Local-length-scale calculation in T-splines meshes for complex geometries

Abstract

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. The direction-dependent expressions introduced earlier target B-splines 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. Here, we extend the local-length-scale calculation method to meshes built from T-splines. T-splines meshes are a superset of B-splines meshes. They provide smooth basis functions in complex geometry and effective refinement without subdividing where we do not need higher resolution. In this article, we focus on the product form T-splines basis functions. They are represented individually in product form, from multiplication of [Formula: see text] 1D basis functions, where [Formula: see text] is the number of parametric dimensions. Each 1D basis function comes from the set of functions associated with one of the parametric directions and the set of functions is defined considering the T-splines nature of the mesh. The product-form basis functions satisfy the partition of unity without using rational functions. For these T-splines, based on the method introduced for B-splines, the local length scales are calculated with Bézier-extraction row operators, which are element-level constants. Using T-splines involves element splitting also for increased integration accuracy. Our local-length-scale expressions are invariant with respect to element splitting performed for integration accuracy but account for the element splitting that is for enhancing the function space.