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

Toshniwal, D.; Hughes, Thomas J.R.

doi:10.1016/j.cma.2020.113576

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

Title

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

Author

Toshniwal, D. (TU Delft Numerical Analysis)
Hughes, Thomas J.R. (Oden Institute for Computational Engineering and Sciences; The University of Texas at Austin)

Date

2021

Abstract

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.

Subject

Optimal approximation
Pointwise incompressibility
Singularly parametrized surfaces
Smooth splines
Surface flows
The de Rham complex

To reference this document use:

http://resolver.tudelft.nl/uuid:ac1c2488-c6d2-49c0-b936-80f44ace0c85

DOI

https://doi.org/10.1016/j.cma.2020.113576

ISSN

0045-7825

Source

Computer Methods in Applied Mechanics and Engineering, 376, 1-44

Part of collection

Institutional Repository