# Higher-order meshing of implicit geometries - part I: Integration and interpolation in cut elements

## TL;DR

This paper introduces a higher-order meshing strategy for implicit geometries using level-set methods, enabling accurate integration and interpolation in cut elements without restrictions on background mesh structure.

## Contribution

It presents an automatic meshing approach for cut elements with higher-order interface meshing and sub-element decomposition for precise integration.

## Key findings

- Effective higher-order interface meshing in 2D and 3D.
- Accurate integration within cut elements.
- Handling of corners and edges in implicit geometries.

## Abstract

An accurate implicit description of geometries is enabled by the level-set method. Level-set data is given at the nodes of a higher-order background mesh and the interpolated zero-level sets imply boundaries of the domain or interfaces within. The higher-order accurate integration of elements cut by the zero-level sets is described. The proposed strategy relies on an automatic meshing of the cut elements. Firstly, the zero-level sets are identified and meshed by higher-order interface elements. Secondly, the cut elements are decomposed into conforming sub-elements on the two sides of the zero-level sets. Any quadrature rule may then be employed within the sub-elements. The approach is described in two and three dimensions without any requirements on the background meshes. Special attention is given to the consideration of corners and edges of the implicit geometries.