scholarly journals The Quasi-Optimal Radial Basis Function Collocation Method: A Technical Note

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Juan Zhang ◽  
Mei Sun ◽  
Enran Hou ◽  
Zhaoxing Ma
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Computational Cost ◽  
Technical Note ◽  
Function Parameter ◽  
Approximation Solution ◽  
Collocation Points ◽  
Radial Basis ◽  
Solution Accuracy ◽  
Parameter Values

The traditional radial basis function parameter controls the flatness of these functions and influences the precision and stability of approximation solution. The coupled radial basis function, which is based on the infinitely smooth radial basis functions and the conical spline, achieves an accurate and stable numerical solution, while the shape parameter values are almost independent. In this paper, we give a quasi-optimal conical spline which can improve the numerical results. Besides, we consider the collocation points in the Chebyshev-type which improves solution accuracy of the method with no additional computational cost.

scholarly journals The Conical Radial Basis Function for Partial Differential Equations

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
J. Zhang ◽  
F. Z. Wang ◽  
E. R. Hou
Keyword(s):  
Radial Basis Function ◽  
Boundary Value Problems ◽  
Basis Function ◽  
Boundary Value ◽  
Computational Cost ◽  
Basis Functions ◽  
Simple Scheme ◽  
Collocation Points ◽  
Radial Basis ◽  
Radial Basis Function Method

The performance of the parameter-free conical radial basis functions accompanied with the Chebyshev node generation is investigated for the solution of boundary value problems. In contrast to the traditional conical radial basis function method, where the collocation points are placed uniformly or quasi-uniformly in the physical domain of the boundary value problems in question, we consider three different Chebyshev-type schemes to generate the collocation points. This simple scheme improves accuracy of the method with no additional computational cost. Several numerical experiments are given to show the validity of the newly proposed method.

Radial basis function surrogate model-based optimization of guardrail post embedment depth in different soil conditions

2019 ◽  
Vol 234 (2-3) ◽  
pp. 739-761
Author(s):  
Sedat Ozcanan ◽  
Ali Osman Atahan
Keyword(s):  
Finite Element ◽  
Radial Basis Function ◽  
Basis Function ◽  
Computational Cost ◽  
Critical Parameters ◽  
Crash Test ◽  
Soil Conditions ◽  
Embedment Depth ◽  
Element Analysis ◽  
Radial Basis

For guardrail designers, it is essential to achieve a crashworthy and optimal system design. One of the most critical parameters for an optimal road restraint system is the post embedment depth or the post-to-soil interaction. This study aims to assess the optimum post embedment depth values of three different guardrail posts embedded in soil with varying density. Posts were subjected to dynamic impact loads in the field while a detailed finite element study was performed to construct accurate models for the post–soil interaction. It is well-known that experimental tests and simulations are costly and time-consuming. Therefore, to reduce the computational cost of optimization, radial basis function–based metamodeling methodology was employed to create surrogate models that were used to replace the expensive three-dimensional finite element models. In order to establish the radial basis function model, samples were derived using the full factorial design. Afterward, radial basis function–based metamodels were generated from the derived data and objective functions performed using finite element analysis. The accuracy of the metamodels were validated by k-fold cross-validation, then optimized using multi-objective genetic algorithm. After optimum embedment depths were obtained, finite element simulations of the results were compared with full-scale crash test results. In comparison with the actual post embedment depths, optimal post embedment depths provided significant economic advantages without compromising safety and crashworthiness. It is concluded that the optimum post embedment depths provide an economic advantage of up to 17.89%, 36.75%, and 43.09% for C, S, and H types of post, respectively, when compared to actual post embedment depths.

scholarly journals Solving the Linear Integral Equations Based on Radial Basis Function Interpolation

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Huaiqing Zhang ◽  
Yu Chen ◽  
Xin Nie
Keyword(s):  
Radial Basis Function ◽  
Integral Equations ◽  
Basis Function ◽  
Approximation Solution ◽  
Integration Schemes ◽  
Radial Basis ◽  
The Matrix ◽  
Linear Integral ◽  
Split Technique ◽  
Linear Integral Equations

The radial basis function (RBF) method, especially the multiquadric (MQ) function, was introduced in solving linear integral equations. The procedure of MQ method includes that the unknown function was firstly expressed in linear combination forms of RBFs, then the integral equation was transformed into collocation matrix of RBFs, and finally, solving the matrix equation and an approximation solution was obtained. Because of the superior interpolation performance of MQ, the method can acquire higher precision with fewer nodes and low computations which takes obvious advantages over thin plate splines (TPS) method. In implementation, two types of integration schemes as the Gauss quadrature formula and regional split technique were put forward. Numerical results showed that the MQ solution can achieve accuracy of1E-5. So, the MQ method is suitable and promising for integral equations.

Pseudo-ARMA Model for Meta-Modeling Extrapolation

2007 ◽  
Author(s):  
Huageng Luo ◽  
Liping Wang ◽  
Don Beeson ◽  
Gene Wiggs
Keyword(s):  
Time Series ◽  
Radial Basis Function ◽  
Basis Function ◽  
Computational Cost ◽  
Arma Model ◽  
Time Space ◽  
Radial Basis ◽  
Meta Modeling ◽  
Input Variables ◽  
Arma Modeling

In spite of exponential growth in computing power, the enormous computational cost of complex and large-scale engineering design problems make it impractical to rely exclusively on original high fidelity simulation codes. Therefore, there has been an increasing interest in the use of fast executing meta-models to alleviate the computational cost required by slow and expensive simulation models — especially for optimization and probabilistic design. However, many state-of-the-art meta-modeling techniques, such as Radial Basis Function (RBF), Gaussian Process (GP), and Kriging can only make good predictions in the case of interpolation. Their ability for extrapolation is not impressive since the models are mathematically constructed for interpolations. Although Multivariate Adaptive Regression Splines (MARS) and Artificial Neural Network (ANN) have been tried for extrapolation problems (forecasting), the results do not always meet accuracy requirements. The autoregressive moving-average (ARMA) model is a popular time series modeling and forecasting tool. It has been widely used in many engineering applications in which all the inputs and outputs are time dependent. Many researchers have tried to extend the time series ARMA modeling technique into so-called spatial ARMA modeling or time-space ARMA modeling. However, the time-space ARMA modeling requires extensive computation in grid data generation as well as in model building, particularly for high dimensional problems. In this paper, a pseudo-ARMA approach is proposed to strengthen meta-modeling extrapolation capability. Each input is randomly sampled at a given mean value and distribution range to form a pseudo-time series. The output variables are evaluated based on input variables, which formulate output variable pseudo time series. The pseudo-ARMA model is built based on the pseudo input and output time series. Using the constructed pseudo-ARMA model, and new input variables generated with extended distribution parameters, such as distribution means and distribution ranges, the output variables can be evaluated to achieve extrapolations. Several numerical examples are presented to demonstrate the proposed approach. The results are compared with Radial Basis Function (RBF) meta-modeling results for both interpolation and extrapolation.

scholarly journals A Hybrid Shapeless Radial Basis Function Applied With the Dual Reciprocity Boundary Element Method

2021 ◽  
Vol 20 ◽  
pp. 159-170
Author(s):  
Krittidej Chanthawara ◽  
Sayan Kaennakham
Keyword(s):  
Boundary Element Method ◽  
Radial Basis Function ◽  
Boundary Element ◽  
Basis Function ◽  
Alternative Scheme ◽  
Good Shape ◽  
Radial Basis ◽  
The Right ◽  
Solution Accuracy ◽  
Element Method

The so-called Dual Reciprocity Boundary Element Method (DRBEM) has been a popular alternative scheme designed to alleviate problems encountered when using the traditional BEM for numerically solving engineering problems that are described by PDEs. The method starts with writing the right-hand-side of Poisson equation as a summation of a pre-chosen multivariate function known as ‘Radial Basis Function (RBF)’. Nevertheless, a common undesirable feature of using RBFs is the appearance of the so-called ‘shape parameter’ whose value greatly affects the solution accuracy. In this work, a new form of RBF containing no shape (so that it can be called ‘shapefree/shapeless’) is invented, proposed and applied in conjunction with DRBEM is validated numerically. The solutions obtained are compared against both exact ones and those presented in literature where appropriate, for validation. It is found that reasonably and comparatively good approximated solutions of PDEs can still be obtained without the difficulty of choosing a good shape for RBF used.

Sign in / Sign up

Export Citation Format

Share Document