# T-Splines for Isogeometric Analysis of the Large Deformation of Elastoplastic Kirchhoff-Love Shells
## 2 T-Splines
### T-Mesh and the T-Spline Basis  
Unlike NURBS, defined by a tensor product structure, the topology of T-splines is defined by T-mesh, as shown in Figure 1a. All the small rectangles in the T-mesh are called _faces_, the corners of the rectangles are called _vertices_, and the line segments connecting the _vertices_ are called _edges_. The _vertices_ marked with red circles in Figure 1 are the T-junctions of T-splines, which provide the flexible topology of T-splines. Each _edge_ of T-mesh is associated with a non-negative real number conveying the parameter information of knot intervals, as shown in Figure 1b. It is worth noting that the sum of knot intervals of opposite _edges_ must be equal. The expression of the T-spline surface is defined as  
\[\mathbf{T}(u,v)=\sum\limits_{i=0}^{n}\mathbf{P}_{i}R_{i}(u,v),(u,v)\in\Omega,R_{i}(u,v) =\frac{w_{i}N_{i}(u)N_{i}(v)}{\sum\limits_{j=0}^{n}w_{i}N_{j}(u)N_{j}(v)}. \tag{1}\]  
where \(\mathbf{P}_{i}\) is the control point for T-splines; \(R_{i}\) is the T-spline basis function associated with \(\mathbf{P}_{i}\); \(w_{i}\) is the weight for \(R_{i}\), as defined in the NURBS; \(N_{i}(u)\) and \(N_{i}(v)\) are the \(u\)-direction and \(v\)-direction B-spline basis functions of the control point \(\mathbf{P}_{i}\). Each control point corresponds \(\mathbf{P}_{i}\) to a _vertex_ in the T-mesh. With the aforementioned definition of T-mesh in the index space, we can obtain the basis function associated with the control point. The cubic T-mesh is widely discussed in the existing literature and is also studied in this work. Figure 2 shows an example of the basis function definition. The basis function associated with the control point \(P_{1}\) is constructed from knot interval sequences inferred from its corresponding _vertex_. The \(u\)-direction knot interval vector is obtained by traveling from the _vertex_ in positive and negative \(u\)-directions until \([p/2+1]\) (when \(p=3\), it is equal to 2) orthogonal _edges_ are encountered, where \([. ]\) represents the integer part of a real number. From Figure 2a,b, we can acquire the \(u\)-direction knot interval vector \(\Xi_{u}=\{u_{1},u_{2},u_{3},u_{4},u_{5}\}\), where the value of \(u_{i}\) is defined by knot intervals in Figure 1b. Figure 1: Example of T-mesh: (**a**) example of T-mesh in index space; (**b**) configuration of knot intervals. This is the same for the \(v\)-direction knot interval vector \(\Xi_{v}=\{v_{1},v_{2},v_{3},v_{4},v_{5}\}\), as shown in Figure 2c. These two knot interval vectors define the rectangular support of the basis function associated to the control point \(P_{1}\) (see the green area in Figure 2d). The B-spline basis functions \(N_{1}(u)\) and \(N_{1}(v)\) associated with the control point \(P_{1}\) are defined by knot interval vectors \(\Xi_{u}\) and \(\Xi_{v}\), respectively. The details of the B-spline basis function are presented in Appendix A. Figure 3 shows a structured bicubic T-spline surface and its corresponding T-mesh. Extraordinary points can further improve the topological flexibility of the T-mesh, and enable the representation of geometries with arbitrary topology using only one T-spline surface. The valence of a _vertex_, denoted by \(\mu\), is the number of _edges_ that emanate from that _vertex_. Extraordinary points are either interior _vertices_ with \(\mu\
eq 4\) that are not T-junctions or boundary _vertices_ with \(\mu>3\). The T-splines with extraordinary points are called unstructured T-splines. An unstructured bicubic T-spline surface is shown in Figure 4, where red circles represent extraordinary points. Figure 2: The knot interval vectors corresponding to the control point \(P_{1}\): **(a)** traveling from the _vertex_; **(b)**\(u\)-direction knot interval vector; **(c)**\(v\)-direction knot interval vector; **(d)** domain of definition associated with the control point \(P_{1}\).