AI 中文总结
研究裂纹尖端和角奇点问题，通过力学训练神经坐标映射对计算半径分级，与多种方法比较，该映射能有效降低误差，在特定情况下神经校正有用
AI 中文摘要
在裂纹尖端或再入角附近，分数径向幂可能具有无界导数并减缓高阶样条的收敛。奇异映射对计算半径进行分级，使拉回场更平滑而不改变物理域。经典映射需要提前知道奇异指数和分级。本文从力学问题中训练径向坐标，它是正神经密度的归一化积分，固定径向边界并确保远离塌陷尖端时 r'(s)>0。通过离散平衡时的伽辽金能量推断径向分级指数和密度校正权重。在标量和平面应变 B 样条公式中测试该映射，并与恒等映射、径向分级节点向量、规定幂映射和自适应富集 B 样条方法进行比较。在 156 个向量自由度下，力学训练映射与恒等映射相比，将相对能量范数误差降低了 33.42 倍。在 3 个轮廓上恢复的混合模式应力强度因子的最大误差为 1.923×10^(-5)。对于直裂纹，学习到的密度校正消失，规定的 r = s^2 映射有相同改进。对于具有训练区间外测试参数的非线性罗宾族，密度校正保持非零，并给出至少 4.839 的固定 q 增量能量增益。因此，对于本文考虑的问题，当所需坐标不由规定幂映射表示时，神经校正很有用。
英文摘要
Near a crack tip or re-entrant corner, fractional radial powers can have unbounded derivatives and slow the convergence of high-order splines. A singular mapping grades the computational radius so that the pulled-back field is smoother without changing the physical domain. Classical maps require the singular exponent and grading in advance. Here the radial coordinate is trained from the mechanics problem. It is the normalized integral of a positive neural density, which fixes both radial boundaries and ensures r'(s)>0 away from the collapsed tip. The radial grading exponent and density-correction weights are inferred from Galerkin energies evaluated at discrete equilibrium, without exponent labels or exact interior fields. We test the mapping in scalar and plane-strain B-spline formulations and compare it with the identity map, radially graded knot vectors, prescribed power maps, and an adaptive enriched B-spline method. At 156 vector degrees of freedom, the mechanics-trained map reduces the relative energy-norm error by a factor of 33.42 compared with the identity map. The maximum error in the mixed-mode stress intensity factors recovered on 3 contours is 1.923 x 10^(-5). For the straight crack, the learned density correction vanishes, and the prescribed r=s^2 map gives the same improvement. For a nonlinear Robin family with test parameters outside the training interval, the density correction remains nonzero and gives a fixed-q incremental energy gain of at least 4.839. Thus, for the problems considered here, the neural correction is useful when the required coordinate is not represented by a prescribed power map.
Comments30 pages, 19 figures
