## 2 Review on the Weighted B-Spline FCM
### The B-Spline Finite Cell Method
\begin{array}{l}N_{i,0}(\xi)=\left\{\begin{array}{l}1\text{ if }\xi_{i}\leq\xi\leq\xi_{i+1}\\ 0\text{ otherwise }\end{array}\right.\\ N_{i,p}(\xi)=\frac{\xi-\xi_{i}}{\xi_{i+p}-\xi_{i}}N_{i,p-1}(\xi)+\frac{\xi_ {i+p+1}-\xi}{\xi_{i+p+1}-\xi_{i+1}}N_{i+1,p-1}(\xi)\\ \text{provide : }\frac{\partial}{\partial}=0\end{array} \right. \tag{10}\]  
In this paper, \(\Xi\), \(\mathbf{H}\), and \(\mathbf{\Psi}\) are uniform and open knot vectors. At the same time, these original knot vectors are all defined as \(\left\{0,0,1,1\right\}\) in parametric space. The quadratic univariate B-spline basis functions defined by this uniform and open knot vector are depicted in Figure 5.  
Let us assume an arbitrary point \(\mathbf{P}=\left\{x,y,z\right\}^{\text{T}}\) inside \(\Omega\); this can be interpolated by the following mapping relation:  
\[\mathbf{P}=\sum_{s=1}^{m\times m\times k}M_{s}(\xi,\eta,\gamma)\mathbf{P}_{s} \tag{11}\]  
where \(\mathbf{P}_{s}=\left\{x_{s},y_{s},z_{s}\right\}^{\text{T}}\) refers to the \(s\)th control point, as illustrated in Figure 4c. The trial functions \(\mathbf{u}^{h}\) and the test functions \(\mathbf{v}^{h}\) are defined over \(R^{h}\), which is a finite-dimensional subspace. The discrete form of the weak form Equation (6) can be stated as  
\[a(\mathbf{u}^{h},\mathbf{v}^{h})=l(\mathbf{v}^{h})\;\forall\mathbf{v}^{h}\in R^{h} \tag{12}\]  
where \(\mathbf{v}\) is spanned by the B-spline basis function \(\mathbf{M}\). A linear combination of shape functions is considered concerning the numerical approximation \(\mathbf{u}^{h}\). \[\mathbf{u}^{h}=\sum_{s=1}^{m\times m\times k}M_{s}u_{s}=\mathbf{M}\mathbf{U} \tag{13}\]  
Figure 5: Quadratic univariate B-spline basis functions (\(q=2\)) defined over the uniform and open knot vector \(\mathbf{H}=\left\{0,0,0,\frac{1}{1},\frac{1}{2},\frac{3}{4},1,1,1\right\}\). with  
\[\mathbf{M}=\left[ \begin{array}{cccccccccccc}M_{1}&0&0&M_{2}&0&0&\cdots&M_{m\times n \times k}&0&0\\ 0&M_{1}&0&0&M_{2}&0&\cdots&0&M_{m\times n\times k}&0\\ 0&0&M_{1}&0&0&M_{2}&\cdots&0&0&M_{m\times n\times k}\end{array} \right] \tag{14}\]  
in which \(\mathbf{U}\) denotes the unknown displacement vector of control points. Based on Ritz-Galerkin approach, inserting Equation (13) into Equation (12), the finite cell formulation, which is identical to the FEM in form, is obtained as  
\[\mathbf{K}\mathbf{U}=\mathbf{F} \tag{15}\]  
where \(\mathbf{K}\) and \(\mathbf{F}\) are the global stiffness matrix and the global load vector that can be assembled in the traditional fashion from all cell stiffness matrices \(\mathbf{K}_{c}\) and load vectors \(\mathbf{F}_{c}\), respectively. \(\mathbf{K}_{c}\) and \(\mathbf{F}_{c}\) are acquired by  
\[\mathbf{K}_{c}=\int\limits_{\Omega_{c}}\mathbf{B}^{\mathrm{T}}\beta\mathbf{D}\mathbf{B}\mathrm{ d}\Omega \tag{16}\]  
\[\mathbf{K}_{c}=\int\limits_{\Omega_{c}}\mathbf{B}^{\mathrm{T}}\beta\mathbf{D}\mathbf{B} \mathrm{d}\Omega(1)\mathbf{F}_{c}=\int\limits_{\Omega_{c}}M^{\mathrm{T}}\beta \mathbf{f}\mathrm{d}\Omega+\int\limits_{\Gamma_{Nc}}\mathbf{M}^{\mathrm{T}}\mathrm{d}\Gamma \tag{17}\]  
in which \(\Omega_{c}\) illustrates one cell domain. \(\Gamma_{Nc}\) is the Neumann boundary related to one cell. \(\mathbf{B}\) is the strain-displacement matrix, written as  
\[\mathbf{B}=\left[ \begin{array}{cccc}\frac{\partial}{\partial x}&0&0\\ 0&\frac{\partial}{\partial y}&0\\ 0&0&\frac{\partial}{\partial z}\\ \frac{\partial}{\partial y}&\frac{\partial}{\partial x}&0\\ 0&\frac{\partial}{\partial z}&\frac{\partial}{\partial y}\\ \frac{\partial}{\partial z}&0&\frac{\partial}{\partial x}\end{array} \right]\mathbf{M} \tag{18}\]