## 3 Finite Element Modelling  
The problem is solved numerically by using the finite element method. In such a context, we approximate the cone in the \(r-z\) plane of revolution by means of shape functions \([N(r,z)^{d}]^{(e)}\). Based on a [PERSON] approximation, we define the thermal distribution for each element \((e)\) as follows  
\[[q]^{(e)}=[N(r,z)^{d}]^{(e)}[\delta]^{(e)} \tag{31}\]  
\[T(r,z,t)^{(e)}=[N(r,z)^{l}]^{(e)}[T(t)]^{(e)},\quad[N(r,z)^{l}]^{(e)}=[N_{i} \ N_{j}\ N_{k}\ N_{l}\ N_{m}\ N_{n}],\]  
\[[T(t)]^{(e)}=\left[ \begin{array}{c}T(t)_{i}\\ T(t)_{j}\\ T(t)_{k}\\ T(t)_{l}\\ T(t)_{m}\\ T(t)_{n}\end{array} \right] \tag{32}\]  
\[[N(r,z)^{d}]^{(e)}=\left[ \begin{array}{cccccccccccc}N_{i}&0&N_{j}&0&N_{k}& 0&N_{l}&0&N_{m}&0&N_{n}&0\\ 0&N_{i}&0&N_{j}&0&N_{k}&0&N_{l}&0&N_{m}&0&N_{n}\end{array} \right]\]  
\[N_{i}=\frac{(r_{23}(z-z_{23})-z_{23}(r-r_{3}))(r_{46}(z-z_{6})-z_{46}(r-r_{6} ))}{(r_{23}z_{13}-z_{23}r_{13})(r_{46}z_{16}-z_{46}r_{16})}\]  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]  
\[N_{k}=\frac{(r_{21}(z-z_{1})-z_{21}(r-r_{1}))(r_{36}(z-z_{6})-z_{56}(r-r_{6} ))}{(r_{21}z_{21}-z_{21}r_{31})(r_{56}z_{36}-z_{56}r_{36})} \tag{33}\]  
\[N_{l}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{32}(z-z_{3})-z_{23}(r-r_{3} ))}{(r_{31}z_{41}-z_{31}r_{41})(r_{32}z_{43}-z_{23}r_{43})}\]  
\[N_{m}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{21}(z-z_{1})-z_{21}(r-r_{1}))} {(r_{31}z_{31}-z_{31}r_{51})(r_{21}z_{51}-z_{21}r_{51})}\]  
\[N_{n}=\frac{(r_{21}(z-z_{1})-z_{21}(r-r_{1}))(r_{32}(z-z_{2})-z_{23}(r-r_{3} ))}{(r_{21}z_{61}-z_{21}r_{61})(r_{23}z_{63}-z_{23}r_{63})}\]  
\[[\delta]^{(e)}=\left[u_{i}\ \ v_{i}\ \ u_{j}\ \ v_{j}\ \ u_{k}\ \ v_{k}\ \ u_{l}\ \ v_{l}\ \ u_{m}\ \ v_{m}\ \ u_{n}\ \ v_{n}\right]^{T}\]The indexes \(d\) and \(t\) refer to the structural and thermal solutions of the problem, respectively. By substitution of Equation (31) into Equation (26), the elastic strain matrix for the arbitrary element \((e)\) can be defined as  
\[\left[e\right]^{(e)}=\left[B^{d}\right]^{(e)}\!\!\left[\delta\right]^{(e)} \tag{34}\]  
\[\text{where }\left[B^{d}\right]^{(e)}=\left[d\right]\!\!\left[N(r,z)^{d}\right]^{(e)}\!\!,\text {or more specifically} \tag{35}\]  
Note that matrix \(\left[N(r,z)\right]^{(e)}\) refers to the second-order interpolation functions in terms of its nodal values for element \((e)\), and \(\left[T(t)\right]\) denotes the corresponding nodal temperature vector. In addition to the displacement components, the inhomogeneity of the material properties of the FG-GPL porous cone can be expressed by means of their nodal values as follows  
\[E=\sum_{i=1}^{6}E_{i}\ N_{i}=\mathbb{N}\hat{E} \tag{36}\]  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]0  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]1  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]2  
\[N_{j}=\frac{(r_{31}(z-z_{1})-z_{31}(r-r_{1}))(r_{34}(z-z_{4})-z_{54}(r-r_{4} ))}{(r_{31}z_{21}-z_{31}r_{21})(r_{54}z_{24}-z_{54}r_{24})}\]3  
where \(E_{i}\), \(\rho_{i}\), \(c_{i}\), \(\alpha_{i}\), \(k_{i}\) refer to the elasticity modulus, mass density, specific heat capacity, thermal expansion coefficient, and heat conduction coefficient related to the arbitrary node \(i\).