Approximate Boundary Research Articles

A Finite Difference Scheme with Improved Boundary Approximation for the Heat Conduction Equation with Third-Type Boundary Conditions

Introduction. The development, analysis, and modification of finite difference schemes tailored to the specific features of the considered problem can significantly enhance the accuracy of modeling complex systems. In simulations of various processes, including hydrodynamic phenomena in shallow water bodies, it has been observed that for problems with thirdtype (Robin) boundary conditions, the theoretical error order of spatial discretization drops from second-order to firstorder accuracy, which in turn decreases the overall accuracy of the numerical solution. The present study addresses the relevant issue of how the approximation of third-type boundary conditions affects the accuracy of the numerical solution to the heat conduction problem. It also proposes a finite difference scheme with improved boundary approximation for the heat conduction equation with third-type boundary conditions and compares the accuracy of the numerical solutions obtained by the authors with known benchmark solutions.Materials and Methods. The paper considers the one-dimensional heat conduction equation with third-type boundary conditions, for which an analytical solution is available. The problem is discretized, and it is shown that under standard boundary approximation, the theoretical order of approximation error for the second-order differential operator in the diffusion equation is O(h). To improve the accuracy of the numerical solution under specific third-type boundary conditions, a finite difference scheme is proposed. This scheme achieves second-order accuracy O(h2), for the differential operator not only at interior nodes but also at the boundary nodes of the computational domain.Results. Test problems were used to compare the accuracy of numerical solutions obtained using the proposed scheme and those based on the standard boundary approximation.Discussion and Conclusion. Numerical experiments demonstrate that the proposed scheme with enhanced boundary approximation for the heat conduction equation under specific third-type boundary conditions exhibits an effective accuracy order close to 2, which corresponds to the theoretical prediction. It is noteworthy that the scheme with standard boundary approximation also demonstrates an effective accuracy order close to 2, despite the lower theoretical order of boundary approximation. Importantly, the numerical error of the proposed scheme decreases significantly faster compared to the scheme with standard boundary treatment.

A full-information numerical simulation method for 3D marine MT based on arbitrary fourier transform

Abstract Abstract There are problems with conventional marine magnetotelluric(MT) numerical simulations, such as large computational requirements, approximate boundary conditions, and difficulty in accurately simulating physical information characterized by three-dimensional(3D) partial differential equations (PDE). Therefore, a novel full-information numerical simulation method for 3D marine MT(HMMT) fields based on an arbitrary Fourier transform(AFFT) scheme is proposed in this paper. The core is based on the true seabed physical model, accurately obtaining the wavenumber spectrum of the electromagnetic field in the spatial-wavenumber domain through AFFT, and then transforming all the spectral information of the electromagnetic field into the spatial domain to obtain the accurate electromagnetic field. The method performs two-dimensional(2D) AFFT along the horizontal direction (-∞, +∞), converting the 3D partial differential equations into one-dimensional(1D) ordinary differential equations with different wave numbers for solving, which has high efficiency and low storage characteristics. The vertical direction is retained as the spatial domain and the wave decomposition is introduced to eliminate the influence of the upper and lower boundaries on the simulation area, and the horizontal mesh corresponding to each depth is arbitrary, which has the advantages of high resolution and flexible meshing, balancing the computational accuracy and efficiency. Each element utilizes the high accuracy solution of the 1D shape function and the high efficiency of the Fourier transform to realize the fast and high accuracy forward simulation. Firstly, a 3D seabed model is designed, and the code is validated by comparing it with the numerical solution of the integral equation method (IE). Secondly, according to the distribution characteristics of the wavenumber spectrum of the anomalies in the Fourier domain, we select the wavenumber sampling method and analyze the computational accuracy and efficiency of the algorithm in this paper. Finally, the combined model is designed to analyze the effect of the boundary on the numerical solutions of the proposed algorithm, standard FFT and Gauss-FFT. The results show that the full-information numerical algorithm based on AFFT does not exhibit truncation effects and can quickly and accurately simulate the physical information represented by 3D partial differential equations.

An Approximate Assessment of Latency in a Computer System with Container Virtualization

The key role in achieving high reliability, security, fault tolerance, and low latency of query service in distributed systems (including cloud computing) is played by the consolidation of data processing and storage resources in clusters, the efficiency of which increases with the use of virtual machine technologies and container virtualization. The complexity of building queuing models for container virtualization systems is caused by the fact that the intensity of query execution in each container is associated with the dynamic division of shared resources between active (performing functional tasks) containers and the costs of supporting all containers deployed in the VM, including inactive containers waiting for service requests to be sent to them. The reduction in service intensity in each container due to shared resource allocation depends on many factors that are difficult to investigate. For clusters with container virtualization, this article provides an approximate boundary estimate of the average request waiting time and the probability of timely service. When building an analytical model, each container is represented as a separate single-channel queuing system with an infinite queue and the simplest input stream. The key feature of the proposed virtual cluster model is the estimation of upper, lower, and average bounds for the potential service intensity reduction in containers, resulting from the allocation of a node's limited computing resources among them. This depends on the number of deployed containers and the dynamically varying count of active containers, which is influenced by the input stream intensity. The study demonstrates the existence of an optimal number of containers per node, minimizing the average request processing time or maximizing the probability of timely request execution. The proposed models can be applied to the structural and parametric optimization of clusters with pipelined virtualization, including in the case of scaling and reconfiguration adaptive to traffic changes by disconnecting or connecting some of the deployed containers depending on changes in the load in the system.

Scattering theory on Riemann surfaces I: Schiffer operators, cohomology, and index theorems

In this paper, we consider a compact Riemann surface [Formula: see text] with a complex of non-intersecting Jordan curves, whose complement is a pair of Riemann surfaces with boundary, each of which may be possibly disconnected. We investigate conformally invariant integral operators of Schiffer, which act on [Formula: see text] anti-holomorphic one-forms on one of these surfaces with boundary and produce holomorphic one-forms on the disjoint union. These operators arise in potential theory, boundary value problems, approximation theory, and conformal field theory, and are closely related to a kind of Cauchy operator. We develop an extensive calculus for the Schiffer and Cauchy operators, including a number of adjoint identities for the Schiffer operators. In the case that the Jordan curves are quasicircles, we derive a Plemelj–Sokhotski jump formula for Dirichlet-bounded functions. We generalize a theorem of Napalkov and Yulmukhametov, which shows that a certain Schiffer operator is an isomorphism for quasicircles. Finally, we characterize the kernels and images, and derive index theorems for the Schiffer operators, which will in turn connect conformal invariants to topological invariants.

Star Formation from Low to High Mass: A Comparative View

Star formation has often been studied by separating the low- and high-mass regimes with an approximate boundary at 8 M⊙. Although some of the outcomes of the star-formation process are different between the two regimes, it is less clear whether the physical processes leading to these outcomes are that different at all. Here, we systematically compare low- and high-mass star formation by reviewing the most important processes and quantities from an observational and theoretical point of view. We identify three regimes in which processes are either similar, quantitatively, or qualitatively different between low- and high-mass star formation. ▪ Similar characteristics can be identified for the turbulent gas properties and density structures of the star-forming regions. Many of the observational characteristics also do not depend that strongly on the environment. ▪ Quantitative differences can be found for outflow, infall, and accretion rates as well as mean column and volume densities. Also, the multiplicity significantly rises from low- to high-mass stars. The importance of the magnetic field for the formation processes appears still less well constrained. ▪ Qualitative differences between low- and high-mass star formation relate mainly to the radiative and ionizing feedback that occurs almost exclusively in regions forming high-mass stars. Nevertheless, accretion apparently can continue via disk structures in ionized accretion flows. Finally, we discuss to what extent a unified picture of star formation over all masses is possible and which issues need to be addressed in the future.

Free- and Forced-Vibration Characteristic Analysis of a Double-Layered Cylindrical Shell with General Boundary Conditions

The double-layered cylindrical shell represents a key structural configuration for underwater vehicles, where its vibration behavior remains a primary concern in engineering design and analysis. This study develops a spectral element method (SEM) for dynamic modeling of multi-component shell systems by extending the vibrational governing equations of conical shells. The methodology is validated through finite element method (FEM) case studies on both conical shells and double-layered cylindrical configurations. Parametric investigations examine ribbed substructures and solid rib plates within the cylindrical shell assembly, while artificial spring techniques model arbitrary boundary conditions—with validation against classical benchmarks confirming their effectiveness for elastic constraints. Numerical demonstrations reveal the following: rib and plate thickness variations exhibit a negligible impact on low-frequency vibrational responses; the natural frequency sensitivity peaks when the elastic boundary stiffness approaches the inherent dynamic stiffness of the shell’s base configuration, while extreme stiffness values approximate clamped or free boundary conditions with engineering significance. The proposed SEM framework demonstrates a superior computational efficiency and accuracy compared to conventional FEM approaches. These findings deliver practical guidance for marine structural engineering, particularly in the boundary condition specifications and performance optimization of composite shell systems.

Fast analysis of MEMS micromirror based on multi-physical electro-mechanical-damping coupling method

Abstract This paper presents an algorithm to improve the computational efficiency of electrostatically driven scanning micromirror with staggered vertical combs. Firstly, a single vertical comb is treated with an approximate periodic boundary. Through mechanical analysis, the mapping relationship between the deflection angle of the comb and the torque is obtained. It is found that when the deflection angle is small, the torque acting on the comb changes little and can be considered as a constant value. Based on the mapping relationship, the electrostatic torque of a single axis electrostatic scanning mirror with different voltages is extracted by considering only the electrostatic field action. In the subsequent deflection analysis of the micromirror, the extracted torque is added to the mechanical field, and the electrostatic field can be ignored. In order to further simplify the model and improve the computational efficiency, an equivalent model was established on the basis of the original single-axis scanning mirror. Immediately after, considering the effect of air damping, solid mechanics and thermal-viscous acoustic coupling calculations of the single-axis electrostatic driven mirror are carried out. Finally, the maximum mechanical deflection angle of the micromirror under different atmospheric pressure is obtained. The model proposed in this paper transforms the electro-mechanical coupling calculation into separate calculation of electrostatic field and mechanical field, which greatly improves the calculation efficiency.

Adversarial detection by approximation of ensemble boundary

Adversarial detection by approximation of ensemble boundary

Assessing the thermohaline coherence of mesoscale eddies as described from in situ data

Abstract. Mesoscale eddies are a ubiquitous feature of the global ocean. According to Lagrangian theory, these eddies often transport a distinct water mass within their cores, making them materially coherent. This study aims to determine if such a distinct water mass exists in eddy cores, thereby verifying their material coherence using in situ data, despite the lack of temporal continuity. We introduce the term “thermohaline coherence” to describe this approach. Identifying such a water mass would signal Lagrangian transport from the eddy formation region. We analyzed the water masses at the cores of various eddies sampled during eight research cruises using high-resolution data (approximately 20 km horizontally and 10 m vertically). We revisited coherence definitions and checked data accuracy. Comparing the horizontal positions of these core anomalies with eddy surface signatures revealed that surface data alone are insufficient for characterizing the eddy material coherence. To calculate eddy volumes, we compare thermohaline anomalies with other criteria, and we present two methods for extrapolating eddy volumes from a single hydrographic section. The results show that the outermost closed contour of the Brunt–Väisälä frequency anomaly at each depth provides a reliable approximation for the eddy boundary.

Soft-Label Supervised Meta-Model with Adversarial Samples for Uncertainty Quantification

Despite the recent success of deep-learning models, traditional models are overconfident and poorly calibrated. This poses a serious problem when applied to high-stakes applications. To solve this issue, uncertainty quantification (UQ) models have been developed to allow the detection of misclassifications. Meta-model-based UQ methods are promising due to the lack of predictive model re-training and low resource requirement. However, there are still several issues present in the training process. (1) Most current meta-models are trained using hard labels that do not allow quantification of the uncertainty associated with a given data sample; and (2) in most cases, the base model has a high test accuracy. Therefore, the samples used to train the meta-model primarily consist of correctly classified samples. This leads the meta-model to learn a poor approximation of the true decision boundary. To address these problems, we propose a novel soft-label formulation that better differentiates between correct and incorrect classifications, thereby allowing the meta-model to distinguish between correct and incorrect classifications with high uncertainty (i.e., low confidence). In addition, a novel training framework using adversarial samples is proposed to explore the decision boundary of the base model and mitigate issues related to training datasets with label imbalance. To validate the effectiveness of our approach, we use two predictive models trained on SVHN and CIFAR10 and evaluate performance according to sensitivity, specificity, an F1-score-style metric, average precision, and the Area Under the Receiver Operating Characteristic curve. We find the soft-label approach can significantly increase the model’s sensitivity and specificity, while the training with adversarial samples can noticeably improve the balance between sensitivity and specificity. We also compare our method against four state-of-the-art meta-model-based UQ methods, where we achieve significantly better performance than most models.

Existence and approximate boundary controllability of some partial functional integrodifferential equations with nonlocal initial condition in Banach spaces

Existence and approximate boundary controllability of some partial functional integrodifferential equations with nonlocal initial condition in Banach spaces

Approximate controllability of neutral fractional stochastic differential systems with control on the boundary

In this paper, the approximate boundary controllability results for neutral fractional stochastic differential system with fractional Brownian motion in Hilbert space is discussed. The existence mild solution of neutral fractional stochastic differential system with fractional Brownian motion is proved by using fractional calculus, compact semigroup, Schauder fixed-point theorem and stochastic analysis. The sufficient conditions for approximate boundary controllability of this system is established under the corresponding linear system is approximately controllable. Finally, an example is given to show the application of our results.

PENGGUNAAN METODE MABAC PADA SISTEM PENDUKUNG KEPUTUSAN PEMILIHAN LOKASI MAGANG TERBAIK

This study aims to develop a Decision Support System (DSS) based on the Multiple Attribute Boundary Approximation Area Comparison (MABAC) method to assist Labuhan Batu University students in choosing the best internship location. The main problem faced is the difficulty in determining a strategic and quality internship location due to limited resources. This study uses a quantitative descriptive method involving five main criteria: Location Distance, Company Reputation, Facilities, Suitability of the Internship Program with the Curriculum, and Work Environment. Data were collected through literature, expert consultation, and primary and secondary data analysis. Based on the results of data processing using the MABAC method in this study, 3 best internship locations were obtained, namely locations A9, A4, and A3. The results of the analysis using the MABAC method show the system's ability to provide internship location recommendations based on objective multicriteria assessments. The resulting system can help make it easier for students to choose an internship location that suits their needs.

WAN Discretization of PDEs: Best Approximation, Stabilization, and Essential Boundary Conditions

WAN Discretization of PDEs: Best Approximation, Stabilization, and Essential Boundary Conditions

Finite element analysis of nanoindentation pile-up and correction of projected contact area

This paper deals with the pile-up phenomenon that can occur during nanoindentation. Finite Element (FE) simulation of X5CrNiCuNb16-4 steel nanoindentation with pyramidal Berkovich indenter was done, and stress and strain beneath the indenter leading to pile-up behavior were analyzed in detail. Pile-up also influences the projected contact area, which should be corrected to include the pile-up into the Oliver-Pharr analysis. Accurate calculation of the projected contact area requires knowledge of its boundary. Few methods of boundary approximation were used, including approximation by the triangle and the semi-ellipse. For more precise approximation, the expression for parabolical approximation was derived. These methods were compared with the projected contact area calculated by finite element method. The most precise results were obtained using semi-elliptical and parabolical correction, which can be used for the determination of the projected contact area and its boundary.

Sustainable Design and Evaluation of Children's Food Packaging from the Perspective of Buyers' Preferences.

Consumer behavior is one of the key factors influencing product sales, especially in food packaging design, where green, organic, sustainable, and human-centered designs are more effective in promoting food sales. This paper aims to develop a sustainability evaluation method for children's food packaging. The study first explores the theoretical foundations of sustainability, establishing a systematic set of quantitative indicators and evaluation criteria. Based on this framework, the research gathers consumption behavior, rating data from 250 parents of various ages, professions, and income backgrounds. Using the CRITIC model, the study performs dimensionless processing and detailed quantitative evaluation of the indicators' comparability, contradictions, and information content to allocate weights for the sustainability evaluation metrics. Furthermore, the MABAC model is applied to construct a weighted decision matrix and boundary approximation area, ranking the sustainability of 20 representative children's food packaging design schemes (S1-S20). The results show that Scheme S1, after calculation using the CRITIC-MABAC model, has a total distance of 0.214 from the boundary approximation area, exhibiting the smallest deviation from the ideal solution across multiple evaluation criteria and achieving the best overall performance. Building on the optimal Scheme S1, this study comprehensively considers key elements such as eco-friendliness, safety, functionality, and educational value in the optimization of a sustainable design for children's fruit puree packaging. The research validates the practicality and effectiveness of the quantitative model through the sustainable design and evaluation of children's food packaging from a consumer behavior perspective, promoting sustainability design and optimization in the children's food packaging sector.

Extension of the local domain-free discretization method to large eddy simulation of compressible flows

Most of the flow problems encountered in practical engineering are wall-bounded turbulent flows at high Reynolds numbers. Wall-modeled large eddy simulation (WMLES) is one of the most viable approaches for predicting these realistic flows. Immersed boundary (IB) approach is an efficient computational technique to solve flow problems involving complex and/or moving geometries. This work extends a sharp-interface IB method, named the local domain-free discretion (DFD), to WMLES of compressible flows at high Reynolds numbers. An equilibrium wall model based on solving the simplified compressible turbulent boundary layer equations is utilized to alleviate the requirement of high near-wall mesh resolution. In conjunction with the approximate boundary conditions prescribed by the modeled wall shear stress and wall heat flux, the tangential velocity and temperature at an exterior dependent node are evaluated. Then, the closure of the discrete form of governing equations at an interior node in the immediate vicinity of the immersed wall is accomplished. A simple non-equilibrium correction of the wall shear stress provided by the equilibrium wall model is introduced explicitly. The WMLES/DFD method is applied to a supersonic zero-pressure-gradient turbulent boundary layer flow, a shock wave/flat-plate boundary layer interaction, a supersonic compression ramp flow and high-speed turbulent Couette flows with various thermal boundary conditions. The influence of grid resolution is investigated in the simulation of zero-pressure-gradient turbulent boundary layer flow. By comparing the computed results with the referenced experimental data and/or numerical results, the accuracy and ability of the WMLES/DFD method to simulate compressible turbulent flows are verified.

Anisotropic Specular Image‐Based Lighting Based on BRDF Major Axis Sampling

Abstract Anisotropic specular appearances are ubiquitous in the environment: brushed stainless steel pans, kettles, elevator walls, fur, or scratched plastics. Real‐time rendering of these materials with image‐based lighting is challenging due to the complex shape of the bidirectional reflectance distribution function (BRDF). We propose an anisotropic specular image‐based lighting method that can serve as a drop‐in replacement for the standard bent normal technique [Rev11]. Our method yields more realistic results with a 50% increase in computation time of the previous technique, using the same high dynamic range (HDR) preintegrated environment image. We use several environment samples positioned along the major axis of the specular microfacet BRDF. We derive an analytic formula to determine the two closest and two farthest points from the reflected direction on an approximation of the BRDF confidence region boundary. The two farthest points define the BRDF major axis, while the two closest points are used to approximate the BRDF width. The environment level of detail is derived from the BRDF width and the distance between the samples. We extensively compare our method with the bent normal technique and the ground truth using the GGX specular BRDF.

Monitoring river discharge from space: An optimization approach with uncertainty quantification for small ungauged rivers

The number of in-situ stations measuring river discharge, one of the Essential Climate Variables (ECV), is declining steadily, and numerous basins have never been gauged. With the aim of improving data availability worldwide, we propose an easily applicable and transferable approach to estimate reach-scale discharge solely using remote sensing data that is suitable for filling gaps in the in-situ network. We combine 20 years of satellite altimetry observations with high-resolution satellite imagery via a hypsometric function to observe large portions of the reach-scale bathymetry. The high-resolution satellite images, which are classified using deep learning image segmentation, allow for detecting small rivers (narrower than 100m) and can capture small width variations. The unobserved part of the bathymetry is estimated using an empirical width-to-depth function. Combined with precise satellite-derived slope measurements, river discharge is calculated at multiple consecutive cross-sections within the reach. The unknown roughness coefficient is optimized by minimizing the discharge differences between the cross-sections. The approach requires minimal input and approximate boundary conditions based on expert knowledge but is not dependent on calibration. We provide realistic uncertainties, which are crucial for data assimilation, by accounting for errors and uncertainties in the different input quantities. The approach is applied globally to 27 river sections with a median normalized root mean square error of 12% and a Nash–Sutcliffe model efficiency of 0.560. On average, the 90% uncertainty range includes 91% of the in-situ measurements.

Effects of fractional derivative and Wiener process on approximate boundary controllability of differential inclusion

One of the core concepts of contemporary control theory is the idea that a dynamical system can be controlled. Several abstract settings have been developed to describe the distributed control systems in a domain in which the control is acted through the boundary. In this manuscript, we investigate the boundary approximate controllability (ABC) of stochastic differential inclusion (SDI) with Hilfer fractional derivative (HFD) and nonlocal condition by implementing the principle of stochastic analysis, the fixed-point theorem, fractional calculus and multi-valued map. Moreover, an example is offered to define the primary results.