## 3 Finite Element Modelling
Moreover, \(\mathbb{N}\), \(\hat{E}\), \(\hat{\rho}\), \(\hat{C}\), \(\hat{\alpha}\), and \(\hat{k}\) refer to the shape functions vector, elasticity modulus, mass density, heat capacity, heat expansion coefficient, and heat conductivity coefficient for each element, defined as  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]4  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]5  
By a mathematical manipulation, Equation (22) can be simplified as  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]6 \[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]7  
where \(V\) is the volume, \(S_{1}\) and \(S_{2}\) are the boundary surfaces of the cone, \(q^{\prime\prime}\), \(Q\), \(T_{\infty}\), and \(h\) stand for the heat flux, rate of energy generation per unit volume, surrounding temperature, and heat convection coefficient, respectively. According to FEM, we redefine Equation (42), in a variational form [69], to yield the following expression for capacitance \({[K_{1}}^{t}]^{(e)}\), stiffness \({[K_{2}}^{t}]^{(e)}\) and \({[K_{3}}^{t}]^{(e)}\), and force \([F^{t}]\) matrices for each element \((e)\)  
\[ \begin{array}{rl}{[K_{1}}^{t}]^{(e)}&=\int\limits_{V(e)}\rho(x)C(x)[N^{t}]^{T}[ N^{t}]dV\\ &=\int\limits_{V(e)}{(\mathbb{N}\hat{\rho})(N\hat{C})[N^{t}]^{T}[N^{t}]dV}\\ \\ {[K_{2}}^{t}]^{(e)}=\int\limits_{V(e)}{[B^{t}]^{T}\Big{[}D(x)^{t}\Big{]}[B^{t}] \,dV}\end{array}  \tag{43}\]  
where  
\[\Big{[}D(x)^{t}\Big{]}=\left[ \begin{array}{cc}rk_{r}(x)&0\\ 0&rk_{z}(x)\end{array} \right]=\left[ \begin{array}{cc}r(\mathbb{N}\hat{R}_{r} )&0\\ 0&r(\mathbb{N}\hat{R}_{z})\end{array} \right] \tag{45}\]  
\[[B^{t}]=\left[ \begin{array}{cc}\frac{\partial N_{i}}{\partial r}\ \frac{\partial N_{j}}{\partial r}\ \frac{\partial N_{k}}{\partial r}\ \frac{\partial N_{l}}{\partial r}\ \frac{\partial N_{m}}{ \partial r}\ \frac{\partial N_{n}}{\partial r}\ \frac{\partial N_{n}}{ \partial r}\ \end{array} \right] \tag{46}\]  
\[{[K_{3}}^{t}]^{(e)}=\int\limits_{S(e)}{h\left[N^{t}\right]^{T}[N^{t}]dS} \tag{47}\]  
\[{[F^{t}]}^{(e)}=\int\limits_{V(e)}{Q[N^{t}]^{T}dV}-\int\limits_{S_{1}(e)}{q^{ \cdot}[N^{t}]^{T}dS}+\int\limits_{S_{2}(e)}{hT_{\infty}[N^{t}]^{T}dS} \tag{48}\]  
in which \(V(e)\) denotes the volume element, \(S_{1}(e)\) and \(S_{2}(e)\) refer to the boundary elements subjected to a heat flux and heat convection, respectively. By assembling the element matrices, the global heat transfer takes the following form  
\[{[K_{1}}^{t}]\Big{[}\hat{\theta}\Big{]}+\big{(}{[K_{2}}^{t}]+{[K_{3}}^{t}] \big{)}[\theta]=[F^{t}] \tag{49}\]  
An unconditionally stable scheme based on a Crank-Nicolson time stepping procedure is thus applied for numerically solving the differential Equation (49), under a suitable time step selection, see Ref. [70]. More details about the algorithm are provided in Appendix A. Once deriving the transient temperature distribution, we solve the thermoelastic problem, while using simplex linear triangular elements to approximate the kinematic field. Based on a Rayleigh-Ritz energy formulation, the structural stiffness and force element matrices owing to a varying temperature and rotational velocity are, thus, obtained as  
\[{[K^{d}]}^{(e)}=\int\limits_{V_{(e)}}{\Big{[}B^{d}\Big{]}^{T}\Big{[}D^{d} \Big{]}\Big{[}B^{d}\Big{]}dV} \tag{50}\]  
\[{[F^{d}]}^{(e)}=\int\limits_{V_{(e)}}{\Big{[}B^{d}\Big{]}\Big{[}D^{d}\Big{]}[ e_{T}]\,dV}+\int_{A^{(e)}}{\bf N}^{T}{\Gamma}dV\]  
\[{[\varepsilon_{T}]}=(\mathbb{N}\Im)\Delta T[1\ \ \ 1\ \ \ 1\ \ \ 0]^{T} \tag{51}\]  
\[{\Big{[}D^{d}\Big{]}}=\frac{(\mathbb{N}\Sigma)}{(1+v)(1-2v)}\left[ \begin{array} []{cccc}1-v&v&v&0\\ v&1-v&v&0\\ v&v&1-v&0\\ 0&0&0&\frac{1-2v}{2}\end{array} \right]\]  
\[\Gamma=\left[