结构可辨识性与基于速度测量的不完整光学图应力重建
Structural identifiability and stress reconstruction from incomplete optical maps with velocimetry
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中文总结 AI 辅助
针对光学覆盖不完整时粘弹性流动应力重建失效问题，提出结合速度通道通过动量平衡约束空间变化，将偏斜误差从50.40%降至27.82%，并证明各向同性应力不可观测。
中文摘要 AI 辅助
从光学测量重建平面粘弹性流动的应力场时，凡光学覆盖中断之处，直接证据即告缺失，且光学精度的任何提升都无法恢复从未进行的观测。我们刻画了第二通道（速度）能增加的信息，以及两个通道都无法提供的信息。两个标定的光学分量逐点确定局部偏斜应力，而速度通过动量平衡约束应力的空间变化，因此两个通道是互补而非冗余的。各向同性部分的应力对两者均不可观测：各向同性场的散度是纯梯度，Leray投影将其消除，因此每个可表示的各向同性模式都位于联合零空间中。当光学场完整时，这恰好解释了零空间；否则，由于有限的不完整采样和孔径零点可能移除更多方向，这构成了零空间的下界。我们在三种离散化上直接验证了这一计数。在具有有限测量孔径、空间相关噪声和光学条纹丢失的成对合成测试中，加入速度将参考噪声为3%时的全域平均偏斜误差从50.40%降至27.82%，且该改进在共享间隙、固定物理采样下的逆网格细化以及本构生成的应力场中均保持。该改进不依赖于正则化参数的选择方式：在期望范数差异准则和广义交叉验证下均成立，我们报告了每种选择及其在搜索区间中的位置，而这正是两种准则的差异所在。
英文摘要
Reconstructing the stress field of a planar viscoelastic flow from optical measurements loses its direct evidence wherever optical coverage is interrupted, and no improvement in optical precision restores an observation that was never made. We characterize what a second, velocity channel adds, and what neither channel can supply. Two calibrated optical components determine the local deviatoric stress pointwise, while velocity constrains spatial stress variation through momentum balance, so the two channels are complementary rather than redundant. The isotropic part of the stress is unobservable to both: the divergence of an isotropic field is a pure gradient, which the Leray projection annihilates, so every representable isotropic mode lies in the joint null space. That accounts for the null space exactly when the optical field is complete, and bounds it from below otherwise, since finite incomplete sampling and aperture zeros can remove further directions. We verify the count directly on three discretizations. In paired synthetic tests with finite measurement apertures, spatially correlated noise and optical stripe dropout, adding velocity reduces the mean whole-domain deviatoric error from 50.40% to 27.82% at 3% reference noise, and the improvement survives shared gaps, inverse-grid refinement at fixed physical sampling, and a constitutively generated stress field. The improvement does not rest on how the regularization parameter is chosen: it holds under both the expected-norm discrepancy rule and generalized cross-validation, and we report each selection with its position in the search interval, which is where the two rules differ.
