几何精确梁的时空形式化
Space--time formulation of geometrically exact beams
AI总结：
本文将几何精确Cosserat梁动力学形式化为非相对论时空定向世界片，提出时间对齐的Q₂/Q₂离散化方法，经数值研究验证其稳定性与协变性，推荐作为该形式化的实际实现。
AI中文摘要：
我们将直参考几何精确Cosserat梁的动力学形式化为非相对论时空中的定向世界片。其中心线历史嵌入在\boldsymbol{\text{R}}^4\boldsymbol{}中，绝对时间保持预设，并非额外的力学自由度。空间力与力矩合力、时间线性动量及本征角动量被组合为统一的时空通量，使得空间Neumann数据、时间流入与流出可由一个共法边界算子表示。混合构型-动量系统通过连续张量积有限元离散，并由未来指向的Petrov-Galerkin摄动稳定。所得时间对齐的\boldsymbol{Q}_2/\boldsymbol{Q}_2\boldsymbol{}方法仅需一个标量稳定参数，并保留内部动量迹作为自由终端流出。数值研究验证了终端通量处理（基于精确刚体运动态），建立了光滑制造的剪切-弯曲解的单调收敛性，量化了延迟加载下的预响应-精度权衡，还验证了在恒定叠加空间转动下的协变性。该稳定并非严格依赖域因果性的证明，它沿未来方向偏置全局时空近似，降低了测得的预激活响应。本文还分析了对非对齐单形面的保守扩展，尽管该扩展是相容且局域保守的，但连续等阶\boldsymbol{P}_2/\boldsymbol{P}_2\boldsymbol{}单形空间表现出强烈依赖网格与取向的逆放大，且未呈现网格一致的稳定性，因此推荐对齐的\boldsymbol{Q}_2/\boldsymbol{Q}_2\boldsymbol{}离散化作为该形式化的实际实现方式。
英文摘要：
We formulate the dynamics of a straight-reference geometrically exact Cosserat beam as a directed world sheet in non-relativistic space--time. The centerline history is embedded in \(\mathbb R^4\), while absolute time remains prescribed and is not an additional mechanical degree of freedom. Spatial force and moment resultants and temporal linear and intrinsic angular momenta are combined into common space--time fluxes, so that spatial Neumann data and temporal inflow and outflow are represented by one co-normal boundary operator. A mixed configuration--momentum system is discretized by continuous tensor-product finite elements and stabilized by a future-directed Petrov--Galerkin perturbation. The resulting time-aligned \(Q_2/Q_2\) method requires one scalar stabilization parameter and retains the interior momentum trace as the free terminal outflow. The numerical study verifies the terminal flux treatment by an exact rigid-motion state, establishes monotone convergence for a smooth manufactured shear--bending solution, quantifies the pre-response--accuracy trade-off under delayed loading, and verifies covariance under constant superposed spatial rotations. The stabilization is not a proof of strict domain-of-dependence causality; it biases the global space-time approximation in the future direction and reduces the measured pre-activation response. A conservative extension to non-aligned simplex facets is also analyzed. Although this extension is consistent and locally conservative, continuous equal-order \(P_2/P_2\) simplex spaces exhibit strongly mesh- and orientation-dependent inverse amplification and do not show mesh-uniform stability. The aligned \(Q_2/Q_2\) discretization is therefore recommended as the practical realization of the formulation.
