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
本文提出余因子参考形式的离散设计理论，分类局部势空间，并证明在特定条件下求积点材料裕度最优值可达，同时给出严格裕度损失示例及保持切线的规范构造。
AI 中文摘要
我们为余因子参考形式建立了一种离散设计理论，该理论能够保持完整的有限元弹性切线。一个固定的连续势生成系数，而变分空间和求积决定兼容性。我们对所有阶张量单元、各向异性空间、三次Serendipity单元和二次四面体单元分类了精确的局部势空间。在无应力Mooney-Rivlin状态下，当材料系数之和为正且公共时，一个凸损失函数表达了可达到的求积点材料裕度。该最优值可以达到；在所述空间和采样假设下，其理想值恰好当物理恒等势兼容时达到。一个一致的固定候选示例证明了在位移富集下存在严格的裕度损失，该损失关于曲率振幅是线性的，且关于单元尺寸是一致的。显式的正候选伴随此限制。一个Gram恒等式给出了在所述范数下指定裕度处的尖锐常系数缺陷。兼容和参考缺陷校正形式通过一个与变形无关的外部边界Hessian保持原始切线；相等的势迹给出相等的精确组装规范。二次张量几何在显式材料、形状、求积和边界假设下，允许一个无应力物理强制界，该界关于网格尺寸和位移阶数一致。经过认证的弯曲四面体剪切族具有零恢复单元压力，确立了有限变形机制，而一个加载的压力-曲率反例则识别了其局限性。该理论和观察区分了局部精确性、材料认证、完全参考强制性和不变的方程。
英文摘要
We develop a discrete design theory for cofactor reference forms that preserve the complete finite-element elasticity tangent. A fixed continuous potential generates the coefficient, while the variation space and quadrature determine compatibility. We classify the exact local potential spaces for all-order tensor elements, anisotropic spaces, cubic serendipity and quadratic tetrahedra. At a stress-free Mooney-Rivlin state with a common positive material-coefficient sum, a convex loss expresses the attainable quadrature-point material margin. The optimum is attained; under stated space and sampling assumptions, its ideal value is attained precisely when the physical identity potential is compatible. A conforming fixed-candidate example proves a strict margin loss, linear in curvature amplitude and uniform in cell size, under displacement enrichment. Explicit positive candidates accompany this restriction. A Gram identity gives the sharp constant-coefficient defect at a prescribed margin in the stated norm. Compatible and reference-defect-corrected forms retain the original tangent through a deformation-independent external-boundary Hessian; equal potential traces give equal exact assembled gauges. Quadratic tensor geometry admits a stress-free physical coercivity bound uniform in mesh size and displacement order under explicit material, shape, quadrature and boundary assumptions. Certified curved tetrahedral shear families with zero recovered cell pressure establish finite-deformation regimes, while a loaded pressure-curvature counterexample identifies their limitation. The theory and observations distinguish local exactness, material certificates, complete-reference coercivity and unchanged equations.
Comments57 pages, 4 figures
