Variational multiscale methods, and their precursors, stabilized methods, have been playing a core-method role in semi-discrete and space–time (ST) flow computations for decades. These methods are sometimes supplemented with discontinuity-capturing (DC) methods. The stabilization and DC parameters embedded in most of these methods play a significant role. Various well-performing stabilization and DC parameters have been introduced in both the semi-discrete and ST contexts. The parameters almost always involve some element length expressions, most of the time in specific directions, such as the direction of the flow or solution gradient. Until recently, stabilization and DC parameters originally intended for finite element discretization were being used also for isogeometric discretization. Recently, element lengths and stabilization and DC parameters targeting isogeometric discretization were introduced for ST and semi-discrete computations, and these expressions are also applicable to finite element discretization. The key stages of deriving the direction-dependent element length expression were mapping the direction vector from the physical (ST or space-only) 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. Targeting B-spline meshes for complex geometries, we introduce here new element length expressions, which are outcome of a clear and convincing derivation and more suitable for element-level evaluation. The new expressions are based on a preferred parametric space and a transformation tensor that represents the relationship between the integration and preferred parametric spaces. The test computations we present for advection-dominated cases, including 2D computations with complex meshes, show that the proposed element length expressions result in good solution profiles.

中文翻译：

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复杂几何的 B 样条网格中的单元长度计算

变分多尺度方法及其前身稳定方法几十年来一直在半离散和时空 (ST) 流计算中发挥核心方法作用。这些方法有时会辅以不连续捕获 (DC) 方法。大多数这些方法中嵌入的稳定性和 DC 参数起着重要作用。在半离散和 ST 环境中都引入了各种性能良好的稳定性和 DC 参数。参数几乎总是涉及一些元素长度表达式，大部分时间是在特定方向上，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稳定性和 DC 参数也被用于等几何离散化。最近，为 ST 和半离散计算引入了针对等几何离散化的单元长度和稳定性以及 DC 参数，这些表达式也适用于有限元离散化。推导与方向相关的元素长度表达式的关键阶段是将方向向量从物理（ST 或仅空间）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，以及将获得的东西映射回物理元素。针对复杂几何的 B 样条网格，我们在此引入了新的元素长度表达式，这些表达式是清晰且令人信服的推导结果，更适合于元素级评估。新表达式基于首选参数空间和表示积分与首选参数空间之间关系的变换张量。我们针对以对流为主的情况进行的测试计算，包括具有复杂网格的 2D 计算，表明所提出的单元长度表达式可产生良好的解决方案剖面。