Isogeometric analysis for thin-walled composite structures

Abstract

The conceptual ideas behind isogeometric analysis (IGA) are aimed at unifying computer aided design (CAD) and finite element analysis (FEA). Isogeometric analysis employs the non-uniform rational B-spline functions (NURBS) used for the geometric description of a structure to approximate its physical response in an isoparametric sense. Due to the tensor product property of multi-variate NURBS, it is difficult to represent complex topological shapes with a single NURBS patch. Multiple, often non-conforming patches are needed to tackle increasing complexity of the geometry. To further facilitate the modeling of complex shapes and geometric features trimming technology is widely used in CAD software, however, the trimmed domain is only visually unseen and the trimming features can not be utilized directly for the analysis. To overcome these difficulties, extra efforts are needed to make isogeometric methods adapted to engineering related cases. Thin-walled structures, such as plates and shells, excel in optimal load-carrying behavior and are of major importance in the design of aerospace components and the automotive engineering. Isogeometric analysis is an ideal candidate for the modeling and simulation of shell structures, especially for rotation-free Kirchhoff-Love type shells, which profit from the exact description of the geometry and from the higher continuity properties of NURBS. Furthermore, it favorably supports continuity requirements for flexible through-the-thickness design of laminate composites. Laminated composite materials are increasingly used in the aerospace industry this asks for reliable and computationally efficient lamina theories. The classical lamination theory belongs to the class of equivalent-single-layer methods (ESL), it is computationally efficient but often fails to capture the 3D stress state accurately. The demand for an accurate 3D stress state within laminates is mainly driven by the need to identify and to evaluate potential damage of lamina structures. While a full 3D layerwise (LW) model is computationally expensive, a combined approach considering both concepts, ESL and LW, seems to be a natural choice to tackle the computational costs of increasing model size and model complexity. In this thesis, a layerwise method for laminated composite structures is proposed in the framework of isogeometric analysis. A highly accurate prediction of the state of stress for thick and moderately thick laminate composite shells including transverse normal and shear stresses is demonstrated. The layerwise theory is successfully extended to linear buckling analysis of delaminated composites where a contact formulation is added to eliminate physically inadmissible buckling states which may result from overlapping plies. Furthermore, a Nitsche type formulation is introduced to enforce both weakly, essential boundary conditions and multi-patch coupling constraints for trimmed and non-conforming isogeometric rotation-free Kirchhoff-Love shell patches. The proposed formulation is variationally consistent and excels in a high level of stability and accuracy. A built-in stabilization, used to ensure coercivity of the formulation, prevents ill-conditioning of the physical problem. The inherent trimming problem is tackled with a fictitious domain extension for the trimming domain following the principles of the finite cell method to facilitate the workflow for geometrically complex structures in engineering practice. Computational efficiency is significantly increased with a blended coupling, taking continuum-like shell elements and thin shells elements, according to the theory of Kirchhoff-Love, into account. The blended approach provides access to the full 3D state of stress within selected subdomains while preserving the computational efficiency of the overall analysis.