Analytical expression of RKPM shape functions

Abstract

In this paper, we derive an analytical expression for reproducing kernel particle method (RKPM) shape functions. Based on this, we propose a necessary and sufficient stability condition for general RKPM in arbitrary function space, and illustrate with degenerate cases. By selecting proper basis vectors and the support of the kernel functions, we demonstrate that the RKPM framework allows generating many kinds of shape functions, including the Lagrangian, B-spline and NURBS shape functions.

Appendices

Appendix A: Illustration for Lemma 1 with $q=2,n=3$

When $q=2, n=3$, we have

$$\begin{aligned}&\varvec{A}=P_1 Q^T_1+P_2 Q^T_2+P_3 Q^T_3, P_I=[P^1_I,P^2_I]^T,\nonumber \\&Q_I=[Q^1_I,Q^2_I]^T,I=1,2,3. \end{aligned}$$

(60)

The determinant is

$$\begin{aligned} \mathrm {det}(\varvec{A})=A_{1,1} A_{2,2} - A_{1,2} A_{2,1}, \end{aligned}$$

(61)

where

$$\begin{aligned} A_{i,j}=\sum _{I=1}^3 P^i_I Q^j_I. \end{aligned}$$

(62)

Expand (61) and assemble the terms according to the same order of components. For example, we take $P^1_1 Q^j_1 (j=1,2)$ from $A_{1,j}$ and take $P^2_2 Q^j_2 (j=1,2)$ from $A_{2,j}$. The summation is

$$\begin{aligned}&(P^1_1 Q^1_1)(P^2_2 Q^2_2)-(P^1_1 Q^2_1)(P^2_2 Q^1_2)=\left| \begin{array}{cc} P^1_1 Q^1_1 &{} P^1_1 Q^2_1 \\ P^2_2 Q^1_2 &{} P^2_2 Q^2_2 \end{array}\right| \nonumber \\&\quad =P^1_1 P^2_2\left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_2 &{} Q^2_2 \end{array}\right| . \end{aligned}$$

(63)

Thus, (61) becomes

$$\begin{aligned} \mathrm {det}(\varvec{A})= & {} P^1_1 P^2_1\left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_1 &{} Q^2_1 \end{array}\right| + P^1_1 P^2_2\left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_2 &{} Q^2_2 \end{array}\right| \nonumber \\&+ P^1_1 P^2_3\left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_3 &{} Q^2_3 \end{array}\right| \nonumber \\&+ P^1_2 P^2_1\left| \begin{array}{cc} Q^1_2 &{} Q^2_2 \\ Q^1_1 &{} Q^2_1 \end{array}\right| \nonumber \\&+ P^1_2 P^2_2\left| \begin{array}{cc} Q^1_2 &{} Q^2_2 \\ Q^1_2 &{} Q^2_2 \end{array}\right| + P^1_2 P^2_3\left| \begin{array}{cc} Q^1_2 &{} Q^2_2 \\ Q^1_3 &{} Q^2_3 \end{array}\right| \nonumber \\&+ P^1_3 P^2_1\left| \begin{array}{cc} Q^1_3 &{} Q^2_3 \\ Q^1_1 &{} Q^2_1 \end{array}\right| + P^1_3 P^2_2\left| \begin{array}{cc} Q^1_3 &{} Q^2_3 \\ Q^1_2 &{} Q^2_2 \end{array}\right| \nonumber \\&+ P^1_3 P^2_3\left| \begin{array}{cc} Q^1_3 &{} Q^2_3 \\ Q^1_3 &{} Q^2_3 \end{array}\right| . \end{aligned}$$

(64)

Obviously, we have

$$\begin{aligned} \left| \begin{array}{cc} Q^1_I &{} Q^2_I \\ Q^1_I &{} Q^2_I \end{array}\right| =0, I=1,2,3. \end{aligned}$$

(65)

Then, we combine the terms corresponding to the same index pair $\mathrm {det}([(\varvec{Q}_I)^T; (\varvec{Q}_J)^T])$,

$$\begin{aligned} \mathrm {det}(\varvec{A})= & {} (P^1_1 P^2_2-P^1_2 P^2_1)\left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_2 &{} Q^2_2 \end{array}\right| \nonumber \\&+ (P^1_2 P^2_3-P^1_3 P^2_2)\left| \begin{array}{cc} Q^1_2 &{} Q^2_2 \\ Q^1_3 &{} Q^2_3 \end{array}\right| \nonumber \\&+ (P^1_1 P^2_3-P^1_3 P^2_1)\left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_3 &{} Q^2_3 \end{array}\right| \nonumber \\= & {} \left| \begin{array}{cc} P^1_1 &{} P^1_2 \\ P^2_1 &{} P^2_2 \end{array}\right| \left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_2 &{} Q^2_2 \end{array}\right| \nonumber \\&+ \left| \begin{array}{cc} P^1_2 &{} P^1_3 \\ P^2_2 &{} P^2_3 \end{array}\right| \left| \begin{array}{cc} Q^1_2 &{} Q^2_2 \\ Q^1_3 &{} Q^2_3 \end{array}\right| + \left| \begin{array}{cc} P^1_1 &{} P^1_3 \\ P^2_1 &{} P^2_3 \end{array}\right| \left| \begin{array}{cc} Q^1_1 &{} Q^2_1 \\ Q^1_3 &{} Q^2_3 \end{array}\right| .\nonumber \\ \end{aligned}$$

(66)

Appendix B: Derivatives of the RKPM analytical expression

We can have direct differentiation of the shape functions and get the corresponding analytical expression:

$$\begin{aligned} \dfrac{\partial }{\partial \alpha }\Psi _I(\varvec{x})= & {} \dfrac{\sum _{1\le l_1<l_2<\cdots<l_{q-1} \le n}H_{l_1 l_2 \cdots l_{q-1} I} \left( \dfrac{\partial }{\partial \alpha } (H_{l_1 l_2 \cdots l_{q-1} p}) \phi _{l_1}\phi _{l_2}\cdots \phi _{l_{q-1}}\phi _I+H_{l_1 l_2 \cdots l_{q-1} p} \dfrac{\partial }{\partial \alpha } (\phi _{l_1}\phi _{l_2}\cdots \phi _{l_{q-1}}\phi _I) \right) }{\sum _{1\le l_1<l_2<\cdots<l_{q}\le n}(H_{l_1 l_2 \cdots l_{q}})^2 \phi _{l_1}\phi _{l_2}\cdots \phi _{l_{q}}}\\ \nonumber&-\dfrac{\left( \sum _{1\le l_1<l_2<\cdots<l_{q-1} \le n}H_{l_1 l_2 \cdots l_{q-1} I} H_{l_1 l_2 \cdots l_{q-1} p} \phi _{l_1}\phi _{l_2}\cdots \phi _{l_{q-1}}\phi _I\right) \left( \sum _{1\le l_1<l_2<\cdots<l_{q}\le n}(H_{l_1 l_2 \cdots l_{q}})^2 \dfrac{\partial }{\partial \alpha }(\phi _{l_1}\phi _{l_2}\cdots \phi _{l_{q}}) \right) }{\left( \sum _{1\le l_1<l_2<\cdots <l_{q}\le n}(H_{l_1 l_2 \cdots l_{q}})^2 \phi _{l_1}\phi _{l_2}\cdots \phi _{l_{q}}\right) ^2}. \end{aligned}$$

(67)

with

$$\begin{aligned} \dfrac{\partial }{\partial \alpha } H_{l_1 l_2 \ldots l_{q-1} p}=\mathrm {det}\left( \left[ \varvec{p}_{l_1},\varvec{p}_{l_2},\ldots ,\varvec{p}_{l_{q-1}}, \dfrac{\partial }{\partial \alpha } \varvec{p}(\varvec{x}) \right] \right) . \end{aligned}$$

(68)

$\alpha$ is an arbitrary variable.

If kernel functions $\phi _I$ are constant, (67) reduces to

$$\begin{aligned}&\dfrac{\partial }{\partial \alpha }\Psi _I(\varvec{x})\nonumber \\&\quad =\dfrac{\sum _{1\le l_1<l_2<\cdots<l_{q-1} \le n}H_{l_1 l_2 \ldots l_{q-1} I} \dfrac{\partial }{\partial \alpha } (H_{l_1 l_2 \ldots l_{q-1} p}) \phi _{l_1}\phi _{l_2}\ldots \phi _{l_{q-1}}\phi _I}{\sum _{1\le l_1<l_2<\cdots <l_{q}\le n}(H_{l_1 l_2 \ldots l_{q}})^2 \phi _{l_1}\phi _{l_2}\ldots \phi _{l_{q}}}. \end{aligned}$$

(69)