Approximation Method Research Articles

Enhanced Subspace Iteration Technique for Probabilistic Modal Analysis of Statically Indeterminate Structures

In structural stochastic dynamic analysis, the consideration of the randomness in the physical parameters of the structure necessitates the establishment of numerous stochastic finite element models and the subsequent computation of their corresponding vibration modes. When the complete analysis is applied to calculate the vibration modes for each sample of the stochastic finite element model, a substantial computational expense is incurred. To enhance computational efficiency, this work presents an extended subspace iteration method aimed at rapidly determining the vibration modal parameters of statically indeterminate structures. The essence of this proposed method revolves around efficiently constructing reduced basis vectors during the subspace iteration process, utilizing flexibility disassembly perturbation and the Krylov subspace. This extended subspace iteration method proves particularly advantageous for the modal analysis of finite element models that incorporate a multitude of random variables. The proposed modal random analysis method has been validated using both a truss structure and a beam structure. The results demonstrate that the proposed method achieves substantial savings in computational time. Specifically, for the truss structure, the calculation time of the proposed method is approximately 1.2% and 65% of that required by the comprehensive analysis method and the combined approximation method, respectively. For the beam structure, on average, the computational time of the proposed method is roughly 2.1% of a full analysis and approximately 48.2% of the Ritz vector method’s time requirement. Compared to existing stochastic modal analysis algorithms, the proposed method offers improved computational accuracy and efficiency, particularly in scenarios involving high-discreteness random analyses.

Application of Digital Technology for Oil and Gas Fields

The paper investigates the problem of isothermal filtration using an approximate method for solving the theory of partial differential equations. The mathematical model under consideration is nonlinear and does not lend itself to analytical methods of solution. The results obtained indicate the need for wide application in the development of oil and gas fields in the Republic of Kazakhstan. In particular, the results of the study make it possible to solve the problems of adapting mathematical models and evaluating changes in technological indicators, which are necessary attributes in the digital technology ""Information System for the Analysis of oil and Gas Field Development"" (ISAR). Many problems and mathematical problems of filtration theory arose while working on specific oil and gas fields in the western region of the Republic of Kazakhstan. The above approximate solution methods have found applications not only in filtration theory, but also in other problems (geophysics, ecology, etc.)"

Quantification and Sensitivity Analysis of the Model Uncertainty in the Projectile-Barrel Coupled Artillery Dynamics System Based on Random Matrix Theory

Simplifications in the engineering of artillery firing dynamics modeling introduce model uncertainty, which seriously affects the model’s predictive performance. To this end, a typical variable-section hollow cylindrical cantilever beam structure in artillery firing dynamics is investigated in this paper. The statistical approximation methods quantify parameter and model uncertainty separately, and their respective confidence intervals were calculated and compared. The results show that the traditional parameter uncertainty approach overestimates the parameter uncertainty ranges, and introducing the model uncertainty approach dramatically improves the model prediction performance. Additionally, a projectile-barrel coupled artillery dynamics model considering model uncertainty is established, and its effectiveness is validated through the actual launch experiments. The sensitivity of the dynamic responses to the system matrix, namely model uncertainty, is further analyzed. This paper provides theoretical references for the study of uncertainty in artillery launch dynamics and proposes effective methods to enhance the predictability of computational models for complex structural systems.

Advancing DFT predictions in Cu-chalcogenides with full-yet-shallow 3d-orbitals: Meta-GGA plus Hubbard-like U correction.

The technologically important Cu-chalcogenides, such as Cu2Se and CuInSe2, usually have relatively small band gaps. Achieving a reliable yet efficient description of the electronic properties has proven to be quite challenging for the popular exchange-correlation functionals of density functional theory, primarily due to the involvement of full-yet-shallow Cu-3d orbitals. In this study, we evaluate the applicability of several meta-generalized gradient approximation (GGA) functionals that have been recently developed. We find that the r2SCAN (regularized-restored strongly constrained and appropriately normed) functional significantly improves upon conventional local density approximation and GGA in terms of geometry and electronic band structure; however, there is still a notable discrepancy with experimental results due to the remaining delocalization error. This error is mitigated by combining r2SCAN with a Hubbard-like U correction applied to the Cu-3d orbitals. For predicting band gaps, both the TASK functional and the mBJ potential, when combined with the U correction, demonstrate similar accuracies with a mean absolute error of 0.17-0.19eV. This accuracy is lower than that achieved with the many-body Hedin's GW approximation method but more accurate than that of hybrid functionals. Moreover, the r2SCAN+U approach well reproduces the phonon dispersion in CuInSe2, revealing a neglected computational problem in previous reports. We conclude that the meta-GGA+U approach represents a significant advancement by striking a balance between reliability and computational effort, and further efforts are still required to describe the Cu-3d orbitals more accurately.

Minimasi Biaya Distribusi pada UMKM Keset Samian dengan Metode VAM dan Modi

Transportation methods are methods used to optimally manage the distribution of goods from suppliers to destinations. The method used is Vogel's Approximate Method (VAM) and Modified Distribution (MODI). This research aims to minimize distribution costs for Keset Samian MSMEs. Samian Doormat UMKM is a small and medium business engaged in the production of doormat crafts. The Samian Doors Umkm experiences problems in managing distribution costs efficiently which has a direct impact on business profits. This research uses primary and secondary data collection related to costs. distribution, demand, and production capacity. The data was analyzed using VAM to determine the initial optimal cost solution in allocating the distribution of doormat products. The results of the VAM method were further optimized using the MODI method to achieve minimal costs in the distribution of doormat products. The research results show that the VAM and MODI methods are able to reduce distribution costs significantly compared to the traditional methods currently used by the Samian Doormat MSMEs. Where from the initial cost of Rp. 4,250,000 to Rp. 2,843,500 which experienced a decrease in transportation costs of Rp. 1,406,500 or 33.09%. So the results of calculating transportation costs for UMKM Keset Samian are optimal and increase business profits.

Implementasi Vogel’s Approximation Method (VAM) dan Modified Distribution (Modi) untuk Minimasi Biaya Distribusi di PT Jimat Perkasa Farm

Efficient and effective distribution management is a key factor in the success of a company's operations. PT Jimat Perkasa Farm, which is engaged in the processing of seaweed waste for animal feed blends, faces significant challenges in minimizing distribution costs. This research aims to implement Vogel's Approximation Method (VAM) and Modified Distribution (MODI) to achieve an optimal solution in reducing distribution costs. The results show that the application of VAM as the initial solution and MODI as the optimal solution is able to produce significant distribution cost savings. The total distribution cost can be minimized from IDR 5,400,000 per month to IDR 5,275,000 per month, which shows a savings of IDR 125,000 / by 2.31% per month. This efficiency in distribution not only reduces operational costs but also improves the timeliness and reliability of delivery, which ultimately improves customer satisfaction and the company's competitiveness in the market

On weighted sum‐rate maximization of full‐duplex systems in presence of STAR‐RIS

AbstractReflecting intelligent surface (RIS) is one of the key enabling technologies for beyond fifth generation wireless networks to further improve coverage, spectral‐ and energy‐efficiency of wireless networks. Recently, a novel simultaneous transmission and reflection RIS (STAR‐RIS) technology is introduced which enables more degrees‐of‐freedom in simultaneously serving users in reflection and transmission regions of RIS. This paper investigates a STAR‐RIS assisted two‐user full‐duplex communication system, wherein multi‐antenna base station (BS) and single‐antenna users are subject to maximum power constraints. Firstly, weighted sum rates are computed for three well‐known STAR‐RIS protocols, e.g. energy splitting, mode selection, and time splitting. Then, for each of the cases, weighted sum rate optimization problem is investigated to find optimal resource allocations, i.e. base station's precoding and combining matrices, coefficients of STAR‐RIS, and power allocations. The optimization problem is transferred into multiple convex sub‐problems using equivalent weighted minimum mean‐square‐error forms. Also, by use of the successive convex approximation method, tunable parameters of the STAR‐RIS are optimized. Although the original problem is a non‐convex one, proposed iterative alternating technique achieves an acceptable sub‐optimal performance. Finally, performance of the system is analysed and compared to some baselines to highlight superiority of the STAR‐RIS compared to the conventional RIS schemes.

A comprehensive study of beam modal functions in the free vibration analysis of cylindrical shells: critical examination on the applicability to the clamped-free boundary condition

Over the past few decades, approximate methods that can provide solutions of sufficient accuracy have received considerable attention in the free vibration analysis of cylindrical shells, where a great deal of studies adopted the beam modal functions as the trial functions for the axial mode shapes of cylindrical shells. Nevertheless, most studies were restricted to the application of single term beam modal function and failed to simulate elastic boundary conditions of cylindrical shells, while the accuracy of the corresponding methods has recently sparked significant controversy, especially for cylindrical shells under the clamped-free boundary condition. This paper presents a comparative study of three forms of beam modal functions in the free vibration analysis of cylindrical shells, one of which is proposed for the first time to simulate elastic boundary conditions of cylindrical shells. A unified model is developed using the general Rayleigh-Ritz method, incorporating the breathing modes with circumferential orders being zero, and four types of commonly used thin shell theories, namely the Donnell, Reissner, Love, and Sanders theories. From both perspectives of natural frequencies and mode shapes, numerical results are validated by comparison with those existing in the literature and those calculated from the finite element method (FEM). The results not only clarify the distinction of different forms of beam modal functions used in the Rayleigh-Ritz method, but also provide explanations for the controversy raised in recent studies. Furthermore, the unified formulations can be extended to vibration analysis of various forms of shell structures, and can also be helpful to the vibration analysis of beams and plates with elastic boundary conditions.

An Efficient Integral Approach for Kinematic Reliability Sensitivity Analysis of Industrial Robots

Abstract Assessment of the reliability and reliability sensitivity of positioning accuracy for industrial robots subject to aleatoric and epistemic uncertainties registers a challenging task. This study proposes a new optimized moment-based method for kinematic reliability analysis and its sensitivity analysis, which incorporates the sparse grid (SPGR) technique and the saddlepoint approximation (SPA) method. To start with, the positioning accuracy reliability and its sensitivity models of industrial robots are established via computational optimization techniques and kinematic criteria. The kinematic accuracy reliability and its sensitivity are then calculated. Specifically, the sparse grid technique is adopted to approach the positioning error statistical moments and moment sensitivities. On this basis, positioning accuracy reliability bounds and reliability sensitivity bounds are obtained by the saddlepoint approximation method and optimization techniques. Finally, two practical examples are implemented to demonstrate the proficiency of the currently proposed method against Monte Carlo simulation (MCS) results. The results show that the currently proposed method exhibits superior computational accuracy and efficiency in kinematic reliability and its sensitivity analyses for industrial robots.

Hydroelastic responses of a multi-modular very large floating structure on a flexible bottom with lateral stress in a two-layer fluid

Based on the small-amplitude water wave theory, an analytical investigation is performed for the interaction between incident waves and a multi-modular very large floating structure (VLFS) in a two-layer fluid with a flexible seabed. A vertical porous flexible plate is fixed in the center of each floating plate to suppress the hydroelastic response. A key innovation is the modeling of both the VLFSs and the seabed as thin elastic plates with the lateral stress. The analytical expressions for dispersion relationships and vertical eigenfunctions are exactly derived in the water regions with a flexible seabed. The series solution of the velocity potential is obtained through the combination of inner product and the least squares approximation method. The study uniquely illustrates the impact of lateral stress and flexible seabeds on wave propagation, revealing that at low frequencies, waves in both open water and plate-covered regions with a flexible seabed exhibit the similar propagation characteristics. At higher frequencies, however, the propagation characteristics of the waves in the open water regions remain unaffected by the presence of a flexible seabed. The lateral stress has significant effects on the roots of dispersion relationships and the hydroelastic responses. The influences of other physical parameters of the fluid and structures are also displayed graphically. The results demonstrate that the increasing number of modules and extending the vertical plates effectively suppress hydroelastic responses and reduce wave transmission. The porous barriers can reduce the stress on the vertical plates and weaken the effect of barriers for suppressing hydroelastic response.

Statistical Properties of SIS Processes with Heterogeneous Nodal Recovery Rates in Networks

The modeling and analysis of epidemic processes in networks have attracted much attention over the past few decades. A major underlying assumption is that the recovery process and infection process are homogeneous, allowing the associated theoretical studies to be conducted in a convenient manner. However, the recovery and infection processes usually exhibit heterogeneous rates in the real world, which makes it difficult to characterize the general relations between the dynamics and the underlying network structure. In this work, we focus on the susceptible–infected–susceptible (SIS) epidemic process with heterogeneous recovery rates in a finite-size complete graph. Specifically, we study the metastable-state statistical properties of SIS epidemic dynamics with two different nodal recovery rates in complete graphs. We propose approximate solutions to the metastable-state expectation and the variance in the number of infected nodes within the framework of the mean-field approximation method. We also derive several upper and lower bounds of the steady-state probability that a node is in the infected state. We verify the proposed approximate solutions of the mean and variance via simulations. This work provides insights into the fluctuations in the statistical properties of epidemic processes with complex dynamical behaviors in networks.

Error estimates for the robust α-stable central limit theorem under sublinear expectation by a discrete approximation method

Error estimates for the robust α-stable central limit theorem under sublinear expectation by a discrete approximation method

Block approximations for probabilistic mixtures of elementary cellular automata

Probabilistic Cellular Automata are a generalization of Cellular Automata. Despite their simple definition, they exhibit fascinating and complex behaviours. The stationary behaviour of these models changes when model parameters are varied, making the study of their phase diagrams particularly interesting. The block approximation method, also known in this context as the local structure approach, is a powerful tool for studying the main features of these diagrams, improving upon Mean Field results. This work considers systems with multiple stationary states, aiming to understand how their interactions give rise to the structure of the phase diagram. Additionally, it shows how a simple algorithmic implementation of the block approximation allows for the effective study of the phase diagram even in the presence of several absorbing states.

Uncertainty-Aware Time Series Anomaly Detection

Traditional anomaly detection methods in time series data often struggle with inherent uncertainties like noise and missing values. Indeed, current approaches mostly focus on quantifying epistemic uncertainty and ignore data-dependent uncertainty. However, consideration of noise in data is important as it may have the potential to lead to more robust detection of anomalies and a better capability of distinguishing between real anomalies and anomalous patterns provoked by noise. In this paper, we propose LSTMAE-UQ (Long Short-Term Memory Autoencoder with Aleatoric and Epistemic Uncertainty Quantification), a novel approach that incorporates both aleatoric (data noise) and epistemic (model uncertainty) uncertainties for more robust anomaly detection. The model combines the strengths of LSTM networks for capturing complex time series relationships and autoencoders for unsupervised anomaly detection and quantifies uncertainties based on the Bayesian posterior approximation method Monte Carlo (MC) Dropout, enabling a deeper understanding of noise recognition. Our experimental results across different real-world datasets show that consideration of uncertainty effectively increases the robustness to noise and point outliers, making predictions more reliable for longer periodic sequential data.

A Novel Analytical Approximate Solution for Strongly Nonlinear Third-Order Jerk Equations Using a Modified Iteration Method

Nonlinear jerk equations, characterized by a third-order time derivative, play a crucial role in modeling various physical phenomena across disciplines like mechanics, circuits, and biology. Accurately solving these equations is essential for understanding and predicting the behavior of such systems. However, obtaining analytical solutions for nonlinear jerk equations can be challenging, necessitating the development of robust and accurate approximation methods. This work explores, for the first time, the application of the modified iteration approach to solve third-order jerk equations. By comparing the obtained approximate solutions with both exact and existing analytical solutions for established engineering problems, we demonstrate the superior accuracy and rapid convergence of the proposed method. The significantly reduced error percentages highlight the effectiveness of the modified iteration approach in providing precise solutions for nonlinear jerk equations, paving the way for its application in a wide range of oscillation problems within nonlinear sciences and engineering.

A deconstruction of methods to derive one-point lensing statistics

Gravitational lensing is a crucial tool for exploring cosmic phenomena, providing insights into galaxy clustering, dark matter, and dark energy. Given the substantial computational demands of N-body simulations, approximate methods like 𝙿𝙸𝙽𝙾𝙲𝙲𝙷𝙸𝙾 and 𝚝𝚞𝚛𝚋𝚘𝙶𝙻 have been proposed as viable alternatives for simulating lensing probability density functions (PDFs). This paper evaluates these methods and their effectiveness across both weak and strong lensing regimes, with a focus in the context where baryonic effects are negligible. Our comparative analysis reveals that these methods are effective for applications where lensing is mild, such as the majority of sources of electromagnetic and gravitational waves. However, both 𝙿𝙸𝙽𝙾𝙲𝙲𝙷𝙸𝙾 and 𝚝𝚞𝚛𝚋𝚘𝙶𝙻 break down for large values of convergence and magnification due to their loss of accuracy in capturing small-scale nonlinear matter fields, owing to oversimplified assumptions about internal halo structures and reliance on perturbation theory. 𝙿𝙸𝙽𝙾𝙲𝙲𝙷𝙸𝙾 yields second-to-fourth moments of the lensing PDFs, which are 6–10% smaller than those resulting from N-body simulations in regimes where baryonic effects are minimal. These findings aim to inform future studies on gravitational lensing of point sources, which are increasingly relevant with upcoming supernova and gravitational wave datasets.

Theory and applications of the sum-of-squares technique

Abstract The sum-of-squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds on the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This note presents an overview of the SOS method. We start with its application in finite-dimensional feature spaces and, subsequently, we extend it to infinite-dimensional feature spaces using reproducing kernels (k-SOS). Additionally, we highlight the utilization of SOS for estimating some relevant quantities in information theory, including the log-partition function.

Plasmonic hemispherical Ag nanoparticles on silicon substrate: A comprehensive study of optical properties

In this work, we demonstrate a comprehensive study of the optical properties of hemispherical Ag nanoparticles (AgNPs) on a single-crystal (c-Si) wafer. The fabricated method involves a galvanic displacement reaction of Ag on c-Si and annealing in oxygen atmosphere at 500 °C. Substrates with different morphological parameters of AgNPs were obtained by varying the volume ratio of Ag in the deposition solution. Using the quasinormal modes (QNMs) theory formalism, the localized surface plasmon resonance (LSPR) multipole positions were determined as a function of the AgNP size and incidence angle. Also, the effective field approximation and QNMs methods were used to calculate the specular reflectance of the disordered hemispherical array on c-Si. Thus, the experimentally obtained LSPR positions in the reflectance spectra were described. Finally, surface-enhanced Raman scattering (SERS) demonstrated reliable detection of brilliant green triphenylmethane dye (1 nM), methyl red dye (10 nM) and hemoglobin from bovine blood (1 mM). The dependence of the SERS enhancement factor on the LSPR position and the absorption band of the analyte was established, the maximum value of which was 1.1 × 107. In summary, this study suggests that the numerical and experimental investigation of the optical properties of hemispherical AgNPs on c-Si contributes to the field of optical sensing.

Analytical Analysis for Space Fractional Helmholtz Equations by Using The Hybrid Efficient Approach

Abstract The Helmholtz equation is an important differential equation. It has a wide range of uses in physics, including acoustics, electro-statics, optics, and quantum mechanics. In this article, a hybrid approach called the Shehu transform decomposition method (STDM) is implemented to solve space-fractional-order Helmholtz equations with initial boundary conditions. The fractional-order derivative is regarded in the Caputo sense. The solutions are provided as series, and then we use the Mittag-Leffler function to identify the exact solutions to the Helmholtz equations. The accuracy of the considered problem is examined graphically and numerically by the absolute, relative, and recurrence errors of the three problems. For different values of fractional-order derivatives, graphs are also developed. The results show that our approach can be a suitable alternative to the approximate methods that exist in the literature to solve fractional differential equations.

A constitutive model based on energy density for power-law variable thickness plates and its application

Variable thickness plates are relatively common structure in large-scale transportation equipment such as aerospace and ships. Due to their high running speed and large structural dimensions, such structures are usually induced high-frequency severe vibrations, which likely lead to structural damage. Therefore, it is crucial to accurately predict the high-frequency dynamic response characteristics during the structural design phase. The energy finite element method (EFEM) is a powerful tool for predicting high-frequency dynamic response. But, in recent EFEM for variable thickness plates, an approximate method (AEFEM) using a large number of constant thickness plate elements to approximate variable thickness plates is used. Due to that AEFEM is based on the constitutive equation for constant thickness plate, it will be imprecise and time-consuming. Aiming at power-law variable thickness plates, the constitutive equation based on energy density is derived and the finite element discrete scheme of the constitutive equation was studied. In this way, the precise new EFEM (NEFEM) for variable thickness plates was established. In addition, the dynamic response of a variable thickness plate was predicted by NEFEM presented and finite element method (FEM) respectively. The results are consistent. It is shown that the constitutive model presented is reliable and time-saving. Comparison of the results from AEFEM, the NEFEM can get more accurate results with fewer elements. The NEFEM based on the constitutive model presented can be used to predict precisely the high-frequency dynamic response for power-law variable thickness plates.