A New Functional for Variational Mesh Generation and Adaptation Based on Equidistribution and Alignment Conditions

TL;DR

This paper introduces a new functional for variational mesh generation that combines equidistribution and alignment conditions into a single, parameterized condition, ensuring mesh quality and stability.

Contribution

It proposes a novel functional unifying key mesh adaptation conditions with a single parameter, and demonstrates its effectiveness through numerical examples.

Findings

Mesh element volumes and altitudes remain bounded away from zero.

The mesh trajectory stays nonsingular if initially so.

The new functional performs comparably to existing methods.

Abstract

A new functional is presented for variational mesh generation and adaptation. It is formulated based on combining the equidistribution and alignment conditions into a single condition with only one dimensionless parameter. The functional is shown to be coercive but not convex. A solution procedure using a discrete moving mesh partial differential equation is employed. It is shown that the element volumes and altitudes of a mesh trajectory of the mesh equation associated with the new functional are bounded away from zero and the mesh trajectory stays nonsingular if it is so initially. Numerical examples demonstrate that the new functional performs comparably as an existing one that is also based on the equidistribution and alignment conditions and known to work well but contains an additional parameter.