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Mesh-free Solution of ADR Equations: A Comparison Between the Application of Hybrid and Standard Radial Basis Functions

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Abstract

Three types of radial basis functions comprising two standard radial basis functions (RBF) and a hybrid Gaussian-cubic (HGC) kernel have been adopted in this paper to solve three-dimensional advection–dispersion-reaction (ADR) equations by an RBF-based mesh-free method. In this context, radial point interpolation method (RPIM) is implemented as a mesh-free discretization tool and the weak form of governing equation is attained by Galerkin weighted residual technique. Since the utilized RBFs contain shape parameter and weight coefficients (in the case of HGC kernel) as the influential parameters on the results accuracy, optimal values of these parameters are assessed via an optimization algorithm based on Particle Swarm Optimization technique. Accuracy/adequacy of the utilized RBFs as well as their stability have been elaborated by solving some examples at the end of the paper. Since the main aim of the present paper is the comparison between performance of different RBFs in ADR equation solution, simple examples with known analytical solutions have been utilized in a unitary cube domain. Accordingly, by solving time-dependent and time-independent problems, it has been concluded that all considered RBFs can be successfully used in RPIM to render results with acceptable accuracy. Besides, the hybrid RBF ended up with the highest stability in comparison with the other RBFs.

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Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Shiraz University of Technology, Shiraz, Iran

    Z. Mohammadinezhad, S. M. Binesh & A. A. Hekmatzadeh

Authors
  1. Z. Mohammadinezhad
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  2. S. M. Binesh
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  3. A. A. Hekmatzadeh
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Correspondence to S. M. Binesh.

Appendix

Appendix

A brief description of the PSO algorithm is given here. In the PSO algorithm, each particle (i.e., particle \(j\)) carries three types of data: (1) position (\({\varvec{x}}^{j}\)), (2) velocity (\(v^{j}\)), and (3) the personal best, which contains the best position (\({\varvec{x}}^{j,best}\)) and the best objective value. Here the objective function is a minimum of the \(L_{2}\) norm of the error which is the function of shape parameter (i.e.,\(s\)) and weight coefficients (i.e., \(\beta_{1}\) and \(\beta_{2}\)). According to the self-organization rules in each iteration for particle \(j\), the velocity and position assessed as:

$$v^{j} (i + 1) = w * v^{j} \left[ i \right] + c_{1} r_{1} ({\varvec{x}}^{{j,{\text{best}}}} (i) - x^{j} (i)) + c_{2} r_{2} ({\varvec{x}}^{{{\text{gbest}}}} (i) - {\varvec{x}}^{j} (i))$$
(50)
$${\varvec{x}}^{j} (i + 1) = {\varvec{x}}^{j} (i) + v^{j} (i + 1)$$
(51)

where \({\varvec{x}}^{{{\text{gbest}}}}\) is the best global position. By comparing the position and cost function obtained for each particle with the best position and best cost function, the cost function will be updated. In the same way, the global best will be updated, which will be the solution.

The flowchart of PSO algorithm is presented in Fig. 14. 

Fig. 14
figure 14

Flow chart of PSO algorithm

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Mohammadinezhad, Z., Binesh, S.M. & Hekmatzadeh, A.A. Mesh-free Solution of ADR Equations: A Comparison Between the Application of Hybrid and Standard Radial Basis Functions. Iran J Sci Technol Trans Civ Eng 48, 589–606 (2024). https://doi.org/10.1007/s40996-023-01176-w

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