Element Length Calculation for Isogeometric Discretization and Complex Geometries

Flow computations with semi-discrete and space–time (ST) methods have been relying on, as core methods, variational multiscale methods and, more generally, stabilized methods. As needed, these methods are supplemented with discontinuity-capturing (DC) methods. Most of these methods have some embedded stabilization and DC parameters. These parameters play an important role and need to be defined carefully. Many well-performing parameters have been introduced over the years 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. In late 2017, element lengths and stabilization and DC parameters targeting isogeometric discretization were introduced for ST and semi-discrete computations, and they are of course 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. In late 2019, targeting B-spline meshes for complex geometries, new element length expressions, which are outcome of a clear and convincing derivation and more suitable for element-level evaluation, were introduced. 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. In this chapter, we provide an overview of these new element length expressions and the test computations performed with them. The test computations, which include advection-dominated problems in 2D and aerodynamics of a tsunami-shelter vertical-axis wind turbine, show that the new element length expressions result in good solution profiles and can be used in complex-geometry flow computations.