In the NI-RPIM, radial basis functions (RBFs) augmented with polynomials are used to construct shape functions that possess the Delta function property.
Chebyshev polynomials are used as basis functions and Chebyshev Gauss Lobatto points are used as grid points.
Although in most cases continuous orthogonal polynomials are used as basis functions for the approximate solution of equations, recently discrete orthogonal polynomials have been noticed for solving stochastic differential equations [36] and in numerical fluid dynamics problems [13] because of the behavior and property of these polynomials [15].
Meanwhile a Fourier series as well as a set of Chebyshev polynomials are used as the basis functions for the main unknown fields.
Many kinds of polynomials are used for interpolation: classical polynomials, trigonometric polynomials, orthogonal polynomials (Tschebyscheff, Legendre, Laguerre), rational polynomials.
Legendre polynomials were used as basis functions.
Unlike the traditional discontinuous Galerkin methods, where either standard Lagrange finite element or hierarchical node-based basis functions are used to represent numerical polynomial solutions in each element, this DG method represents the numerical polynomial solutions using a Taylor series expansion at the centroid of the cell.
where b ij are the coefficients of the power basis that are used to determine the respective Bernoulli polynomials.
For example, an overabundance of freeform terms on a surface can lead to unnecessarily large freeform departures and when low-order basis functions are used to describe an optical surface, such as the Zernike polynomials used herein, freeform departure correlates to local slope.
The regression variables consist of the uncertain parameters θ α and the initial length of the cracks l 0. Polynomials of degree five are used as basis functions and determined using linear regression.
