Isogeometric discrete differential forms: Non-uniform degrees, Bézier extraction, polar splines and flows on surfaces

• Published in: Computer methods in applied mechanics and engineering Vol. 376; p. 113576
• Main Authors: Toshniwal, Deepesh, Hughes, Thomas J.R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.04.2021; Elsevier BV
• Subjects: Algorithms; Bezier patches; Computational fluid dynamics; Conservation laws; Degree elevation; Fluid flow; Fluid mechanics; Incompressible flow; Mathematical analysis; Numerical methods; Optimal approximation; Partial differential equations; Pointwise incompressibility; Robustness (mathematics); Singularly parametrized surfaces; Smooth splines; Stokes flow; Surface flows; Tensors; The de Rham complex; Pointwise incompressibility; Optimal approximation; Smooth splines; Surface flows; The de Rham complex; Singularly parametrized surfaces
• ISSN: 0045-7825; 1879-2138
• DOI: 10.1016/j.cma.2020.113576

Spaces of discrete differential forms can be applied to numerically solve the partial differential equations that govern phenomena such as electromagnetics and fluid mechanics. Robustness of the resulting numerical methods is complemented by pointwise satisfaction of conservation laws (e.g., mass conservation) in the discrete setting. Here we present the construction of isogeometric discrete differential forms, i.e., differential form spaces built using smooth splines. We first present an algorithm for computing Bézier extraction operators for univariate spline differential forms that allow local degree elevation. Then, using tensor-products of the univariate splines, a complex of discrete differential forms is built on meshes that contain polar singularities, i.e., edges that are singularly mapped onto points. We prove that the spline complexes share the same cohomological structure as the de Rham complex. Several examples are presented to demonstrate the applicability of the proposed methodology. In particular, the splines spaces derived are used to simulate generalized Stokes flow on arbitrarily curved smooth surfaces and to numerically demonstrate (a) optimal approximation and inf–sup stability of the spline spaces; (b) pointwise incompressible flows; and (c) flows on deforming surfaces. •We build discrete differential form spaces using univariate and bivariate splines.•In 1D, Bezier extraction algorithms for multi-degree 0- and 1-form splines are provided.•In 2D, we build 0-, 1- and 2-form splines on meshes with polar singularities.•Pointwise incompressible flows on stationary and deforming genus 0-surfaces are simulated.•The examples demonstrate optimal approximation by and inf–sup stability of the spline spaces.