Variational multiscale methods and their precursors, stabilized methods, which are sometimes supplemented with discontinuity-capturing (DC) methods, have been playing their core-method role in flow computations increasingly with isogeometric discretization. The stabilization and DC parameters embedded in most of these methods play a significant role. The parameters almost always involve some local-length-scale expressions, most of the time in specific directions, such as the direction of the flow or solution gradient. Until recently, local-length-scale expressions originally intended for finite element discretization were being used also for isogeometric discretization. The direction-dependent expressions introduced in [Y. Otoguro, K. Takizawa and T. E. Tezduyar, Element length calculation in B-spline meshes for complex geometries, Comput. Mech. 65 (2020) 1085–1103, https://doi.org/10.1007/s00466-019-01809-w ] target B-spline meshes for complex geometries. The key stages of deriving these expressions are mapping the direction vector from the physical element to the parent element in the parametric space, accounting for the discretization spacing along each of the parametric coordinates, and mapping what has been obtained back to the physical element. The expressions are based on a preferred parametric space and a transformation tensor that represents the relationship between the integration and preferred parametric spaces. Element splitting may be a part of the computational method in a variety of cases, including computations with T-spline discretization and immersed boundary and extended finite element methods and their isogeometric versions. We do not want the element splitting to influence the actual discretization, which is represented by the control or nodal points. Therefore, the local length scale should be invariant with respect to element splitting. In element definition, invariance of the local length scale is a crucial requirement, because, unlike the element definition choices based on implementation convenience or computational efficiency, it influences the solution. We provide a proof, in the context of B-spline meshes, for the element-splitting invariance of the local-length-scale expressions introduced in the above reference.

中文翻译：

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复杂几何的 B 样条网格中的元素分裂不变局部长度尺度计算

变分多尺度方法及其前身、稳定方法，有时辅以不连续性捕获 (DC) 方法，在等几何离散化的情况下，越来越多地在流量计算中发挥其核心方法的作用。大多数这些方法中嵌入的稳定性和直流参数起着重要作用。参数几乎总是涉及一些局部长度尺度表达式，大部分时间在特定方向上，例如流动方向或溶液梯度。直到最近，最初用于有限元离散化的局部长度尺度表达式也被用于等几何离散化。在 [Y. Otoguro、K. Takizawa 和 TE Tezduyar，复杂几何的 B 样条网格中的元素长度计算，Comput。机甲。65 (2020) 1085–1103, https://doi.org/10.1007/s00466-019-01809-w ] 针对复杂几何形状的 B 样条网格。推导这些表达式的关键阶段是将方向向量从物理元素映射到参数空间中的父元素，考虑沿每个参数坐标的离散化间距，并将获得的内容映射回物理元素。这些表达式基于首选参数空间和表示积分和首选参数空间之间关系的变换张量。在多种情况下，单元分割可能是计算方法的一部分，包括使用 T 样条离散化和浸入边界的计算以及扩展的有限元方法及其等几何版本。我们不希望元素分裂影响由控制点或节点表示的实际离散化。因此，局部长度尺度对于元素分裂应该是不变的。在元素定义中，局部长度尺度的不变性是一个关键要求，因为与基于实现方便或计算效率的元素定义选择不同，它会影响解决方案。我们在 B 样条网格的上下文中为上述参考文献中介绍的局部长度尺度表达式的元素分割不变性提供了一个证明。局部长度尺度的不变性是一个关键要求，因为与基于实现便利性或计算效率的元素定义选择不同，它会影响解决方案。我们在 B 样条网格的上下文中为上述参考文献中介绍的局部长度尺度表达式的元素分割不变性提供了一个证明。局部长度尺度的不变性是一个关键要求，因为与基于实现便利性或计算效率的元素定义选择不同，它会影响解决方案。我们在 B 样条网格的上下文中为上述参考文献中介绍的局部长度尺度表达式的元素分割不变性提供了一个证明。