发表机构
The University of Melbourne; Shanghai Jiao Tong University; Southern University of Science and Technology（墨尔本大学; 上海交通大学; 南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限元中体积求积对非线性约束的影响，揭示行列式缺陷与可行集几何的关系，并证明在特定高斯点数量下体积精确性条件，提出修正方法以保持单元体积。
AI 中文摘要
体积求积可以改变非线性有限元约束，同时保持其参考状态导数。我们将显式行列式缺陷与可行集几何和奇异力学响应联系起来。对于坐标次数为$p\ge3$且每个坐标有$n\ge p+1$个高斯点的仿射张量单元，当且仅当$2n\ge3p$时，行列式体积才是精确的。低于该阈值时，我们在每个阶数下构造一个边界固定的三次缺陷。相同的方向在显式单元支撑、系数和物理范数假设下，产生全空间立方根残差-距离界，并在固定阶数下具有网格一致的上常数。在保留所有单元压力方程的情况下，尽管在静止状态下二阶导数一致，体积雅可比矩阵在附近的可行状态获得秩；立方根指数在每个固定网格上都是尖锐的。一个一般的局部最小化定理表明，在联合小载荷、大体积极限下，第一个约化相容性项将其加权平方贡献给主能量。因此，三次和二次缺陷产生六次和四次项。完整的三次单元内部空间具有精确的标准型和尖锐的局部误差指数。弯曲二次四面体提供了二阶对比、稀疏有理见证和精确的四雅可比体积公式。有限应变张量计算说明了归一化响应分离，并具有显式的驻点、极端体积和压力恢复限定条件。构造性修正保持了精确的单元体积，因此求积可行性仍然不同于物理体积保持。
英文摘要
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 $p\ge3$ with $n\ge p+1$ Gauss points per coordinate, determinant volume is exact precisely when $2n\ge3p$. 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.
Comments57 pages, 6 figures
