Irslinger, J. (2013).

*Mechanische Grundlagen und Numerik dreidimensionaler Schalenelemente*. Doktorarbeit, Bericht Nr. 61, Institut für Baustatik und Baudynamik, Universität Stuttgart.

https://doi.org/10.18419/opus-511
#### Zusammenfassung

The present thesis covers several significant aspects of the simulation of shell structures. First, the mechanical fundamentals of shell theories are discussed with the primary objective to establish a link between the assumptions of three-dimensional finite shell element formulations and the shell theories they are based on. Subsequently, the formulation of a robust three-dimensional solid-like shell element is presented. Finally, it is analysed, to which extent several approaches for the setup of preconditioners based on substitute problems are suitable for an efficient iterative solution of ill-conditioned equation systems arising from shell simulations.
The discussion of model assumptions of classical shell theories is the first important topic of this work. Therefore, the consistent kinematic and constitutive equations of a shear-deformable shell formulation (Reissner-Mindlin kinematics) and a shear-rigid shell formulation (Kirchhoff-Love kinematics) are derived from the continuum in a first step. Kinematic assumptions and modification of the material law, which is mandatory for the asymptotic correctness of those theories, are discussed in detail. The shell theories of Love and Koiter as well as the theory of shallow shells, which can be considered historic from today’s perspective, are presented and compared with regard to the kinematic and constitutive equations. Additionally, numerical analyses of the properties of these formulations based on the Method of Manufactured Solutions are performed (see for instance Roache (2002)). It turns out, that the shell theory of Love and the theory of shallow shells are not asymptotically correct. Although this is true for the formulation of Koiter, so-called „best formulations“, as for example the one of Naghdi (1963b), often show a faster convergence and are therefore asymptotically better.
In contrast to classical shell theories, three-dimensional shell theories are able to represent complete three-dimensional strain and stress states, and thus modifications of the material law can be omitted. Shell finite elements discretising these kinds of theories are nowadays usually derived on the basis of the concept of degeneration. Alternatively, standard continuum elements are optimised for shell simulations by additional element technology. Therefore, many of the assumptions made in element formulations lack the link to shell theories. Hence, a further essential objective of this work is to identify these relations in order to figure out the influence of those assumptions and thus to gain knowledge for the design of three-dimensional shell element formulations. For a 7-parameter shell kinematics it can be shown in numerical experiments, that if the accuracy of classical shell formulations shall be increased beyond the inclusion of thickness change, accounting for the quadratic strain components is crucial. Only in this case the incorporation of curvature in the material equations and in the shifter has a positive effect on the accuracy. To ensure that this additional information enters the shell formulation, it must be detected by the numerical thickness integration, which in this thesis is considered as part of the shell theory and not of an element formulation. It becomes apparent, that although a 2-point Gauss quadrature is sufficient for asymptotic correctness, this additional information is mainly lost. Only with three or more Gauss points across the thickness asymptotically better shell theories are obtained.
Special focus of this work is on the development and implementation-related documentation of a robust three-dimensional solid-like shell element, whose topology is identical to a trilinear 8-node continuum element. Rank-sufficient integration substantially contributes to the robustness of the element formulation and precludes numerical instabilities (hourglassing). The major locking effects are eliminated by assumed natural strain approaches (Hughes und Tezduyar (1981), Bathe und Dvorkin (1986), Betsch und Stein (1995)) and the enhanced assumed strain method (Simo und Rifai 1990). The element is not formulated in its natural coordinate system, but in accordance with Sze und Yao (2000) in an alternative convective system. This considerably improves the performance in configurations, where the element edges in thickness direction are not orthogonal to the fictitious mid-surface. Furthermore, a method to stabilise the transverse shear first proposed by Lyly u.a. (1993) is used to optimise the performance of elements which are distorted in-plane. An adaptive approach is introduced, that determines the size of the stabilisation parameter for each element depending on its geometry. In geometrically linear and nonlinear benchmark problems from literature as well as practical problems from aircraft industry the performance of the presented element formulation is analysed and compared with existing elements of the same topology. With respect to the quality of both displacements and stresses, the new element formulation is superior in many cases and is able to provide reliable results already with relatively coarse finite element meshes. Especially the stabilisation of the transverse shear for in-plane distorted elements improves the quality of the transverse shear strains and stresses.
The iterative solution of the ill-conditioned equation systems arising from the finite element simulation of shell structures represents a particular challenge. Besides elaborate multigrid methods, up to now no preconditioners are known, which enable an efficient iterative solution of such systems for arbitrary configurations. It is analysed to which extend approaches for setting up preconditioners according to Shklarski und Toledo (2008) and Avron u.a. (2009) can be adapted to finite element simulations of shell structures. Therein, substitute problems are created, which have to be fully factorised for preconditioning. This approach is in contrast to an incomplete factorisation of the initial problem, which many popular preconditioners are based on. In the first approach presented herein, the substitute problem is created by specific elimination of finite elements from the initial discretisation, taking into account that no singular systems arise. In the second approach, the stiffness matrix is first approximated by a symmetric diagonally dominant matrix, which is simplified by means of combinatorial procedures such that a similar matrix evolves, for which factorisation is cheaper. The algorithms necessary for both approaches are developed on the basis of the 8-node solid-like shell element presented in this thesis and the resulting preconditioners are analysed for problems of varying complexity. It turns out that the preconditioners based on modification of the initial discretisation have a positive impact on the convergence characteristics of the iterative solution process with respect to monotony. However, the computational effort with all presented preconditioners was in most cases significantly larger than with well-established preconditioners from literature. This approach therefore leads to no substantial increase of efficiency for the iterative solution of equation systems arising from the simulation of shell structures.#### BibTeX