## 3 Method  
### The Proposed Isogeometric Boundary Element Method  
The guidelines to use IGA-BEM were established by [3]. In such a way, Equation (13) remains the same, except that the parameter \(\xi\) replaces the term \(\zeta\) and the shape functions are the B-spline basis functions \(N\) with order \(k\):  
\[ \begin{array}{ll}C(P)u_{i}(P)&+\sum\limits_{e=1}^{n_{e}}\sum\limits_{c=1}^{ k}\left[\int_{-1}^{1}T_{ij}\ (P,Q(\xi))N_{c}(\xi)F^{e}(\xi)d\zeta\right]u_{i}^{e}\\ &=\sum\limits_{e=1}^{n_{e}}\sum\limits_{c=1}^{k}\left[\int_{-1}^{1}U_{ij}\ (P,Q(\xi))N_{c}(\xi)F^{e}(\xi)d\xi\right]u_{i}^{e} \end{array}  \tag{14}\]  
However, unlike the method proposed by [3], our model generates the body boundary through a single NURBS, that is, with one \"macro-element\". In such a way, it is not necessary to integrate  
Figure 2: Nodes and elements using in the conventional BEM. the kernels. Nevertheless, it is required to integrate the Jacobian because the element represents the whole-body contour. The result is the body perimeter. In addition to the boundary conditions, three sets define the problem to be analyzed:  
* Parameter values \(\xi\) to get the nodes \(P\) and \(Q\) on the boundary and generate the values of the kernel. * The control points (\(B\)) that define the shape of the body. * Parameter values \(\xi\) to calculate the Jacobian of the macro-element. The proposed integral boundary equation is given by Equation (15), where \(\eta\) represents the \(\xi\) values that are used to calculate the Jacobian, which are different from the \(\xi\) values to find the \(P\) and \(Q\) nodes. \[ \begin{array}{ll}C(P(\xi))u_{i}(P(\xi))&+T_{ij}\;(P(\xi),Q(\xi))u_{i}(Q(\xi ))\int_{\Gamma}J(\eta)d\eta\\ &=U_{ij}\;(P(\xi),Q(\xi))t_{i}(Q(\xi))\int_{\Gamma}J(\eta)d\eta\end{array}  \tag{15}\]  
In conventional BEM, the jump term \(C\) was calculated indirectly, using rigid body considerations [25]. However, as [PERSON] et al. [4] described, these considerations are no longer valid, because of the nature of NURBS so the jump term has to be calculated explicitly [29]. However, because the geometry of the body-under-analysis is represented by a single element, the jump term equations are reduced to Equation (16). This matrix only affects the diagonal of the traction kernel when the load point, \(P\) coincides with the field point, \(Q\). \[C(P)=\left[ \begin{array}{cc}\frac{1}{2}&0\\ 0&\frac{1}{2}\end{array} \right], \tag{16}\]  
On the other hand, because the boundary variable, \(\Gamma\), is a function of \(\xi\), the term of Jacobian transformation, \(J(\eta)\), is described by the following:  
\[J(\eta)=\frac{d\Gamma}{d\eta}=\sqrt{\left(\frac{dx(\xi)}{d\xi}\right)^{2}+ \left(\frac{dy(\xi)}{d\xi}\right)^{2}}, \tag{17}\]  
The general flowchart of the isogeometric boundary method proposed is shown in Figure 3. The first step is to read the properties of the material-under-analysis and the NURBS geometry data. That is, the Young modulus (\(E\)), the Poisson coefficient (\(v\)), the control points (\(B\)), the knot vector (\(\Xi\)), and the curve order (\(k\)). Then, a quantity \(N\) of load/field points is chosen. The coordinates of the load points on boundary are generated by replacing a value of \(\xi\) in the NURBS function. The next step is to calculate the kernels for each combination of load points, \(P\), and field points, \(Q\). Every Kernel is saved in a global matrix of displacement (\(U\)) and traction (\(T\)) kernels with size \((N\times 2)\times(N\times 2)\). When the load point matches with the field point, the calculation of that traction kernel is skipped and replaced by the jump term of Equation (14). In the case of the displacement kernel, a tolerance is added or subtracted from the field point, to prevent the term (1/r) from going to infinity. The obtained global kernels are multiplied by the integral of the body boundary Equation (17); if a more exact value of the integral is desired, more \(\xi\) values may be added and may be different from the \(\xi\) values for the load and field points.