## 2 Review on the Weighted B-Spline FCM
### The B-Spline Finite Cell Method  
The finite cell method [20; 21] is an extension to the core principle and concept using high-order basis functions to approximate the unknown physical domain. Without losing generality, this paper focuses on 3D linear elasticity problems. The original physical domain \(\varOmega_{p}\) with boundary \(\partial\varOmega_{p}\) is embedded into a cubic domain \(\varOmega\), as shown in Figure 3.  
Figure 3: The original physical domain \(\varOmega_{p}\) is extended to the embedded domain \(\varOmega\). Figure 2: Illustration of the level-set function. (**a**) The physical domain is embedded to the computational domain (**b**) The level set function. The strong form of the boundary value problem for embedded domain \(\Omega\) can be expressed as  
\[\left\{ \begin{array}{ll}-\text{div}\sigma(\mathbf{u})=\beta\mathbf{f}&\text{in }\ \Omega\\ \mathbf{u}=\mathbf{g}&\text{on }\ \Gamma_{D}\\ \sigma(\mathbf{u})\cdot\mathbf{n}=\mathbf{t}&\text{on }\ \Gamma_{N}\end{array} \right. \tag{4}\]  
where \(\sigma\) denotes the stress tensor, \(\mathbf{u}\) is the displacement field, \(\mathbf{t}\) expresses the traction vector defining on \(\Gamma_{N}\), \(\mathbf{g}\) is the prescribed displacement vector defining Dirichlet boundary condition (DBC) on \(\Gamma_{D}\), \(\mathbf{f}\) denotes the body force on \(\Omega_{p}\), and \(\mathbf{n}\) is the unit outward normal vector on \(\Gamma_{N}\). The scalar factor \(\beta\) can be defined as  
\[\beta=\left\{ \begin{array}{ll}1\ \text{in}\ \Omega_{p}\\ 0\ \text{in}\ \Omega\backslash\Omega_{p}\end{array} \right. \tag{5}\]  
The weak form of Equation (4) can be stated as  
\[a(\mathbf{u},\mathbf{v})=l(\mathbf{v})\ \mathbf{u},\mathbf{v}\in H^{1}(\Omega) \tag{6}\]  
with the bilinear form being  
\[a(\mathbf{u},\mathbf{v})=\int_{\Omega}\varepsilon(\mathbf{v}):\beta\mathbf{D}:\varepsilon( \mathbf{u})\text{d}\Omega \tag{7}\]  
and the load function being  
\[l(\mathbf{v})=\int_{\Omega}\mathbf{v}\cdot\beta\mathbf{f}\text{d}\Omega+\int_{\Gamma_{N} }\mathbf{v}\cdot\text{d}\Omega \tag{8}\]  
in which \(H^{1}(\Omega)\) expresses the Sobolev space of order 1. \(\mathbf{v}\) denotes the admissible test displacement in kinematics. \(\varepsilon\) and \(\mathbf{D}\) are the strain tensor and the constitutive elasticity tensor of \(\Omega_{p}\), respectively. Similar to the finite element method, the embedded domain \(\Omega\) is discretized by a general Cartesian grid. As illustrated in Figure 3(a), cells are simply hexahedrons, and can be classified into physical cells, fictitious cells, and boundary cells, according to the implicit level set function of \(\Omega_{p}\). Figure 4: The outline of B-spline FCM: (**a**) cells in \(\Omega\) and finite cell classification; (**b**) parametric grid; (**c**) grid of control points in \(\Omega\) with polynomial degree \(p=q=r=2\). In the B-spline FCM [22; 23; 24], the B-spline basis functions are used as the interpolation functions over the hexahedral cells. The standard parametric domain (see Figure 4b) is mapped to the discretization of the embedded domain. For 3D problem mapping, it can be easily constructed by taking the tensor product of corresponding univariate B-spline basis functions. \[M_{s}(\xi,\eta,\gamma)=N_{i,p}(\xi)\cdot N_{j,q}(\eta)\cdot N_{h,r}(\gamma)\;s= i+n\cdot(j-1)+m\cdot n\cdot(h-1) \tag{9}\]  
in which \(N_{i,p}(\xi)\), \(N_{j,q}(\eta)\) and \(N_{h,r}(\gamma)\) are three univariate B-spline basis functions defined over \(\Xi=\left\{\xi_{1},\xi_{2},\ldots,\xi_{n+p+1}\right\}\), \(\mathbf{H}=\left\{\eta_{1},\eta_{2},\ldots,\eta_{m+q+1}\right\}\) and \(\mathbf{\Psi}=\left\{\gamma_{1},\gamma_{2},\ldots,\gamma_{k+r+1}\right\}\), respectively. \(n\), \(m\), and \(k\) are numbers of \(N_{i,p}(\xi)\), \(N_{j,q}(\eta)\), and \(N_{h,r}(\gamma)\), respectively. \(p\), \(q\), and \(r\) denote the polynomial degree, in this paper \(p=q=r=2\). The following recurrence relation (10) is included by the above expression. \[\left\{