RUPA: Nonlinear volume consistency, constraint geometry and singular penalty limits in finite elements
Abstract
Volume quadrature can change nonlinear finite-element constraints while preserving their reference-state derivatives. We connect an explicit determinant defect to feasible-set geometry and singular mechanical response. For affine tensor elements of coordinate degree p≥3 with n≥ p+1 Gauss points per coordinate, determinant volume is exact precisely when 2n≥3p. Below that threshold we construct a boundary-fixed cubic defect at every order. The same directions yield a full-space cube-root residual--distance bound under explicit cell-support, coefficient and physical-norm assumptions, with mesh-uniform upper constants at fixed order. With all cell-pressure equations retained, the volume Jacobian gains rank at nearby feasible states despite agreement through second derivatives at rest; the cube-root exponent is sharp on each fixed mesh. A general localized-minimum theorem shows that the first reduced compatibility term contributes its weighted square to the leading energy in a joint small-load, large-bulk limit. Cubic and quadratic defects therefore produce sextic and quartic terms. The full cubic-element interior space has an exact normal form and sharp local error exponents. Curved quadratic tetrahedra supply the second-order contrast, a sparse rational witness and an exact four-Jacobian volume formula. Finite-strain tensor calculations illustrate normalized response separation, with explicit stationary-point, extreme-bulk and pressure-recovery qualifications. The constructive correction preserves exact cell volumes, so quadrature feasibility remains distinct from physical volume preservation.
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BibTeX
@misc{liu2026rupa,
title = {{RUPA: Nonlinear volume consistency, constraint geometry and singular penalty limits in finite elements}},
author = {Yanlin Liu and Chao Huang and Kaixiang Yao and Yao Shen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13446},
eprint = {2609.13446},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13446},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - RUPA: Nonlinear volume consistency, constraint geometry and singular penalty limits in finite elements
AU - Yanlin Liu
AU - Chao Huang
AU - Kaixiang Yao
AU - Yao Shen
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13446
UR - https://arxiv.org/abs/2609.13446
AB - Volume quadrature can change nonlinear finite-element constraints while preserving their reference-state derivatives. We connect an explicit determinant defect to feasible-set geometry and singular mechanical response. For affine tensor elements of coordinate degree p≥3 with n≥ p+1 Gauss points per coordinate, determinant volume is exact precisely when 2n≥3p. Below that threshold we construct a boundary-fixed cubic defect at every order. The same directions yield a full-space cube-root residual--distance bound under explicit cell-support, coefficient and physical-norm assumptions, with mesh-uniform upper constants at fixed order. With all cell-pressure equations retained, the volume Jacobian gains rank at nearby feasible states despite agreement through second derivatives at rest; the cube-root exponent is sharp on each fixed mesh. A general localized-minimum theorem shows that the first reduced compatibility term contributes its weighted square to the leading energy in a joint small-load, large-bulk limit. Cubic and quadratic defects therefore produce sextic and quartic terms. The full cubic-element interior space has an exact normal form and sharp local error exponents. Curved quadratic tetrahedra supply the second-order contrast, a sparse rational witness and an exact four-Jacobian volume formula. Finite-strain tensor calculations illustrate normalized response separation, with explicit stationary-point, extreme-bulk and pressure-recovery qualifications. The constructive correction preserves exact cell volumes, so quadrature feasibility remains distinct from physical volume preservation.
ER -