An IGA Discontinuous Galerkin Method on the union of overlapped patches

In Isogeometric Analysis (IGA), the Discontinuous Galerkin method (IGA DG) is often applied when the domain is composed of multiple Non-Uniform Rational B-splines (NURBS) patches. These patches are often required to be C0-continuous to make the IGA DG available, and this requirement brings some problems to the CAD process. This paper extends the IGA DG method to solve problems on the union of the overlapped patches. With this method (IGAODG), a problem on the overlapped domain is seen as multiple sub-problems on the non-trimmed patches, and then the continuities of traces of the numerical solutions are penalized on the boundaries of non-trimmed elements, with the coefficients deduced from the classical DG method. Comparing with the IGA methods which apply the CAD Boolean operations to deal with the overlapped domains, the presented method carries out the integrations in the non-trimmed patches, thus no additional integration scheme needs to be considered to achieve the Gaussian integration precision. Discussions on the presented discretization scheme and its convergence analysis show that this method is available on the domain created by the union of the matched, mismatched and overlapped patches. As predicted in the convergence analysis, the presented method gives the optimal convergence rates in our numerical examples. Finally, through one of its engineering applications, this paper shows how to extend IGAODG to solve problems that are defined on the trimmed patches.