Isogeometric Analysis of Wrinkling

Wrinkles are ubiquitous in the world around us. In our daily lives, we encounter wrinkles in various forms, whether in our clothes or on our skin. Wrinkles emerge as a result of a delicate interplay between bending, membrane, and foundation stiffness contributions within membranes. While experimental investigations provide insights into the physics underlying wrinkling, numerical investigations find their purpose in the design, analysis, and optimisation of membranes subjected to wrinkling. Nevertheless, the numerical simulation of membrane wrinkling presents several challenges. Firstly, wrinkling constitutes a buckling phenomenon in membranes with low bending stiffness. Wrinkles have the potential to evolve into folds, creases, or other wrinkling patterns as loads or displacements increase. Secondly, the wavelengths of wrinkling can be orders of magnitude smaller than the overall geometry, requiring a small resolution of the numerical simulation and hence increasing computational costs. Overall, the question arises of how to design robust and accurate numerical models for the analysis of wrinkled membranes. This dissertation is subdivided into four parts and aims to provide answers to this question.

The first theme considers hyperelastic material modelling, with a focus on developing wrinkling models under large strains. The shell model employed in this dissertation is based on the isogeometric analysis paradigm. Specifically, the Kirchhoff--Love shell model is used, which leverages the higher-order continuity of underlying spline spaces. Chapter 3 extends hyperelastic material formulations to stretch-based materials, enabling the use of the isogeometric analysis paradigm for rubber-like shells. Since the modelling of wrinkling patterns imposes physical scales limiting element mesh sizes, chapter 4 introduces a hyperelastic isogeometric membrane element that incorporates an implicit wrinkling model, thus avoiding explicit modelling of wrinkling amplitudes.

The second theme addresses adaptive methods. On the one hand, spatial adaptivity enhances the local detail in a numerical simulation. Chapter 5 presents an adaptive isogeometric analysis framework based on intuitive goal functions, such as wrinkling amplitudes, to guide adaptive meshing routines. On the other hand, temporal or quasi-temporal adaptivity serves to enhance the efficiency of dynamic or quasi-static simulations. Chapter 6 introduces an adaptive parallel arc-length method. The method's adaptivity arises as a by-product of parallelisation efforts aimed at reducing computational times for quasi-static simulations.

The advantage of the smoothness inherent in the spline spaces used in isogeometric analysis is limited to simple topologies. To benefit from this smoothness in complex geometries, the third theme of this dissertation focuses on complex domain modelling. Chapter 7 presents a qualitative and quantitative comparison of unstructured spline constructions for multi-patch modelling using isogeometric analysis. This chapter offers insights and suggestions for future developments related to unstructured spline constructions.

The final theme of this dissertation concerns the reproducibility of the developed methods. In this section, design considerations are presented for an open-source software library, along with small examples, aimed at ensuring easy reproducibility and supporting future research in the three themes mentioned earlier.

In summary, this dissertation offers a wide range of methods for the isogeometric analysis of structural instabilities in thin-walled structures, including the modelling of wrinkling. The concepts developed in terms of hyperelasticity expand the applicability of wrinkling models to encompass large strains. The concepts developed in terms of adaptivity provide intuitive error estimators that drive local refinement in space, as well as a novel continuation method that eliminates the inherently serial arc-length methods. Through the use of unstructured splines, complex domains become accessible for the analysis of structural stabilities. By creating an open-source, forward-compatible software library, these concepts are made available for future developments in the field of isogeometric analysis of wrinkling.