Scaled boundary isogeometric analysis with advanced for trimmed objects, higher order continuity, and structural dynamics
The research project deals with an isogeometric analysis of solids described by complex geometries, such as trimmed objects. The method is based on the isogeometric scaled boundary formulation, which was developed in the first phase of the project and is consistent with the isogeometric concept. In the scaled boundary method, solids are defined by their surfaces. It is conceptually different from the usual 3D patch definition. With reference to the surfaces, the solids are divided into sections.
Adjacent sections can be discretised conformally or non-conformally. The approximation of the displacements within a section can be of higher order, along the boundaries of adjacent sections it is only C0-steady. Therefore, this project aims to develop a method that will provide higher continuity between sections. Solids are described in CAD by their surfaces. I. Generally, the surface descriptions intersect, with the body defined as the kernel of the enveloping surfaces. A challenge can be the treatment of the degrees of freedom of adjacent surfaces when there are no common control points and thus the degrees of freedom along the intersection are not coupled. Based on the isogeometric scaled boundary method, a method is aimed at which allows a higher continuity of the displacement approximation along the intersection. For this purpose, a relation between the degrees of freedom acting on the intersection is derived. Different possibilities for ensuring continuity, such as the collocation and mortar methods, are discussed. Ck continuity along the intersections is to be achieved by a modification of the NURBS approach functions. Starting from the continuity condition, a master-slave relation is to be derived, which is used to determine the modification. The method is to be designed for different cases, such as conformal, hierarchical and non-conformal discretisations. One advantage of a Ck-continuous discretisation is the reduction of optical branches, which occur e.g. in the finite element method for dynamic investigations. The method is to be extended to time-dependent problems in order to be able to assess the advantages of the higher continuity. The differential-algebraic extension of the α-method serves as time integration method. The project shall contribute to develop a general approach for isogeometric analysis applicable to a wide class of geometric features and complex multi-patch constellations. The aim is to take advantage of the increased continuity to provide an accurate and robust numerical method for the analysis of dynamic problems in solid mechanics.