Scaled boundary isogeometric analysis with advanced features for trimmed objects, higher order continuity, and structural dynamics

• Applicants: Professor Dr.-Ing. Sven Klinkel; Professor Dr. Bernd Simeon
• Subject Area: Applied Mechanics, Statics and Dynamics ; Mathematics
• Term: from 2015 to 2021
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 285973342

This follow-up research proposal is concerned with the analysis of advanced geometry representations of solids, such as trimmed objects, by employing the isogeometric scaled boundary approach. Developed in the first phase of the project, this approach enables a boundary oriented modelling of solids, which is in full accordance with the isogeometric paradigm. In the scaled boundary approach, the solid is split into sections in relation to its boundary surfaces and the scaling centre. This is conceptually different to the standard 3D-patch definition where a tri-variate tensor product representation is assumed. While the displacement interpolation on the interface between the sections is only C0-continuous, the discretization within adjacent sections can be of higher order and conforming or non-conforming. Therefore, we seek for a general method to couple adjacent sections while preserving higher-order continuity on the interface. In CAD, solids are defined through the definition of their bounded surfaces. Commonly these surfaces overlap and the kernel of all surfaces represents the solid. An issue that may occur here affects the degrees of freedom of adjacent surfaces. Due to the lack of shared control points, these degrees of freedom might not be coupled along the intersection. Using the isogeometric scaled boundary approach, we are aiming for a method which provides higher continuity for the displacement approximation on the intersections. Based on the continuity requirement, a relation is derived between the degrees of freedom acting on the intersection. Different approaches for the enforcement of the continuity constraint are discussed, such as a collocation or the mortar approach. Furthermore, the derivation of a higher-order coupling approach implies a modification of the basis functions. To this end, a master-slave framework is suggested that allows the derivation of a modification matrix in a least square setting. The derived methodology is applied to different types of sections such as conforming, hierarchical and non-conforming sections. One advantage of the Ck-continuous formulation lies in the reduction of optical branches present in structural dynamics, e.g. the finite element analysis. Higher-order continuity is in particular promising for problems in structural dynamics. The extension of the methodology to time-dependent problems enables the assessment of higher-continuity in the spatial dimension while the differential-algebraic extension of the α-method forms the basis for the time integration scheme.The methodology contributes towards providing a general approach for isogeometric analysis that can handle a wide class of geometric features and complex multi-patch constellations. A further aim of this project is to exploit the benefits of increased continuity in order to provide an accurate and robust numerical analysis framework for dynamic problems in solid mechanics.

• DFG Programme: Research Grants