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

Subject Area Applied Mechanics, Statics and Dynamics
Mathematics
Term from 2015 to 2025
Project identifier Deutsche Forschungsgemeinschaft (DFG) - Project number 285973342
 
Final Report Year 2024

Final Report Abstract

The aim of the two-phase research project was to develop a robust numerical method that combines isogeometric analysis (IGA) and the Scaled Boundary Finite Element Method (SB-FEM) to precisely model complex geometries in linear and nonlinear solid mechanics and to close the gap between the geometric model and numerical analysis. This issue has already been addressed by isogeometric analysis. Nevertheless, the definition of suitable parametrizations, especially in complex domains, remains challenging and time-consuming. In the first phase of the project, the Scaled Boundary Isogeometric Analysis (SB-IGA) for CAD geometry descriptions was developed within the framework of the Boundary Representation Modelling Technique (BRMT). Non-Uniform Rational B-Splines (NURBS) were used for the geometry representation in both the radial and the scaling direction. It was shown how the SB-IGA compares to the classical Galerkin-based Finite Element Method (FEM), the stability was analysed and convergence investigations were carried out. It was shown that the developed methods can be applied to a broad class of problems, even taking into account sophisticated material laws of solid mechanics. Furthermore, the feasibility for non-conforming surface sections was demonstrated and a decomposition of involved domains into star-shaped blocks was analysed. In the second phase of the project, special requirements for the Scaled Boundary (SB) parametrization were investigated to ensure a higher smoothness across patch boundaries, a central aspect in particular for trimmed surfaces. Special attention was paid to the C1 continuity across the patch interfaces by special treatment of the NURBS test functions. The overall goal was to create a universal approach for isogeometric analysis that is applicable to a broad class of geometric features and complex multi-patch constellations. The applicability of the trimmed and C1-coupled SB parametrizations was demonstrated in the context of shells and plates with Kirchhoff theory. In addition to the question of higher regularity, an analysis framework was created for 3D solids in boundary representation for problems in nonlinear structural dynamics. The approach of Scaled Boundary Isogeometric Analysis presented here represents a userfriendly method for the construction of finite element discretizations on curved surfaces by focusing on boundary-oriented modelling with the help of NURBS. Due to the interdisciplinary design in two working groups from civil engineering and applied mathematics, the vision for the complete simulation process could be considered.

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