A New Adaptive Algorithm for the Optimal Distribution of Computational Centers in the Meshless Multiquadric Method

The computational centers in the multiquadric radial basis functions meshless method have high adaptability considering the lack of geometric and physical connection between the centers. In this research, a new adaptive algorithm is proposed based on the gradients of the physical variables of the problem with the aim of creating an optimal distribution. The resulted adaptive distribution generated by this algorithm improves significantly the accuracy and speed of the multiquadric method compared to the uniform distribution in steady and unsteady problems. In this approach, firstly, the domains with low and high physical variations are identified in a known time step, then the number of computational centers decreases and increases in these areas, respectively. Thus, the centers will be distributed more compact where needed and will be eliminated where not. Facing another important challenge of the multiquadric method, i.e. determining the optimal shape parameter, a simple and efficient method is introduced in such a way that there is no need to optimize the shape parameter at each time step and the computational costs are controlled. Finally, the effectiveness of the proposed method is shown by solving examples of diffusion, convection and convection-diffusion equations. The results are compared to their uniform distributions by measuring their efficiency and to the exact solution by evaluating the accuracy.