Checkerboard Shells: A Position-Only Thin-Shell Discretization with Scale-Compatible Completion
Abstract
Checkerboard edge-midpoint geometry provides an exact planar Varignon parallelogram for every spatial quadrilateral, allowing a local tangent frame and normal to be recovered directly from nodal positions even when the raw quadrilateral is warped. Building on this property, we develop a position-only thin-shell discretization with no independent director, rotation, or strain variables. The connected edge-midpoint surface is taken as the physical midsurface: the first fundamental form is evaluated on planar B faces, while a W-centered second fundamental form is constructed from variations of neighboring B-face normals, so membrane and bending share the same geometric carrier. A variational-kernel analysis shows that smooth second-order consistency does not eliminate lattice-scale blind modes. After quotienting out the raw checkerboard gauge, the B metric has one physical membrane blind direction and the symmetric W curvature has two curvature blind directions. We introduce a quotient-minimal membrane compatibility coordinate XM and an objective reference-relative curvature coordinate XWrel, placed consistently in the O(t) membrane and O(t3) bending sectors. The formulation admits an explicit midpoint quotient, complete flat blind-mode classification, rigid-motion objectivity, reference-state consistency, and an O(h2) near-isometry approximation result for aligned generalized cylinders. Numerical tests show second-order curvature convergence, targeted removal of the membrane defect, and a sub-percent, refinement-decaying influence of XWrel. Linear and nonlinear shell benchmarks further demonstrate flat bending, curved-shell membrane-bending coupling, thickness sensitivity, large rotation, nonlinear pinching, and localized ovalization within a single position-only framework.
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BibTeX
@misc{song2026checkerboard,
title = {{Checkerboard Shells: A Position-Only Thin-Shell Discretization with Scale-Compatible Completion}},
author = {Junjie Song and Xuanyu Wu and Zhifeng Zhang and Bingtao Hu and Zhaoxi Hong and Xiuju Song and Yixiong Feng and Jianrong Tan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30195},
eprint = {2608.30195},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30195},
}
RIS
TY - UNPB
TI - Checkerboard Shells: A Position-Only Thin-Shell Discretization with Scale-Compatible Completion
AU - Junjie Song
AU - Xuanyu Wu
AU - Zhifeng Zhang
AU - Bingtao Hu
AU - Zhaoxi Hong
AU - Xiuju Song
AU - Yixiong Feng
AU - Jianrong Tan
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30195
UR - https://arxiv.org/abs/2608.30195
AB - Checkerboard edge-midpoint geometry provides an exact planar Varignon parallelogram for every spatial quadrilateral, allowing a local tangent frame and normal to be recovered directly from nodal positions even when the raw quadrilateral is warped. Building on this property, we develop a position-only thin-shell discretization with no independent director, rotation, or strain variables. The connected edge-midpoint surface is taken as the physical midsurface: the first fundamental form is evaluated on planar B faces, while a W-centered second fundamental form is constructed from variations of neighboring B-face normals, so membrane and bending share the same geometric carrier. A variational-kernel analysis shows that smooth second-order consistency does not eliminate lattice-scale blind modes. After quotienting out the raw checkerboard gauge, the B metric has one physical membrane blind direction and the symmetric W curvature has two curvature blind directions. We introduce a quotient-minimal membrane compatibility coordinate X_M and an objective reference-relative curvature coordinate X_W^rel, placed consistently in the O(t) membrane and O(t^3) bending sectors. The formulation admits an explicit midpoint quotient, complete flat blind-mode classification, rigid-motion objectivity, reference-state consistency, and an O(h^2) near-isometry approximation result for aligned generalized cylinders. Numerical tests show second-order curvature convergence, targeted removal of the membrane defect, and a sub-percent, refinement-decaying influence of X_W^rel. Linear and nonlinear shell benchmarks further demonstrate flat bending, curved-shell membrane-bending coupling, thickness sensitivity, large rotation, nonlinear pinching, and localized ovalization within a single position-only framework.
ER -