scholarly journals A well-conditioned and efficient implementation of dual reciprocity method for Poisson equation

2021 ◽  
Vol 6 (11) ◽  
pp. 12560-12582
Author(s):  
Suliman Khan ◽  
◽  
M. Riaz Khan ◽  
Aisha M. Alqahtani ◽  
Hasrat Hussain Shah ◽  
...  
Keyword(s):  
Radial Basis Functions ◽  
Poisson Equation ◽  
Large Scale ◽  
Basis Functions ◽  
Dual Reciprocity Method ◽  
Reciprocity Method ◽  
Interpolation Matrix ◽  
Large Scale Problems ◽  
Radial Basis ◽  
Governing Differential Equation

<abstract><p>One of the attractive and practical techniques to transform the domain integrals to equivalent boundary integrals is the dual reciprocity method (DRM). The success of DRM relies on the proper treatment of the non-homogeneous term in the governing differential equation. For this purpose, radial basis functions (RBFs) interpolations are performed to approximate the non-homogeneous term accurately. Moreover, when the interpolation points are large, the global RBFs produced dense and ill-conditioned interpolation matrix, which poses severe stability and computational issues. Fortunately, there exist interpolation functions with local support known as compactly supported radial basis functions (CSRBFs). These functions produce a sparse and well-conditioned interpolation matrix, especially for large-scale problems. Therefore, this paper aims to apply DRM based on multiquadrics (MQ) RBFs and CSRBFs for evaluation of the Poisson equation, especially for large-scale problems. Furthermore, the convergence analysis of DRM with MQ and CSRBFs is performed, along with error estimate and stability analysis. Several experiments are performed to ensure the well-conditioned, efficient, and accurate behavior of the CSRBFs compared to the MQ-RBFs, especially for large-scale interpolation points.</p></abstract>

Some comments on the use of Radial Basis Functions in the Dual Reciprocity Method

1998 ◽  
Vol 21 (2) ◽  
pp. 141-148 ◽  
Author(s):  
M. A. Golberg ◽  
C. S. Chen ◽  
H. Bowman ◽  
H. Power
Keyword(s):  
Radial Basis Functions ◽  
Basis Functions ◽  
Dual Reciprocity Method ◽  
Reciprocity Method ◽  
Radial Basis

Some comments on the use of radial basis functions in the dual reciprocity method

1998 ◽  
Vol 22 (1) ◽  
pp. 61-69 ◽  
Author(s):  
M. A. Golberg ◽  
C. S. Chen ◽  
H. Bowman ◽  
H. Power
Keyword(s):  
Radial Basis Functions ◽  
Basis Functions ◽  
Dual Reciprocity Method ◽  
Reciprocity Method ◽  
Radial Basis

Solving PDEs with radial basis functions

Acta Numerica ◽  
2015 ◽  
Vol 24 ◽  
pp. 215-258 ◽  
Author(s):  
Bengt Fornberg ◽  
Natasha Flyer
Keyword(s):  
Radial Basis Functions ◽  
Finite Differences ◽  
Large Scale ◽  
Numerical Approach ◽  
Basis Functions ◽  
Integration Time ◽  
Small Scale ◽  
Major Step ◽  
Mesh Free ◽  
Radial Basis

Finite differences provided the first numerical approach that permitted large-scale simulations in many applications areas, such as geophysical fluid dynamics. As accuracy and integration time requirements gradually increased, the focus shifted from finite differences to a variety of different spectral methods. During the last few years, radial basis functions, in particular in their ‘local’ RBF-FD form, have taken the major step from being mostly a curiosity approach for small-scale PDE ‘toy problems’ to becoming a major contender also for very large simulations on advanced distributed memory computer systems. Being entirely mesh-free, RBF-FD discretizations are also particularly easy to implement, even when local refinements are needed. This article gives some background to this development, and highlights some recent results.

scholarly journals An Approach Towards Generating Surrogate Models by Using RBFN With a Priori Bias

2014 ◽  
Author(s):  
Kaveh Amouzgar ◽  
Niclas Stromberg
Keyword(s):  
Radial Basis Functions ◽  
Large Scale ◽  
A Priori ◽  
Surrogate Models ◽  
Basis Functions ◽  
Normal Equation ◽  
A Posteriori ◽  
Mathematical Functions ◽  
Radial Basis ◽  
Price Functions

In this paper, an approach to generate surrogate models constructed by radial basis function networks (RBFN) with a priori bias is presented. RBFN as a weighted combination of radial basis functions only, might become singular and no interpolation is found. The standard approach to avoid this is to add a polynomial bias, where the bias is defined by imposing orthogonality conditions between the weights of the radial basis functions and the polynomial basis functions. Here, in the proposed a priori approach, the regression coefficients of the polynomial bias are simply calculated by using the normal equation without any need of the extra orthogonality prerequisite. In addition to the simplicity of this approach, the method has also proven to predict the actual functions more accurately compared to the RBFN with a posteriori bias. Several test functions, including Rosenbrock, Branin-Hoo, Goldstein-Price functions and two mathematical functions (one large scale), are used to evaluate the performance of the proposed method by conducting a comparison study and error analysis between the RBFN with a priori and a posteriori known biases. Furthermore, the aforementioned approaches are applied to an engineering design problem, that is modeling of the material properties of a three phase spherical graphite iron (SGI). The corresponding surrogate models are presented and compared.

Axial Compressor Rotor Optimization Using a Novel Ensemble of Surrogates-Based Infill Criterion

2017 ◽  
Author(s):  
Jan Kamenik ◽  
Michele Stramacchia ◽  
David J. J. Toal ◽  
Andy J. Keane ◽  
Ron Bates
Keyword(s):  
Radial Basis Functions ◽  
Large Scale ◽  
Axial Compressor ◽  
Correlation Coefficients ◽  
Basis Functions ◽  
Prediction Errors ◽  
Compressor Rotor ◽  
Radial Basis ◽  
Ensemble Of Surrogates ◽  
Update Strategy

A novel infill criterion for so-called ensemble of surrogates-based optimization is proposed and applied in practice for an aerodynamic compressor rotor design optimization. The ensemble uses a combined approach based on different radial basis functions and aims to reduce prediction errors through weighted linear combinations of radial basis functions. The update strategy uses a new hybrid custom metric termed α, which incorporates information about each surrogate’s local agreement through correlation coefficients and also information about the global accuracy of each ensemble combination through the root-mean-square error. Surrogate models are searched using a hybrid optimizer, i.e., with a genetic algorithm and sequential quadratic programming, and proposed update points are evaluated using the high-fidelity black box function. The results are compared with established optimization approaches and the best design is analyzed further in terms of the flow physics. Results show that α-based ensemble of surrogates approaches are particularly efficient for large-scale cases, where other types of surrogates such as Kriging models are onerous to construct.

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