Morley Type Virtual Element Method for Von K\'{a}rm\'{a}n Equations

TL;DR

This paper develops and analyzes a Morley type virtual element method for the von Kármán equations, providing error estimates, existence and uniqueness results, and numerical validation for bending thin elastic plates.

Contribution

It introduces a nonconforming virtual element approach for the von Kármán equations with rigorous error analysis and practical solution procedures.

Findings

Error estimates in energy, H^1, and L^2 norms are established.

Existence and uniqueness of discrete solutions are proven.

Numerical results confirm theoretical accuracy.

Abstract

This paper analyses the nonconforming Morley type virtual element method to approximate a regular solution to the von K\'{a}rm\'{a}n equations that describes bending of very thin elastic plates. Local existence and uniqueness of a discrete solution to the non-linear problem is discussed. A priori error estimate in the energy norm is established under minimal regularity assumptions on the exact solution. Error estimates in piecewise $H^1$ and $L^2$ norm are also derived. A working procedure to find an approximation for the discrete solution using Newtons method is discussed. Numerical results that justify theoretical estimates are presented.