Approximate Boundary Research Articles

A C0 linear finite element method for a second‐order elliptic equation in non‐divergence form with Cordes coefficients

AbstractIn this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear FEM for second‐order elliptic equations in non‐divergence form. The elliptic equation is casted into a symmetric non‐divergence weak formulation, in which second‐order derivatives of the unknown function are involved. We use gradient and Hessian recovery operators to calculate the second‐order derivatives of linear finite element approximations. Although, thanks to low degrees of freedom of linear elements, the implementation of the proposed schemes is easy and straightforward, the performances of the methods are competitive. The unique solvability and the seminorm error estimate of the GRBL scheme are rigorously proved. Optimal error estimates in both the norm and the seminorm have been proved when the coefficient is diagonal, which have been confirmed by numerical experiments. Superconvergence in errors has also been observed. Moreover, our methods can handle computational domains with curved boundaries without loss of accuracy from approximation of boundaries. Finally, the proposed numerical methods have been successfully applied to solve fully nonlinear Monge–Ampère equations.

Thin Dielectric Sheet Approximation-Based Characteristic Mode Formulations for Lossless Dielectric Coated Conducting Bodies

The thin dielectric sheet (TDS) approximation-based theory of characteristic modes (TCM) is formulated for lossless thin dielectric-coated conducting bodies. The proposed formulations are derived from the TDS-electric, magnetic, and combined field integral operators (TDS-EFIO, TDS-MFIO, and TDS-CFIO). Similar to the cases of perfect electric conductor (PEC) and impedance surface, the TDS-CFIO-based formulation can avoid modes corruption by spurious solutions at the internal resonant frequencies of the PEC cavity filled with the material of the background. Due to the utilization of TDS approximation and PEC boundary conditions, the eigencurrents are expressed only by electric currents on the surface of PEC. Therefore, the proposed formulations are more efficient than the conventional formulations based on the surface integral operator (SIO) or volume-surface integral operator (VSIO). The coating thickness of TDS approximation is recommended to be less than one tenth of a wavelength in the dielectric for accurate analysis of characteristic mode (CM). Several numerical examples are given to validate the proposed formulations.

Adaptive Neural-Network-Based Fault-Tolerant Control for a Flexible String With Composite Disturbance Observer and Input Constraints

We propose an adaptive neural-network-based fault-tolerant control scheme for a flexible string considering the input constraint, actuator gain fault, and external disturbances. First, we utilize a radial basis function neural network to compensate for the actuator gain fault. In addition, an observer is used to handle composite disturbances, including unknown approximation errors and boundary disturbances. Then, an auxiliary system eliminates the effect of the input constraint. By integrating the composite disturbance observer and auxiliary system, adaptive fault-tolerant boundary control is achieved for an uncertain flexible string. Under rigorous Lyapunov stability analysis, the vibration scope of the flexible string is guaranteed to remain within a small compact set. Numerical simulations verify the high control performance of the proposed control scheme.

On the validity of numerical models for viscothermal losses in structural optimization for micro-acoustics

The increasing interest in miniaturizing acoustic devices has made accurate and efficient models of acoustic viscous and thermal losses progressively more important. This is especially the case in micro-acoustic devices such as hearing aids, condenser microphones and MEMS devices. Using the full linearized Navier Stokes equations to numerically model losses comes at a high computational cost. An approximate boundary layer impedance boundary condition representing acoustic losses has therefore become popular due to its high computational efficiency. This is especially true in the context of optimization where an efficient numerical method is required due to the many repeated analyses needed. However, the boundary layer impedance is only valid in the computational region where boundary layers are non-overlapping. Applying the boundary layer impedance can therefore lead to poor optimization results or limit the possible design space if the optimization violates this limitation. Therefore, the benefit of losses in narrow regions cannot be exploited if the boundary layer impedance is used. This work investigates two shape optimization test cases for maximizing the absorption properties of Helmholtz-like geometries based on the Boundary Element Method. The test cases are used to compare and validate the boundary layer impedance against a full viscothermal implementation revealing the benefits of the boundary layer impedance but also its limitations in a structural optimization setting. Based on the numerical experiments it is recommend to avoid the use of the boundary layer impedance in cases where any theoretical boundary layer overlap exists or at least verify simulation and optimization results with a full-losses implementation.

Fast prototype selection algorithm based on adjacent neighbourhood and boundary approximation

The unceasing increase of data quantity severely limits the wide application of mature classification algorithms due to the unacceptable execution time and the insufficient memory. How to fast incrementally obtain high decision reference set and adapt to incremental data environment is urgently needed in incremental environments, large dataset, etc. This paper proposes a novel prototype selection algorithm by integrating the strategies between condensing method and editing method. To an unlearned pattern, this algorithm extends the references scope from its single nearest neighbour to its k nearest neighbourhood that can expand the judgment information to obtain its detailed neighbour relationship. Then a pattern was determined whether it is a prototype using its neighbour relationship and classification boundary asymptotically strategy. To maintain the higher reference set, this algorithm periodically updates those prototypes that locates in the non-boundary zone or is long-time unlearned. The empirical study shows that this algorithm obtains the smaller and higher boundary prototypes without decreasing classification accuracy and reduction rate than the compared algorithms.

Unconventional pairing in few-fermion systems at finite temperature

Attractively interacting two-component mixtures of fermionic particles confined in a one-dimensional harmonic trap are investigated. Properties of balanced and imbalanced systems are systematically explored with the exact diagonalization approach, focusing on the finite-temperature effects. Using single- and two-particle density distributions, specific non-classical pairing correlations are analyzed in terms of the noise correlations—quantity directly accessible in state-of-the-art experiments with ultra-cold atoms. It is shown that along with increasing temperature, any imbalanced system hosting Fulde–Ferrel–Larkin–Ovchinnikov pairs crossovers to a standard Bardeen-Cooper-Schrieffer one characterized by zero net momentum of resulting pairs. By performing calculations for systems with different imbalances, the approximate boundary between the two phases on a phase diagram is determined.

Implementation and Validation of an Algebraic Wall Model for LES in Nek5000

Turbulent flows are most often wall-bounded, rendering the treatment of the wall essential. In this work, the near-wall layer is modelled in large eddy simulations which enables simulating high Reynolds number flows. An algebraic wall model has been implemented in the spectral element code Nek5000. It consists of an approximate boundary condition that relates the wall shear stress to the velocity measured close to the wall, on the upper edge of the first spectral element. The wall shear stress model approximates the law of the wall for hydraulically smooth cases. The model is applied to channel flow cases at Re_{tau }=1000 and at Re_{tau }=5200. The wmLES results obtained with the present implementation are seen to compare very well with those of reference direct numerical simulations in the resolved region. They also remain remarkably close to the reference results for a large part of the under-resolved region; which is not necessarily the case when using low order implementations and even other types of high order discretizations, as found in the literature. Various parameters are studied such as the time averaging, the height of the near wall under-resolved element, and the mesh requirements. The obtained results indicate that accurate results can be obtained with Nek5000 at a reduced cost thanks to this newly implemented wmLES model. This work provides the necessary guidelines for simple flows, and it will serve as a first basis for simulating more complex flows at high Reynolds numbers.

Approximate boundary conditions for electromagnetic fields in electrodmagnetics

The results of an analytical review of literature sources on the use of approximate boundary conditions for electromagnetic fields of impedance type in solving boundary value problems of electromagnetism for more than 80 recent years are presented. During this period, the impedance approach was generalized to various electrodynamic problems, in which its use made it possible to significantly expand the limits of mathematical modeling, which adequately considers the physical properties of real boundary surfaces. More than eighty years have passed since the publication of approximate boundary conditions for electromagnetic fields. The meaning and value of these conditions lies in the fact that they allow solving diffraction problems about fields outside well-conducting bodies without considering the fields inside them, which greatly simplifies the solution. Since then, numerous publications have been devoted to the application of impedance boundary conditions, the main of which (according to the authors) are presented in this paper. Particular attention is paid to the characteristics of electrically thin impedance vibrators and film-type surface structures as a personal contribution of the authors to the theory of impedance boundary conditions in electromagnetism. The subject of research in this article is the analysis of the limits and conditions for the correct application of impedance boundary conditions. The goal is to systematize the results of using the concept of approximate impedance boundary conditions for electromagnetic fields in problems of electrodynamics based on an analytical review of literature sources. The following results were obtained. The types of metal-dielectric structures are presented, for which methods of theoretical determination of the values of surface impedances for film-type structures are currently known, which are the most promising for creating technological control elements on their basis in centimeter and millimeter wavelength devices. Conclusions. The materials of this paper do not pretend to be a complete reference book covering all the results and aspects of the development of the concept of approximate impedance type boundary conditions in problems of electromagnetism over the past decades. Simultaneously, the authors hope that the information presented in this paper will be useful to specialists in the field of theoretical and applied electrodynamics, as well as graduate students, young scientists and students who are just mastering radiophysics and radio engineering specialties.

A Multi-Fidelity Approach for Reliability Assessment Based on the Probability of Classification Inconsistency

Abstract Most multi-fidelity schemes for optimization or reliability assessment rely on regression surrogates, such as Gaussian processes. Contrary to these approaches, we propose a classification-based multi-fidelity scheme for reliability assessment. This technique leverages multi-fidelity information to locally construct failure boundaries using support vector machine (SVM) classifiers. SVMs are subsequently used to estimate the probability of failure using Monte Carlo simulations. The use of classification has several advantages: It can handle discontinuous responses and reduce the number of function evaluations in the case of a large number of failure modes. In addition, in the context of multi-fidelity techniques, classification enables the identification of regions where the predictions (e.g., failure or safe) from the various fidelities are identical. At the core of the proposed scheme is an adaptive sampling routine driven by the probability of classification inconsistency between the models. This sampling routine explores sparsely sampled regions of inconsistency between the models of various fidelity to iteratively refine the approximation of the failure domain boundaries. A lookahead scheme, which looks one step into the future without any model evaluations, is used to selectively filter adaptive samples that do not induce substantial changes in the failure domain boundary approximation. The model management strategy is based on a framework that adaptively identifies a neighborhood of no confidence between the models. The proposed scheme is tested on analytical examples of dimensions ranging from 2 to 10, and finally applied to assess the reliability of a miniature shell and tube heat exchanger.

The effect of prior exposure on the lightfastness of early synthetic dyes on textiles

Many studies have investigated light-induced damage to colourants in heritage collections using prepared samples of materials such as artist paints and dyed textiles. The body of research focuses primarily on the response of virgin materials, where colour change is assessed with respect to the original colour as light dose increases. From a practical perspective, most objects in museum collections have accumulated a significant light dose from illumination before acquisition, and subsequent years of exhibit lighting. When considering the risk of further degradation, it is often stated by heritage professionals that fugitive colourants with past light exposure are no longer as sensitive due to the slowing rate of visual damage. This is evident by studying ‘fading curves’, where the rate of colour change typically diminishes with increasing light dose. It remains unclear, however, to what degree the lightfastness of remaining colour changes with ongoing exposure. To address the issue, the light sensitivity of residual colour was investigated as a function of prior dose using a published dataset of colour measurements from textiles dyed with early synthetic colourants in the period of 1874–1905. The CIELAB colour values (D65/2°) for each material and dose increment were used to determine the future dose that causes a just-noticeable difference (JND), ΔE00 = 1.7, starting at different amounts of past exposure. This involved resetting the reference CIELAB values to that of the residual colour after each prior dose. The analysis provides an extension of our earlier work, where only the initial lightfastness was reported. A summary of the findings illustrates the shifting distribution of lightfastness, toward higher Blue Wool (BW) ratings with increasing exposure. As the dose progresses, dyes that start at BW1 sensitivity progress to BW2, then BW3 and so on. The findings from this work may assist with boundary approximations of object sensitivity when prior light exposure is known, or a reasonable estimate is available. An analysis of experimental data is summarised as a tool for this type of decision-making process.

Effect of Contact Angle Hysteresis on Evaporation Dynamics of a Sessile Drop on a Heated Surface

Contact angle hysteresis (CAH) is a significant factor affecting the drop motion on solid substrates. A model of CAH is introduced to explore the influence of CAH on the dynamics of a sessile drop on a uniformly heated surface, and a two-dimensional evolution equation of the drop thickness is established using the lubrication approximation and Navier slip boundary conditions. A numerical simulation is performed to examine the dynamic behaviors of an evaporating drop, and the drop profile, contact angle, contact line, and moving speed are investigated. Simulated results indicate that the drop evolution process involves drop spreading, pinning, and depinning of the contact line. In the drop spreading stage, when the hysteresis angle increases, the spreading period is shortened, and the spreading radius and spreading speed are reduced; in contrast, the pinning period is raised, and the mass of the drop is apparently reduced with increasing hysteresis angle. In the depinning stage, the CAH declines the contact angle, and a flatter pattern is evolved, thereby improving the heat transfer performance, promoting drop evaporation, and shortening the depinning time. The presence of CAH can speed up the drying of the drop, and the large hysteresis angle leads to faster evaporation. Regulating the CAH is an effective way to manipulate the motion of the contact line for an evaporating drop.

Asymptotic analysis of long‐term investment with two illiquid and correlated assets

AbstractWe consider a long‐term portfolio choice problem with two illiquid and correlated assets, which is associated with an eigenvalue problem in the form of a variational inequality. The eigenvalue and the free boundaries implied by the variational inequality correspond to the portfolio's optimal long‐term growth rate and the optimal trading strategy, respectively. After proving the existence and uniqueness of viscosity solutions for the eigenvalue problem, we perform an asymptotic expansion in terms of small correlations and obtain semi‐analytical approximations of the free boundaries and the optimal growth rate. Our leading order expansion implies that the free boundaries are orthogonal to each other at four corners and have C1 regularity. We propose an efficient numerical algorithm based on the expansion, which proves to be accurate even for large correlations and transaction costs. Moreover, following the approximate trading strategy, the resulting growth rate is very close to the optimal one.

Numerical simulation of a relativistic magnetron using a fluid electron model

An approach to numerically modeling relativistic magnetrons, in which the electrons are represented with a relativistic fluid, is described. A principal effect in the operation of a magnetron is space-charge-limited (SCL) emission of electrons from the cathode. We have developed an approximate SCL emission boundary condition for the fluid electron model. This boundary condition prescribes the flux of electrons as a function of the normal component of the electric field on the boundary. We show the results of a benchmarking activity that applies the fluid SCL boundary condition to the one-dimensional Child–Langmuir diode problem and a canonical two-dimensional diode problem. Simulation results for a two-dimensional A6 magnetron are then presented. Computed bunching of the electron cloud occurs and coincides with significant microwave power generation. Numerical convergence of the solution is considered. Sharp gradients in the solution quantities at the diocotron resonance, spanning an interval of three to four grid cells in the most well-resolved case, are present and likely affect convergence.

Ion channel model reduction using manifold boundaries.

Mathematical models of voltage-gated ion channels are used in basic research, industrial and clinical settings. These models range in complexity, but typically contain numerous variables representing the proportion of channels in a given state, and parameters describing the voltage-dependent rates of transition between states. An open problem is selecting the appropriate degree of complexity and structure for an ion channel model given data availability. Here, we simplify a model of the cardiac human Ether-à-go-go related gene (hERG) potassium ion channel, which carries cardiac IKr, using the manifold boundary approximation method (MBAM). The MBAM approximates high-dimensional model-output manifolds by reduced models describing their boundaries, resulting in models with fewer parameters (and often variables). We produced a series of models of reducing complexity starting from an established five-state hERG model with 15 parameters. Models with up to three fewer states and eight fewer parameters were shown to retain much of the predictive capability of the full model and were validated using experimental hERG1a data collected in HEK293 cells at 37°C. The method provides a way to simplify complex models of ion channels that improves parameter identifiability and will aid in future model development.

Macroscopic Lattice Boltzmann Method for Shallow Water Equations

The lattice Boltzmann method (LBM) is characterised by its simplicity, parallel processing and easy treatment of boundary conditions. It has become an alternative powerful numerical method in computational physics, playing a more and more important role in solving challenging problems in science and engineering. In particular, the lattice Boltzmann method with the single relaxation time (SLBM) is the simplest and most popular form of the LBM that is used in research and applications. However, there are two long-term unresolved problems that prevent the SLBM from being an automatic simulator for any flows: (1) stability problem associated with the single relaxation time and (2) no method of direct implementation of physical variables as boundary conditions. Recently, the author has proposed the macroscopic lattice Boltzmann method (MacLAB) to solve the Navier–Stokes equations for fluid flows, resolving the aforementioned problems; it is unconditionally stable and uses physical variables as boundary conditions at lower computational cost compared to conventional LBMs. The MacLAB relies on one fundamental parameter of lattice size δx, and is a minimal version of the lattice Boltzmann method. In this paper, the idea of the MacLAB is further developed to formulate a macroscopic lattice Boltzmann method for shallow water equations (MacLABSWE). It inherits all the advantages from both the MacLAB and the conventional LBM. The MacLABSWE is developed regardless of the single relaxation time τ. Physical variables such as water depth and velocity can directly be used as boundary conditions, retaining their initial values for Dirichlet’s boundary conditions without updating them at each time step. This makes not only the model to achieve the exact no-slip boundary condition but also the model’s efficiency superior to the most efficient bounce-back scheme for approximate no-slip boundary condition in the LBMs, although the scheme can similarly be implemented in the proposed model when it is necessary. The MacLABSWE is applied to simulate a 1D unsteady tidal flow, a 2D steady wind-driven flow in a dish-shaped lake and a 2D steady complex flow over a bump. The results are compared with available analytical solutions and other numerical studies, demonstrating the potential and accuracy of the model.

High accuracy power series method for solving scalar, vector, and inhomogeneous nonlinear Schrödinger equations

We develop a high accuracy power series method for solving partial differential equations with emphasis on the nonlinear Schrödinger equations. The accuracy and computing speed can be systematically and arbitrarily increased to orders of magnitude larger than those of other methods. Machine precision accuracy can be easily reached and sustained for long evolution times within rather short computing time. In-depth analysis and characterisation for all sources of error are performed by comparing the numerical solutions with the exact analytical ones. Exact and approximate boundary conditions are considered and shown to minimise errors for solutions with finite background. The method is extended to cases with external potentials and coupled nonlinear Schrödinger equations.

Bifurcation and stability analysis of temperature patterns in shallow-bed catalytic reactors

Complex temperature patterns have been observed in shallow packed-bed catalytic reactors operated under autothermal conditions. These patterns are not desired in practice, and their elimination requires a fundamental understanding of the conditions leading to the emergence and disappearance of lower and higher mode stable patterns. This work analyzes the formation of transverse temperature patterns and their stability by utilizing cell models. Our results indicate that the minimum effective bed thermal conductivity required to eliminate all the stable patterned states scales with the reactor diameter. We also present transient simulations that elucidate the impact of initial conditions and the basins of attraction of various patterned and homogeneous states. Our results indicate that unstable patterned states can provide a good approximation of the basin boundaries of the initial conditions that lead to stable patterns.

Theoretical analysis and comparison of transient-extracting transform and time-reassigned synchrosqueezing transform

Transient-extracting transform (TET) and time-reassigned synchrosqueezing transform (TSST) developed under the framework of short-time Fourier transform (STFT) can effectively enhance the time–frequency (TF) resolution of nonstationary signal composed of multicomponent with impulse-like components. Nevertheless, the TET and TSST methods lack relevant theoretical support to ensure their practical application. In this paper, we supplement the theoretical analysis of these two methods and provide a new view to compare them. We first review the original definitions of TET and TSST proposed in the previous literatures. Then, we demonstrate the performance and properties of TSST and TET in detail, e.g., group delay (GD) approximation error, reconstruction error and boundary analysis, extraction principle of TET, and property derivation. Finally, the performance of TET and TSST is further compared in the analysis of the numerical and real signals. Through complete theoretical and property analysis, the application of TET and TSST in reality can be largely enhanced.

Polygonal approximation for digital image boundary using firefly algorithm

Polygonal approximation (PA) is a challenging problem and has its wide applications in computer vision, electronic imaging, image classification, image visualization, pattern recognition and image analysis. This paper proposes a stochastic technique-based firefly algorithm (FA) for PA. FA customizes a kind of randomization by searching a set of solutions where PA requires more combinations of approximation to find an optimal solution. The significant feature of the FA is real random sample generation and attractive features. The attractiveness and brightness of the firefly have been used efficiently to solve the approximation problem by detecting the dominant points (DPs) with fewer iterations to produce better results. Thus, this technique achieves the main goal of PA that is a minimum error value with a lesser number of DPs. While compared with other similar algorithms, FA is independent of velocities which are considered as an advantage for this algorithm. Subsequently, the multiswarm nature of FA allows finding multiple optimal solutions concurrently. The experimental results show that the proposed algorithm generates better solutions compared with other algorithms.

Transverse oscillations and damping of magnetic flux tubes with a thick transitional layer

ABSTRACT In this paper, the magnetohydrodynamic kink waves and their damping are investigated in a thin coronal flux tube model. The thickness of the inhomogeneous transitional layer is of the order of the tube’s radius. The governing equations are solved analytically in all regions of the tube except for the thin dissipative layer. The dispersion relation is derived and solved numerically. The oscillation frequency does not vary much when varying the thickness of the transitional layer. When the transitional layer is thin, the damping rate of the kink wave increases significantly with the increase in the thickness of the transitional layer; however, when the transitional layer is thick, the damping rate does not vary much with the increase in the thickness of the transitional layer. For small values of the thickness of the transitional layer, our results are in agreement with results obtained in the thin tube, thin boundary approximation. In flowing tubes, the effect of flow on the damping rate of the waves depends on the direction of the wave propagation and plasma flow. The resonant instability of the kink waves in flux tubes with flowing transitional layer has been investigated.