A Trace--Logarithmic Variational Functional for Equidistribution and Alignment in Moving Mesh Adaptation

Abstract

Existing variational mesh functionals often require an empirical equidistribution--alignment weight or use a strongly nonlinear density. We propose a calibrated trace--logarithmic functional in the inverse pullback tensor $ A= J^{-1}M^{-1}J^{-T}$,with the logarithmic coefficient fixed by the local target $A=θI$. We prove strict convexity in the symmetric positive-definite (SPD) tensor variable, inverse-Jacobian polyconvexity and coercivity, and weak minimizer existence under explicit admissibility assumptions. The density also satisfies the standard coercivity condition for semi-discrete physical-coordinate mesh nonsingularity; the geometric discretization yields a compact computational-coordinate residual for the reported direct-secant implementation. Numerical tests for metric-induced mesh adaptation, a Burgers benchmark, a Rayleigh--Taylor instability simulation, and a matched high-anisotropy stress test show robust mesh redistribution; in the stress test, the trace--logarithmic runs complete over a broader fixed-protocol range of the compression parameter than the scanned Huang benchmark.