Mesh adaptivity is a technique to provide detail in numerical solutions without the need to refine the mesh over the whole domain. Mesh adaptivity in isogeometric analysis can be driven by Truncated Hierarchical B-splines (THB-splines) which add degrees of freedom locally based on finer B-spline bases. Labeling of elements for refinement is typically done using residual-based error estimators. In this paper, an adaptive meshing workflow for isogeometric Kirchhoff–Love shell analysis is developed. This framework includes THB-splines, mesh admissibility for combined refinement and coarsening and the Dual-Weighted Residual (DWR) method for computing element-wise error contributions. The DWR can be used in several structural analysis problems, allowing the user to specify a goal quantity of interest which is used to mark elements and refine the mesh. This goal functional can involve, for example, displacements, stresses, eigenfrequencies etc. The proposed framework is evaluated through a set of different benchmark problems, including modal analysis, buckling analysis and non-linear snap-through and bifurcation problems, showing high accuracy of the DWR estimator and efficient allocation of degrees of freedom for advanced shell computations.

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In order to perform isogeometric analysis with increased smoothness on complex domains, trimming, variational coupling or unstructured spline methods can be used. The latter two classes of methods require a multi-patch segmentation of the domain, and provide continuous bases along patch interfaces. In the context of shell modelling, variational methods are widely used, whereas the application of unstructured spline methods on shell problems is rather scarce. In this paper, we therefore provide a qualitative and a quantitative comparison of a selection of unstructured spline constructions, in particular the D-Patch, Almost-C¹, Analysis-Suitable G¹ and the Approximate C¹ constructions. Using this comparison, we aim to provide insight into the selection of methods for practical problems, as well as directions for future research. In the qualitative comparison, the properties of each method are evaluated and compared. In the quantitative comparison, a selection of numerical examples is used to highlight different advantages and disadvantages of each method. In the latter, comparison with weak coupling methods such as Nitsche's method or penalty methods is made as well. In brief, it is concluded that the Approximate C¹ and Analysis-Suitable G¹ converge optimally in the analysis of a bi-harmonic problem, without the need of special refinement procedures. Furthermore, these methods provide accurate stress fields. On the other hand, the Almost-C¹ and D-Patch provide relatively easy construction on complex geometries. The Almost-C¹ method does not have limitations on the valence of boundary vertices, unlike the D-Patch, but is only applicable to biquadratic local bases. Following from these conclusions, future research directions are proposed, for example towards making the Approximate C¹ and Analysis-Suitable G¹ applicable to more complex geometries.

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Modelling nonlinear phenomena in thin rubber shells calls for stretch-based material models, such as the Ogden model which conveniently utilizes eigenvalues of the deformation tensor. Derivation and implementation of such models have been already made in Finite Element Methods. This is, however, still lacking in shell formulations based on Isogeometric Analysis, where higher-order continuity of the spline basis is employed for improved accuracy. This paper fills this gap by presenting formulations of stretch-based material models for isogeometric Kirchhoff–Love shells. We derive general formulations based on explicit treatment in terms of derivatives of the strain energy density functions with respect to principal stretches for (in)compressible material models where determination of eigenvalues as well as the spectral basis transformations is required. Using several numerical benchmarks, we verify our formulations on invariant-based Neo-Hookean and Mooney–Rivlin models and with a stretch-based Ogden model. In addition, the model is applied to simulate collapsing behaviour of a truncated cone and it is used to simulate tension wrinkling of a thin sheet. @en