Approximate Function Research Articles

New method for predicting the free vibration characteristics of composite laminates based on generalized cell and spectral‐Tchebychev methods

AbstractThis paper presents a new method for analyzing the free vibration characteristics of composite laminates under arbitrary boundary conditions by integrating the generalized method of cells (GMC) and the Two‐Dimensional Spectral‐Tchebychev (2D‐ST) method. First, a meso‐macro correlation matrix is constructed based on the GMC, and the macroscopic performance parameters of the composite laminates are predicted. Then, virtual boundary springs are introduced to simulate the arbitrary boundary conditions, and the energy equation of the laminate is derived based on the first‐order shear deformation theory (FSDT) and Hamilton's principle. Furthermore, an approximate displacement function of the laminate is derived by using the 2D‐ST Tchebychev polynomials, and the Gauss‐Lobatto points are selected for meshless discretization of the structure to obtain the discretized characteristic equation, which is solved to obtain the free vibration frequency of the laminate. Based on Matlab programming, the free vibration frequencies of the laminate under different conditions are calculated, and the convergence, accuracy and computational efficiency of the calculation method are discussed, and the effects of boundary conditions, stacking angle, geometrical parameters, and material parameters on the free vibration frequencies of the laminate are analyzed. The numerical results show that the method has the advantages of fast convergence, high accuracy and high efficiency, and the method can be convenient and fast for parametric studies.Highlights The free vibration characteristics of composite laminates are studied. Combining generalized cell method with spectral‐Tchebychev method. The method has the characteristics of fast convergence, high accuracy and high efficiency. A parametric study is conducted on the free vibration of laminated panels.

Assessing many-body methods on the potential energy surface of the (H2)2 hydrogen dimer.

The anisotropic potential energy surface of the (H2)2 dimer represents a challenging problem for many-body methods. Here, we determine the potential energy curves of five different dimer configurations (T, Z, X, H, and L) using the lattice regularized diffusion Monte Carlo method and a number of approximate functionals within density functional theory (DFT), including advanced orbital-dependent functionals based on the random phase approximation (RPA). We assess their performance in describing the potential wells, bond distances, and relative energies. The repulsive potential wall is studied by looking at the relative stability of the different dimer configurations as a function of an applied force acting along the intermolecular axis. It is shown that most functionals within DFT break down at finite compression, even those that give an accurate description around the potential well minima. Only by including exchange within RPA, a qualitatively correct description along the entire potential energy curve is obtained. Finally, we discuss these results in the context of solid molecular hydrogen at finite pressures.

Fast and Efficient Lunar Finite Element Gravity Model

In this paper, the finite element method (FEM) is integrated with orthogonal polynomial approximation in high-dimensional spaces to innovatively model the Moon’s surface gravity anomaly. The aim is to approximate solutions to Laplace’s classical differential equations of gravity, employing classical Chebyshev polynomials as basis functions. Using classical Chebyshev polynomials as basis functions, the least-squares approximation was used to approximate discrete samples of the approximation function. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. The first application of this project is to substitute the globally valid classical spherical harmonic series of approximations with locally valid series of orthogonal polynomial approximations (i.e., using the FEM approach). With an error tolerance set at 10−9ms−2, this method is used to adapt the gravity model radially upwards from the lunar surface. The results showcase a need for a higher degree of approximation on and near the lunar surface, with the necessity decreasing as the radius increases. Notably, this method achieves a computational speedup of five orders of magnitude when applying the method to radial adaptation. More intrinsically, the second application involves using the methodology as an effective tool in solving boundary value problems. Specifically, this approach is implemented to solve classical differential equations involved with high-precision, long-term orbit propagation. This application provides a four-order-of-magnitude speedup in computational time while maintaining an error within the 10−10ms−2 error range for various orbit propagation tests. Alongside the advancements in orthogonal approximation theory, the FEM enables revolutionary speedups in orbit propagation without compromising accuracy.

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.

DeepPipe: A Multi-Stage Knowledge-Enhanced Physics-Informed Neural Network for Hydraulic Transient Simulation of Multi-Product Pipeline

In the chemical pipelining industry, owing to the high-pressure transportation process, an accurate hydraulic transient simulation tool plays a central role in preventing the slack line flow and overpressure from causing pipeline operation treacherous. Nevertheless, the current model-driven method often faces challenges in balancing computational efficiency with accuracy, and the existing data-driven models struggle to produce explainable results from the physics perspectives since insufficient theoretical principles are incorporated into the model training. Additionally, the existing physics-informed learning architecture fails to achieve a gradient-balanced training, resulting from the significant magnitude difference in outputs and multiple loss terms. Consequently, a Multi-Stage Knowledge-Enhanced Physics-Informed Neural Network (MS-KE-PINN) is proposed for the hydraulic transient simulation of multi-product pipelines. To enforce the neural network producing simulation results with high consistency to physical laws, the governing equations, boundary, and initial condition are incorporated into the training process for an efficient mesh-free simulation. Then, considering that the significant magnitude difference between outputs can easily lead to deficient performance in the gradient descent, the magnitude conversion on the outputs and the equivalent conversion of the governing equations are implemented to enhance the training effect of the neural network. Subsequently, to tackle the imbalanced gradient of multiple loss terms with fixed weights, a multi-stage hierarchical training strategy is designed to improve the approximation capacity of the neural network. Numerical simulation cases demonstrate a better approximation function of the proposed model than the state-of-art models, while the mean absolute percentage errors yielded by MS-KE-PINN are reduced by 77.4 %, 88.7 %, and 87.8 % in three simulation operation conditions for pressure prediction. Furthermore, experimental investigations from a real-world multi-product pipeline suggest that the proposed model can still draw accurate simulation results even under complex and dynamic hydraulic transient scenarios in practice, with root mean squared errors reduced by 94.8 % and 80 % than that of the physics-informed neural network. To this end, the proposed model can conduct accurate and effective hydraulic transient analysis, thus ensuring the safe operation of the pipeline.

The Influence of the Ionic Core on Structural and Thermodynamic Properties of Dense Plasmas

In this paper, a new ion–ion screened potential was numerically calculated, which takes into account the ion core effect, i.e., the influence of strongly bound electrons. The pseudopotential model describing the shielding of ion cores and the screening using the density response function in the long wavelength approximation were used. To study the influence of this ion core effect on dense plasma’s structural and thermodynamic properties, the integral Ornstein–Zernike equation was solved in the hypernetted chain approximation. Our results show that the ion core has a significant impact on ionic radial distribution functions and thermodynamic properties when compared to the results obtained for the Yukawa potential, which does not take the ion core into account. Increasing the steepness of the core edge or decreasing the depth of the minimum leads to more pronounced screening due to bound electrons.

Identification of characteristics of conceptual prototype of microprocessor resource-saving relay protection system

The object of the study is the conceptual prototype of a microprocessor resource-saving relay protection system. Currently, relay protection ensures electrical networks reliable and efficient work, however, the traditional architecture is proprietary, not allowing to fix and replace damaged parts without a company specialist. Therefore, the open-architecture relay protection is a very pressing issue, but the problem lies in meeting the relay protection requirements. The data transmission protocols nRF and ESP-NOW, Hall sensors evaluation for AC current measurement determination and sensor accuracy improvement was implemented. Experimental validation demonstrated that nRF and ESP-NOW protocols meet the delay and reliability requirements, however, the nRF protocol is more suitable due to its flexibility and obstacle penetration. The data demonstrated that the most effective conditions are without obstacles at 15 meters from the modem and with obstacles 5 meters from the modem. The experiment of Hall sensors characteristics determination demonstrated the accuracy of current measurement with the set values of the opening and closing currents. Nevertheless, it is not accurate (12.45 %) for the relay protection application. Therefore, the application of the changing values of the opening and closing currents is more effective and accuracy reaches 6.92 %. As a result, the service life of the Hall sensor was determined, and even after 10 million openings, the open state time remained unchanged. Therefore, the approximation function for current amplitude determination depending on open state time was found. On the other hand, Hall sensors may suffer from temperature drift and require further optimization to be fully reliable. The study limitation is the current range from 0 to 800 A

Fast Nonlinear Model Predictive Control Combining Online Trajectory Optimization and Value Function Regression

ABSTRACTModel predictive control (MPC) is a well‐developed method capable of handling complex control tasks. Implementation of an MPC requires solution of a deterministic finite horizon nonlinear optimal control (OC) problem. OC problems can be solved globally, explicitly expressing the optimal policy (and value function) as a function of the present state, or, locally, using online trajectory optimization, generating a solution that is only valid for the present state. For nonlinear problems however, neither is possible analytically, and the latter, finding a local solution online, is usually preferred. The difficulty with online trajectory optimization is that the solution must be available within a single sampling period. The main parameter affecting the computational demand of MPC is the length of the prediction horizon. The goal in this work is to reduce the length of the prediction horizon of a model predictive controller (MPC) to reduce computation time whilst preserving optimality guarantees. To this end, we propose approximation of the trajectory optimization problem for MPC by learning a finite horizon value function. The approximated value function is inserted into a truncated trajectory optimization problem so that the MPC can be attained with a reduced prediction horizon, ergo reduced computational load. By sampling the value function in a goal oriented way, we show that an effective approximate value function can be found by including both the function value and the gradients of the value function. The result is an accurate approximate MPC which leverages learning methodologies to reduce computational cost while still accounting for constraints.

НЕЙРОСЕТЕВОЙ ПОДХОД К КОГНИТИВНОМУ МОДЕЛИРОВАНИЮ

Some features of cognitive modeling are presented, including the prerequisites for a cognitive approach to solving complex problems. Cognitive modeling involves the use of various artificial neural networks, including convolutional neural networks. The classification of artificial neural networks according to various characteristics is given. The features of self-organizing neural networks and networks using deep learning methods are considered. The artificial neural network, which is a three-layer unidirectional direct propagation network, the interface of a computer program used to approximate functions using the specified neural network, as well as the solution of the image recognition problem using an artificial convolutional neural network, in which the neural network parameters are adjusted for each recognizable image fragment in order to adaptively filter the image, are considered in detail. The analysis of images in video surveillance systems in order to detect fires allows them to be detected at an early stage and, thus, prevent the fire propogation.

Finite element analysis of a generalized Robin boundary value problem in curved domains based on the extension approach

Abstract A theoretical analysis of the finite element method for a generalized Robin boundary value problem, which involves a second-order differential operator on the boundary, is presented. If $\varOmega $ is a general smooth domain with a curved boundary, we need to introduce an approximate domain $\varOmega _{h}$ and to address issues owing to the domain perturbation $\varOmega \neq \varOmega _{h}$. In contrast to the lift approach used in existing studies, we employ the extension approach, which need not assume that boundary nodes of $\partial \varOmega _{h}$ lie exactly on $\partial \varOmega $. Assuming that approximate domains and function spaces are given by isoparametric finite elements of order $k$, we prove the optimal rate of convergence in the $H^{1}$- and $L^{2}$-norms. A numerical example is given for the piecewise linear case $k = 1$.

Physical information-guided multidirectional gated recurrent unit network fusing attention to solve the Black-Scholes equation

Reasonable option pricing is crucial in the financial derivatives market. Finding analytical solutions for the Black-Scholes (BS) equation, particularly for American options or with fluctuating volatility and interest rates, is challenging. BS equations exhibit strong time-series characteristics, with asset prices typically adhering to geometric Brownian motion. To address the BS equations, we propose a sequence-to-sequence model guided by physical information (PI), called PiMGA. The PiMGA fuses a multidirectional gated recurrent unit (GRU) network with an attention module, where multidirectional GRU enhances the coding performance of the input sequences and the attention module balances the feature weights of the hidden variables. Prior physical knowledge in BS equations is jointly used as a constraint, forming the penalty function for objective optimization. This allows PiMGA to serve as an efficient approximation function in the learning paradigm of physically informed machine learning to solve BS equations. BS equations with various complexities illustrate the accuracy and feasibility of PiMGA for numerical solutions. Furthermore, the out-of-distribution generalization ability of PiMGA is verified by predicting the Nasdaq 100 index.

Influence maximization based on discrete particle swarm optimization on multilayer network

Influence maximization (IM) aims to strategically select influential users to maximize information propagation in social networks. Most of the existing studies focus on IM in single-layer networks. However, we have observed that individuals often engage in multiple social platforms to fulfill various social needs. To make better use of this observation, we consider an extended problem of how to maximize influence spread in multilayer networks. The Multilayer Influence Maximization (MLIM) problem is different from the IM problem because information propagation behaves differently in multilayer networks compared to single-layer networks: users influenced on one layer may trigger the propagation of information on another layer. Our work successfully models the information propagation process as a Multilayer Independent Cascade model in multilayer networks. Based on the characteristics of this model, we introduce an approximation function called Multilayer Expected Diffusion Value (MLEDV) for it. However, the NP-hardness of the MLIM problem has posed significant challenges to our work. To tackle the issue, we devise a novel algorithm based on Discrete Particle Swarm Optimization. Our algorithm consists of two stages: 1) the candidate node selection, where we devise a novel centrality metric called Random connectivity Centrality to select candidate nodes, which assesses the importance of nodes from a connectivity perspective. 2)the seed selection, where we employ a discrete particle swarm algorithm to find seed nodes from the candidate nodes. We use MLEDV as a fitness function to measure the spreading power of candidate solutions in our algorithm. Additionally, we introduce a Neighborhood Optimization strategy to increase the convergence of the algorithm. We conduct experiments on four real-world networks and two self-built networks and demonstrate that our algorithms are effective for the MLIM problem.

Bridging electronic and classical density-functional theory using universal machine-learned functional approximations.

The accuracy of density-functional theory (DFT) calculations is ultimately determined by the quality of the underlying approximate functionals, namely the exchange-correlation functional in electronic DFT and the excess functional in the classical DFT formalism of fluids. For both electrons and fluids, the exact functional is highly nonlocal, yet most calculations employ approximate functionals that are semi-local or nonlocal in a limited weighted-density form. Machine-learned (ML) nonlocal density-functional approximations show promise in advancing applications of both electronic and classical DFTs, but so far these two distinct research areas have implemented disparate approaches with limited generality. Here, we formulate a universal ML framework and training protocol to learn nonlocal functionals that combine features of equivariant convolutional neural networks and the weighted-density approximation. We prototype this new approach for several 1D and quasi-1D problems and demonstrate that functionals with exactly the same hyperparameters achieve excellent accuracy for a diverse set of systems, including the hard-rod fluid, the inhomogeneous Ising model, the exact exchange energy of electrons, the electron kinetic energy for orbital-free DFT, as well as for liquid water with 1D inhomogeneities. These results lay the foundation for a universal ML approach to approximate exact 3D functionals spanning electronic and classical DFTs.

Marginal Effect-aware Multiple-Vehicle Scheduling for Road Data Collection: A Near-optimal Result

Vehicles equipped with abundant sensors offer a promising way for large-scale, low-cost road data collection. To realize this potential, a well-designed vehicle scheduling scheme is essential for deploying the recruited drivers efficiently. Nevertheless, existing works fail to consider the marginal effect among drivers’ collections. Different from them, this article introduces a, to the best of our knowledge, new multiple-vehicle scheduling problem that jointly optimizes task allocation and vehicle trajectory planning to maximize the overall collection utility by accounting for the marginal effect in drivers’ data collections. However, solving this problem is non-trivial due to its involvement with multiple coupled NP-hard problems. To this end, we propose MeSched, a marginal effect-aware multiple-vehicle scheduling scheme designed for road data collection. Specifically, we first present a greedy-based auxiliary graph construction method to disentangle the initial problem into multiple independent single-vehicle scheduling subproblems. Furthermore, we build an approximate surrogate function that transforms each subproblem into a tractable form involving only a single variable. The theoretical analysis proves that MeSched can achieve a 1-(1/ e ) ¼ -approximation ratio in polynomial time. Comprehensive evaluations based on a real-world trajectory dataset of 12,493 vehicles demonstrate that MeSched can significantly improve the collection utility by 104.5% on average compared with four baselines.

Very-Large-Scale GPU-Accelerated Nuclear Gradient of Time-Dependent Density Functional Theory with Tamm-Dancoff Approximation and Range-Separated Hybrid Functionals.

Modern graphics processing units (GPUs) provide an unprecedented level of computing power. In this study, we present a high-performance, multi-GPU implementation of the analytical nuclear gradient for Kohn-Sham time-dependent density functional theory (TDDFT), employing the Tamm-Dancoff approximation (TDA) and Gaussian-type atomic orbitals as basis functions. We discuss GPU-efficient algorithms for the derivatives of electron repulsion integrals and exchange-correlation functionals within the range-separated scheme. As an illustrative example, we calculate the TDA-TDDFT gradient of the S1 state of a full-scale green fluorescent protein with explicit water solvent molecules, totaling 4353 atoms, at the ωB97X/def2-SVP level of theory. Our algorithm demonstrates favorable parallel efficiencies on a high-speed distributed system equipped with 256 Nvidia A100 GPUs, achieving >70% with up to 64 GPUs and 31% with 256 GPUs, effectively leveraging the capabilities of modern high-performance computing systems.

A DFT study of the effect of hydrostatic pressure on the structure and electronic properties of sarcosine crystal

ContextWe perform density functional theory calculations to study the dependence of the structural and electronic properties of the amino acid sarcosine crystal structure on hydrostatic pressure application. The results are analyzed and compared with the available experimental data. Our findings indicate that the crystal structure and properties of sarcosine calculated using the Grimme dispersion-corrected PBE functional (PBE-D3) best agree with the available experimental results under hydrostatic pressure of up to 3.7 GPa. Critical structural rearrangements, such as unit cell compression, head-to-tail compression, and molecular rotations, are investigated and elucidated in the context of experimental findings. Band gap energy tuning and density of state shifts indicative of band dispersion are presented concerning the structural changes arising from the elevated pressure. The calculated properties indicate that sarcosine holds great promise for application in electronic devices that involve pressure-induced structural changes.MethodsThree widely used generalized gradient approximation functionals—PBE, PBEsol, and revPBE—are employed with Grimme’s D3 dispersion correction. The non-local van der Waals density functional vdW-DF is also evaluated. The calculations are performed using the projector-augmented wave method in the Quantum Espresso software suite. The geometry optimization results are visualized using VMD. The Multiwfn and NCIPlot programs are used for wavefunction and intermolecular interaction analyses.

Numerical solution of the forward Kolmogorov equations in population genetics using Eta functions

This paper introduces a new numerical method for solving forward Kolmogorov equations in population genetics. Since there's no simple analytical expression for the distribution of allele frequencies (DAF), we use these equations to derive it. The accuracy of solving these equations depends on the choice of base functions, so we use Eta-based functions for better approximations. By employing the operational matrix of integral, we simplify the partial differential equation to an algebraic one. The method's error bounds, stability, and validity are demonstrated through numerical examples. Finally, we apply this method to analyze the behavior of forward Kolmogorov equations under various evolutionary forces.

Approximate probability density function for nonlinear surging in irregular following seas

Approximate probability density function for nonlinear surging in irregular following seas

Density functional theory study of cobalt sulfide as a counter electrode material for dye-sensitized solar cells

Abstract Dye-sensitized solar cells (DSSCs) are viewed as potential substitutes for traditional silicon-based solar cells due to their affordability, impressive performance in low-light conditions, eco-friendly energy production, and adaptability in solar product integration. The use of noble metals such as platinum as counter electrodes in DSSCs was initially limited by their high cost; however, cobalt sulfide has been recognized as a viable alternative due to its abundance, non-toxic properties, and cost-effectiveness. In this work, the crystal structure, electronic, optical characteristics and electrocatalytic activity of the tetragonal phases of cobalt sulfide are investigated through density functional theory with a quantum espresso package. The results obtained for the lattice parameter are a = b = 3.53 Å and c = 4.80 Å. The generalized gradient approximation and hybrid exchange correlation function yield bandgaps of 1.63 and 1.69 eV, respectively, which are consistent with the reported experimental values. An analysis of the density of states and the projected density was carried out to validate the accuracy of the calculated band gaps. Additionally, significant information was obtained from the optical properties through the calculation of the dielectric function. The findings reveal real and imaginary static dielectric constants of 11.85 and 0.13, respectively. Furthermore, the measured absorption and conductivity spectra exhibit promising attributes in the UV–visible range and good electrical conductivity. Moreover, the electrocatalytic activity was studied to analyze the adsorption energy of CoS and the electrolyte. Generally, the calculated electronic and optical properties of CoS crystals indicate their potential application as counter electrodes in dye-sensitized solar cells.

Matched power-frequency-modulated signal transform and its application in bat call signal analysis.

Bat call signal analysis is an important research topic, which is meaningful for bat species identification, and the design of various biomimetic systems. In addition to the commonly used methods in the time-frequency domain, the fractional Fourier transform (FRFT) is a valuable signal processing tool, as it is a generalization of the Fourier transform. However, the FRFT is constrained to the analysis of the linear frequency modulated-like bat call signal, while the modulation of the harmonics in a bat call is often nonlinear. For this reason, this paper proposes an integral transform, named matched power-frequency-modulated (PFM) signal transform (MPST), which is also the generalization of the Fourier transform, more precisely, a time-warping Fourier transform. As with the limitation of FRFT, the MPST is constrained to the analysis of the PFM-like bat call with the instantaneous frequency defined as an approximate power function abut time, in which the power can be an arbitrary positive integer or a fraction. The applications of MPST on the PFM-modeled bat call analysis are mainly parameter estimation and harmonic separation, and the performance is fully validated using the recordings of the feeding buzzes, social calls, and distress calls from the European bats.