$C^1$ isogeometric spline space for trilinearly parameterized multi-patch volumes

TL;DR

This paper develops a general framework for constructing $C^1$ isogeometric spline spaces on trilinearly parameterized multi-patch volumes, enabling improved geometric modeling and analysis of complex 3D shapes.

Contribution

It introduces a unified method for designing $C^1$ spline spaces on multi-patch volumes, extending previous two-patch approaches to more complex configurations.

Findings

Successfully constructed a $C^1$ spline basis for multi-patch volumes.

Numerically computed the dimension of the $C^1$ spline space.

Explored approximation properties through $L^2$ tests.

Abstract

We study the space of $C^1$ isogeometric spline functions defined on trilinearly parameterized multi-patch volumes. Amongst others, we present a general framework for the design of the $C^1$ isogeometric spline space and of an associated basis, which is based on the two-patch construction [7], and which works uniformly for any possible multi-patch configuration. The presented method is demonstrated in more detail on the basis of a particular subclass of trilinear multi-patch volumes, namely for the class of trilinearly parameterized multi-patch volumes with exactly one inner edge. For this specific subclass of trivariate multi-patch parameterizations, we further numerically compute the dimension of the resulting $C^1$ isogeometric spline space and use the constructed $C^1$ isogeometric basis functions to numerically explore the approximation properties of the $C^1$ spline space by performing $L^2$ approximation.