Abstract
Recently a novel solution procedure for the scaled boundary method was developed based on the theory of matrix functions and the real Schur decomposition. It has been proven that the base functions obtained from the Schur decomposition are weighted block-orthogonal. A reduced set of base functions can be constructed by retaining the terms with the smallest real parts of the eigenvalues, which requires only a partial Schur decomposition (a subset of the eigenvectors). Significant reduction in computation time is achieved without significant loss of accuracy. However, this approach has so far only been applied to unbounded domains where all the base functions automatically satisfy both Dirichlet and Neumann boundary conditions, and these boundary conditions are only applied on the side faces and at infinity. For bounded domains the boundary conditions are applied at the discretized boundary, and the method cannot be applied directly. To extend the reduced base function method, this paper proposes the use of a penalty approach. The approach is used to apply boundary conditions at the discretized boundary of bounded domains analyzed using the scaled boundary method with a reduced set of base functions. Greater advantages can be taken of the properties of the scaled boundary method if a sub-structuring approach is adopted, including improved modeling of complex geometries and more accurate modeling of boundary conditions. A similar penalty approach is used at the interface between scaled boundary sub-domains, allowing the reduced set of base functions procedure to be used in sub-structured domains. Results are presented for a number of benchmark problems to demonstrate the effectiveness of the method.