# Robust and efficient validation of the linear hexahedral element

## TL;DR

This paper introduces a robust and efficient method for validating linear hexahedral mesh elements by computing Bezier coefficients of the Jacobian determinant, enabling fast validation of millions of elements per second.

## Contribution

It presents a simplified, computationally efficient algorithm for hexahedral validity checking based on Jacobian Bezier coefficients, improving upon previous curvilinear methods.

## Key findings

- Validates about 6 million hexahedra per second on a single core
- Uses only 20 Jacobian determinants to compute 27 Bezier coefficients
- Provides a reproducible method with straightforward implementation

## Abstract

Checking mesh validity is a mandatory step before doing any finite element analysis. If checking the validity of tetrahedra is trivial, checking the validity of hexahedral elements is far from being obvious. In this paper, a method that robustly and efficiently compute the validity of standard linear hexahedral elements is presented. This method is a significant improvement of a previous work on the validity of curvilinear elements. The new implementation is simple and computationally efficient. The key of the algorithm is still to compute B\'ezier coefficients of the Jacobian determinant. We show that only 20 Jacobian determinants are necessary to compute the 27 B\'ezier coefficients. Those 20 Jacobians can be efficiently computed by calculating the volume of 20 tetrahedra. The new implementation is able to check the validity of about 6 million hexahedra per second on one core of a personal computer. Through the paper, all the necessary information is provided that allow to easily reproduce the results, \ie write a simple code that takes the coordinates of 8 points as input and outputs the validity of the hexahedron.