Efficient analysis of shell structures defined in highly complex CAD models
Final Report Abstract
The main objective of this project is developing an efficient shell element for the large deformation analysis of complex thin-walled structures based on the geometry description of Computer-Aided design (CAD) software. The intended field of application are CAD models of built structures, but applications in other fields of engineering are akin possible. The basic idea is to combine exact geometry description by Non-Uniform Rational B-splines (NURBS) with the interesting properties of the spectral element method (SEM), where nodal points, i.e. discrete points where all unknown quantities are defined, and integration points, i.e. points where the relevant equations are evaluated, coincide. The location of these points and the nodal normal vectors are computed exactly from the CAD model. The discretization effort is kept minimal since the patch-wise definition of the CAD domains is retained. Interpolation of unknown quantities and derivatives thereof are obtained using Lagrange basis functions, which provide a proper and stable base also for high order computations, while the use of NURBS basis functions for the description of the shape of the elements removes the geometrical approximation error. The implemented SEM shell formulation overcomes the major shortcomings of pure isogeometric (IGA) shell formulations, where both geometry and unknowns are interpolated by NURBS functions. The numerical examples show that the developed SEM shell yields stable condition numbers and significantly alleviates locking effects for high orders of basis functions. As expected, a simple rotational formulation is sufficient in SEM to obtain results with comparable accuracy as an IGA shell with complicated rotational formulation. In contrast to the initial hypothesis, we discovered that the intended non-isoparametric formulation using both Lagrange and NURBS basis functions is not required to obtain high accuracy results. The specially developed benchmark parcour has shown that the geometrical approximation error is much smaller than the interpolation error. Thus, we adapted the work programme of the project and implemented an isoparametric spectral shell element (SEMI) formulation. The results of the benchmark parcour revealed that for all examples, the SEMI shell performed as least as good as the IGA shell. Surprisingly, we discovered that especially for complicated geometries, i.e. high and highly changing curvature, the SEMI formulation performed more robust and accurate as the IGA shell. By exploiting the Kronecker-Delta property of SEM, we could show that significant gains in computational costs of element stiffness matrices are possible in comparison to IGA. The project provided a deep understanding of the significance of exact geometry representation, which are of high interest to the shell formulation community. In combination with the shown efficiency of SEM, this paves the way for high efficiency computations of complex geometries using SEM formulations.
Link to the final report
https://nbn-resolving.org/urn:nbn:de:hbz:386-kluedo-91304
Publications
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A spectral finite element Reissner–Mindlin shell formulation with NURBS-based geometry definition. Computational Mechanics, 74(3), 537-559.
Azizi, Nima & Dornisch, Wolfgang
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Vergleich zwischen isogeometrischen und spektralen Reissner-Mindlin Schalenelementen. In B. Oesterle, A. Bögle und W. Weber (eds.): Berichte der Fachtagung Baustatik–Baupraxis 15, 333-340, 2024
W. Dornisch & N. Azizi
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A rotation-based geometrically nonlinear spectral Reissner–Mindlin shell element. Finite Elements in Analysis and Design, 251, 104416.
Azizi, Nima & Dornisch, Wolfgang
