Efficient analysis of shell structures defined in highly complex CAD models

• Applicant: Professor Dr.-Ing. Wolfgang Dornisch
• Subject Area: Applied Mechanics, Statics and Dynamics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 503246947

Thin-walled structures are ubiquitous both in nature and in the built environment due to their advantageous ratio of own weight to payload. The analysis of such structures is commonly performed using shell formulations in the frame of the finite element method. The basic assumption for shell formulations is a dimensional reduction from the three-dimensional continuum to a two-parametrical surface in space. The thickness of this surface is defined by a vector field, which is referred to as director vector. The definition of complex thin-walled structures in current design software bases on this dimensional reduction. However, with non-uniform rational B-splines (NURBS) a different geometry description than in the finite element method is used. Isogeometric analysis tries to unify design and analysis by using the NURBS geometry description as a basis for the analysis with the ultimate goal of obtaining more efficient planning and analysis processes. The numerous publications about this method show the high potential of isogeometric analysis, but in some details, robust and efficient solutions are still lacking. Examples to be mentioned are the interpolation of the director vector or the analysis of trimmed surfaces. In this project, a combination of the basis functions of the spectral element method with a NURBS geometry description is proposed in order to overcome these shortcomings. The exact geometry description of isogeometric analysis is retained, while unknown quantities are interpolated using the Lagrange basis functions of the spectral element method, where integration points and nodal points always coincide. This simplifies the interpolation of the deformed director vector significantly and precludes artificial thinning of the structure. Trimmed subdomains are meshed by quadrilateral and triangular elements in the immediate vicinity of the trimming curve, which is directly used as element edge. The small support issue and the resulting ill-conditioning of the global stiffness matrix is entirely avoided by the reparametrization with the spectral basis functions, which have regular support. The subdomain structure of the NURBS geometry model is retained and the patches are meshed independently from each other. This yields a straightforward meshing procedure, which requires only very low computational resources. The coupling of the individual subdomains is established using a mortar method, which enforces the equality of mutual deformations along shared edges in a weak manner. Extensive verification using relevant benchmarks as well as comparisons to isogeometric and conventional shell elements will reveal the robustness and efficiency of the proposed simulation framework.

• DFG Programme: Research Grants