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.

翻译：分析适用$G^1$（AS-$G^1$）多片样条曲面[4]是一类特殊的$G^1$光滑多片样条曲面，这类曲面对于构建具有最优多项式再生特性的$C^1$光滑多片样条空间[16]是必要的。本文提出了一种基于拉格朗日乘子法的局部化设计方法，用于构造AS-$G^1$多片样条曲面。该方法简洁易行，通过对给定$G^1$光滑但非AS-$G^1$的多片曲面进行逼近，生成AS-$G^1$多片样条曲面。多个数值算例展示了所提技术在构造AS-$G^1$多片样条曲面方面的潜力，并通过求解双调和问题（一种四阶偏微分方程）证明了这类曲面特别适用于等几何分析，可在此类曲面上实现最优收敛速率。