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VirtualElementMethods

VEM × バイオフィルム力学 — 任意多角形/多面体メッシュによるバイオフィルムの機械的解析

VEM (Virtual Element Method) は FEM の一般化で、任意多角形/多面体要素上で弾性問題を解ける。形状関数を明示構成しない ("virtual") 代わりに射影演算子で剛性行列を近似する。

pipeline

schematic

File	Description
`vem.py`	2D Poisson (Sutton 2017 base)
`vem_elasticity.py`	2D plane-stress elasticity (patch test 1e-18)
`vem_3d.py`	3D elasticity on polyhedra (patch test 1e-19)
`vem_3d_advanced.py`	3D Voronoi + sparse solver + VTK export
`vem_apple.py`	Apple-shaped Voronoi demo

File	Idea
`vem_spacetime.py`	Space-Time VEM: anisotropic (x,t) Voronoi, SLS 粘弾性
`vem_growth_coupled.py`	Growth-Coupled VEM: 5-species replicator + VEM + cell division
`vem_confocal_pipeline.py`	Confocal → VEM: 5ch confocal → Voronoi → VEM (2-step)
`vem_phase_field.py`	Phase-field VEM: Aldakheel 2018 style 破壊, DI→G_c
`vem_nonlinear.py`	Neo-Hookean VEM: 大変形超弾性, Newton-Raphson
`vem_error_estimator.py`	A posteriori error: ZZ-type, Dörfler marking
`process_heine_fish.py`	Heine 2025 FISH: real image → VEM

File	Purpose
`pipeline_3d_real.py`	3D TIFF → colony segment → Voronoi → VEM
`benchmark_3d_vem_vs_tet.py`	VEM Voronoi vs FEM tet comparison
`phase_field_real_3d.py`	Phase-field fracture on real SAPA biofilm
`vem_phase_field_3d.py`	3D Phase-field VEM on polyhedral mesh (NEW)

Metric	Value
E range	10–1000 Pa (literature: 20–14,000 Pa)
E ratio CS/DS	up to 28×
Convergence L²	2.14 (Voronoi)
3D Real PA	65 cells, E 42–237 Pa
3D Real SAPA	131 cells, DI 0.18–0.79, E 74–682 Pa
Phase-field 2D	69 elements, d_max=0.85, 57/82 cracked
Phase-field 3D	208 polyhedra, catastrophic at step 14, d_max=0.56 pre-cat
Tests	70 passing

grand showcase