Comparison of Bending-Energy Discretization Methods for Anisotropic Meshes in Morphogenetic Simulations

Accurately modeling bending energy in morphogenetic simulations is crucial, especially when dealing with anisotropic meshes where remeshing is infeasible due to the biologically meaningful entities of vertex positions (e.g., cells). This study addresses the underexplored question of which bending-energy discretization methods are most accurate and suitable for such simulations. The evaluation consists of two stages: First, the accuracy of each method is tested by comparing predicted bending energy and force against theoretical values for two benchmark cases--a wrinkled planar sheet and a smooth spherical sheet. Second, we simulate the formation of wrinkles in a planar sheet caused by anisotropic cell division, analyzing the resulting wavenumber patterns for two division orientations: uniaxial and random. The results highlight that the choice of the optimal discretization method depends on the application. For simulations requiring precise quantitative predictions, the Hamann model demonstrates superior accuracy. Conversely, for simulations where qualitative trends in morphology are of primary interest, the J"ulicher model provides a computationally efficient alternative. These findings provide guidance for selecting appropriate bending-energy discretization methods in morphogenetic simulations, ultimately leading to more accurate and efficient modeling of complex biological forms.