title: C1finite elements on non-tensor-product 2d and 3d manifolds.
abstract: Geometrically continuous (Gk) constructions naturally yield families of finite elements for isogeometric analysis (IGA) that areCkalso for non-tensor-product layout. This paper describes and analyzes one such concreteC1geometrically generalized IGA element (short: gIGA element) that generalizes bi-quadratic splines to quad meshes with irregularities. The new gIGA element is based on a recently-developedG1surface construction that recommends itself by its a B-spline-like control net, low (least) polynomial degree, good shape properties and reproduction of quadratics at irregular (extraordinary) points. Remarkably, for Poisson's equation on the disk using interior vertices of valence 3 and symmetric layout, we observeO(h3) convergence in theL∞norm for this family of elements. Numerical experiments confirm the elements to be effective for solving the trivariate Poisson equation on the solid cylinder, deformations thereof (a turbine blade), modeling and computing geodesics on smooth free-form surfaces via the heat equation, for solving the biharmonic equation on the disk and for Koiter-type thin-shell analysis.
authors.affiliation: Department CISE, University of Florida, USA.
authors.forename: Thien
authors.initials: T
authors.lastname: Nguyen
authors.affiliation: Department of Mathematics, Vilnius University, Lithuania.
authors.forename: Kęstutis
authors.initials: K
authors.lastname: Karčiauskas
authors.affiliation: Department CISE, University of Florida, USA.
authors.forename: Jörg
authors.initials: J
authors.lastname: Peters
keywords: Bi-quadratic; Extraordinary point; Geometric continuity; Irregular quad mesh; Isogeometric finite element; Isoparametric method; Multi-sided facets