# Refined Meshless Local Strong Form solution of Cauchy-Navier equation on an irregular domain

## TL;DR

This paper introduces a meshless local strong form method for solving the Cauchy-Navier equation in linear elasticity on irregular domains, demonstrating flexibility in node placement and refinement techniques with promising accuracy and convergence.

## Contribution

It presents a novel meshless local strong form approach that handles irregular domains and node placement flexibility for solving elasticity problems.

## Key findings

- Accurate solutions for irregular domain problems.
- Effective node placement and refinement strategies.
- Comparable or improved convergence rates.

## Abstract

This paper considers a numerical solution of a linear elasticity problem, namely the Cauchy-Navier equation, using a strong form method based on a local Weighted Least Squares (WLS) approximation. The main advantage of the employed numerical approach, also referred to as a Meshless Local Strong Form method, is its generality in terms of approximation setup and positions of computational nodes. In this paper, flexibility regarding the nodal position is demonstrated through two numerical examples, i.e. a drilled cantilever beam, where an irregular domain is treated with a relatively simple nodal positioning algorithm, and a Hertzian contact problem, where again, a relatively simple h-refinement algorithm is used to extensively refine discretization under the contact area. The results are presented in terms of accuracy and convergence rates, using different approximations and refinement setups, namely Gaussian and monomial based approximations, and a comparison of execution time for each block of the solution procedure.