A Unified Variational Functional for Equidistribution and Alignment in Moving Mesh Adaptation

TL;DR

This work introduces a parameter-free variational moving-mesh functional built on an $A$-pullback, $\boldsymbol A = \boldsymbol J^{-1}\boldsymbol M^{-1}\boldsymbol J^{-T}$, combining a trace-based term with a $-\ln\det$ term to balance mesh size and anisotropy. The authors prove key analytical properties—scale invariance, polyconvexity, and geodesic convexity—and establish coercivity and the existence of minimizers, ensuring a well-posed mesh adaptation framework. They develop a concise geometric discretization and a gradient-flow-based moving-mesh algorithm with a Newton-Krylov solution strategy, along with discrete simplifications that yield compact, affine-structure gradients. Numerical experiments on function-induced meshes, Burgers’ equation, and Rayleigh–Taylor instability demonstrate robust equidistribution and alignment, competitive accuracy, and superior efficiency relative to parameterized or conventional approaches. The method provides a theoretically grounded, efficient, and adaptable tool for anisotropic mesh generation with potential extensions to broader element types and higher-order discretizations.

Abstract

Existing variational mesh functionals often suffer from strong nonlinearity or dependence on empirical parameters.We propose a new variational functional for adaptive moving mesh generation that enforces equidistribution and alignment through an $\boldsymbol A$-pullback formulation, where $\boldsymbol A=\boldsymbol J^{-1}\boldsymbol M^{-1}\boldsymbol J^{-T}$. The functional combines a trace-based term with a logarithmic determinant term, achieving balanced control of mesh size and anisotropy without empirical parameters. We establish coercivity, polyconvexity, existence of minimizers, and geodesic convexity with respect to the inverse Jacobian, and derive a simplified geometric discretization leading to an efficient moving mesh algorithm. Numerical experiments confirm the theoretical properties and demonstrate robust adaptive behavior for function-induced meshes and Rayleigh-Taylor instability simulations.

Figures (13)

• Figure 1: Element-wise affine approximation.
• Figure 2: Mesh distribution induced by the function in Example 5.1 using the proposed functional and Huang's functional with 25600 elements.
• Figure 3: Cell mesh quality distributions induced by the function in Example 5.1 using the proposed functional and Huang's functional with 25600 elements.
• Figure 4: Evolution of the mesh functional value and minimum element volume for the function in Example 5.1 using the proposed functional and Huang's functional with 25600 elements.
• Figure 5: Mesh distribution induced by the function in Example 5.2 using the proposed functional and Huang's functional with 25600 elements.

Theorems & Definitions (32)

• Example 2.1: Huang's functional
• Example 2.2: Kolasinski-Huang functional
• Lemma 2.1
• Lemma 2.2
• Theorem 3.1: Scale Invariance of the Minimizer
• proof
• Remark 3.1
• Theorem 3.2: Polyconvexity
• proof
• Theorem 3.3: Geodesical Convexity