M2P 2023

Spline-Based Parameterisation Techniques and Applications to Networks

  • Hinz, Jochen (EPFL)
  • Buffa, Annalisa (EPFL)
  • Antolin, Pablo (EPFL)

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The field of computational engineering (CE) has seen major efforts into the consolidation of geometry and simulation by seeking rigorous numerical frameworks that are compatible with NURBS-based geometry representations. This has culminated in the advent of Isogeometric Analysis (IGA), a variant of FEM that utilises NURBS for both the geometrical and the numerical aspects of the CE workflow. Hereby, the NURBS-based Surface-to-Volume (StV) problem becomes the IGA-analogue to the classical meshing step. The purpose of this talk is discussing the recent advances in NURBS-based parameterisation techniques, focusing mainly on the generalisation of the classical step based on Harmonic Maps, a staple approach from the classical realm, to its NURBS-based counterpart . For this, this talk proposes robust PDE-based numerical algorithms for approximating harmonic maps based on the core-principles of IGA. Hereby, the PDE-based formulation enables an effortless integration of the StV step into a well-developed IGA codebase while furthermore enabling the parameterisation of domains that are topologically equivalent to any convex polygon, rather than only the unit quadrilateral. This generalisation is made possible by tackling the PDE-based StV problem based on its weak formulation, which is agnostic toward the connectivity properties of the division the parametric domain into patches (structured or unstructured). To validate the approach, this talk applies the IGA-based StV pipeline to complex spline networks which are collections of spline curves forming loops that fence-off various faces. Hereby, each face needs to be parameterised individually in order to, for instance, locally impose differing material parameters. We discuss strategies that retain element conformity across the faces and techniques to build desired features into the harmonic map, such as boundary layers and parametric continuity across patch interfaces. The results of applying these techniques to a real-world industrial application are presented and discussed in detail.