Shape-Newton Method for Isogeometric Discretizations of Free-Boundary Problems

We derive Newton-type solution algorithms for a Bernoulli-type freeboundary problem at the continuous level. The Newton schemes are obtained by applying Hadamard shape derivatives to a suitable weak formulation of the freeboundary problem. At each Newton iteration, an updated free boundary position is obtained by solving a boundary-value problem at the current approximate domain. Since the boundary-value problem has a curvature-dependent boundary condition, an ideal discretization is provided by isogeometric analysis. Several numerical examples demonstrate the apparent quadratic convergence of the Newton schemes on isogeometric-analysis discretizations with C1-continuous discrete free boundaries.