A geometric discretization and a simple implementation for variational mesh generation and adaptation

TL;DR

This paper introduces a straightforward geometric discretization method for variational mesh generation that preserves the functional's structure and simplifies implementation, enabling effective mesh adaptation and smoothing.

Contribution

It proposes a direct geometric discretization approach for variational mesh functionals that maintains geometric properties and simplifies computational implementation.

Findings

Preserves geometric structure of the continuous functional.

Allows simple and effective implementation of mesh adaptation.

Numerical examples demonstrate the method's effectiveness.

Abstract

We present a simple direct discretization for functionals used in the variational mesh generation and adaptation. Meshing functionals are discretized on simplicial meshes and the Jacobian matrix of the continuous coordinate transformation is approximated by the Jacobian matrices of affine mappings between elements. The advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional, which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional. Moreover, the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form, which allows a simple implementation of variational mesh generation and adaptation on computer. Since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods, the result in this work can be used to develop simple implementations of those methods. Numerical examples are given.