arXiv subjects

Zhenghua Duan

Publications and source records attributed to Zhenghua Duan.

3 recordsLinked to original sources

arXiv2026

Adaptive moving mesh methods for the planar Willmore flow

In this paper, we propose adaptive moving mesh methods for the planar Willmore flow by incorporating a tangential velocity into the original geometric evolution. The tangential velocity is designed based on a monitor function constructed from the curvature and its variation, enabling dynamic mesh redistribution along the evolving interface. This adaptive redistribution enhances spatial resolution in regions of high geometric complexity while preserving mesh regularity. The resulting moving mesh formulation is discretized using the kth-order backward differentiation formula (BDFk) in time and finite difference methods in space. Moreover, a class of new relaxed Lagrange multiplier approaches is further incorporated into the adaptive moving mesh framework to construct energy-stable adaptive moving mesh methods. Furthermore, to enhance the adaptivity and flexibility of the proposed framework, we develop an adaptive strategy for selecting the monitor function and introduce an alternative redistribution approach. Finally, extensive numerical experiments demonstrate that the proposed BDFk-based adaptive scheme accurately captures the geometric evolution of the planar Willmore flow and exhibits excellent robustness and computational efficiency for problems involving complex interface geometries.

math.NA↗

arXiv2025

Solid-state dewetting of axisymmetric thin film on axisymmetric curved-surface substrates: modeling and simulation

In this work, we consider the solid-state dewetting of an axisymmetric thin film on a curved-surface substrate, with the assumption that the substrate morphology is also axisymmetric. Under the assumptions of axisymmetry, the surface evolution problem on a curved-surface substrate can be reduced to a curve evolution problem on a static curved substrate. Based on the thermodynamic variation of the anisotropic surface energy, we thoroughly derive a sharp-interface model that is governed by anisotropic surface diffusion, along with appropriate boundary conditions. The continuum system satisfies the laws of energy decay and volume conservation, which motivates the design of a structure-preserving numerical algorithm for simulating the mathematical model. By introducing a symmetrized surface energy matrix, we derive a novel symmetrized variational formulation. Then, by carefully discretizing the boundary terms of the variational formulation, we establish an unconditionally energy-stable parametric finite element approximation of the axisymmetric system. By applying an ingenious correction method, we further develop another structure-preserving method that can preserve both the energy stability and volume conservation properties. Finally, we present extensive numerical examples to demonstrate the convergence and structure-preserving properties of our proposed numerical scheme. Additionally, several interesting phenomena are explored, including the migration of 'small' particles on a curved-surface substrate generated by curves with positive or negative curvature, pinch-off events, and edge retraction.

math.NA↗

arXiv2021

A Marching Cube Algorithm Based on Edge Growth

Marching Cube algorithm is currently one of the most popular 3D reconstruction surface rendering algorithms. It forms cube voxels through the input image, and then uses 15 basic topological configurations to extract the iso-surfaces in the voxels. It processes each cube voxel in a traversal manner, but it does not consider the relationship between iso-surfaces in adjacent cubes. Due to ambiguity, the final reconstructed model may have holes. We propose a Marching Cube algorithm based on edge growth. The algorithm first extracts seed triangles, then grows the seed triangles and reconstructs the entire 3D model. According to the position of the growth edge, we propose 17 topological configurations with iso-surfaces. From the reconstruction results, the algorithm can reconstruct the 3D model well. When only the main contour of the 3D model needs to be organized, the algorithm performs well. In addition, when there are multiple scattered parts in the data, the algorithm can extract only the 3D contours of the parts connected to the seed by setting the region selected by the seed.

cs.GR↗