  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.
