# T-Splines for Isogeometric Analysis of the Large Deformation of Elastoplastic Kirchhoff-Love Shells
## 4 Adaptive Refinement  
In this section, we introduce the adaptive refinement method for isogeometric analysis of elastoplastic shells built with T-splines. The purpose of adaptive refinement is to improve analysis efficiency by refining the mesh locally. The classical ingredients of the adaptive method are 1. setup of a reliable error estimator; 2. generation of the new mesh according to the error estimator. A simple and accurate a posteriori error estimation was developed by [PERSON] (the so-called ZZ estimate), employing post-processed stress fields to evaluate the error [59; 60; 61]. The ZZ estimate framework has been successfully applied to many fields of traditional finite element analysis, such as nonlinear analysis, dynamic problems, and fluid mechanics [62; 63; 64]. For isogeometric analysis, the a posteriori error estimator has also been applied to T-splines [65; 66] and to hierarchical splines [67; 68]. Subsequently, it has been extended to KL shells [69] and to trimmed models [70]. In our work, we employ the projection process to obtain the post-processed stress. \(\hat{\mathbf{\sigma}}\) is the stress result computed from the isogeometric analysis program. According to the assumption that the post-processed stress and the displacement are interpolated with the same function \(\mathbf{R}\):  
\[\mathbf{\sigma}^{*}=\mathbf{R}\hat{\mathbf{\sigma}}^{*}, \tag{56}\]  
and  
\[\int_{\Omega}\mathbf{R}^{\mathsf{T}}(\mathbf{\sigma}^{*}-\hat{\mathbf{\sigma}})d\Omega =0. \tag{57}\]  
Then, the post-processed stress can be obtained from Equations (56)-(57):  
\[\mathbf{\sigma}^{*}=A^{-1}\int_{\Omega}\mathbf{R}^{\mathsf{T}}\hat{\mathbf{\sigma}}d\Omega, \tag{58}\]  
where  
\[A=\int_{\Omega}\mathbf{R}^{\mathsf{T}}\mathbf{R}d\Omega. \tag{59}\]  
As explained in the reference [59], \(\mathbf{\sigma}^{*}\) is a more accurate approximation than \(\hat{\mathbf{\sigma}}\). Therefore, the error \(\mathbf{e}_{\sigma}\) is estimated as  
\[\mathbf{e}_{\sigma}=\mathbf{\sigma}^{*}-\hat{\mathbf{\sigma}}. \tag{60}\]  
\(\|\mathbf{e}_{\sigma}\|_{i}\) denotes the local error of the element \"\(i\)\", which can be defined in the \(L_{2}\) norm:  
\[\|\mathbf{e}_{\sigma}\|_{i}=(\int_{\Omega_{i}}(\mathbf{e}_{\sigma})^{\mathsf{ T}}(\mathbf{e}_{\sigma})d\Omega_{i})^{1/2}. \tag{61}\]  
Another method to compute the local error in the so-called energy norm is introduced in the reference [71]. The global error can then be obtained by summing the local contributions of every element as  
\[\|\mathbf{e}_{\sigma}\|=(\sum_{i=1}^{ne}\|\mathbf{e}_{\sigma}\|_{i}^{2})^{1/2}, \tag{62}\]  
where \"\(ne\)\" represents the total number of elements. The estimate of relative error is obtained as  
\[\eta=\frac{\|\mathbf{e}_{\sigma}\|}{\|\mathbf{\sigma}^{*}\|}\times 100\%. \tag{63}\]  
Naturally, the refinement strategy depends on a predefined accuracy criterion. Thus, during the whole analysis process, the error needs to satisfy the condition  
\[\eta\leq\bar{\eta}, \tag{64}\]  
where \(\bar{\eta}\) is the prescribed error. If this condition is violated, the mesh is updated, and a recalculation for the current step is performed. We can judge which elements need to berefined by local error \(\left\|\mathbf{e}_{\sigma}\right\|_{i}\), and this local error is in fact computed in all elements. The local refinement parameter for each element is  
\[\xi_{i}=\frac{\left\|\mathbf{e}_{\sigma}\right\|_{i}\sqrt{n\epsilon}}{\left\| \mathbf{e}_{\sigma}\right\|}. \tag{65}\]  
The adaptive method will perform Quad-tree refinement on the elements of T-mesh, where the parameter \(\xi_{i}>1\), as shown in \[\int_{\Omega}\mathbf{R}^{\mathsf{T}}(\mathbf{\sigma}^{*}-\hat{\mathbf{\sigma}})d\Omega =0. \tag{57}\]0  
The linear independence of the T-mesh is an important consideration for applying the Galerkin method. Our proposed adaptive refinement method does not contain multiple knots since the T-mesh is obtained from Quad-tree refinement by inserting knots into the middle of adjacent ones. According to the reference [72] (Theorem 3.