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Checkerboard Shells: A Position-Only Thin-Shell Discretization with Scale-Compatible Completion

Junjie Song, Xuanyu Wu, Zhifeng Zhang, Bingtao Hu, Zhaoxi Hong, Xiuju Song, Yixiong Feng, Jianrong Tan

arXiv:2608.30195v1math.NA

TL;DR

The paper addresses whether a position-only shell discretization can obtain reliable membrane and bending observability from nodal positions while avoiding checkerboard gauge redundancy and lattice-scale blind modes. It uses the Varignon midpoint surface with B-face metrics, W-centered curvature, quotient analysis, and scale-compatible completion. The resulting formulation provides second-order geometric convergence, targeted membrane-defect removal, small refinement-decaying curvature-completion effects, and broad linear and nonlinear benchmark coverage.

  • Problem

    Geometric consistency does not exclude lattice-scale blind modes: after removing the checkerboard gauge, the B metric and symmetric W curvature still miss physical membrane and curvature directions.

  • Method

    The formulation uses the connected edge-midpoint surface as physical midsurface, derives metric and curvature from B faces and neighboring normals, and adds quotient-minimal, scale-compatible completion coordinates.

  • Results

    Second-order curvature convergence, targeted removal of the membrane defect, and sub-percent refinement-decaying influence of X_W^rel are reported across shell tests and benchmarks.

  • Takeaways & Limitations

    A single position-only framework combines unified membrane-bending geometry, physical-quotient analysis, blind-mode completion, objectivity, and broad finite-deformation shell responses.

  • Takeaways & Limitations

    Extremely thin shells may require higher-order near-isometry approximation and more local membrane observations for further improvement.

Abstract

from arXiv · show

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.

1 Introduction

The paper motivates a position-only shell discretization from exactly planar Checkerboard midpoint patches, using one midpoint surface for membrane and bending geometry. It addresses the distinct problem that geometric convergence can coexist with lattice-scale physical blind modes.

  • Motivation: Varignon midpoint patches are exactly planar for arbitrary spatial quadrilaterals, avoiding diagonal-dependent tangent planes, normals, and curvatures.This supplies a local geometric frame directly from nodal positions.
  • Motivation: Checkerboard geometry enables finite rotations using updated nodal positions without independent director or rotation variables.The formulation exploits strict planarity, a position-only state, and shared midpoint connectivity.
  • Problem: Smooth geometric convergence does not guarantee lattice-scale observability because the raw checkerboard gauge coexists with genuine physical deformations invisible to discrete metric or curvature observations.The paper distinguishes representation redundancy from zero-energy or anomalously soft physical mechanisms.
  • Method: The connected edge-midpoint surface serves as the physical midsurface, with planar B faces carrying the metric and neighboring B-face normals defining the W-centered second form.Membrane and bending consequently use the same discrete geometric carrier.
  • Problem: After quotienting the raw gauge, the B metric has one membrane blind direction and symmetric W curvature has two curvature blind directions.The formulation targets these defects with scale-compatible completion in the membrane and bending sectors.
  • Contribution: The completion restores missing mechanical observability while retaining a position-only state and a unified physical midsurface for membrane and bending.Quotient analysis separates physical deformation from checkerboard representation redundancy.

2 Checkerboard Representation and Discrete Shell Geometry

The formulation treats the connected edge-midpoint surface as the physical midsurface, using exact planar B faces for the metric and neighboring B-face normals for W-centered curvature. It removes raw checkerboard redundancy before analyzing physical observability and completes the geometry with a shared position-only carrier.

  • Checkerboard representation: The raw-to-midpoint map has one three-dimensional alternating black–white gauge degree of freedom that leaves the physical midpoint surface unchanged.This redundancy is removed by quotienting rather than by assigning it material stiffness or a penalty.
  • Checkerboard representation: Edge midpoints of each spatial quadrilateral form an exactly planar Varignon B face, giving a tangent frame and normal for arbitrary current nodal positions.Neighboring B faces connect through shared midpoint nodes, producing the continuous Checkerboard surface used for mechanical measurements.
  • Discrete first fundamental form: The B-face Gram matrix is the discrete first fundamental form, recording squared directional lengths and relative angle without independent directors or strain variables.For smooth regular immersions, the corresponding continuous first fundamental form is approximated to second order.
  • Observability and consistency: Smooth second-order consistency does not rule out lattice-scale blind modes, so physical deformation must be checked against the linearized kernels after quotienting the raw gauge.The curvature construction remains second-order consistent for smooth fields, including edges and corners with one-sided reconstruction and tensor-product corner extension.
  • W-centered second fundamental form: The W-centered second form compares relative rotations of neighboring B-face normals at staggered W sites without fitting an additional plane or introducing a second curvature carrier.This keeps metric, curvature, gauge, and compatibility analysis in the same position-only state space.

3 Physical Blind Modes and Variational Observability

The paper distinguishes raw checkerboard representation redundancy from physical blind modes by analyzing variational kernels on the midpoint quotient. The B metric has one physical membrane defect, while symmetric W curvature has two physical curvature-blind directions.

  • Variational observability: A physical blind mode changes the midpoint surface while remaining invisible to the corresponding discrete metric or curvature observation.Smooth consistency probes long wavelengths, whereas lattice-scale alternating modes can remain unobserved and acquire zero or anomalously soft energy directions.
  • Variational observability: The kernel analysis removes rigid motions and the raw gauge, then classifies surviving directions by their physical midpoint response.The quotient is represented as ker Jh/(Rh + Gh), with shared-node compatibility determining the lattice kernel.
  • First fundamental form: The B metric has a one-dimensional physical quotient defect: zM moves the midpoint surface but satisfies DaB[zM] = 0.The pure checkerboard carrier is only a representation gauge because it leaves all edge midpoints fixed.
  • Second fundamental form: The flat symmetric W-curvature kernel contains two non-rigid, non-gauge blind modes, ip and jp, alongside rigid motions and the scalar checkerboard gauge.These modes move the physical midpoint surface while satisfying DbW = 0, so the W observation supplies no first-order curvature response.
  • Physical consequences: A bending energy using only bW would retain two zero-stiffness curvature mechanisms, complementing the one membrane defect identified for aB.The complete missing information is therefore one membrane direction and two curvature directions.

4 Scale-Compatible Completion and Finite-Configuration Shell Energy

The completion strategy adds only the missing membrane information at O(t) and restores curvature compatibility within the O(t^3) bending sector. XM is quotient-minimal and objective, while XWrel uses reference-relative surface-polar transport for finite rotations.

  • Design requirements: Membrane completion is constrained by thin-shell scaling because unnecessary O(t) stiffness is amplified relative to bending as t approaches zero.The added membrane term must remove only zM, preserve rigid motions and the raw gauge, and avoid shrinking near-isometric shell deformations.
  • Design requirements: Curvature completion remains in the O(t^3) bending sector, so it does not acquire the membrane-side t^-2 amplification relative to bending.It must respond to ip and jp, remain objective under finite rotations, preserve smooth consistency, and avoid dominating low-frequency bending.
  • Membrane completion: XM is a single topology-fixed M-character moment that detects the missing alternating microrotation compatibility without measuring additional stretch or shear.It is designed to vanish on homogeneous affine membrane deformation, remain invariant under rigid motions and the raw gauge, and respond to zM.
  • Membrane completion: XM is quotient-minimal because the one-dimensional defect requires at least one linear observation, and XM supplies a rank-one update.Its coefficient normalizes the missing M-point shear channel to the isotropic trace-free shear modulus without benchmark fitting.
  • Curvature completion: XWrel compares current and transported reference curvature compatibility vectors using the W-centered surface-polar rotation, preserving objectivity under superposed rigid rotations.For smooth surface families, XWrel = O(h2).
  • Finite-configuration energy: The completion terms repair discrete geometric observability while preserving the continuum membrane and bending scalings in the finite-configuration shell energy.The resulting potential uses nodal positions as the only independent unknowns and evaluates internal geometry on the midpoint/B-face surface.

5 Analytical Properties

The formulation establishes gauge-invariant, rigid-objective shell mechanics and preserves an O(h^2) near-isometric approximation for aligned generalized cylinders, while clarifying residual ultra-thin locking.

  • Objectivity and consistency: Rigid translations and rotations leave the internal energy unchanged, while raw checkerboard gauge changes preserve the represented physical state.Gauge invariance and Euclidean objectivity are distinct requirements, not interchangeable boundary-fixing choices.
  • Consistency properties: The metric and curvature observations are second-order consistent, X_W^rel is O(h^2), and X_M vanishes for homogeneous affine membrane deformation.These properties prevent artificial response in rigid-motion, reference, and uniform affine-strain states.
  • Near-isometry approximation: An aligned generalized-cylinder family admits exact completed-membrane-kernel fields approximating continuous infinitesimal isometries with O(h^2) discrete L2 and scaled H1 errors.The construction uses row-independent kernel fields, arc-length chord expansions, shape-regular meshes, and nondegenerate section chords.
  • Near-isometry approximation: The theorem is limited to aligned, row-independent generalized-cylinder infinitesimal isometries and does not establish arbitrary-shell solution convergence or uniform locking-free convergence.It shows preservation of a nontrivial curved-surface near-isometric branch, not a general curved-shell convergence theorem.
  • Residual ultra-thin locking: At fixed mesh resolution, any positive near-kernel membrane eigenvalue attenuates thin-shell bending response as t approaches zero; refinement shifts this locking onset to smaller thicknesses.Exact blind modes have zero eigenvalues, whereas residual locking arises from load-bearing near-kernel modes with positive finite-resolution eigenvalues.

6 Numerical Results

Numerical tests verify second-order curvature convergence, targeted membrane completion, and consistent linear and nonlinear shell responses using the same position-only midpoint geometry.

  • Shell benchmarks: Linear and nonlinear benchmarks cover flat bending, curved-shell coupling, thickness sensitivity, nearly 360° rotation, nonlinear pinching, and localized ovalization with one positional formulation.The study uses the same midpoint geometry and position-based mechanics across these tests.
  • Geometric convergence: The B-face/W-centered second form converges at approximately second order on cylinder, sphere, twist, and skew surfaces.The construction preserves smooth curvature consistency while retaining the shared physical geometric carrier.
  • Membrane completion: Adding X_M reduces flat-patch nullity from 6 to 5, matching the rigid-plus-gauge dimension, while preserving the B-side low-end positive spectrum.The full B/W observation instead shifts the same soft spectrum upward by several orders of magnitude.
  • Membrane completion: The maximum B-to-B + X_M displacement difference is 1.19 × 10^-10, and the maximum X_M membrane-energy fraction is 5.41 × 10^-12 in the M3 response.These values indicate that the targeted completion changes the removed mechanism without materially altering active load-bearing behavior.
  • Curvature completion: X_W^rel changes plate, Scordelis–Lo, and M3 displacements by 0.0136%, 0.0210%, and 0.2983%, respectively, with effects decreasing under refinement.All reported shifts remain below 0.3% in these linear shell tests.

7 Discussion

The discussion emphasizes a shared midpoint-surface carrier, compact positional state, low-rank completion, and finite-rotation capability, while identifying thin-shell refinement as an ongoing boundary.

  • Checkerboard geometry: The midpoint surface remains the physical carrier for both membrane and bending, with planar B faces and neighboring B-face normals supplying the geometric measures.This avoids switching to a separate curvature representation.
  • State complexity: Checkerboard uses 867 positional scalars versus MidedgeTan’s 1667 on the n = 16 plate, with errors of 1.488% and 1.248%, respectively.On M3 at n = 12, Checkerboard reaches 2.20% error with 507 states versus 3.74% with 963 states for MidedgeTan.
  • Targeted completion: X_M adds only a rank-one membrane channel, while X_W^rel restores missing curvature information with below-0.3% displacement shifts that decay under refinement.The completion therefore targets absent lattice directions rather than broadly stiffening the normal response.
  • Finite deformation: Finite-deformation tests reach 359.841° cantilever rotation, a 1.736% nonlinear-hemisphere joint RMS path error, and full open-cylinder ovalization with displacement reversal.The geometry is updated directly from current positions without changing kinematic descriptions for large rotations.
  • Scope and extensions: At fixed mesh resolution, residual membrane influence still limits the ultra-thin regime; further improvement is associated with higher-order near-isometry approximation and more local membrane observations.Refinement moves this influence toward progressively thinner shells rather than eliminating the finite-resolution effect outright.

8 Conclusions

The formulation uses the compatible midpoint surface as a shared carrier for position-only membrane and bending mechanics, completing the physical blind modes while preserving objectivity and reference consistency. Analytical and numerical tests support accurate shell behavior across linear and finite-deformation benchmarks.

  • Conclusions: The connected edge-midpoint surface supplies both planar B-face membrane geometry and W-centered curvature geometry from nodal positions alone.No independent director or rotational degrees of freedom are introduced.
  • Conclusions: The physical quotient identifies one membrane blind direction and two curvature blind directions beyond the raw checkerboard gauge.XM targets the membrane defect, while XrelW restores the missing curvature compatibility information.
  • Conclusions: The completion terms enter the finite-configuration Koiter-type energy at the physically appropriate O(t) membrane and O(t3) bending scales.The membrane and curvature terms are assigned to their respective shell-energy sectors.
  • Conclusions: The analytical results establish quotient and blind-mode classifications, rigid-motion objectivity, reference-state consistency, and O(h2) near-isometry approximation for aligned generalized cylinders.The near-isometry result preserves a natural curved-shell bending branch.
  • Conclusions: Numerically, W-centered curvature converges at second order, XM removes the target membrane defect, and XrelW remains a small correction that decreases under refinement.The benchmarks cover flat bending, curved-shell coupling, thickness sensitivity, large rotation, nonlinear pinching, and localized ovalization.
  • Conclusions: The resulting framework unifies Checkerboard geometry, the physical quotient, variational observability, and finite-deformation shell energy in one position-only discretization.The benchmark suite spans both classical linear-shell and finite-deformation responses.

A Midpoint Quotient and Gauge Identities

The midpoint quotient formalizes the compatible physical midpoint image of raw vertex positions and isolates the fixed checkerboard gauge. Gauge choices alter only the raw representative, not the midpoint surface or gauge-invariant mechanics.

  • A Midpoint Quotient and Gauge Identities: The midpoint map is defined from the undirected edge-sum operator and applies componentwise to three-dimensional raw positions.The representation-level proof starts from scalar fields on the raw vertex graph.
  • A Midpoint Quotient and Gauge Identities: The construction requires only a connected bipartite raw graph, not rectangular regularity.This broadens the representation result beyond regular rectangular grids.
  • A Midpoint Quotient and Gauge Identities: For a connected bipartite graph, the scalar edge-sum kernel is the one-dimensional alternating coloring mode, yielding a three-dimensional raw-position gauge.Applying the scalar result independently to Cartesian components gives ker P = {pA : A∈R3}.
  • A Midpoint Quotient and Gauge Identities: The quotient image consists of compatible midpoint assignments; arbitrary edge-vector assignments generally violate alternating cycle-closure relations.Raw positions automatically enforce midpoint compatibility.
  • A Midpoint Quotient and Gauge Identities: Physical boundary conditions and gauge-fixing constraints serve different roles: one constrains the midpoint surface, while the other selects a unique raw representative.The reduced solve satisfies both requirements simultaneously.
  • A Midpoint Quotient and Gauge Identities: Changing the gauge slice changes only the raw display; the midpoint surface, internal energy, external work, residual, and gauge-invariant observables remain unchanged.This invariance holds under finite-amplitude alternating-gauge perturbations.

B Flat First- and Second-Form Kernel Proofs

The flat-kernel analysis separates rigid motions and the raw checkerboard gauge from genuine physical blind modes. It identifies one in-plane metric defect and two transverse curvature defects, then shows how minimal observations remove them.

  • B.1 In-Plane Quotient Defect of the B Metric: On a flat grid, the B-metric kernel condition requires each local tangent increment to be a planar infinitesimal rotation, Fij = ωijJ.Shared raw nodes impose a recurrence on the cell microrotations.
  • B.1 In-Plane Quotient Defect of the B Metric: The recurrence yields rigid translations, rigid yaw, the two-dimensional raw gauge, and one additional membrane mode, giving nullity 6.The additional mode is not a raw gauge because it changes edge-midpoint positions and cannot be represented by an affine rigid field.
  • B.1 In-Plane Quotient Defect of the B Metric: The single scalar coordinate XM removes exactly the one-dimensional physical B-metric quotient defect and attains the minimum possible added observation size.Its derivative vanishes on the rigid and gauge subspaces while detecting the missing membrane mode.
  • B.2 Transverse Kernel of the Symmetric W Curvature: The symmetric W-curvature stencil acts as a parity-preserving discrete Hessian on diagonal sublattices, whose kernel is span{1, i, j, p, ip, jp}.The six-dimensional form follows from affine branches on the even and odd parity sublattices.
  • B.2 Transverse Kernel of the Symmetric W Curvature: The modes 1, i, j represent transverse translation and infinitesimal rotations, p is the raw gauge, and ip and jp are physical curvature blind modes.The latter two move edge midpoints while remaining invisible to first-order symmetric W curvature.
  • B.2 Transverse Kernel of the Symmetric W Curvature: Fourier analysis places the rigid, gauge, and curvature-blind modes in second-order zeros at Γ and M, while stripe points remain observed by symmetric curvature.The alternating normal observation detects the first jets at M.
  • C.1 Finite Gauge Invariance and Rigid Covariance of the B/W Geometry: The B and W geometric operators are exactly invariant under raw checkerboard gauge changes and covariant under superposed rigid motions.Consequently, the observed metric and curvature do not assign artificial responses to representation redundancy or rigid motion.

C.2 Reference-Relative Objectivity of XM

Reference-relative transport makes the curvature completion objective for curved references while preserving the alternating normal information needed to repair the W-curvature defects. The membrane coefficient is fixed by an explicit constitutive normalization rather than benchmark fitting.

  • C.2 Reference-Relative Objectivity of XM: For a rigid image of the reference, the neighboring-cell connections reduce to the reference-relative identity and XM=0; homogeneous affine membrane strain is not itself measured by XM.The coordinate responds to connection differences rather than ordinary affine strain.
  • C.4 Determination of the Membrane and Curvature Completion Coefficients: The membrane completion is normalized so a unit alternating microrotation matches a full-area trace-free shear channel, with βM=8µ.This coefficient is an a priori constitutive convention, not benchmark fitting or a unique consequence of the continuum Gram law.
  • C.2 Reference-Relative Objectivity of XM: The current/reference W deformation map uses its orientation-preserving right polar factor, which transforms covariantly under a superposed rigid rotation.On the admissible set FW∈GL+(3), RW(Qq+c,q0)=QRW(q,q0).
  • C.2 Reference-Relative Objectivity of XM: Transporting the reference alternating normal with one W-centered polar rotation makes the Euclidean relative curvature norm objective for a curved reference.A direct current-minus-reference difference would fail objectivity under a current-only rigid motion.
  • C.2 Reference-Relative Objectivity of XM: The same transport preserves the alternating normal component required for curvature compatibility, whereas independently transporting all four B normals would remove that information.This is why the completion retains precisely the missing observable.
  • C.4 Determination of the Membrane and Curvature Completion Coefficients: The curvature completion coefficient is obtained by interpreting the alternating B-normal mode as an auxiliary-Q1 normal field and exactly integrating its reference-metric gradient energy.Symmetry removes the integrated g12 cross term, and the curvature energy inherits the Koiter bending-sector shear modulus µ.

D Aligned Generalized-Cylinder Approximation

The appendix constructs a row-independent discrete near-isometry for aligned generalized cylinders and proves that the midpoint-based correction has O(h^2) nodal accuracy while preserving the raw-gauge quotient.

  • Construction: The generalized-cylinder setting uses an arc-length curve c(x), constant generator coordinate z, and a continuous row-independent infinitesimal isometry.The field v combines tangential, normal, and generator-direction components subject to u′(x) + κ(x)w(x) = 0.
  • Construction: The discrete B-metric zero condition reduces to one scalar recurrence for each circumferential chord.The recurrence is formulated using chord vectors, normalized chord directions, and nodal differences.
  • Discrete compatibility: The constructed field is row-independent, satisfies the chord condition exactly, and has zero axial and mixed B-metric variations.Consequently, the linearized mixed moment vanishes on every complete plaquette without requiring boundary-parity cancellation.
  • Boundary conditions: Essential boundary data must match the constructed trace, whereas a free patch requires no additional boundary condition.
  • Approximation estimate: O(h^3) correction steps accumulate over O(h^-1) circumferential intervals to produce an O(h^2) nodal correction with O(h^3) first differences.Boundary moment reconstruction and tensor-product closure retain second-order local consistency without mirror ghost data imposing a mechanical Neumann condition.
  • Approximation estimate: The resulting physical-midpoint norm has the same-order bound and exactly annihilates the raw checkerboard gauge.The constant is independent of h and depends on regularity, domain length, chord nondegeneracy, and mesh shape regularity.

F.1 Nonlinear Solution Procedure and Reference Data

The nonlinear procedure uses a consistent reference-state Hessian for linear problems and automatic differentiation of one scalar potential for nonlinear residuals and tangents. Benchmark paths combine analytical and digitized references with normalized RMS and maximum-error measures.

  • Solution procedure: Linear problems use the consistent reference-state Hessian, while nonlinear residuals and tangents are obtained by automatic differentiation of the same scalar potential.
  • Solution procedure: Newton corrections, an energy-aware line search, adaptive load increments, rollback, and consistent tolerances support the nonlinear solves.None of the three complete nonlinear benchmarks requires arc-length continuation.
  • Reference data: The cantilever uses an analytical constant-curvature path, while hemisphere and open-cylinder paths are digitized from verified literature graphics.Independent tabulated endpoints serve only as cross-checks, not as exact representations of the digitized curves.
  • Reference data: Path errors are measured using RMS and maximum absolute measures normalized by the reference path norm or maximum reference value.

F.2 Spectral Properties and Finite-Mesh Contribution of the Curvature Completion

Spectral tests show that X_M removes one active curved nullity, while full B/W observations eliminate the active null space; X_W^rel has small, refinement-decaying effects on standard responses.

  • Spectral properties: For a free sphere patch, numerical nullities are 51, 50, and 9 for B-only, B + X_M, and full B/W observations, respectively.On the active n = 16 M3 space, the corresponding nullities are 10, 9, and 0.
  • Spectral properties: X_M changes only one curved active nullity, whereas full B/W observation strongly compresses the original soft space.
  • Finite-mesh contribution: 0.0136%, 0.0210%, and 0.2983% are the fine-mesh displacement shifts induced by X_W^rel for the plate, Scordelis–Lo, and M3 problems, respectively.
  • Finite-mesh contribution: The influence of X_W^rel on standard load-bearing responses is small and decreases further with refinement.
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