## 2 Problem and Equations
### Approximation of Displacement and Temperature Fields  
The discrete form of the integral Equations (16) and (17) can be obtained in the XFEM using the test and trial functions for the displacement and temperature fields. The trial functions \(u(X,t)\) are defined as follows [15].  
\[\begin{array}{l}u(X,t)\approx u^{h}(X,t)=\sum\limits_{I\in N}N_{ul}(X) \overline{u}_{I}(t)+\sum\limits_{J\in N^{slix}}N_{ul}(X)(H(X)-H(X_{J})) \overline{a}_{J}(t)\\ +\sum\limits_{K\in N^{tip}}\sum\limits_{a=1}^{4}N_{uk}(X)(B_{ua}(X)-B_{ua}(X_{ K}))\overline{b}_{aK}(t)\end{array} \tag{19}\]  
The variables \(\overline{a}_{J},\overline{b}_{aK}\) are the additional degree of freedom (DOF) according to enrichment functions. \(N_{u}\) is the standard shape functions of displacement fields. \(B(X)\) is the additional asymptotic enrichment functions depending on the tip singularity of the displacement or temperature field. \(H(x)\) is the Heaviside step function.  
The discretization of (17) in terms of the temperature is similar to Equation (16). The main difference is the number of the branch functions. The other difference is the introduction of the temperature as body forces [12]. The trial functions \(T(X,t)\) can be defined as follows.  
\[\begin{array}{l}T(X,t)\approx T^{h}(X,t)=\sum\limits_{I\in N}N_{Tl}(X) \overline{T}_{I}(t)+\sum\limits_{J\in N^{slis}}N_{Tl}(X)(H(X)-H(X_{J})) \overline{e}_{J}(t)\\ +\sum\limits_{K\in N^{tip}}N_{TK}(X)(B_{T}(X)-B_{T}(X_{K}))\overline{f}_{K}(t) \end{array} \tag{20}\]  
The variables \(\overline{e}_{J}\) and \(\overline{f}_{K}\) are the additional DOF according to enrichment functions. \(N_{T}\) are the standard shape functions of displacement and temperature fields. It is noted that \(B_{T}(X)\) is different in the case of an adiabatic or an isothermal crack analysis.  
(1) Adiabatic crack  
The adiabatic crack is defined by a zero flux normal to the crack surface that causes a discontinuity in the temperature field. Therefore, the signed distance function \(\psi(X)\) is used as an enrichment function in this case [12], since it ensures the discontinuity of the temperature and the continuity of its derivative through the crack.  
\[\left\{\begin{array}{l}\Psi(X)=sign[n\cdot(X-X_{I})]\\ sign(x)=\left\{\begin{array}{cc}1&\mbox{if }x\;>0\\ -1&\mbox{if }x\;<0\end{array}\right.\end{array}\right. \tag{21}\]The tip nodes are enriched by the branch function that is given by  
\[B_{T}(r,\theta)=\sqrt{r}\sin\biggl{(}\frac{\theta}{2}\biggr{)} \tag{22}\]  
(2) Isothermal crack  
The crack is considered as isothermal when the temperature has a constant value on the crack surface, which causes a discontinuous heat flux with the continuous temperature field. The enrichment function that can satisfy these conditions is the one proposed by [PERSON] et al. (2016).  
\[\left\{\begin{array}{c}\psi(X)=\sum\limits_{I}N_{TI}|f_{1}(X)|-\left|\sum \limits_{I}N_{TI}f_{1}(X)\right|\\ f_{1}(X)=sign[n\cdot(X-X_{I})]\min\|X-X^{T}\|\end{array}\right. \tag{23}\]  
The near tip nodes are enriched by  
\[B_{T}(r,\theta)=\sqrt{r}\cos\biggl{(}\frac{\theta}{2}\biggr{)} \tag{24}\]  
The enriched \(FE\) approximation of the displacement (19) and temperature (20) can be symbolically written in the following form.  
\[u^{h}(X,t) =N_{u}^{std}(X)\overline{u}(t)+N_{u}^{Hex}(X)\overline{a}(t)+N_{u }^{tip}(X)\overline{b}(t) \tag{25}\] \[T^{h}(X,t) =N_{T}^{std}(X)\overline{T}(t)+N_{T}^{Hex}(X)\overline{e}(t)+N_{ T}^{tip}(X)\overline{f}(t) \tag{26}\]  
\(N_{u}^{std}(X)\) is the matrix of the standard displacement shape functions. \(N_{u}^{Hex}(X)\) and \(N_{u}^{tip}(X)\) are the matrices of the enriched displacement shape functions associated with the Heaviside and asymptotic tip functions. \(N_{T}^{std}(X)\) is the matrix of the standard temperature shape functions, and \(N_{T}^{Hex}(X)\) and \(N_{T}^{tip}(X)\) are the matrices of the enriched temperature shape functions associated with the Heaviside and asymptotic tip functions.