Types Of Basis Functions Research Articles

Spherical Shepard-Bernoulli operator

AbstractWe consider the problem of interpolating large sets of scattered data on the sphere using Shepard-Bernoulli operators constructed based on two types of basis functions. We study the interpolation properties and the approximation errors of the methods proposed here. We evaluate the efficiency using several test functions and the practical applicability through two real-data problems.

Development of a software solution to an applied mechanics problem based on numerical methods

The present work delves into the significance of material selection in the manufacturing of tools for milling operations and determines their suitability through a detailed analysis of strength and behavior during material processing. To enhance longevity and optimize production, the utilization of mathematical models and numerical methods, notably the least squares method and the solution of systems of linear algebraic equations (SLAE) employing the Gaussian elimination method with pivot selection, is proposed. These methods are applied for the approximation of experimental data and the analysis of material characteristics, ensuring precision in the evaluation of its properties. Situations where only partial data is known are investigated, along with simplifications in the computation of known functions. The study encompasses software for numerical computation and visualization of various problem types, effectively addressed through the aforementioned methods. The software algorithm for data approximation involves storing information in a text file, user input of variables, and selection of the quantity and type of basis functions. After the user inputs of parameters, the program formulates a system of equations based on selected functions, determines approximation coefficients, and generates a graph for an objective assessment of results. With a user-friendly interface, users can easily interact with the program, input values, and analyze results through graphical representation, streamlining the workflow and facilitating the comprehension of obtained data. The approximation of functions using numerical methods proves to be efficiently applicable in diverse fields for addressing applied mechanics problems.

Novel Numerical Investigations of Some Problems Based on the Darcy–Forchheimer Model and Heat Transfer

In this study, we introduced a new type of basis function and subsequently a Chebyshev delta shaped collocation method (CDSC). We then use this method to numerically investigate both the natural convective flow and heat transfer of nanofluids in a vertical rectangular duct on the basis of a Darcy–Brinkman–Forchheimer model and the electroosmosis-modulated Darcy–Forchheimer flow of Casson nanofluid over stretching sheets with Newtonian heating problems. The approximate solution is represented in terms of Chebyshev delta shaped basis functions. Novel error estimates for interpolating polynomials are derived. Computational experiments were carried out to corroborate the theoretical results and to compare the present method with the existing Chebyshev pseudospectral method. To demonstrate our proposed approach, we also compared the numerical solutions with analytic solutions of the Poisson equation. Computer simulations show that the proposed method is computationally cheap, fast, and spectrally accurate and more importantly the obtained approximate solution can easily be used by researchers in this field.

Functional Bayesian Model Average Regression Based on Derivative Information

Functional data regression models based on basis function expansion have been widely used in economic, financial, and environmental fields. However, determining the appropriate type and quantity of basis functions, such as KL expansion and B-spline, has remained a challenge to overcome. Additionally, when two curves are similar, constructing an effective regression model can prove to be difficult. On top of that, this paper suggests a functional Bayesian model average regression model, which is based on the KL expansion of the derivative function. On the one hand, the method uses the first and second-order derivative function information as predictors to capture the curve's fluctuation trend more comprehensively. On the other hand, the method applies model averaging to dynamically determine the appropriate number of basis functions to use, which can effectively alleviate the overfitting and underfitting problems of the model. The experiments and corresponding data analysis demonstrate that the proposed method has higher prediction accuracy and robustness than the comparative methods.

Mapped material point method for large deformation problems with sharp gradients and its application to soil‐structure interactions

AbstractThe material point method (MPM) is often applied to large deformation problems that involve sharp gradients in the solution field. Representative examples in geomechanics are interactions between soils and various “structures” such as foundations, penetrometers, and machines, where the displacement fields exhibit sharp gradients around the soil‐structure interfaces. Such sharp gradients should be captured properly in the MPM discretization to ensure that the numerical solution is sufficiently accurate. In the MPM literature, several types of locally refined discretizations have been developed and used for this purpose. However, these local refinement schemes are not only quite complicated but also restricted to certain types of basis functions or update schemes. In this work, we propose a new MPM formulation, called the mapped MPM, that can efficiently capture sharp gradients with a uniform background grid compatible with every standard MPM basis function and scheme. The mapped MPM is built on the method of auxiliary mapping that reparameterizes the given problem in a different domain whereby sharp gradients become much smoother. Because the reparameterized problem is free of undesirably sharp gradients, it can be well solved with the standard MPM ingredients including a uniform background grid. We verify and demonstrate the mapped MPM through several numerical examples, with particular attention to soil‐structure interaction problems.

An efficient ECG signals denoising technique based on the combination of particle swarm optimisation and wavelet transform

During the recording time, electrocardiogram (ECG) signals are subject to multiple artefact noises, such as muscle activity, white Gaussian noise (WGN), baseline wander, and power line interference (PLI). Therefore, pre-processing of ECG signals is essential to eliminate these artefacts and to obtain efficient ECG features. Many approaches have been proposed for removing ECG noises, including ECG signal denoising using wavelet transform (WT). However, the effectiveness and performance of the WT technique are strongly related to the configuration of its control parameters, which are typically fine-tuned through a laborious and time-consuming series of experiments. This paper introduces a technique that combines particle swarm optimisation (PSO) with WT for ECG signal denoising. The key contribution of this research is the use of PSO to determine the optimum settings for all WT parameters for ECG signal denoising (type of wavelet basis function Φ, thresholding function β, level of decomposition L, rule for threshold selection λ, and rescaling method ρ). The efficiency of the proposed method is evaluated using the percentage root mean square difference (PRD) and the signal-to-noise ratio (SNR), employing various ECG signals available online from the MIT-BIH Arrhythmia database. Experimental results show that the proposed PSO-WT technique yields better results than state-of-the-art techniques in terms of SNR, particularly for PLI measured at 60 Hz and still acceptable at 50 Hz. For example, a denoised ECG signal resulting from the proposed technique at an SNR input of 10 dB corresponds to an SNRoutput of 27.47 dB at 60 Hz, improving the quality of the denoised ECG signal and making it more appropriate for clinical diagnosis. Furthermore, the proposed method also shows promising efficiency in the presence of WGN, making it highly relevant for IoT applications and RF transmission.

Numerical solution of Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems using fractional-order Legendre collocation method

The collocation method is one of the most powerful and effective techniques to solve nonlinear differential equations. This method reduces the original problem to a system of algebraic equations. Awareness of the nature of the problem and the analytical behavior of the solution has an important influence on the choice of type and number of basis functions required by the collocation method for solving the different classes of differential equations. In general, the polynomial basis functions such as the Legendre polynomials are not always an appropriate choice to approximate the functions that can be expanded based on monomial basis functions with real powers. In such cases, a combination of polynomial basis functions with some maps to replace the natural powers with arbitrary real powers can be a suitable alternative. This paper presents a collocation method based on the shifted fractional-order Legendre functions to solve the linear and nonlinear Hamilton–Jacobi–Bellman PDEs in stochastic optimal control problems. The convergence analysis of the proposed method has also been investigated. Moreover, the method is used to solve some practical problems with different solution structures such as resource extraction and Merton’s portfolio selection models. Numerical results for four different examples indicated that the collocation method based on the fractional-order Legendre functions are provided. Due to the non differentiability of the solution of HJB PDEs at x = 0 for resource extraction and Merton’s portfolio selection models, the Legendre collocation method cannot produce solutions with the desired accuracy. In fact, it is observed that the collocation method based on the fractional-order Legendre functions can be more efficient than the standard Legendre collocation method. Also, the sample paths of the control and state processes are obtained together with the estimate given by the Monte Carlo simulation.

Grounded source transient electromagnetic 3D forward modeling with the spectral-element method and its application in hydraulic fracturing monitoring

A long wire with large current source transient electromagnetic (TEM) monitoring, with a large detection depth, low cost, safety, and environmental protection, has unique advantages in the testing and identification of unconventional reservoir fluid and the evaluation of stimulated reservoir volume. So, the TEM 3D forward modeling method has become a research hotspot. Although the finite-element method (FEM) is a type of numerical algorithm that has been widely applied in three-dimensional (3D) electromagnetic field forward modeling, the efficiency and accuracy of FEM require further improvement in order to meet the demand of fast 3D inversion. By increasing the order of the basis function and adjusting the principle of mesh discretization, the precision of the mixed-order spectral-element (SEM) result will be increased. The backward Euler scheme is an unconditionally stable technique which can ignore the impact of the scale of the time step. To achieve a better description of the nonlinear electromagnetic (EM) response of the grounded source TEM method and to optimize the efficiency and accuracy/precision of the 3D TEM forward modeling method significantly, we proposed the use of 3D TEM forward modeling based on the mixed-order SEM and the backward Euler scheme, which can obtain more accurate EM results with fewer degrees of freedom. To check its accuracy and efficiency, the 1D and 3D layered models are applied to compare the SEM results with the semi-analytical and FEM solutions. In addition, we analyzed the accuracy and efficiency of the SEM method for different types of order basis functions. Finally, we calculated the long-wire source TEM response for a practical 3D earth model of a shale gas reservoir for fracturing monitoring and tested the feasibility of the TEM method in a hydraulic fracturing monitoring area to further demonstrate the flexibility of the SEM method.

Algebraic multiscale grid coarsening using unsupervised machine learning for subsurface flow simulation

Subsurface flow simulation is vital for many geoscience applications, including geoenergy extraction and gas (energy) storage. Reservoirs are often highly heterogeneous and naturally fractured. Therefore, scalable simulation strategies are crucial to enable efficient and reliable operational strategies. One of these scalable methods, which has also been recently deployed in commercial reservoir simulators, is algebraic multiscale (AMS) solvers. AMS, like all multilevel schemes, is found to be highly sensitive to the types (geometries and size) of coarse grids and local basis functions. Commercial simulators benefit from a graph-based partitioner; e.g., METIS to generate the multiscale coarse grids. METIS minimizes the amount of interfaces between coarse partitions, while keeping them of similar size which may not be the requirement to create a coarse grid. In this work, we employ a novel approach to generate the multiscale coarse grids, using unsupervised learning methods which is based on optimizing different parameter. We specifically use the Louvain algorithm and Multi-level Markov clustering. The Louvain algorithm optimizes modularity, a measure of the strength of network division while Markov clustering simulates random walks between the cells to find clusters. It is found that the AMS performance is improved when compared with the existing METIS-based partitioner on several field-scale test cases. This development has the potential to enable reservoir engineers to run ensembles of thousands of detailed models at a much faster rate.

Shape Preserving Properties of Parametric Szász Type Operators on Unbounded Intervals

The Be´zier-type operator has become a powerful tool in operator theory, neural networks, curve and surface design and representation because of its good shape-preserving properties. Motivated by the improvements of the operator in computational disciplines, we investigate some elementary properties of two kinds of modified Sza´sz type basis functions, depending on non-negative parameters. Using the derivative, the symmetry of variables, the modulus of continuity and the concave continuous modulus, we study some shape preserving properties of these operators concerning monotonicity, convexity, starshapedness, semi-additivity and the preservation of smoothness. Moreover, some illustrative examples are provided to demonstrate the approximation behavior of the proposed operators and the classical ones.

Adaptive Basis Function Selection Enhanced Multisurrogate-Assisted Evolutionary Algorithm for Production Optimization

Summary Surrogate-assisted evolutionary algorithms (SAEAs) have become a popular approach for solving reservoir production optimization problems. The radial-basis-function network (RBFN) is a robust surrogate model technology suitable for reservoir development with numerous wells and a long production lifetime. There are several types of basis functions for constructing RBFN models. However, existing research shows that selecting the basis function with competitive performance for the current optimization problem is challenging without prior knowledge. In conventional SAEAs, the basis function is often predetermined, but its prediction accuracy for the problem at hand cannot be guaranteed. Furthermore, canonical SAEAs usually employ only one surrogate model for the entire optimization process. However, relying on a single surrogate model for optimization increases the probability of search direction misdirection due to prediction deviations. In this paper, a novel method named adaptive basis function selection enhanced multisurrogate-assisted evolutionary algorithm (ABMSEA) is introduced for production optimization. This method mainly includes two innovations. First, by training and testing different types of basis functions, the one with the best prediction performance is adaptively selected. Second, the ensemble model is constructed using the bootstrap sampling method, comprising multiple global surrogate models based on the selected best basis function. To search for a set of solutions that perform well on multiple surrogates, we employ an efficient multiobjective optimization (MOO) algorithm called nondominated sorting genetic algorithm II (NSGA-II). This algorithm uses the surrogates themselves as objective functions, aiming to find solutions that yield favorable results across multiple surrogates. The proposed method improves the efficiency of production optimization while enhancing global search capabilities. To evaluate the effectiveness of ABMSEA, we conduct tests on four 100D benchmark functions, a three-channel model, and an egg model. The obtained results are compared with those obtained from differential evolution (DE) and three other surrogate-model-based methods. The experimental results demonstrate that ABMSEA exhibits an accurate selection of competitive basis functions for the current optimization period while maintaining high optimization efficiency and avoiding local optima. Consequently, our method enables optimal well control, leading to the attainment of the highest net present value (NPV).

Radial basis function based finite element method: Formulation and applications

In the study, a method is proposed and developed for solid analysis that is a hybrid of the radial basis function and the finite element method (RBF-FEM). Based on the finite nodes, the method employs the radial basis functions to produce the shape functions with simplicity, especially when increasing the element node number. The formulation and applications of the method in the analysis of solids are examined. Several numerical examples are carried out to analyze the convergence, accuracy, and computational time cost. Its comeouts are then compared with that by the finite element method. The study also shows the factors that affect the accuracy of the method, such as the radial basis function types, radial basis function shape parameter, node number, and node position in an element. It is shown that the method quickly produces better results with large-node-number elements (seven and eight nodes) in comparison to the conventional numerical method. Discussion on the method and the extensibility of the approach are addressed in the conclusion.

Rapid empirical battery electromotive-force and overpotential modelling using input–output linear parameter-varying methods

In this paper, battery overpotential model identification approaches based on local and global Linear Parameter-Varying (LPV) input–output models are developed. Key features such as model structure, number of local models, and type and order of basis functions are considered. The LPVcore toolbox (Boef, 2021) has been used to solve the global identification problems. Furthermore, an iterative scheme is proposed which identifies a complete empirical battery model, i.e., both the ElectroMotive Force (EMF), also known as open-circuit voltage, and the overpotential model. This is achieved by iteratively obtaining an EMF realisation by (1) subtracting the modelled overpotential from a measured terminal voltage resulting from Constant-Current (CC) (dis)charging, and (2) using this EMF to calculate the overpotential from dynamic (dis)charging data and identifying an overpotential model using the LPV methods. This approach results in an empirical battery model with a precision similar (around 4 mV root-mean-square error in the range between 100% and 20% SoC) to models identified through a common cascaded approach in which the EMF is obtained separately from, e.g., pulse-(dis)charge data, but requires less measurement data resulting in a reduction factor in the order of 7 to 35 in terms of required experiment time.

Characterization of excited states in time-dependent density functional theory using localized molecular orbitals.

Localized molecular orbitals are often used for the analysis of chemical bonds, but they can also serve to efficiently and comprehensibly compute linear response properties. While conventional canonical molecular orbitals provide an adequate basis for the treatment of excited states, a chemically meaningful identification of the different excited-state processes is difficult within such a delocalized orbital basis. In this work, starting from an initial set of supermolecular canonical molecular orbitals, we provide a simple one-step top-down embedding procedure for generating a set of orbitals, which are localized in terms of the supermolecule but delocalized over each subsystem composing the supermolecule. Using an orbital partitioning scheme based on such sets of localized orbitals, we further present a procedure for the construction of local excitations and charge-transfer states within the linear response framework of time-dependent density functional theory (TDDFT). This procedure provides direct access to approximate diabatic excitation energies and, under the Tamm-Dancoff approximation, also their corresponding electronic couplings-quantities that are of primary importance in modeling energy transfer processes in complex biological systems. Our approach is compared with a recently developed diabatization procedure based on subsystem TDDFT using projection operators, which leads to a similar set of working equations. Although both of these methods differ in the general localization strategies adopted and the type of basis functions (Slaters vs Gaussians) employed, an overall decent agreement is obtained.

Fast Training of Neural Affine Models for Model Predictive Control: An Explicit Solution

This work describes a nonlinear affine model developed especially for prediction in Model Predictive Control (MPC). A multi-model structure is used in which independent sub-models compute predictions for the consecutive sampling instants. All sub-models are affine with respect to the future values of the manipulated variable that are calculated in MPC. Model coefficients are time-varying; they are determined online by a neural network of the Radial Basis Function type. It is proved that the described model configuration makes it possible to formulate the training task as a least squares problem that has an explicit solution (the global optimum). A chemical reactor benchmark is considered to show advantages of the discussed modeling approach and the resulting MPC scheme. As a result of model affinity, MPC requires solving online simple quadratic optimization tasks.

Generalized Multiscale Finite Element Method for scattering problem in heterogeneous media

In this paper, we consider the scattering problem in the heterogeneous domain. The mathematical model is described by the Helmholtz equation related to wave propagation with absorbing boundary conditions. For the solution of the problem using the finite element method, we construct an unstructured fine grid that resolves perforation on the grid level. Such approximations lead to a large system of equations. To reduce the size of the discrete system, we use a multiscale approximation on a coarse grid using the Generalized Multiscale Finite Element Method. We construct a multiscale space using the solution of local spectral problems on the snapshot space in each local domain. Two types of multiscale basis functions are presented and investigated. We present numerical results for the Helmholtz problem in a heterogeneous domain with heterogeneous properties on obstacles. The proposed method is studied for different wavenumbers and numbers of the multiscale basis functions.

Unified Control Parameterization Approach for Finite-Horizon Feedback Control With Trajectory Shaping

This article presents control parameterization as a unifying framework for designing a linear feedback control law that achieves the finite-time transfer of output as well as trajectory shaping. Representing control input as a linear combination of independent basis functions allows wide variability in the resultant feedback control laws through the selection of the number and types of basis functions. Given an array of basis functions that meets the trajectory shaping necessities, the unified design approach proceeds with the determination of the coefficients so that the predicted trajectory attains the desired output at the final time. The input evaluated with the coefficients found at each instance essentially turns out to be a linear state feedback policy with an additional feedforward term and time-dependent gains, which is appropriate for practical use. The unified control parameterization approach lends itself well to missile guidance applications with the expandability and direct trajectory shaping capability that it provides. To emphasize the expandability of the framework, this study revisits the trajectory shaping guidance laws from the control parameterization viewpoint and shows how the notion of specifying input basis functions not only generalizes various existing methods but also enables further extensions. Furthermore, an application to integrated guidance and control illustrates the strength of the design process in handling the shaping requirements more directly through the construction of appropriate basis.

Trefftz-Type Functions Applied for Subcooled Flow Boiling Heat Transfer Calculations in a Minichannel of Different Spatial Orientation

ABSTRACT The paper describes the results of subcooled flow boiling heat transfer in an asymmetrically heated group of five minichannels 1 mm deep. The heated element for the cooling fluid (FC-72) flowing in the minichannels was a foil made of Haynes-230 alloy. The time-dependent temperature distribution of the outer heated foil surface was recorded as a result of infrared thermography. The test section was oriented in several spatial orientations, that is, 30°, 45°, 60° and 75° inclinations to the horizontal plane. The objective of the numerical calculations, referring to the experimental results, was to determine the time–dependent heat transfer coefficient on the wall-fluid contact surface from the Robin boundary condition. Both the foil and the fluid temperatures were the result of solving the inverse and direct two-dimensional heat transfer problem in two neighboring domains: the heated foil and flowing fluid. The problem was solved using the Trefftz method and the FEM with Trefftz type basis functions. The heat transfer coefficient obtained by two numerical methods appears to be similar. During subcooled flow boiling, at higher minichannel inclination angles, higher values of the heated foil temperature were obtained, as well as higher values of the heat transfer coefficient.

A linear smoothed meshfree method with intrinsic enrichment functions for 2D crack analysis

PurposeThis article aims to develop an accurate and efficient meshfree Galerkin method based on the strain smoothing technique for linear elastic continuous and fracture problems.Design/methodology/approachThis paper proposed a generalized linear smoothed meshfree method (LSMM), in which the compatible strain is reconstructed by the linear smoothed strains. Based on the idea of the weighted residual method and employing three linearly independent weight functions, the linear smoothed strains can be created easily in a smoothing domain. Using various types of basic functions, LSMM can solve the linear elastic continuous and fracture problems in a unified way.FindingsOn the one hand, the LSMM inherits the properties of high efficiency and stability from the stabilized conforming nodal integration (SCNI). On the other hand, the LSMM is more accurate than the SCNI, because it can produce continuous strains instead of the piece-wise strains obtained by SCNI. Those excellent performances ensure that the LSMM has the capability to precisely track the crack propagation problems. Several numerical examples are investigated to verify the accurate, convergence rate and robustness of the present LSMM.Originality/valueThis study provides an accurate and efficient meshfree method for simulating crack growth.

Correction: Yalcin, F.; Simsek, Y. A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions. Symmetry 2020, 12, 779

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