Radial Basis Function Approximation Research Articles

Stable and Accurate Least Squares Radial Basis Function Approximations on Bounded Domains

Stable and Accurate Least Squares Radial Basis Function Approximations on Bounded Domains

Numerical analysis of small-strain elasto-plastic deformation using local Radial Basis Function approximation with Picard iteration

In this paper, we discuss a von Mises plasticity model with nonlinear isotropic hardening assuming small strains in a plane strain example of internally pressurised thick-walled cylinder subjected to different loading conditions. The elastic deformation is modelled using the Navier-Cauchy equation. In regions where the von Mises stress exceeds the yield stress, corrections are made locally through a return mapping algorithm. We present a novel method that uses a Radial Basis Function-Finite Difference (RBF-FD) approach with Picard iteration to solve the system of nonlinear equations arising from plastic deformation. This technique eliminates the need to stabilise the divergence operator and avoids special positioning of the boundary nodes, while preserving the elegance of the meshless discretisation and avoiding the introduction of new parameters that would require tuning. The results of the proposed method are compared with analytical and Finite Element Method (FEM) solutions. The results show that the proposed method achieves comparable accuracy to FEM while offering significant advantages in the treatment of complex geometries without the need for conventional meshing or special treatment of boundary nodes or differential operators.

Geodesic metrics for RBF approximation of some physical quantities measured on sphere

The radial basis function (RBF) approximation is a rapidly developing field of mathematics. In the paper, we are concerned with the measurement of scalar physical quantities at nodes on sphere in the 3D Euclidean space and the spherical RBF interpolation of the data acquired. We employ a multiquadric as the radial basis function and the corresponding trend is a polynomial of degree 2 considered in Cartesian coordinates. Attention is paid to geodesic metrics that define the distance of two points on a sphere. The choice of a particular geodesic metric function is an important part of the construction of interpolation formula.We show the existence of an interpolation formula of the type considered. The approximation formulas of this type can be useful in the interpretation of measurements of various physical quantities. We present an example concerned with the sampling of anisotropy of magnetic susceptibility having extensive applications in geosciences and demonstrate the advantages and drawbacks of the formulas chosen, in particular the strong dependence of interpolation results on condition number of the matrix of the system considered and on round-off errors in general.

Research on ride comfort optimization of the vehicle considering the subframe.

When the vehicle is in motion, the elastic deformation of the flexible subframe significantly influences ride comfort. Therefore, it is crucial to investigate the impact of flexible subframes on vehicle ride comfort. In order to enhance the reliability and optimization efficiency of our research, this paper incorporates the concept of elastic deformation in the flexible subframe into the investigation of vehicle ride comfort, and proposes a multi-objective optimization approach to enhance the overall vehicle ride comfort. The vibration mathematical model elucidates how flexible subframes affect vehicle ride comfort and establishes a rigid-flexible coupling model for a specific vehicle with a flexible subframe to analyze the impact of its elastic deformation on vehicle ride comfort through simulation experiments. Subsequently, a radial basis function approximation model is established, and the multi-objective particle swarm optimization and non-dominated sorting genetic algorithm II algorithms are employed to conduct multi-objective optimization of the stiffness of the subframe bushing with the aim of enhancing vehicle ride comfort. The findings indicate that the flexible subframe has a significant impact on vehicle ride comfort. Specifically, on bump roads, peak values of vertical and longitudinal seat accelerations decrease while lateral seat acceleration increases. On random roads, peak values of longitudinal and lateral seat accelerations increase while vertical acceleration decreases. Furthermore, the stiffness of the subframe bushing optimized by the non-dominated sorting genetic algorithm II algorithm further enhances vehicle ride comfort and aligns more closely with the optimization requirements in this study.

A modified lattice Boltzmann approach based on radial basis function approximation for the non‐uniform rectangular mesh

AbstractWe have presented a novel lattice Boltzmann approach for the non‐uniform rectangular mesh based on the radial basis function approximation (RBF‐LBM). The non‐uniform rectangular mesh is a good option for local grid refinement, especially for the wall boundaries and flow areas with intensive change of flow quantities. Which allows, the total number of grid cells to be reduced and so the computational cost, therefore improving the computational efficiency. But the grid structure of the non‐uniform rectangular mesh is no longer applicable to the classic lattice Boltzmann method (CLBM), which is based on the famous BGK collision‐streaming evolution. This is why the present study is inspired by the idea of the interpolation‐supplemented LBM (ISLBM) methodology. The ISLBM algorithm is improved in the present manuscript and developed into a novel LBM approach through the radial basis function approximation instead of the Lagrangian interpolation scheme. The new approach is validated for both steady states and unsteady periodic solutions. The comparison between the radial basis function approximation and the Lagrangian interpolation is discussed. It is found that the novel approach has a good performance on computational accuracy and efficiency. Proving that the non‐uniform rectangular mesh allows grid refinement while obtaining precise flow predictions.

2D temperature field reconstruction using optimized Gaussian radial basis function networks

Monitoring temperature fields by sensors and mapping the reconstructed temperature field holds great importance in numerous contexts. Concurrently, as sensing technologies continue to advance, there is a parallel development of temperature reconstruction algorithms. In this study, we present an enhanced Quadratic Optimization Gaussian Radial Basis Function approximation (QO-GRBF). We demonstrate the effectiveness of this algorithm by reconstructing a 10 cm x 10 cm temperature field using distributed optical fiber sensing system. The absolute error across four different temperature stages from 35 °C to 65 °C ranged from 0.21 °C to 0.45 °C, indicating a strong reconstruction ability. Furthermore, we compared the performance of QO-GRBF with GRBF in different point densities. The results reveal an over 30 % improvement in scenarios with denser data points, highlighting the efficacy of proposed algorithm. The data size chosen was quantified as well. There was a 10.22 % improvement observed when transitioning from selecting 4 points to selecting 6 points.

Small dataset augmentation with radial basis function approximation for causal discovery using constraint‐based method

AbstractCausal analysis involves analysis and discovery. We consider causal discovery, which implies learning and discovering causal structures from available data, owing to the significance of interpreting causal relationships in various fields. Research on causal discovery has been primarily focused on constraint‐ and score‐based interpretable methods rather than on methods based on complex deep learning models. However, identifying causal relationships in real‐world datasets remains challenging. Numerous studies have been conducted using small datasets with established ground truths. Moreover, constraint‐based methods are based on conditional independence tests. However, such tests have a lower statistical power when applied to small datasets. To solve the small sample size problem, we propose a model that generates a continuous function from available samples using radial basis function approximation. We address the problem by extracting data from the generated continuous function and evaluate the proposed method on both real and synthetic datasets generated by structural equation modeling. The proposed method outperforms constraint‐based methods using only small datasets.

Applying a New Trigonometric Radial Basis Function Approximation in Solving Nonlinear Vibration Problems

Applying a New Trigonometric Radial Basis Function Approximation in Solving Nonlinear Vibration Problems

Parallel multi-objective optimization for expensive and inexpensive objectives and constraints

Expensive objectives and constraints are key characteristics of real-world multi-objective optimization problems. In practice, they often occur jointly with inexpensive objectives and constraints. This paper presents the Inexpensive Objectives and Constraints Self-Adapting Multi-Objective Constraint Optimization algorithm that uses Radial Basis function Approximations (IOC-SAMO-COBRA) for such problems. This is motivated by the recently proposed Inexpensive Constraint Surrogate-Assisted Non-dominated Sorting Genetic Algorithm II (IC-SA-NSGA-II). These algorithms and their counterparts that do not explicitly differentiate between expensive and inexpensive objectives and constraints are compared on 22 widely used test functions. The IOC-SAMO-COBRA algorithm finds significantly better (identical/worse) Pareto fronts in at least 78% (6%/16%) of all test problems compared to IC-SA-NSGA-II measured with both the hypervolume and Inverted Generational Distance+ performance metric. The empirical cumulative distribution functions confirm this advantage for both algorithm variants that exploit the inexpensive constraints. In addition, the proposed method is compared against state-of-the-art practices on a real-world cargo vessel design problem. On this 17-dimensional two-objective practical problem, the proposed IOC-SAMO-COBRA outperforms SA-NSGA-II as well. From an algorithmic perspective, the comparison identifies specific strengths of both approaches and indicates how they should be hybridized to combine their best components.

Radial basis function‐based Pareto optimization of an outer rotor brushless DC motor

AbstractThis paper presents the development of an optimization and modeling method for the objective functions of output power, efficiency and weight of an outer rotor permanent magnet brushless DC (BLDC) motor based on radial basis function (RBF) approximation technique. The proposed RBF‐based Pareto optimization method requires less knowledge about electric/magnetic formulas and can replace conventional optimizations based on these equations with higher accuracy. To apply the proposed optimization method, the initial design should be developed using such equations. Therefore, RBFs are used to model and predict engine behavior. To optimize the objective functions, we used a genetic algorithm optimization technique with nonlinear electric and magnetic constraints to find the Pareto front set. The design obtained by the proposed radial basis function Pareto optimization (RBFPO) method was finally verified by Ansoft Maxwell. The results of optimal design using the RBFPO method have higher output power and efficiency. Also, in addition to the advantage of a favorable accuracy, RBF‐based models are significantly faster than models available in simulation tools.

A Novel Radial Basis Function Description of a Smooth Implicit Surface for Musculoskeletal Modelling

As musculoskeletal illnesses continue to increase, practical computerised muscle modelling is crucial. This paper addresses this concern by proposing a mathematical model for a dynamic 3D geometrical surface representation of muscles using a Radial Basis Function (RBF) approximation technique. The objective is to obtain a smoother surface while minimising data use, contrasting it from classical polygonal (e.g. triangular) surface mesh models or volumetric (e.g. tetrahedral) mesh models. The paper uses RBF implicit surface description to describe static surface generation and dynamic surface deformations based on its spatial curvature preservation during the deformation. The novel method is tested on multiple data sets, and the experiments show promising results according to the introduced metrics.

A novel global RBF direct collocation method for solving partial differential equations with variable coefficients

Conventional radial basis function (RBF) approximation techniques face two fundamental challenges: determining the optimal shape parameter and solving the ill-conditioned linear system, which result from the use of typical kernel functions such as the Multiquadric (MQ) RBF. The objective of this study is to address these challenges by employing a novel RBF kernel, termed the coupled MQ RBF (CMQ-RBF). The new kernel combines the MQ with an m-order conical spline, where m denotes the optimal value determined by the so-called back check method (BCM) introduced in this study. Leveraging the CMQ-RBF kernel, we present a novel global RBF collocation method that generates a well-conditioned linear system, enabling direct solver utilization and delivering highly accurate numerical results without the need for laborious shape parameter selection. Consequently, the previously challenging issues can be effectively circumvented by switching to the "better kernel" offered by the CMQ-RBF. The proposed methodology is characterized by its mathematical simplicity and ease of numerical implementation. Several benchmark examples are investigated to verify its performance.

Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation

In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including L2, L∞, and Lrms.

Bayesian approach for radial kernel parameter tuning

In this paper we present a new fast and accurate method for Radial Basis Function (RBF) approximation, including interpolation as a special case, which enables us to effectively find the optimal value of the RBF shape parameter. In particular, we propose a statistical technique, called Bayesian optimization, that consists in modeling the error function with a Gaussian process, by which, through an iterative process, the optimal shape parameter is selected. The process is step by step self-updated resulting in a relevant decrease in search time with respect to the classical leave one out cross validation technique. Numerical results deriving from some test examples and an application to real data show the performance of the proposed method.

The Impact of Inclined Magnetic Field on Streamlines in a Constricted Lid-driven Cavity

The influence of oriented magnetic field on the incompressible and electrically conducting flow is investigated in a square cavity with a moving top wall and a no-slip constricted bottom wall. Radial basis function (RBF) approximation is employed to velocity-stream function-vorticity formulation of MHD equations. Numerical results are shown in terms of streamlines for different values of Hartmann number M, orientation angle of magnetic field ϴ and the height of the constricted bottom wall hc with a fixed Reynolds number. It is obtained that the number of vortices increases as either hc or M increases. However, the increase in ϴ leads to decrease the number of vortices. Formation of vortices depends on not only the strength and the orientation of the magnetic field but also the constriction of the bottom wall.

Meshfree methods for the nonlinear variable-order fractional advection–diffusion equation

The nonlinear variable-order fractional advection–diffusion equation is applied to simulate complex engineering problem that exhibits time memory and space global dependence, for example, anomalous diffusion. In this paper, Kansa’s method is used to solve the nonlinear time–space variable-order fractional advection–diffusion equation, in which the variable order of space fractional derivative relies on both time and space while the variable order of the time fractional derivative is determined by space. Moreover, the convection coefficients, diffusion coefficients and source term are nonlinear functions that depend on exact solution. The finite difference method is contracted for the time fractional derivative and Wendland’s C6 compactly supported radial basis functions are used to substitute the space fractional derivative in the problem. The nonlinear terms are approximated by using an explicit difference scheme. The Gauss-Jacobi quadrature rule is used to approximate the weakly singular integral of space fractional derivative after radial basis function approximation. To illustrate the effectiveness and accuracy of the method, three numerical examples are solved, in which 2D experiments include the cases of regular and irregular regions. The results show that Kansa’s method has obvious advantages for variable-order fractional advection–diffusion equation.

A new variable shape parameter strategy for RBF approximation using neural networks

The choice of the shape parameter highly effects the behaviour of radial basis function (RBF) approximations, as it needs to be selected to balance between the ill-conditioning of the interpolation matrix and high accuracy. In this paper, we demonstrate how to use neural networks to determine the shape parameters in RBFs. In particular, we construct a multilayer perceptron (MLP) trained using an unsupervised learning strategy, and use it to predict shape parameters for inverse multiquadric and Gaussian kernels. We test the neural network approach in RBF interpolation tasks and in a RBF-finite difference method in one and two-space dimensions, demonstrating promising results.

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

Solution of space–time fractional diffusion equation involving fractional Laplacian with a local radial basis function approximation

The analysis and optimization of thermal sensation of train drivers under occupational thermal exposure.

Prolonged exposure of train drivers to thermal discomfort can lead to occupational safety and health (OSH) risks, causing physical and mental injuries. Traditional method of treating human skin as a wall surface fail to observe accurate skin temperature changes or obtain human thermal comfort that adapts to the thermal environment. This study employs the Stolwijk human thermal regulation model to investigate and optimize the thermal comfort of train drivers. To minimize the time-consuming design optimization, a pointer optimization algorithm based on radial basis function (RBF) approximation was utilized to optimize the train cab ventilation system design and enhance drivers' thermal comfort. The train driver thermal comfort model was developed using Star-CCM+ and 60 operating conditions were sampled using an Optimal Latin Hypercube Design (Opt LHD). We analyzed the effects of air supply temperature, air supply volume, air supply angle, solar radiation intensity and solar altitude angle on the local thermal sensation vote (LTSV) and overall thermal sensation vote (OTSV) of the train driver. Finally, the study obtained the optimal air supply parameters for the Heating Ventilation and Air Conditioning (HVAC) in the train cabin under extreme summer conditions, effectively improving the thermal comfort of the driver.

A localized meshfree technique for solving fractional Benjamin–Ono equation describing long internal waves in deep stratified fluids

This article proposes a local radial basis function partition of unity method to find the numerical solution of the time-fractional Benjamin–Ono equation. The proposed method is based on partitioning the original domain into several overlapping subdomains and employing the radial basis function approximation on each local subdomain. First, the time-discrete scheme of this equation is obtained by utilizing a finite difference approach. Then, spatial discretization is established by using the local radial basis function partition of unity method. Furthermore, the convergence and stability analysis of the proposed time-discrete scheme are investigated thoroughly. Also, numerical experiments are executed through some illustrated problems, and the results are compared with those acquired by the tanh approach and Kudryashov approach solutions to show the high accuracy and plausibility of the proposed method. Moreover, to demonstrate the physical significance, the conservation constants known as mass, momentum, and energy for this equation are also calculated by the proposed method.