An approximate C1 multi-patch space for isogeometric analysis with a comparison to Nitsche’s method

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 [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 $C^0$. 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 [2], [3], [4], [5], [6]. 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 [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 $C^1$ 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.