Radial Basis Function Approximation Research Articles

Coupling of the Crank–Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow

This paper proposes an efficient localized meshless technique for approximating the viscoelastic wave model. This model is a significant methodology to explain wave propagation in solids modeled with a wide collection of viscoelastic laws. In the first method, a difference scheme with the second-order accuracy is implemented to obtain a semi-discrete scheme. Then, a localized radial basis function partition of unity scheme is adopted to get a full-discrete scheme. This localization technique consists of decomposing the initial domain into several sub-domains and constructing a local radial basis function approximation over every sub-domain. A well-conditioned resulting linear system and a low computational burden are the main merits of this technique compared to global collocation methods. Further, the stability and convergence analysis of the temporal discretization scheme are deduced using discrete energy method. Numerical results are shown to validate the accuracy and effectiveness of the proposed method.

Crashworthiness design for multi-cell circumferentially corrugated thin-walled tubes with sub-sections under multiple loading conditions

Multi-cell, multi-corner and adding edge-junctions structures are widely used approaches to enhance the crash characteristic of the thin-walled structures. In this study, the crashworthiness of twenty-one structures combining these three structures is examined under axial and oblique loading angles. The finite element models under axial loading are validated by experimental data from the literature and theoretical approach. In the theoretical approach, removing the corner elements in the inner structure from the theoretical calculation in multi-cell tubes has increased the accuracy. With the validations performed in axial loadings, it is predicted that the finite element model will be accurate also in oblique loadings. On the other hand, rupture strain has not been used in finite element models, which may cause some errors. Crashworthiness performance has improved as the cell number increases under all loading conditions except the C2 tube under 20-degree oblique loading. Also, the sub-sections added to the inner wall corners of the tubes significantly increase the energy absorption capacity. The complex proportion assessment (COPRAS) method and the technique for order of preference by similarity to ideal solution (TOPSIS) are utilized to get the tube with the best crashworthiness performance. The entropy method is used for weighting to avoid human intervention. The best tube varies depending on the weighting and selection method. Finally, the radial basis function (RBF) approximation approach and four multiobjective optimization methods are used to obtain optimum sizes of the C4O tube. The results of the optimizations show that the optimum structure does not differ depending on the optimization method, and the results are very close to each other.

A greedy non-intrusive reduced order model for shallow water equations

In this work, we develop Non-Intrusive Reduced Order Models (NIROMs) that combine Proper Orthogonal Decomposition (POD) with a Radial Basis Function (RBF) interpolation method to construct efficient reduced order models for time-dependent problems arising in large scale environmental flow applications. The performance of the POD-RBF NIROM is compared with a traditional nonlinear POD (NPOD) model by evaluating the accuracy and robustness for test problems representative of riverine flows. Different greedy algorithms are studied in order to determine a near-optimal distribution of interpolation points for the RBF approximation. A new power-scaled residual greedy (psr-greedy) algorithm is proposed to address some of the primary drawbacks of the existing greedy approaches. The relative performances of these greedy algorithms are studied with numerical experiments using realistic two-dimensional (2D) shallow water flow applications involving coastal and riverine dynamics.

Numerical extrapolation method for complex conductivity of disordered metals

Recently, quantum corrections to optical conductivity of disordered metals up to the UV region were observed. Although this increase of conductivity with frequency, also called anti-Drude behaviour, should disappear at the electron collision frequency, such transition has never been observed or described theoretically. Thus, the knowledge of optical conductivity in a wide frequency range is of great interest. It is well known that the extrapolation of complex conductivity is ill-posed - a solution of the analytic continuation problem is not unique for data with finite accuracy. However, we show that assuming physically appropriate properties of the searched function $\sigma(\omega)$, such as: symmetry, smoothness, and asymptotic solution for low and high frequencies, one can significantly restrict the set of solutions. We present a simple numerical method utilizing the radial basis function approximation and simulated annealing, which reasonably extrapolates the optical conductivity from visible frequency range down to far infrared and up to ultraviolet region. Extrapolation obtained on MoC and NbN thin films was checked by transmission measurement across a wide frequency range.

Energy absorption characteristics of axially varying thickness lateral corrugated tubes under axial impact loading

A new type of axially varying thickness lateral corrugated tube (AVTLCT) is proposed. The crashworthiness of the AVTLCT under axial impact has been systematically studied. The studies show that the initial peak crushing force (PCF) increases as the amplitude increases and axial variation coefficient (k) decreases. The starting positions of all folds of the AVTLCT with axial variation coefficient k>0 are near the impact end. As the number of corrugations increases, the number of folds decreases and the fluctuation of the AVTLCT increases during the impact. Compared with the AVTLCT with k=0, the AVTLCT with k>0 has absolute advantages in the two key crashworthiness indicators, PCF and crush force efficiency (CFE). PCF is reduced by up to 44.53%, and CFE is increased by up to 81.72%. Finally, a surrogate model of the AVTLCT is constructed using the radial basis function approximation, and the interactive effects of structural parameters on crashworthiness indicators are analyzed. The nondominated sorting genetic algorithm II is used to perform multiobjective optimization under unconstrained and constrained conditions.

A random search for discrete robust design optimization of linear-elastic steel frames under interval parametric uncertainty

This study presents a new random search method for solving discrete robust design optimization (RDO) problem of planar linear-elastic steel frames. The optimization problem is formulated with an explicit objective function of discrete design variables, unknown-but-bounded uncertainty in material properties and external loads, and black-box constraint functions of the structural responses. Radial basis function (RBF) models serve as approximations of structural responses. The anti-optimization problem is approximated by an RBF-based optimization problem that is solved using an adaptive strategy coupled with a difference-of-convex functions algorithm. The adaptive strategy is embedded in an iterative process for solving the upper-level optimization problem. This process starts with a set of candidate solutions and iterates through selecting the best candidate among the available candidates and generating new promising candidates by performing a small random perturbation around the best solution found so far to refine the RBF approximations. It terminates when the number of iterations reaches an upper bound, and outputs the optimal solution that is the best solution obtained through the optimization process. Two test problems and two design examples demonstrate that the exact optimal or a good approximate solution can be found by the proposed method with a few trials of the algorithm.

Optimization design of the reactor coolant pump flywheel comprised of multi-ring packing heavy metal alloy

To improve energy storage performance of the multi-ring RCP flywheel comprised of inner hub, tungsten alloy ring and outer retainer, optimization design process for the radial thicknesses of components and interference fit assembly is demonstrated. Radial thicknesses of these components and magnitude of interference fit are set as design parameters to construct parametric analysis model. Design of experiments (DOE) technique is adopted to generate design matrix for sensitive analysis. Subsequently, radial basis function (RBF) approximation is employed to search for the optimal design considering structural integrity requirement. Finally, a proper range of the magnitude of interference fit is obtained which can not only prevent neither slippage nor separation between the inner hub and tungsten alloy ring but also ensure the safety of structural strength. In addition, it is found that to ensure a high moment of inertia and energy density, a reasonable large radial thickness of tungsten alloy ring and a reasonable small radial thickness of outer retainer are required.

Efficient representation of supply and demand curves on day-ahead electricity markets

We model the supply and demand curves of electricity day-ahead auctions in a parsimonious way by building an appropriate algorithm to present the information about electricity prices and demand with far fewer parameters than the existing algorithm. We represent each curve using mesh-free interpolation techniques based on radial basis function approximation. We describe the results of this method for the day-ahead IPEX spot price of Italy and then use these representations to forecast supply and demand and find the intersection of the predicted supply and demand curves in order to obtain the market clearing price.

Location of the Leakage From a Simulated Water-Cooling Wall Tube Based on Acoustic Method and an Artificial Neural Network

Water-cooling wall tube leakage accident is an important factor affecting the safe and economic operation of power station boilers. In this article, the acoustic method is applied to locate the leakage from a simulated water-cooling wall tube, and several key technologies are studied in order to identify the precise location of leakage source. Sound wave propagation path in ‘nonuniform temperature field is obtained to overcome the shortcoming that the traditional location technology always assumed that sound waves propagate along the straight line. Considering the refraction effect of sound waves, the reconstruction method based on radial basis function approximation with polynomial reproduction (RBF-PR) is used to reconstruct the temperature field. A generalized cross correlation with second correlation is applied to estimate the time delay of arrival (TDOA). According to the temperature field measured by acoustic method and the propagate time of sound waves, the location fingerprint method including the statistical correlation characteristics between the coordinate of fingerprint points and TDOA are constructed. Radial basis function artificial neural network (RBF ANN) is used to estimate the coordinate of leakage point. Numerical simulations and experimental study are implemented to evaluate the effectiveness of the proposed location system. An acoustic location test platform is constructed to study the location accuracy in different working conditions. The results indicate that considering the refraction effect, the reconstruction performance of temperature field is superior than that without the refraction effect, and acoustic method with RBF ANN algorithm can gain the leakage location with high accuracy and good antinoise ability in the above high-quality reconstructed nonuniform temperature distribution.

Distributed Online Dispatch For Microgrids Using Hierarchical Reinforcement Learning Embedded With Operation Knowledge

This paper considers the problem of distributed online economic dispatch (DOED) from sequential data using reinforcement learning. Learning operation behavior in high-dimension environments with constraints is a major challenge for the DOED of networked microgrids (MGs), where insufficient exploration disables agents to build complex policies. Therefore, this paper develops a hierarchical reinforcement learning (HRL) to handle the DOED problem, where radial basis function (RBF) approximation is incorporated to make policies in continuous space. Based on the hierarchical framework, the HRL algorithm increases the training efficiency and reduces computational cost. The online HRL achieves distributed selfadaption and better performance of real-time dispatch by a modest number of interacting variables. In addition, guided by domain knowledge, the HRL algorithm avoids onerous additional learning beyond feasible action space. In the case of an actual networked MGcluster in Qingdao with real operation data, simulation is conducted to verify that the proposed learning framework can reduce longterm operation costs and enhance operation stability. To explore the learning process, we also provide its convergence condition andanalyze the sensitivity of the learning parameters.

A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

This research work investigates the use of Artificial Neural Network (ANN) based on models for solving first and second order linear constant coefficient ordinary differential equations with initial conditions. In particular, we employ a feed-forward Multilayer Perceptron Neural Network (MLPNN), but bypass the standard back-propagation algorithm for updating the intrinsic weights. A trial solution of the differential equation is written as a sum of two parts. The first part satisfies the initial or boundary conditions and contains no adjustable parameters. The second part involves a feed-forward neural network to be trained to satisfy the differential equation. Numerous works have appeared in recent times regarding the solution of differential equations using ANN, however majority of these employed a single hidden layer perceptron model, incorporating a back-propagation algorithm for weight updation. For the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using statistical regression. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD software, which enables us to slightly adjust the intrinsic biases. We compare the convergence and the accuracy of our results with analytic solutions, as well as well-known numerical methods and obtain satisfactory results for our example ODE problems.

Multiquadric Radial Basis Function Approximation Scheme for Solution of Total Variation BasedMultiplicative Noise Removal Model

This article introduces a fast meshless algorithm for the numerical solution nonlinear partial differential equations (PDE) by Radial Basis Functions (RBFs) approximation connected with the Total Variation (TV)-based minimization functional and to show its application to image denoising containing multiplicative noise. These capabilities used within the proposed algorithm have not only the quality of image denoising, edge preservation but also the property of minimization of staircase effect which results in blocky effects in the images. It is worth mentioning that the recommended method can be easily employed for nonlinear problems due to the lack of dependence on a mesh or integration procedure. The numerical investigations and corresponding examples prove the effectiveness of the recommended algorithm regarding the robustness and visual improvement as well as peak-signal-to-noise ratio (PSNR), signal-to-noise ratio (SNR), and structural similarity index (SSIM) corresponded to the current conventional TV-based schemes.

Numerical Solution of Fractional Black-Scholes Equation by Using Radial Basis Function (RBF) Approximation Method

Numerical Solution of Fractional Black-Scholes Equation by Using Radial Basis Function (RBF) Approximation Method

P.871Relationship between therapeutic compliance with antipsychotics and total antioxidant status in acute paranoid schizophrenia

Constrained optimization of high-dimensional numerical problems plays an important role in many scientific and industrial applications. Function evaluations in many industrial applications are severely limited and often only little analytical information about objective function and constraint functions is available. For such expensive black-box optimization tasks, the constraint optimization algorithm COBRA (Constrained Optimization By Radial Basis Function Approximation) was proposed, making use of RBF (radial basis function) surrogate modeling for both objective and constraint functions. COBRA has shown remarkable success in solving reliably complex benchmark problems in less than 500 function evaluations. Unfortunately, COBRA requires careful adjustment of parameters in order to do so.In this work we present a new algorithm SACOBRA (Self-Adjusting COBRA), which is based on COBRA and capable of achieving high-quality results with very few function evaluations and no parameter tuning. It is shown with the help of performance profiles on a set of benchmark problems (G-problems, MOPTA08) that SACOBRA consistently outperforms COBRA algorithms with different fixed parameter settings. We analyze the importance of the new elements in SACOBRA and show that each element of SACOBRA plays a role to boost up the overall optimization performance. We discuss the reasons and get in this way a better understanding of high-quality RBF surrogate modeling.

Radial basis function and multi-level 2D vector field approximation

We propose a new approach for meshless multi-level radial basis function (ML-RBF) approximation which offers data-sensitive compression and progressive details visualization. It leads to an analytical description of compressed vector fields, too. The proposed approach approximates the vector field at multiple levels of detail. The low-level approximation removes minor flow patterns while the global character of the flow remains unchanged. And conversely, the higher level approximation contains all small details of the vector field. The ML-RBF has been tested with a numerical forecast data set and 3D tornado data set to prove its ability to handle data with complex topology. Comparison with the Fourier vector field approximation has been made and significant advantages, i.e. high compression ratio, accuracy, extensibility to a higher dimension etc., of the proposed ML-RBF were proved.

A meshless BEM for solving transient non-homogeneous convection-diffusion problem with variable velocity and source term

In this paper, a meshless BEM based on the radial integration method is developed to solve transient non-homogeneous convection-diffusion problem with spatially variable velocity and time-dependent source term. The Green function served as the fundamental solution is adopted to derive the boundary domain integral equation about the normalized field quantity. The two-point backward finite difference technique is utilized to discretize the time-dependent terms in the integral equation, which results in that the final integral equation formulation is only related with the normalized field quantity at the current time and has three domain integrals. Both two domain integrals regarding the normalized field quantity at the current and previous times are transformed into boundary integrals by using radial integration method and radial basis function approximation. For domain integral about the source term being known function of time and coordinate, radial basis functions approximation is still adopted to make the transformed boundary integral be evaluated only once, not at each time level. A pure boundary element algorithm with boundary-only discretization and internal points is established and the system of equations is assembled like the corresponding steady problem. Four numerical examples are given to demonstrate the accuracy and effectiveness of the present method.

The RBF partition of unity method for solving the Klein-Gordon equation

In this paper, a localized radial basis function (RBF) method is applied to obtain a global approximation of the solution of two dimensional Klein-Gordon equation on a given bounded domain. We use the RBF partition of unity (RBF-PU) method which is based on partitioning the original domain to several patches and using the RBF approximation on each local domain. Low computational cost and well conditioned final linear system are the main advantages of this method comparing with the original (global) RBF techniques. Numerical experiments show that the given problem could be solved successfully with a reasonable accuracy.

A stable computation on local boundary-domain integral method for elliptic PDEs

Many local integral methods are based on an integral formulation over small and heavily overlapping stencils with local Radial Basis Functions (RBFs) interpolations. These functions have become an extremely effective tool for interpolation on scattered node sets, however the ill-conditioning of the interpolation matrix – when the RBF shape parameter tends to zero corresponding to best accuracy – is a major drawback. Several stabilizing methods have been developed to deal with this near flat RBFs in global approaches but there are not many applications to local integral methods. In this paper we present a new method called Stabilized Local Boundary Domain Integral Method (LBDIM-St) with a stable calculation of the local RBF approximation for small shape parameter that stabilizes the numerical error. We present accuracy results for some Partial Differential Equations (PDEs) such as Poisson, convection–diffusion, thermal boundary layer and an elliptic equation with variable coefficients.

Radial basis functions and improved hyperparameter optimisation for gaussian process strain estimation

Over the past decade, a number of algorithms for full-field elastic strain estimation from neutron and X-ray measurements have been published. Many of the recently published algorithms rely on modelling the unknown strain field as a Gaussian Process (GP) – a probabilistic machine-learning technique. Thus far, GP-based algorithms have assumed a high degree of smoothness and continuity in the unknown strain field. In this paper, we propose three modifications to the GP approach to improve performance, primarily when this is not the case (e.g. for high-gradient or discontinuous fields); hyperparameter optimisation using k-fold cross-validation, a radial basis function approximation scheme, and gradient-based placement of these functions.

Optimization of Slots-Groove Coupled Casing Treatment for an Axial Transonic Compressor

Abstract A multi-objective optimization of a coupled casing treatment (CCT) for an axial transonic compressor is performed in this study. The coupled casing treatment is the basis axial slots with a circumferential groove located at various positions along the slots. During the optimization stage, five important parameters to control the geometry are used as the optimal variables. The stall margin and the peak efficiency are selected as the optimal objectives. Non-dominated sorting genetic algorithm II coupled with radial basis function (RBF) approximation is used to search for Pareto-optimal solutions. Then, four optimal configurations are selected from Pareto-front for further analysis. As shown in the simulation results with and without the coupled casing treatments, the leakage flow is reenergized and the blocking region near the blade leading edge at rotor tip is decreased by the use of these structures under the low flowrate condition, which is the main reason for stability enhancement. Besides, a coupled casing treatment with the groove settled near the end of the basis slots have the potential to generate more injection flow and extend the operating range of compressor further.