Element length calculation in B-spline meshes for complex geometries

Yuto Otoguro, Kenji Takizawa*, Tayfun E. Tezduyar

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

19 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)1085-1103
Number of pages19
JournalComputational Mechanics
Issue number4
Publication statusPublished - 2020 Apr 1


  • Advection–diffusion equation
  • Discontinuity-capturing parameter
  • Element length
  • Isogeometric discretization
  • Preferred parametric space
  • Space–time computational methods
  • Stabilization parameter

ASJC Scopus subject areas

  • Computational Mechanics
  • Ocean Engineering
  • Mechanical Engineering
  • Computational Theory and Mathematics
  • Computational Mathematics
  • Applied Mathematics


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