# Stabilized Conforming Nodal Integration (SCNI)

## 0. Post Details

- Reference: [Chen et al., A stabilized conforming nodal integration for Galerkin mesh-free methods, 2001.](https://onlinelibrary.wiley.com/doi/abs/10.1002/1097-0207%2820010120%2950%3A2%3C435%3A%3AAID-NME32%3E3.0.CO%3B2-A)
- Post by: Jedd Yang
- Date: 2025-05-12
- Keywords: Computational Mechanics, SCNI

## 1. Mesh-free Approximation
2 most commonly used approximation theories in mesh-free methods are the MLS approximation in the EFG method by Belytschko 1994, and RK approximation by RKPM in Liu 1995. They lead to identical approximation when monomial Basis Functions are used. Here, we consider RK.

## 2. Approximation Function

For RK approximation,

$u^h = \sum^{NP}_{I = 1} \Psi_I (\textbf{x})d_I$

$\Psi_I (\textbf{x}) = \textbf{H}^{[n]^{T}}(\textbf{0})\textbf{M}^{[n]^{-1}}(\textbf{x})\textbf{H}^{[n]}(\textbf{x} - \textbf{x}_I)w_a(\textbf{x} - \textbf{x}_I)$

## 3. Completeness (Consistency)

"Can your basis functions represent a straight line or a parabola exactly?"

- Regards ==basis functions==
- Higher-order completeness improves the accuracy of approximations.
- Ensuring convergence and proper behavior under rigid body motions and polynomial fields.

$\Psi_I (\textbf{x})$ satisfies n-th order completeness—exactly reproduce all polynomials up to degree n
$\sum^{NP}_{I = 1} \Psi_I (\textbf{x}) x_{1I}^p x_{2I}^q = x_{1}^p x_{2}^q, \forall p+q=0,...,n$
$\sum^{NP}_{I = 1} D_{ij}(\Psi_I (\textbf{x})) x_{1I}^p x_{2I}^q = D_{ij} (x_{1}^p x_{2}^q), \forall p+q=0,...,n$

$D_{ij} \equiv \partial^{i+j} / \partial x^{i}_{1} \partial x^{j}_{2}$

## 4. Boundary Condition

$\Psi_I (\textbf{x})$ does not bear Kronecker delta properties, therefore requires extra work on EBC imposition.

## 5. Exactness (Integration Constraints)

"Can your numerical method correctly integrate that straight line or parabola over a domain?"

- Regards ==numerical integration methods==
- Ensuring that numerical integration of weak form yields the exact values when integrating polynomial functions up to a desired degree over a domain.