An approximate $C^1$ multi-patch space for isogeometric analysis with a comparison to Nitsche's method

Abstract

We present an approximately C 1 -smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). The construction extends the one presented in [42] for two-patch domains. A key property of IGA is that it is simple to achieve high order smoothness within a single patch. However, to represent more complex geometries one often uses a multi-patch construction. In this case, the global continuity for the basis functions is in general only C 0 . Therefore, to obtain C 1 -smooth isogeometric functions, a special construction for the basis is needed. Such spaces are of interest when solving numerically fourth-order problems, such as the biharmonic equation or Kirchhoff-Love plate/shell formulations, using an isogeometric Galerkin method.Isogeometric spaces that are globally C 1 over multi-patch domains can be constructed as in [9,[19][20][21][22]. The constructions require geometry parametrizations that satisfy certain constraints along the interfaces, so-called analysis-suitable G 1 parametrizations. To allow C 1 spaces over more general multi-patch parametrizations, one needs to increase the polynomial degree and/or to relax the C 1 conditions. Thus, we define function spaces that are not exactly C 1 but only approximately. We adopt the construction for two-patch domains, as developed in [42], and extend it to more general multi-patch domains.We employ the construction for a biharmonic model problem and compare the results with Nitsche's method. We compare both methods over complex multi-patch domains with non-trivial interfaces. The numerical tests indicate that the proposed construction converges optimally under h-refinement, comparable to the solution using Nitsche's method. In contrast to weakly imposing coupling conditions, the approximate C 1 construction is explicit and no additional terms need to be introduced to stabilize the method/penalize the jump of the derivative at the interface. Thus, the new proposed method can be used more easily as no parameters need to be estimated.