# T-Splines for Isogeometric Analysis of the Large Deformation of Elastoplastic Kirchhoff-Love Shells
## 1 Introduction
The isogeometric KL shell is firstly extended to the large deformation elastoplasticity problem in the reference [34], with the application of the stress-based three-dimensional constitutive model, where the stress results are computed by Gaussian integral points through the thickness direction. The same constitutive model is also implemented for multi-patch KL shells based on the bending strip method [35]. A velocity-based updated Lagrangian framework has been developed for isogeometric KL shell analysis using the stress-based three-dimensional model in a time-rate form in the reference [36]. All the abovementioned studies on plastic isogeometric analysis are based on NURBS. Although NURBS has many advantages as the basis of isogeometric analysis, it also has drawbacks in many applications due to its tensor product structure. It is well known that adaptive mesh methods can increase the reliability and efficiency of numerical simulation, especially for strain localization in elastoplastic large deformation [37; 38; 39]. Nevertheless, the h-refinement of NURBS requires knot insertions on the global mesh, which will generate many redundant knots. Moreover, trimmed NURBS surfaces are often used to design thin-walled structures in engineering applications, and additional work should be employed to deal with the trimming situations before analysis. In order to remedy these problems, T-splines were introduced to isogeometric analysis [40; 41]. The flexible topology of T-splines makes it possible to represent a multi-trimmed NURBS surface model with an untrimmed watertight T-spline surface [42; 43; 44]. In addition, benefiting from the existence of the T-junction, T-splines support local h-refinement in the regions requiring higher resolution. Regarding the complexity of shell geometries and the demand for the adaptive method, T-splines provide an alternative to the NURBS-based IGA. Applying T-spline-based IGA to shell element analysis is obviously an attractive topic [43; 45]. An isogeometric large deformation continuum shell formulation including finite strain elastoplasticity is presented in [46]. However, to the best of the authors' knowledge, the extension of T-spline-based IGA to the elastoplastic large deformation of shell elements has not been well studied. In this work, we develop an efficient computational shell model using T-spline discretization for both geometry and displacement field. The classical Kirchhoff-Love assumption is employed due to its superior computational efficiency (only three displacement degrees of freedom are required). The stress-based three-dimensional plastic constitutive is applied to our shell analysis frame. The applicability of our approach is validated by several benchmarks. To explore the local refinement capability of T-splines, we apply the adaptive analysis to the KL shells, guided by an a posteriori error estimator. Compared with the uniformly refined mesh, our numerical experiments show a better efficiency of the adaptive refinement. Moreover, using the D-patch framework, we perform large deformation elastoplastic analysis on relatively complex models built with unstructured T-splines.