TY - JOUR

T1 - Isogeometric discrete differential forms: Non-uniform degrees, Bezier extraction, polar splines and flows on surfaces

T2 - Non-uniform degrees, Bézier extraction, polar splines and flows on surfaces

AU - Toshniwal, Deepesh

AU - Hughes, Thomas J.R.

PY - 2021

Y1 - 2021

N2 - 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.

AB - 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.

KW - Optimal approximation

KW - Pointwise incompressibility

KW - Singularly parametrized surfaces

KW - Smooth splines

KW - Surface flows

KW - The de Rham complex

UR - http://www.scopus.com/inward/record.url?scp=85099615392&partnerID=8YFLogxK

U2 - 10.1016/j.cma.2020.113576

DO - 10.1016/j.cma.2020.113576

M3 - Article

VL - 376

SP - 1

EP - 44

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0045-7825

M1 - 113576

ER -