An approximate [formula omitted] multi-patch space for isogeometric analysis with a comparison to Nitsche’s method

Abstract

We present an approximately C1-smooth multi-patch spline construction which can be used in isogeometric analysis (IGA). The construction extends the one presented in [1] 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 C0. Therefore, to obtain C1-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 C1 over multi-patch domains can be constructed as in [2–6]. The constructions require geometry parametrizations that satisfy certain constraints along the interfaces, so-called analysis-suitable G1 parametrizations. To allow C1 spaces over more general multi-patch parametrizations, one needs to increase the polynomial degree and/or to relax the C1 conditions. Thus, we define function spaces that are not exactly C1 but only approximately. We adopt the construction for two-patch domains, as developed in [1], 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 C1 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.