The present study provides an overview of modeling and discretization aspects in finite element analysis of thin‐walled structures. Shell formulations based upon derivation from three‐dimensional continuum mechanics, the direct approach, and the degenerated solid concept are compared, highlighting conditions for their equivalence. Rather than individually describing the innumerable contributions to theories and finite element formulations for plates and shells, the essential decisions in modeling and discretization, along with their consequences, are discussed. It is hoped that this approach comprises a good amount of the existing literature by including most concepts in a generic format. The contribution focuses on nonlinear finite element formulations for large displacements and rotations in the context of elastostatics. Although application to dynamics and problems involving material nonlinearities is straightforward, these subjects are not taken into account explicitly.

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Starting from a rigorous mechanical formulation, a numerical procedure is developed for the form finding of minimal surfaces and pre-stressed membrane structures. The method is based on an iteratively adapted regularisation of the stiffness matrix which gives the method its name: the updated reference strategy. The method can be applied to any finite element discretization of cable and membrane structures subjected to pre-stress as well as lateral pressure or other external loading. It is very robust and reliable as is shown by many illustrative examples. Further analysis states that the well known force density method is the special case of applying the updated reference strategy to cable nets.

Computational modeling of thin biological membranes can aid the design of better medical devices. Remarkable biological membranes include skin, alveoli, blood vessels, and heart valves. Isogeometric analysis is ideally suited for biological membranes since it inherently satisfies the C1-requirement for Kirchhoff-Love kinematics. Yet, current isogeometric shell formulations are mainly focused on linear isotropic materials, while biological tissues are characterized by a nonlinear anisotropic stress-strain response. Here we present a thin shell formulation for thin biological membranes. We derive the equilibrium equations using curvilinear convective coordinates on NURBS tensor product surface patches. We linearize the weak form of the generic linear momentum balance without a particular choice of a constitutive law. We then incorporate the constitutive equations that have been designed specifically for collagenous tissues. We explore three common anisotropic material models: Mooney-Rivlin, May Newmann-Yin, and Gasser-Ogden-Holzapfel. Our work will allow scientists in biomechanics and mechanobiology to adopt the constitutive equations that have been developed for solid three-dimensional soft tissues within the framework of isogeometric thin shell analysis.

This article presents a staggered approach to couple the interaction of very flexible tension structures with large deformations, described with the finite element method (FEM), and objects undergoing large, complex, and arbitrary motions discretized with particle methods, in this case the discrete element method (DEM). The quantitative solution quality and convergence rate of this partitioned approach is highly time step dependent. Thus, the strong coupling approach is presented here, where the convergence is achieved in an iterative manner within each time step. This approach helps increase the time step size significantly, decreases the overall computational costs, and improves the numerical stability. Moreover, the proposed algorithm enables the application of two independent, standalone codes for simulating DEM and structural FEM as blackbox solvers. Systematic evaluations of the newly proposed iterative coupling scheme with respect to accuracy, robustness, and efficiency as well as cross comparisons between strong and weak FEM-DEM coupling approaches are performed. Additionally, the approach is validated against the rest position of an impacting object, and further examples with objects impacting highly flexible protection structures are presented. Here, the protection nets are described with nonlinear structural finite elements and the impacting objects as DEM elements. To allow the interested reader to independently reproduce the results, detailed code and algorithm descriptions are included in the appendix.

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