A meshfree unification: reproducing kernel peridynamics

Abstract

This paper is the first investigation establishing the link between the meshfree state-based peridynamics method and other meshfree methods, in particular with the moving least squares reproducing kernel particle method (RKPM). It is concluded that the discretization of state-based peridynamics leads directly to an approximation of the derivatives that can be obtained from RKPM. However, state-based peridynamics obtains the same result at a significantly lower computational cost which motivates its use in large-scale computations. In light of the findings of this study, an update to the method is proposed such that the limitations regarding application of boundary conditions and the use of non-uniform grids are corrected by using the reproducing kernel approximation.

Appendix: Least squares based meshfree methods

The LS approximation is obtained as a special case of Eq. (11) when using the weight function $\omega =1$, and when considering that point $\mathbf {x}$ is fixed in space which leads to constant coefficients $\mathbf {a}(\mathbf {x})$. The fixed points in the domain coincide with the grid points of the discretization $\mathbf {x} \equiv \mathbf {X}_I$, so for convenience we can drop the prime in the notation of the arbitrary coordinate $\mathbf {x}' \equiv \mathbf {X}$:

$$\begin{aligned} f^h(\mathbf {X}) = \mathbf {p}^\mathrm {T}(\mathbf {X}_I-\mathbf {X})\mathbf {a}(\mathbf {X}_I) \end{aligned}$$

(49)

where the constant coefficients $\mathbf {a}(\mathbf {X}_I)$ are given by Eq. (10) for the particular conditions of a LS approximation:

$$\begin{aligned} \mathbf {a}(\mathbf {X}_I) = \left[ \,\,\sum _{J=1}^{L}\mathbf {p}(\mathbf {X}_I-\mathbf {X}_J)\mathbf {p}^\mathrm {T}(\mathbf {X}_I-\mathbf {X}_J)\right] ^{-1} \sum _{J=1}^{L}\mathbf {p}(\mathbf {X}_I-\mathbf {X}_J) f_J \end{aligned}$$

(50)

Then, the approximations to the derivatives of function $f$ are simple to obtain since the coefficients are constant:

$$\begin{aligned} \frac{\partial f^h(\mathbf {X})}{\partial X} = \frac{\partial \mathbf {p}^\mathrm {T}(\mathbf {X}_I-\mathbf {X})}{\partial X} \mathbf {a}(\mathbf {X}_I) \end{aligned}$$

(51)

As was already noted in Sect. 2, the above approximation method based on LS can be used to obtain several fundamental methods in numerical analysis: (1) classic and generalized finite difference formulas; (2) discretized Lanczos derivatives; (3) Savitzky-Golay filters for uniform and non-uniform grids.

Finite difference methods

The classic and generalized finite difference formulas for uniform and arbitrarily spaced grids, respectively, are obtained when the number of points $L$ in the local approximation matches the number of monomials $m$ in the polynomial basis. The constant coefficients of the polynomial basis $\mathbf {a}(\mathbf {X}_I)$ can easily be determined according to Eq. (50). Fornberg [26] proposed an algorithm to calculate these coefficients, that are usually called “weights” in the finite differences literature despite being completely different from the weight function $\omega$ in WLS or MLS/RKPM.

Note that the polynomial approximation that is constructed to obtain the finite difference formulas interpolates every point $J$ of the domain because the system is fully determined. This is equivalent to finding the approximation function $f^h$ by overfitting the discretized points using a polynomial approximation, i.e. the polynomial is forced to pass through every discretized point. For this reason, finite difference formulas have a large sensitivity to noise.

Lanczos derivative

Lanczos [27] was perhaps the first to notice that it was possible to approximate the derivatives of a function in a different way:

$$\begin{aligned} D_h f(X) = \lim _{h \rightarrow 0^+} \frac{3}{2h^3}\int _{-h}^h uf\left( X+u\right) \mathrm {d}u \end{aligned}$$

(52)

in what is nowadays called the Lanczos derivative.

Remarkably, Lanczos also proposed a discretized version of the above derivative for the case of having a one-dimensional function $f(X)$ that is evaluated at uniformly spaced points at a distance $\varDelta X$ of each other:

$$\begin{aligned} f_J = f\left( X_J\right) \;,\quad J=-\frac{L-1}{2},\ldots ,\frac{L-1}{2} \end{aligned}$$

(53)

with $X_J = X_0 + J\cdot \varDelta X$, and $L$ being the odd number of discretization points.

Lanczos used a quadratic polynomial, Eq. (3) with $k=2$ (or $m=3$), and minimized the least squares approximation so that he could obtain the coefficients of that polynomial. This means that he used the simplest version of Eq. (50) to determine those coefficients. Calculating the coefficients and determining the derivatives from Eq. (51), Lanczos found simple formulas for the approximation of the first derivative of function $f$ due to the use of a uniform grid, as can be seen in Table 1.

Table 1 Lanczos numerical differentiators [27]

These formulas can be generalized for any number of grid points $L$ as:

$$\begin{aligned} \frac{\partial f_0}{\partial X_0} = \frac{3}{\varDelta X}\sum _{J=1}^{M}\frac{J\left( f_{J^+} - f_{J^-}\right) }{M\left( M+1\right) \left( 2M+1\right) } \;,\quad M=\dfrac{L-1}{2} \end{aligned}$$

(54)

Savitzky–Golay filters

After the discovery of the Lanczos derivative, with the seminal contribution of Savitzky and Golay [28], it was possible to generalize this procedure for polynomials of higher degree and for multidimensions by using Eqs. (49) and (50). The impact of this discovery was tremendous, not only in mathematical and statistical analysis but also in many other fields because it gave the ability to extract smooth data from noisy experimental measures. Moreover, it was also found that these derivatives when plotted in frequency space suppress high-frequency signals (noise) which opened the way to the design of digital filters [48]. In digital filters literature the differentiation formulas using $k=2$ presented in Table 1 are called “low-noise Lanczos differentiators”, and if $k=4$ they are called “super low-noise Lanczos differentiators”.

From this brief historical perspective it is obvious that subsequent discoveries followed natural steps, as was shown in Sect. 2: (1) generalizing the Savitzky-Golay filters to arbitrarily spaced grids; (2) using WLS; (3) and finally using MLS. Thus, it is not surprising that MLS/RKPM represents the general case that encompasses all the meshfree methods considered in this article.