The Shifted Boundary Collocation Immersogeometric Method: an Application to Poisson Problems

Classical mesh-based numerical methods, such as the finite element methods via the Galerkin weak formulation, suffer from complex and computationally expensive meshing procedures, especially for irregular geometries with holes, trimmed boundary surfaces and singularities. In contrast, immersed boundary methods use a background regular mesh that does not fit the boundary representation, eliminating classical conforming meshing. A problem arises for cut finite elements due to the demand for numerical integration, the imposition of essential Dirichlet boundary conditions, and the stabilization of the small cut elements or the supports of the used basis functions. These drawbacks are removed with the application of the shifted boundary finite element method where the cut elements are discarded from the analysis, but imposition of boundary conditions is replaced with internal surrogate boundaries. In this paper, we present a novel form of the shifted boundary method applying the procedure based on strong formulation. A system of equations is formed by the collocation method where differential equation is satisfied in internal Greville collocation points on regular background grid, and all the boundary conditions are satisfied only in the collocation points of the internal surrogate boundaries via a Taylor series expansion or an equivalent cut element polynomial up to the order of the Fup (spline) basis functions used. The methodology is demonstrated on two-dimensional Poisson examples, attaining the same convergence rate as other collocation procedures, which proves that a shifted imposition of boundary conditions does not reduce the accuracy or efficiency of the proposed immersogeometric method.