Isogeometric analysis with [formula omitted]-smooth functions over multi-patch surfaces

Abstract

We present a framework for the construction of a globally C1-smooth isogeometric spline space over a particular class of G1-smooth multi-patch surfaces called analysis-suitable G1 (in short AS-G1) multi-patch surfaces. The class of AS-G1 multi-patch surfaces consists of those G1-smooth multi-patch surfaces which allow the construction of C1-smooth isogeometric spline spaces with optimal polynomial reproduction properties, see [1]. Our method extends the work [2], which is limited to the case of planar AS-G1 multi-patch parameterizations, to the case of AS-G1 multi-patch surfaces. The C1-smooth isogeometric spline space is generated as the span of locally supported and explicitly given basis functions of three different types that correspond to the patches, interfaces and vertices of the considered AS-G1 multi-patch surface. We further present simple and practical methods for the design of AS-G1 multi-patch surfaces and demonstrate the potential of the C1-smooth spline space for solving fourth order partial differential equations over AS-G1 multi-patch surfaces on the basis of the biharmonic equation. The obtained numerical results indicate convergence rates of optimal order in the L2-norm and in the H1- and H2-seminorms.