A locally based construction of analysis-suitable $G^1$ multi-patch spline surfaces

TL;DR

This paper introduces a local method using Lagrange multipliers to construct analysis-suitable G1 multi-patch spline surfaces, enabling better isogeometric analysis of complex PDEs with improved convergence.

Contribution

A novel local approach for designing analysis-suitable G1 multi-patch spline surfaces using Lagrange multipliers, approximating non-AS-G1 surfaces effectively.

Findings

Method successfully constructs AS-G1 surfaces from non-AS-G1 surfaces.

Surfaces demonstrate optimal convergence rates in biharmonic problem solutions.

Approach is simple and effective for isogeometric analysis applications.

Abstract

Analysis-suitable $G^1$ (AS-$G^1$) multi-patch spline surfaces [4] are particular $G^1$-smooth multi-patch spline surfaces, which are needed to ensure the construction of $C^1$-smooth multi-patch spline spaces with optimal polynomial reproduction properties [16]. We present a novel local approach for the design of AS-$G^1$ multi-patch spline surfaces, which is based on the use of Lagrange multipliers. The presented method is simple and generates an AS-$G^1$ multi-patch spline surface by approximating a given $G^1$-smooth but non-AS-$G^1$ multi-patch surface. Several numerical examples demonstrate the potential of the proposed technique for the construction of AS-$G^1$ multi-patch spline surfaces and show that these surfaces are especially suited for applications in isogeometric analysis by solving the biharmonic problem, a particular fourth order partial differential equation, with optimal rates of convergence over them.