Implementation and validation of an isogeometric hierarchic shell formulation

Within this thesis, thin walled shell structures are discussed with modern element formulationsin the context of the Isogeometric Analysis (IGA). IGA was designed to achieve a directinterface from CAD to analysis. According to the concept of IGA, Non-Uniform RationalB-Splines (NURBS) are used as shape functions in the design and the analysis. Dependingon the polynomial order, NURBS can come along with a high order continuity. Therefore,the curvature of a shell surface can be described directly by the shape function derivativeswhich is not possible within the classical Finite Element Analysis (FEA) using linear meshes.This description of the curvature gives rise to the application of the Kirchho-Love shellformulation, which describes the curvature stiness with the dierentiation of the spatialdegrees of freedom. Based upon this, the formulation can be enhanced with further kinematicalexpressions as the shear dierence vector, which leads to a 5-parameter Reissner-Mindlinformulation. This kinematic formulation is intrinsically free from transverse shear lockingdue to the split into Kirchho-Love and additional shear contributions. The formulation canbe further extended to a 7-parameter three-dimensional shell element, which considers volumetriceects in the thickness direction. Two additional parameters are engaged to describethe related thickness changes under load and to enable the use of three-dimensional materiallaws. In general, three-dimensional shell elements suer from curvature thickness and Poisson'sthickness locking. However, these locking phenomena are intrinsically avoided by thehierarchic application of the shear dierence vector and the 7th parameter respectively. The3-parameter Kirchho-Love, the 5-parameter Reissner-Mindlin and the 7-parameter 3D shellelement build a hierarchic family of model-adaptive shells.This hierarchic family of shell elements is presented and discussed in the scope of this thesis.The concept and the properties of the single elements are elaborated and the dierences arediscussed. Geometrically linear and non-linear benchmark examples are simulated. Convergencestudies are performed and the results are validated against analytical solutionsand solutions from literature, taking into account deections and internal forces. Furthermore,the dierent locking phenomena which occur in analyses with shell formulations areexamined. Several test cases are designed to ensure a validated implementation of the hierarchicshell elements. The element formulations and further pre- and postprocessing featuresare implemented and validated within the open-source software environment Kratos Multi-physics