## 3 NURBS-Based Isogeometric Analysis
### Discrete Governing Equations  
The analysis of the FGM plate is simplified to a 2D problem using the process outlined in Section 2. The midplane geometry of the FGM plate is modeled based on 2D NURBS basis functions, which can be expressed as  
\[\Omega=\Big{\{}\mathbf{x}=(x,y)\Big{|}\mathbf{x}(\zeta)=\sum_{[\hat{1}]=1}^{Q} \mathbf{x}_{[\hat{1}]}\mathcal{R}_{[\hat{1}]}(\zeta),\ \ \ \zeta=(\zeta,\eta)\in\hat{\Omega}\Big{\}} \tag{17}\]  
where \(\mathbf{x}_{[\hat{1}]}\) are the geometric control points, and \(\hat{\Omega}\) is the mapping of the midplane \(\Omega\) in the parameter space. The displacement fields are discretized using 2D NURBS basis functions calculated as  
\[\mathbf{q}(\zeta)=\sum_{[\hat{1}]=1}^{Q}\mathbf{q}_{[\hat{1}]}\mathcal{R}_{[ \hat{1}]}(\zeta) \tag{18}\]  
where \(\mathbf{q}=[u_{0},\dots,u_{N},v_{0},\dots,v_{N},w_{0},\dots,w_{M-1}]^{\text{T}}\) is a vector that collects all the basic unknowns. \(\mathbf{q}_{[\hat{1}]}\) are the control vectors. Combined with Equations (4) and (6), the displacement and the strain fields can be expressed as  
\[ \begin{array}{l}\vec{\mathbf{u}}(\zeta)=\sum_{[\hat{1}]=1}^{Q}\mathbf{U}_{[ \hat{1}]}(\zeta)\mathbf{q}_{[\hat{1}]}\\ \vec{\mathbf{\epsilon}}(\zeta)=\sum_{[\hat{1}]=1}^{Q}\mathbf{E}_{[\hat{1}]}( \zeta)\mathbf{q}_{[\hat{1}]}\end{array}  \tag{19}\]  
Figure 3: Two-dimensional NURBS plane. where \(\mathbf{U}_{[\bar{1}]}\) and \(\mathbf{E}_{[\bar{1}]}\) are sparse block coefficient matrices, the nonzero elements of which are listed as  
\[ \begin{array}{l}\left(\mathbf{U}_{[\bar{1}]}\right)_{11}=\left(\mathbf{U}_{[ \bar{1}]}\right)_{2N+2}=\mathcal{R}_{[\bar{1}]};\\ \left(\mathbf{U}_{[\bar{1}]}\right)_{3i+1;i+1}=\left(\mathbf{U}_{[\bar{1}]} \right)_{3i+2,N+i+2}=\left(\mathbf{U}_{[\bar{1}]}\right)_{3i+3,2N+i+2}= \mathcal{R}_{[\bar{1}]},\\ \left(\mathbf{U}_{[\bar{1}]}\right)_{3i+1,2N+i2}=-\mathcal{R}_{[\bar{1}],x^{ \prime}}\ \ \left(\mathbf{U}_{[\bar{1}]}\right)_{3i+2,2N+i+2}=-\mathcal{R}_{[\bar{1}],y^{ \prime}}\ \ 1\leq i\leq M;\\ \left(\mathbf{U}_{[\bar{1}]}\right)_{3i+1;i+1}=\left(\mathbf{U}_{[\bar{1}]} \right)_{3i+2,N+i+2}=\mathcal{R}_{[\bar{1}]},\ \ \ M+1\leq i\leq N.\end{array}  \tag{20}\]  
and  
\[ \begin{array}{l}\left(\mathbf{E}_{[\bar{1}]}\right)_{11}=\left(\mathbf{E}_{[ \bar{1}]}\right)_{6i,N+2}=\mathcal{R}_{[\bar{1}],x^{\prime}}\left(\mathbf{E}_{[ \bar{1}]}\right)_{2,N+2}=\left(\mathbf{E}_{[\bar{1}]}\right)_{61}=\mathcal{R}_ {[\bar{1}],y^{\prime}};\\ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+1,i+1}=\left(\mathbf{E}_{[\bar{1}]} \right)_{6i+6,N+i+2}=\mathcal{R}_{[\bar{1}],x^{\prime}}\ \ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+1,2N+i+2}=- \mathcal{R}_{[\bar{1}],xx^{\prime}}\\ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+2,N+i+2}=\left(\mathbf{E}_{[\bar{1}]} \right)_{6i+6,i+1}=\mathcal{R}_{[\bar{1}],y^{\prime}}\ \ \left(\mathbf{E}_{[\bar{1}]} \right)_{6i+2,2N+i+2}=-\mathcal{R}_{[\bar{1}],yy^{\prime}}\\ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+3,2N+i+2}=\left(\mathbf{E}_{[\bar{1}] }\right)_{6i+4,N+i+2}=\left(\mathbf{E}_{[\bar{1}]}\right)_{6i+5,i+1}=\mathcal{ R}_{[\bar{1}]},\\ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+6,2N+i+2}=-2\mathcal{R}_{[\bar{1}],xy^{ \prime}}\ 1\leq i\leq M;\\ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+1,i+1}=\left(\mathbf{E}_{[\bar{1}]} \right)_{6i+6,N+i+2}=\mathcal{R}_{[\bar{1}],x^{\prime}}\\ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+2,N+i+2}=\left(\mathbf{E}_{[\bar{1}]} \right)_{6i+6,i+1}=\mathcal{R}_{[\bar{1}],y^{\prime}}\\ \left(\mathbf{E}_{[\bar{1}]}\right)_{6i+4,N+i+2}=\left(\mathbf{E}_{[\bar{1}]} \right)_{6i+5,i+1}=\mathcal{R}_{[\bar{1}]},\ \ \ M+1\leq i\leq N.\end{array}  \tag{21}\]  
Substituting Equation (19) into Equation (12), the discrete governing equations of the FGM plate can be obtained as  
\[