## 2 Background of Modeling with NURBS and Boundary elements
### Boundary Element Method  
The boundary element method uses [PERSON]'s theorem, which says that the work done by a force system \((a)\) on the displacements on a system \((b)\) is equal to the work done by the forces of \((b)\) on the system displacements of \((a)\), in a body in equilibrium for two different groups of stresses and strains, Equation (6). \[\int_{V}\sigma_{i}^{a}\varepsilon_{i}^{b}=\int_{V}\sigma_{i}^{b}\varepsilon_{ i}^{a}, \tag{6}\]  
Using the differential equations of [PERSON] for displacements, the traction definition, and the divergence theorem, Equation (6) turns into (7). \[\int_{\Gamma}t_{i}^{a}u_{i}^{b}dS+\int_{V}f_{i}^{a}u_{i}^{b}dV=\int_{\Gamma}t_ {i}^{b}u_{i}^{a}dS+\int_{V}f_{i}^{b}u_{i}^{a}dV, \tag{7}\]  
where \(t_{i}^{a}\) and \(t_{i}^{b}\) are tractions; \(u_{i}^{b}\) and \(u_{i}^{a}\) are displacements; and \(f_{i}^{a}\) and \(f_{i}^{b}\) are loads from set \(a\) and \(b\), respectively. Since the problem to be solved is about solid mechanics, the [PERSON]'s solution was applied in the previous equation, to arrive at the Somigliana identity for displacements, Equation (8). \[u_{i}(p)=-\int_{\Gamma}T_{ij}\left(p,Q\right)u_{i}(Q)dS+\int_{\Gamma}U_{ij} \left(p,Q\right)t_{i}(Q)dS+\int_{V}U_{ij}\left(p,Q\right)f_{i}(q)dV, \tag{8}\]  
\(T_{ij}\) and \(U_{ij}\) represent the traction and displacement kernels from the Kelvin solution, which gives us tractions and displacements on any point of the surface \(Q\) when a load is applied to an interior point, \(p\), and is applicable to any geometry. The variables \(u_{i}\), \(t_{i}\), and \(f_{i}\) are the displacements, tractions, and forces to be found. Equations (9) and (10) represent the displacement and traction kernels for the two-dimensional elastostatic problem. These kernels depend on the values of the shear modulus (\(\mu\)), the Poisson coefficient (\(v\)), and the distance between the load point \((p)\) and field point, \(Q\), \(r(p,\,Q)\), described by Equation (11). For each load point and field point, displacement and traction kernels are calculated. \[U_{ij}\left(p,Q\right)=\frac{1}{8\pi\mu(1-v)}(3-4v)ln\left[\frac{1}{r(p,Q)} \right]\delta_{ij}+\frac{dr(p,Q)}{dx_{i}}\frac{dr(p,Q)}{dx_{j}}, \tag{9}\]  
\[T_{ij}\left(p,Q\right)=-\frac{1}{8\pi\mu(1-v)r(p,Q)}\frac{dr(p,Q)}{dn}x\left[( 1-2v)\delta_{i,j}+2\frac{dr(p,Q)}{dx_{i}}\frac{dr(p,Q)}{dx_{j}}\right]+ \tag{10}\]  
\[r(p,Q)=\sqrt{\left(X_{p}-x_{Q}\right)^{2}+\left(Y_{p}-y_{Q}\right)^{2}}, \tag{11}\]  
As Equations (8)-(11) have the load point, \(p\), in the interior of body and field point, \(Q\), in the boundary body, it is necessary to move the load point, \(p\), to the boundary. Thus, Equation (7) is transformed into Equation (12), with the absence of body forces:  
\[C(P)u_{i}(P)+\int_{\Gamma}T_{ij}\left(P,Q\right)u_{i}(Q)dS=\int_{\Gamma}U_{ij} \left(P,Q\right)t_{i}(Q)dS, \tag{12}\]where \(C(P)\) is called \"Jump Term\". This element moves the internal load point (\(p\)) to the boundary, and now the load point is denoted as \(P\). The numerical way to solve Equation (12) is to divide the body contour under analysis into \"elements\". Each element is composed of \"nodes\" (Figure 2). Shape functions describe the geometry and displacement and traction variables. To generate the kernels, a node is taken as the load point, P, and the field point, Q, is generated by each element using shape functions. The diagonal of these kernels corresponds to the point P being equal to the point Q, or P equal to any node of the element. When this happens, the values of the kernels are singular or very close to being singular. A correct approximation of these values is detailed in [25]. Then, a method as the Gaussian quadrature is applied to integrate the kernels on each element. Given these conditions, Equation (12) can be rewritten as follows:  
\[