# Isogeometric analysis using G-spline surfaces with arbitrary unstructured quadrilateral layout

## TL;DR

This paper introduces novel G-spline surface constructions with extraordinary points for isogeometric analysis, enabling flexible, high-quality surface representations with global C^1 continuity on arbitrary unstructured quadrilateral layouts, suitable for complex structural analysis.

## Contribution

The work presents the first G-spline EP constructions for IGA that use only vertex control points, avoid singularities, and support arbitrary EP placement, enhancing flexibility and surface quality.

## Key findings

- G-splines outperform D-patch based EP constructions in convergence and surface quality.
- Successful application to B-pillar structural analysis with excellent agreement to finite element results.
- Flexible handling of interior and boundary EPs without element splitting or singularities.

## Abstract

G-splines are a generalization of B-splines that deals with extraordinary points by imposing G^1 constraints across their spoke edges, thus obtaining a continuous tangent plane throughout the surface. Using the isoparametric concept and the Bubnov-Galerkin method to solve partial differential equations with G-splines results in discretizations with global C^1 continuity in physical space. Extraordinary points (EPs) are required to represent manifold surfaces with arbitrary topological genus. In this work, we allow both interior and boundary EPs and there are no limitations regarding how close EPs can be from each other. Reaching this level of flexibility is necessary so that splines with EPs can become mainstream in the design-through-analysis cycle of complex thin-walled structures. To the authors' knowledge, the two EP constructions based on imposing G^1 constraints proposed in this work are the first two EP constructions used in isogeometric analysis (IGA) that combine the following distinctive characteristics: (1) Only vertex-based control points are used and they behave as geometric shape handles, (2) any control point of the control net can potentially be an EP, (3) global C^1 continuity in physical space is obtained without introducing singularities, (4) faces around EPs are not split into multiple elements, and (5) good surface quality is attained. The studies of convergence and surface quality performed in this paper suggest that G-splines are more suitable for IGA than EP constructions based on the D-patch framework. Finally, we have represented the stiffener, the inner part, and the outer part of a B-pillar with G-spline surfaces and solved eigenvalue problems using both Kirchhoff-Love and Reissner-Mindlin shell theories. The results are compared with bilinear quadrilateral meshes and excellent agreement is found between G-splines and conventional finite elements.