications.

Observation of multifunctional robust topological states based on asymmetric C4 photonic crystals

Recent advancements in high-order topological insulators have heralded new opportunities for the innovation and utilization of optical devices. This paper presents a composite asymmetric C4 photonic crystal to achieve multifunctional, robust topological states. Through detailed analysis of the fine changes in the topological bandgap induced by distortion parameters, we facilitate the realization of topological edge states in wavelength division multiplexing applications. We utilize both trivial and nontrivial properties of the topological bandgap to precisely manipulate zero-dimensional angular states, one-dimensional topological boundary states, and two-dimensional body states. Through simulations and experimental results, our advanced asymmetric C4 photonic crystal structure demonstrates superior robustness for the transmission of topological edge states. Our research paves the way for the deployment of more robust topological boundary state transmission systems and advances the application potential of higher-order topological states.

Convex co-compact groups with one-dimensional boundary faces

In this paper, we consider convex co-compact subgroups of the projective linear group. We prove that such a group is relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2 if and only if each open face in the ideal boundary has dimension at most one. We also introduce the “coarse Hilbert dimension” of a subset of a convex set and use it to characterize when a naive convex co-compact subgroup is word hyperbolic or relatively hyperbolic with respect to a collection of virtually Abelian subgroups of rank 2.

Modeling the measurement accuracy of one-dimensional boundary subsets in digital image correlation

Boundary subsets, which contain unwanted pixels from background image or other regions, generally occur at areas of vital mechanical importance. In this work, we establish a theoretical model characterizing the measurement accuracy of 1-dimensional boundary subsets, which explicitly expresses the retrieved displacement in terms of subset size, invalid pixels, shape function order, and underlying deformation field. Particularly, the transfer function of digital image correlation at boundary subsets is derived and verified, indicating possible amplification of amplitude and the existence of phase shift, which are quite different from standard full square subsets. Finally, the retrieved deformation near boundary is theoretically analyzed, and closed-form formulae are deduced for polynomial underlying displacements in consideration of the Weierstrass approximation theorem. To the author's knowledge, this is the first general mathematical model for boundary subsets, and paves the way for enhanced measurement accuracy.

High multiplicity of positive solutions in a superlinear problem of Moore–Nehari type

In this paper we consider a superlinear one-dimensional elliptic boundary value problem that generalizes the one studied by Moore and Nehari in Moore and Nehari (1959). Specifically, we deal with piecewise-constant weight functions in front of the nonlinearity with an arbitrary number κ≥1 of vanishing regions. We study, from an analytic and numerical point of view, the number of positive solutions, depending on the value of a parameter λ and on κ.Our main results are twofold. On the one hand, we study analytically the behavior of the solutions, as λ↓−∞, in the regions where the weight vanishes. Our result leads us to conjecture the existence of 2κ+1−1 solutions for sufficiently negative λ. On the other hand, we support such a conjecture with the results of numerical simulations which also shed light on the structure of the global bifurcation diagrams in λ and the profiles of positive solutions.Finally, we give additional numerical results suggesting that high multiplicity also holds true for a much larger class of weights, even arbitrarily close to situations where there is uniqueness of positive solutions.

Local Spectral Multiplicity of Selfadjoint Couplings with General Interface Conditions

We consider selfadjoint operators obtained by pasting a finite number of boundary relations with one-dimensional boundary space. A typical example of such an operator is the Schrödinger operator on a star-graph with a finite number of finite or infinite edges and an interface condition at the common vertex. A wide class of “selfadjoint” interface conditions, subject to an assumption which is generically satisfied, is considered. We determine the spectral multiplicity function on the singular spectrum (continuous as well as point) in terms of the spectral data of decoupled operators.

Impact of surface current and temperature feedback on kinetic energy over the North-East Atlantic from a coupled ocean / atmospheric boundary layer model

A one-dimensional Atmospheric Boundary Layer (ABL1D) model is coupled with the NEMO ocean model and implemented over the Iberian–Biscay–Ireland (IBI) area at 1/36° resolution to investigate the damping effect of the current and the thermal feedback on the kinetic energy (KE) at the mesoscale. This type of coupling between an ocean model and an ABL1D is a newly proposed approach as an alternative of intermediate complexity between bulk forcing and full coupling with an atmosphere model. In ABL1D, the prognostic tracers are nudged toward large-scale variables and the wind is guided by a low-frequency geostrophic wind provided from the ERA-Interim reanalyses. First, the ABL1D is successfully validated against satellite observations regarding the wind, and the dynamic coupling coefficient (linking the near surface wind and wind-stress to the of the surface currents) are consistent with the literature, over the period 2016–2017. Our results show that the thermal feedback has a negligible impact on kinetic energy (KE) and does not influence the strength of the current feedback in the region. Given the ABL1D physics, this further indicates that the changes in the vertical wind structure caused by CFB are primarily governed by local mechanical mechanisms associated with surface wind-stress condition, rather than by thermodynamic or non-local processes within the planetary boundary layer. The induced KE reduction by the current feedback amounts to 14% at the surface and propagates down to 2000 m, indicating that it can modify the vertical distribution of KE throughout the water column. KE reductions in the surface boundary layer (0 – 300 m) and in the interior (300 – 2000 m) are attributed to a reduction of the surface wind work by 4%, and of the pressure work by 7%, respectively. The Ekman pumping anomalies induced by the current feedback tend to attenuate eddy activity and horizontal pressure gradients at depth, illustrating the potential of the current feedback to induce a geostrophic adjustment on the water column. These results illustrate the relevance of the proposed ABL1D coupling approach for reproducing the wind-current coupling (a.k.a. current feedback effect) which cannot be taken into account straightforwardly with simple bulk forcing.

Bifurcation delay and front propagation in the real Ginzburg-Landau equation on a time-dependent domain.

This work analyzes bifurcation delay and front propagation in the one-dimensional real Ginzburg-Landau equationwith periodic boundary conditions on isotropically growing or shrinking domains. First, we obtain closed-form expressions for the delay of primary bifurcations on a growing domain and show that the additional domain growth before the appearance of a pattern is independent of the growth time scale. We also quantify primary bifurcation delay on a shrinking domain; in contrast with a growing domain, the time scale of domain compression is reflected in the additional compression before the pattern decays. For secondary bifurcations such as the Eckhaus instability, we obtain a lower bound on the delay of phase slips due to a time-dependent domain. We also construct a heuristic model to classify regimes with arrested phase slips, i.e., phase slips that fail to develop. Then, we study how propagating fronts are influenced by a time-dependent domain. We identify three types of pulled fronts: homogeneous, pattern spreading, and Eckhaus fronts. By following the linear dynamics, we derive expressions for the velocity and profile of homogeneous fronts on a time-dependent domain. We also derive the natural "asymptotic" velocity and front profile and show that these deviate from predictions based on the marginal stability criterion familiar from fixed domain theory. This difference arises because the time dependence of the domain lifts the degeneracy of the spatial eigenvalues associated with speed selection and represents a fundamental distinction from the fixed domain theory that we verify using direct numerical simulations. The effect of a growing domain on pattern spreading and Eckhaus front velocities is inspected qualitatively and found to be similar to that of homogeneous fronts. These more complex fronts can also experience delayed onset. Lastly, we show that dilution-an effect present when the order parameter is conserved-increases bifurcation delay and amplifies changes in the homogeneous front velocity on time-dependent domains. The study provides general insight into the effects of domain growth on pattern onset, pattern transitions, and front propagation in systems across different scientific fields.

Localization of Chiral Edge States by the Non-Hermitian Skin Effect

Localization of Chiral Edge States by the Non-Hermitian Skin Effect

Analysis of Varying Temperature Regimes in a Conductive Strip during Induction Heating under a Quasi-Steady Electromagnetic Field

Transition processes in a steel conductive strip are analyzed during its induction heating under a quasi-steady electromagnetic field. In particular, the temperature field in the strip is studied. A method of solving corresponding initial boundary problems in a two-dimensional mathematical model for differential equations of electrodynamics and heat conduction is developed. The Joule heat and the temperature are determined with a high level of accuracy. The defining functions are the temperature and component of the magnetic field intensity vector tangent to the bases and end planes of the strip. To find them, we use cubic approximation of the defining functions’ distribution along the thickness coordinate. The original two-dimensional initial boundary value problems for the defining functions are reduced to one-dimensional initial boundary value problems on their integral characteristics. General solutions for these problems are obtained using the finite integral transformation by the transverse variable and the Laplace transform of the integral by time. Integral characteristics’ expressions are represented as convolutions for functions that describe homogeneous solutions of one-dimensional initial boundary value problems and limiting values of defining functions on the bases and end planes of the strip. The change of temperature under a varying regime in the dimensionless Fourier time and temperature distribution over the strip cross-section in a steady state depending on the parameters of induction heating and the Biot number are numerically analyzed. Varying and constant temperature regimes of the strip under conditions of the near-surface and continuous induction heating are studied.

Robust error estimates of PINN in one-dimensional boundary value problems for linear elliptic equations

<p>In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving $ L^2 $-contraction estimates, we show that the error, defined as the mean square of the differences between the true solution and our trial function at the sample points, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients.</p>

Stress state modeling of non-circular orthotropic hollow cylinders under different types of loading

Based on a spatial model of the linear theory of elasticity, using an unconventional approach of the reduction of the original three-dimensional boundary value problem described by a system of partial differential equations with variable coefficients to a one-dimensional boundary value problem for a system of ordinary differential equations with constant coefficients, the problem of finding the dimensional stress of hollow elliptic orthotropic cylinders under the influence of various types of loading has been solved under certain boundary conditions at the orientation plane. Reducing the dimensionality of the original problem is carried out using analytical methods of separating variables in two coordinate directions in combination with the method of approximating functions by discrete Fourier series. The one-dimensional boundary value problem is solved by the stable numerical method of discrete orthogonalization.

DIRECT SOLUTION OF THE DYNAMICAL ELASTICITY PROBLEM FOR A QUARTER SPACE

The wave field of an elastic quarter space is constructed when one face is rigidly fixed and a dynamic normal compressive load is concentrated at the point on another face. The problem was solved by the direct application of the integral Laplace and Fourier transforms to the motion equations and the boundary conditions. This operation leads to the one-dimensional vector inhomogeneous boundary value problem with respect to unknown displacement’s transformants. The problem was solved using the matrix differential calculus. A fundamental matrix and a decreasing solution to the corresponding homogenous matrix equation were constructed with a basic residue theorem. A singular integral equation was obtained in the process by satisfying unrealized boundary condition. Weakly convergent part of the equation was summed up. The behavior of the unknown function had been analyzed based on its mechanical sense. The form of unknown function was expressed as a series on Laguerre polynomials. The original displacements’ field was found after an application the inverse integral transforms.

Boundary value problems of the dynamics of thermoelastic rods and their solutions

We consider spatially one-dimensional boundary value problems of uncoupled thermoelasticity, which can be used to study various rod structures under thermal heating conditions. As is known, rod structures are connecting and transmission links of various parts of the machines and mechanisms. Here we propose a unified technique for solving various boundary value problems typical for practical applications. The problems of determining the thermally stressed state of a thermoelastic rod under various boundary conditions at its ends and acting power and heat sources along the entire length of the rod are considered. Based on the method of generalized functions, generalized solutions of non-stationary and stationary direct and semi-inverse boundary value problems under the action of power and heat sources of various types, including stationary sources of periodic oscillations, are constructed. Acting sources can also be specified by singular generalized functions, under various boundary conditions at the ends of the rod. Shock elastic waves that arise in such structures under the action of shock loads are considered. Regular integral representations of generalized solutions are obtained, which provide an analytical solution to the posed boundary value problems. The peculiarity of the constructed solutions makes them convenient for studying network thermoelastic systems that can be modeled by thermoelastic graphs.

Визначення двовимірних нестаціонарних температурних полів у плитах та панелях з плоскопаралельними межами за наявності джерел тепла

Non-stationary two-dimensional problems of thermal conductivity for plates and panels with plane-parallel boundaries in the presence of volume-distributed non-stationary heat sources are formulated. A method of constructing a solution to the formulated heat conduction problems for the considered bodies is proposed. The technique uses the approximation of the temperature distribution in both elements by the thickness variable by a cubic polynomial. The coefficients of the approximation polynomial are given through the integral over the thickness variable temperature characteristics and the conditions for the boundary values of the temperature on the outer surfaces of the plate and panel. As a result, the initial two-dimensional initial boundary value problems for the temperature for the plate and panel are reduced to one-dimensional initial boundary value problems for the integral temperature characteristics. To construct the solution of the initial-boundary value problem for the integral characteristics of the temperature in the case of a plate infinite in longitudinal and transverse coordinates, the integral Laplace transform in time and the integral Fourier transform in the longitudinal coordinate were used. The solution of the problem on the integral temperature characteristics in the case of the panel is found using the integral Laplace transform in time and the finite integral transform in the transverse coordinate. Expressions of integral temperature characteristics for the plate and panel are obtained in the form of convolutions of functions corresponding to homogeneous solutions of initial-boundary value problems for integral temperature characteristics and functions describing the available non-stationary heat sources in these bodies and given surface temperature values. The general solutions of the two-dimensional initial boundary value problems of thermal conductivity for slabs and panels are recorded for the presence of arbitrarily variable spatial coordinates of non-stationary heat sources and the conditions of convective heat exchange with the external environment on the surfaces of the considered bodies.

MATHEMATICAL MODELING OF DIFFUSION BOUNDARY PROBLEMS WITH PERIODIC BOUNDARY CONDITIONS USING MATLAB AND C++ FOR NUMERICAL AND ANALYTICAL SOLUTIONS

The article examines a second-order parabolic partial differential equation of a three-dimensional (3D) non-stationary boundary problem with constant diffusion coefficients and periodic boundary conditions in the x and y directions. The method for reducing the (3D) non-stationary boundary problem to the corresponding one-dimensional (1D) non-stationary boundary problem using periodic boundary conditions in the x and y directions is discussed. The stationary (analytical) solution of the obtained (1D) stationary boundary problem is also obtained. The numerical solutions of the 1D boundary problem are obtained using the Matlab package "pdepe" and the C++ programming language. As a practical application of the developed mathematical model, the article discusses calculating the concentration of heavy metal Ca in a peat layer based on the obtained experimental data (measurements).

The dynamic analysis of axisymmetric bodies with damping effects using the modified radial integration boundary elements method (MRIBEM)

This paper investigates the elastodynamic analysis of axisymmetric bodies under axisymmetric dynamic loads using the modified radial integration boundary elements method (MRIBEM). Purposely, the governing integral equations have been developed in cylindrical coordinates by employing the three-dimensional (3D) static fundamental solution. The axisymmetric bodies can be modeled as a two-dimensional (2D) problem, which is much simpler than 3D analysis and only requires a one-dimensional boundary mesh. In the proposed formulation, the effects of inertial and damping forces appear as 2D domain integrals. The domain variables are approximated using global radial basis functions (RBFs), and the modified radial integration method (MRIM) is employed to deal with the domain integral as the main idea of the present research. Various sample problems are provided in order to illustrate the capability of the presented method for analyzing axisymmetric bodies with concave geometry and damping effect under different types of transient loads. The Newmark and Houbolt methods were applied for time marching, and their efficiency was examined. The results of the present study were also validated by the axisymmetric FEM formulation and other available references.

Molecular Structures and Vibrational Spectra of trans- and cis-Polyacetylene and Their Oligoenes Revisited Using Density Functional Theory Calculations.

Polyacetylene, the most representative synthetic conducting polymer, has attracted much attention because it exhibits high conductivity upon doping. In this paper, molecular structures, electronic excitation energies, and Raman and infrared spectra were calculated using density functional theory for trans- and cis-oligoenes with various chain lengths up to the number of C═C bonds (n) of 100 and trans- and cis-polyacetylenes under one-dimensional periodic boundary condition. The harmonic vibrational frequencies obtained at the B3LYP/6-311G(d,p) level were scaled by the scaling factors determined with respect to the anharmonic vibrational frequencies using the B2PLYP method, in which the coefficients of the functional were optimized for trans-oligoenes. The calculated infrared and Raman frequencies reproduce reasonably well the observed frequencies for trans- and cis-polyacetylene. Based on the chain-length dependence of the calculated Raman spectra of trans-oligoenes, we proposed the possibility of longer conjugated trans-segments observed in the resonance Raman spectra of trans-polyacetylene excited at longer wavelengths of 647.1 and 1064 nm. We also elucidated the origin of the excitation-wavelength dependence of the resonance Raman spectra of trans-polyacetylene and the structure of isomerization intermediates from cis-form to trans-form. In addition, the previous assignments of Raman and infrared spectra of trans- and cis-polyacetylene were reexamined in the present study based on the chain-length dependence of the spectra.

A method for real-time mechanical characterisation of microcapsules

Characterising the mechanical properties of flowing microcapsules is important from both fundamental and applied points of view. In the present study, we develop a novel multilayer perceptron (MLP)-based machine learning (ML) approach, for real-time simultaneous predictions of the membrane mechanical law type, shear and area-dilatation moduli of microcapsules, from their camera-recorded steady profiles in tube flow. By MLP, we mean a neural network where many perceptrons are organised into layers. A perceptron is a basic element that conducts input–output mapping operation. We test the performance of the present approach using both simulation and experimental data. We find that with a reasonably high prediction accuracy, our method can reach an unprecedented low prediction latency of less than 1 millisecond on a personal computer. That is the overall computational time, without using parallel computing, from a single experimental image to multiple capsule mechanical parameters. It is faster than a recently proposed convolutional neural network-based approach by two orders of magnitude, for it only deals with the one-dimensional capsule boundary instead of the entire two-dimensional capsule image. Our new approach may serve as the foundation of a promising tool for real-time mechanical characterisation and online active sorting of deformable microcapsules and biological cells in microfluidic devices.

Numerical simulation of Bratu’s problem using a new form of the Adomian decomposition technique

Purpose This paper aims to discuss a new form of the Adomian decomposition technique for the numerical treatment of Bratu’s type one-dimensional boundary value problems (BVPs). Moreover, the author also addresses convergence and error analysis for the completeness of the proposed technique. Design/methodology/approach First, the author discusses the standard Adomian decomposition method and an algorithm based on Duan’s corollary and Rach’s rule for the fast calculation of the Adomian polynomials. Then, a new form of the Adomian decomposition technique is present for the numerical simulation of Bratu’s BVPs. Findings The reliability and validity of the proposed technique are examined by calculating the absolute errors of Bratu’s problem for some different values of Bratu parameter λ. Numerical simulation demonstrates that the proposed technique yields higher accuracy than the Bessel collocation and other known methods. Originality/value Unlike the other methods, the proposed technique does not need linearization, discretization or perturbation to handle the non-linear problems. So, the results obtained by the present technique are more physically realistic.