# Cell-based Maximum Entropy Approximants for Three Dimensional Domains: Application in Large Strain Elastodynamics using the Meshless Total Lagrangian Explicit Dynamics Method

## TL;DR

This paper introduces Cell-based Maximum Entropy (CME) approximants for 3D domains, enhancing meshless elastodynamics simulations by improving accuracy, stability, and boundary condition enforcement in large strain problems.

## Contribution

The paper develops a novel meshfree approximation method combining maximum entropy properties with compact support, specifically tailored for 3D elastodynamics applications.

## Key findings

- CME provides accurate boundary condition imposition without complex constraints.
- The method maintains numerical stability for large time steps in explicit dynamics.
- CME outperforms traditional meshless methods in large-scale 3D elastodynamics simulations.

## Abstract

We present the Cell-based Maximum Entropy (CME) approximants in E3 space by constructing the smooth approximation distance function to polyhedral surfaces. CME is a meshfree approximation method combining the properties of the Maximum Entropy approximants and the compact support of element-based interpolants. The method is evaluated in problems of large strain elastodynamics for three-dimensional (3D) continua using the well-established Meshless Total Lagrangian Explicit Dynamics (MTLED) method. The accuracy and efficiency of the method is assessed in several numerical examples in terms of computational time, accuracy in boundary conditions imposition, and strain energy density error. Due to the smoothness of CME basis functions, the numerical stability in explicit time integration is preserved for large time step. The challenging task of essential boundary conditions imposition in non-interpolating meshless methods (e.g., Moving Least Squares) is eliminated in CME due to the weak Kronecker-delta property. The essential boundary conditions are imposed directly, similar to the Finite Element Method. CME is proven a valuable alternative to other meshless and element-based methods for large-scale elastodynamics in 3D.