The temperature governing equation can be reduced to a simpler form than that given in Eq. (3) by using Kirchhoff transformation which is defined as follows [14]
$$ \Theta = \mathop \smallint \limits_{{T_{0} }}^{T} \frac{{\lambda \left( {\overline{T}} \right)}}{{\lambda_{0} }}d\overline{T} $$
(7)
By using Kirchhoff transformation, Eq. (3) can be written as follows
$$ \nabla^{2} \Theta \left( {X,t} \right) + \frac{1}{{\lambda_{0} }}h\left( {X,\Theta ,{\text{t}}} \right) = \frac{\rho \left( \Theta \right) c\left( \Theta \right)}{{\lambda \left( \Theta \right)}}\frac{{\partial \Theta \left( {X,t} \right)}}{\partial t} $$
(8)
where \(\Theta\) and \(\lambda_{0}\) are temperature function and thermal conductivity at \({\text{T}}_{0}\), respectively.
Now, the right-hand side of (8) can be decomposed into linear and nonlinear parts as
$$ \nabla^{2} \Theta \left( {X,t} \right) + \frac{1}{{\lambda_{0} }}h\left( {X,\Theta ,{\text{t}}} \right) = \frac{{\rho_{0} c_{0} }}{{\lambda_{0} }}\frac{{\partial \Theta \left( {X,t} \right)}}{\partial t} + Nl\left( {X,\Theta , \dot{\Theta }} \right) $$
(9)
where \(\lambda_{0} , \;\rho_{0} \;{\text{and}}\;c_{0}\) are thermal conductivity, density and specific heat, respectively, at \(T_{0}\).
where the nonlinear term can be written as
$$ Nl\left( {X,\Theta , \dot{\Theta }} \right) = \left[ {\frac{\rho \left( \Theta \right) c\left( \Theta \right)}{{\lambda \left( \Theta \right)}} - \frac{{\rho_{0} c_{0} }}{{\lambda_{0} }}} \right]\dot{\Theta } $$
(10)
According to [15, 16], Eq. (9) can be written as
$$ \nabla^{2} \Theta \left( {X,t} \right) + \frac{1}{{\lambda_{0} }}h_{Nl} \left( {X,\Theta , \dot{\Theta }, {\text{t}}} \right) = \frac{{\rho_{0} c_{0} }}{{\lambda_{0} }}\frac{{\partial \Theta \left( {X,t} \right)}}{\partial t} $$
(11)
where
$$ h_{Nl} \left( {X,\Theta , \dot{\Theta }, {\text{t}}} \right) = h\left( {X,\Theta , {\text{t}}} \right) + \left[ {\rho_{0} c_{0} - \frac{{\lambda_{0} }}{\lambda \left( \Theta \right)}\rho \left( \Theta \right) c\left( \Theta \right)} \right]\dot{\Theta } $$
(12)
The integral equation which corresponds to Eq. (11) can be expressed as
$$ \begin{aligned} & C\left( P \right)\Theta \left( {P,t_{n + 1} } \right) + a_{0} \int\limits_{\Gamma } {\mathop \int \limits_{{t_{n} }}^{{t_{n + 1} }} \Theta \left( {Q, \tau } \right)q^{*} \left( {P,t_{n + 1} ;Q,\tau } \right)d\tau d\Gamma } \\ & = a_{0} \int\limits_{\Gamma } {\mathop \int \limits_{{t_{n} }}^{{t_{n + 1} }} q\left( {Q, \tau } \right)\Theta^{*} \left( {P,t_{n + 1} ;Q,\tau } \right)d\tau d\Gamma } \\ & \quad + \frac{{a_{0} }}{{\lambda_{0} }}\int\limits_{\Omega } {\mathop \int \limits_{{t_{n} }}^{{t_{n + 1} }} h_{Nl} \left( {Q, \Theta , \dot{\Theta }, \tau } \right)\Theta^{*} \left( {P,t_{n + 1} ;Q,\tau } \right)d\tau d\Omega } \\ & \quad + \int\limits_{\Omega } {\Theta \left( {Q, t_{n} } \right)\Theta^{*} \left( {P,t_{n + 1} ;Q,t_{n} } \right)d\Omega } \\ \end{aligned} $$
(13)
The fundamental solution and its normal derivative, respectively, can be written as
$$ {\Theta }^{*} \left( {P,t_{n + 1} ;Q,{\uptau }} \right) = \frac{1}{{4\pi a_{0} \left( {t - \tau } \right)}}exp\left[ {\frac{{ - r^{2} }}{{4a_{0} \left( {t - \tau } \right)}}} \right]H\left( {t - \tau } \right) $$
(14)
$$ {\text{q}}^{*} \left( {P,t;Q,{\uptau }} \right) = \frac{\partial }{\partial n}{\Theta }^{*} \left( {P,t;Q,{\uptau }} \right) = \frac{ - r}{{8\pi a_{0}^{2} \left( {t - \tau } \right)^{2} }}exp\left[ {\frac{{ - r^{2} }}{{4a_{0} \left( {t - \tau } \right)}}} \right]H\left( {t - \tau } \right)\frac{\partial r}{{\partial n}} $$
(15)
where \(a_{0} = \frac{{\lambda_{0} }}{{\rho_{0} { }c_{0} }}\) and \(H\) is the Heaviside function.
The time integrals which corresponds to (14) and (15) can be computed analytically as
$$ \mathop \int \limits_{{t_{n} }}^{{t_{n + 1} }} {\Theta }^{*} \left( {P,t_{n + 1} ;Q,{\uptau }} \right)d{\uptau } = \frac{1}{{4\pi a_{0} }}Ei\left( {\frac{{r^{2} }}{{4a_{0} \Delta t}}} \right) $$
(16)
$$ \mathop \int \limits_{{t_{n} }}^{{t_{n + 1} }} {\text{q}}^{*} \left( {P,t_{n + 1} ;Q,{\uptau }} \right)d{\uptau } = \frac{ - 1}{{2\pi a_{0} r}}exp\left( {\frac{{ - r^{2} }}{{4a_{0} \Delta t}}} \right)\frac{\partial r}{{\partial n}} $$
(17)
where the exponential integral function \(Ei\left( \right)\) can be defined as
$$ Ei\left( \alpha \right) = \mathop \int \limits_{\alpha }^{\infty } \frac{{exp\left( { - x} \right)}}{x}dx $$
(18)
The first domain integral of Eq. (13) contains the nonlinear term, that is
$$ I = \frac{{a_{0} }}{{\lambda_{0} }}\int\limits_{\Omega } {\mathop \smallint \limits_{{t_{n} }}^{{t_{n + 1} }} h_{Nl} \left( {Q, \Theta , \dot{\Theta }, \tau } \right)\Theta^{*} \left( {P,t_{n + 1} ;Q,\tau } \right)d\tau d\Omega } $$
(19)
or
$$ I = \frac{{a_{0} }}{{\lambda_{0} }}\int\limits_{\Omega } {\int\limits_{{t_{n} }}^{{t_{n + 1} }} {\left\{ {h\left( {Q, \Theta , \tau } \right) + \left[ {\rho_{0} c_{0} - \frac{{\lambda_{0} }}{\lambda \left( \Theta \right)}\rho \left( \Theta \right) c\left( \Theta \right)} \right]\dot{\Theta }} \right\}\Theta^{*} \left( {P,t_{n + 1} ;Q,\tau } \right)d\tau {\text{d}}\Omega } } $$
(20)
By substituting the midpoint value of \(h_{Nl}\) and finite difference expression of \({\dot{\Theta }}\), we can write
$$ I_{Nl} = \frac{1}{{4\pi \lambda_{0} }}\int\limits_{\Omega } {h_{NI} \left( {Q, \Theta_{n + 0.5} , \dot{\Theta }_{n + 0.5} , t_{n + 0.5} } \right){\text{Ei}}\left( {\frac{{r^{2} }}{{4a_{0} \Delta t}}} \right)d\Omega } $$
(21)
where
$$ h_{Nl} \left( {Q, {\Theta }_{n + 0.5} , t_{n + 0.5} } \right) = h\left( {Q, {\Theta }_{n + 0.5} , t_{n + 0.5} } \right) + \left[ {\rho_{0} c_{0} - \frac{{\lambda_{0} }}{{\lambda \left( {{\Theta }_{n + 0.5} } \right)}}\rho \left( {{\Theta }_{n + 0.5} } \right)c\left( {{\Theta }_{n + 0.5} } \right)} \right]{\dot{\Theta }}_{n + 0.5} $$
(22)
$$ \begin{aligned} & 2C\left( P \right)\Theta \left( {{\text{P}},t_{n + 0.5} } \right) - \frac{1}{2\pi }\int\limits_{\Gamma } {\mathop \smallint \limits_{{t_{n} }}^{{t_{n + 1} }} \frac{{\Theta \left( {Q, t_{n + 0.5} } \right)}}{r}exp\left[ {\frac{{ - r^{2} }}{{4a_{0} \Delta t}}} \right]\frac{\partial r}{{\partial n}} {\text{d}}\Gamma } \\ & = \frac{1}{4\pi }\int\limits_{\Gamma } {{\text{q}}\left( {Q, t_{n + 0.5} } \right) {\text{Ei}}\left( {\frac{{r^{2} }}{{4a_{0} \Delta t}}} \right){\text{d}}\Gamma } \\ & \frac{1}{{4\pi a_{0} }}\int\limits_{\Omega } {h_{NI} \left( {Q, \Theta_{n + 0.5} , \dot{\Theta }_{n + 0.5} , t_{n + 0.5} } \right){\text{Ei}}\left( {\frac{{r^{2} }}{{4a_{0} \Delta t}}} \right){\text{d}}\Omega } \\ & \frac{1}{{4\pi a_{0} \Delta t}}\int\limits_{\Omega } {\Theta \left( {{\text{Q}}, t_{n} } \right)\exp \left( {\frac{{ - r^{2} }}{{4a_{0} \Delta t}}} \right)d\Omega + C\left( P \right)\Theta \left( {{\text{P}}, t_{n} } \right)} \\ \end{aligned} $$
(23)
Now, we implement the CTM without domain discretisation to evaluate the domain integrals of (23). Thus, the unknown values at \(M^{\prime}\) boundary nodes can be computed directly from the following system of matrix equations
$$ H{\Theta }^{{\Gamma }} = GQ^{{\Gamma }} + F + F_{NI} $$
(24)
where \({\Theta }^{{\Gamma }}\) and \(Q^{{\Gamma }}\) are \(M^{\prime}\) dimension vectors contain boundary nodal values \({\Theta }\) and \(q\), \(F\) a vector depends on previous time step, \(F_{NI}\) is a nonlinear term vector depends on unknown internal values, \(H\) and \(G\) are \(M^{\prime} \times M^{\prime}\) dimension coefficient matrices. Also, the unknown values at \(M^{\prime\prime}\) internal points may be calculated from the following system of matrix equations
$$ {\Theta }^{{\Omega }} = \hat{G}Q^{{\Gamma }} - \hat{H}{\Theta }^{{\Gamma }} + \hat{F} + \hat{F}_{NI} $$
(25)
where \(H\), \(G\), \(\hat{H}\) and \(\hat{G}\) can be computed for all time steps. Also, \(F,\;F_{NI} ,\; \hat{F}\;{\text{and}}\; \hat{F}_{NI}\) can be computed using CTM for all time steps.
The CTM method can be implemented to transform several domain integrals into boundary ones [47].
Now, we consider the following two-dimensional regular domain integral
$$ I = \int\limits_{\Omega } {p\left( {x_{1} , x_{2} } \right){\text{d}}\Omega } $$
(26)
By implementing Green’s theorem as
$$ \int\limits_{\Omega } {\frac{{\partial u\left( {x_{1} , x_{2} } \right)}}{{\partial x_{1} }}{\text{d}}\Omega } = \int\limits_{\Gamma } {u\left( {x_{1} , x_{2} } \right){\text{d}}x_{2} } $$
(27)
Now, we can write
$$ I = \int\limits_{{\Gamma }} {P_{1} \left( {x_{1} , x_{2} } \right){\text{d}}x_{2} } $$
(28)
where
$$ P_{1} \left( {x_{1} , x_{2} } \right) = \int\limits_{{\Gamma }} {p\left( {x_{1} , x_{2} } \right){\text{d}}x_{1} } $$
(29)
Since the integral in (29) cannot be determined analytically, so, we evaluate it numerically by the following integral equation
$$ P_{1} \left( {x_{1} , x_{2} } \right) = \mathop \int \limits_{{\upalpha }}^{{x_{1} }} p\left( {x^{\prime}_{1} , x_{2} } \right){\text{d}}x^{\prime}_{1} $$
(30)
According to Khosravifard and Hematiyan [49], and using (30), the domain integral of (26) can be expressed as
$$ I = \int\limits_{{\Gamma }} {\left( {\mathop \int \limits_{{\upalpha }}^{{x_{1} }} p\left( {x^{\prime}_{1} , x_{2} } \right){\text{d}}x^{\prime}_{1} } \right){\text{d}}x_{2} } $$
(31)
where
$$ \alpha = \frac{{x_{1min} + x_{1max} }}{2} $$
(32)
where \(x_{1min}\) and \(x_{1max}\) are minimum \(x_{1}\) and maximum \(x_{1}\) values, respectively.
The composite Gaussian quadrature method is applied to (26) yields
$$ I = \sum\limits_{k = 1}^{K} {\int\limits_{{{\Gamma }_{k} }} {\int\limits_{{\upalpha }}^{{x_{1} }} {\int\limits_{{\upalpha }}^{{x_{1} }} {p\left( {x^{\prime}_{1} , x_{2} } \right){\text{d}}x^{\prime}{\text{d}}x_{2} } } } } $$
(33)
Equation (33) can be expressed as
$$ I = \mathop \sum \limits_{k = 1}^{K} J_{k} \mathop \sum \limits_{i = 1}^{N} w_{i} \mathop \sum \limits_{l = 1}^{L} J_{l} \mathop \sum \limits_{j = 1}^{J} w_{j} p\left( {x_{1} \left( {\eta_{j} } \right), x_{2} \left( {\eta_{i} } \right)} \right) $$
(34)
where \(J_{k}\) and \(J_{l}\) are the transformation Jacobian for the \(k{\text{th}}\) interval \(l{\text{th}}\) interval, respectively, \(K\) is the boundary elements number, \(N\) and \(J\) are the Gaussian integration points numbers of (33) for the outer integral and inner integral, respectively, \(w_{i}\) and \(w_{j}\) are Gauss points weights.
If \(p\) described over a domain-boundary grid with irregularly spaced data. Then, by using the radial point interpolation method (RPIM) [50], the approximation using two-dimensional interpolation of the function \(p\) may be written as
$$ p\left( {x_{1} , x_{2} } \right) = \mathop \sum \limits_{i = 1}^{M} \phi_{i} \left( {x_{1} , x_{2} } \right)p_{i} = {{\varvec{\Phi}}}^{T} {\mathbf{\rm P}} $$
(35)
in which \(\left( {x_{1} , x_{2} } \right)\) is any arbitrary point, \(p_{i}\) is the \(p\) value at \(i\) and \(\phi_{i}\) its shape function, \(M \left( {\text{total}\; \text{ number}} \right) = M^{\prime}\left( {\text{boundary}\; \text{nodes}\; \text{number}} \right) + M^{\prime\prime}\left( {\text{internal}\; \text{grid}\; \text{points}\; \text{number}} \right)\). In the considered RPIM, the consistent shape functions are constructed using the radial basis functions. According to [50], the function \(p\left( {x_{1} , x_{2} } \right)\) can be approximated as
$$ p\left( {\mathbf{x}} \right) = \mathop \sum \limits_{i = 1}^{n} \alpha_{i} \psi_{i} \left( {\mathbf{x}} \right) + \mathop \sum \limits_{j = 1}^{{\overline{m}}} b_{j} u_{j} \left( {\mathbf{x}} \right) = {{\varvec{\Psi}}}^{{\varvec{T}}} \left( {\mathbf{x}} \right){\mathbf{a}} + {\mathbf{u}}^{{\varvec{T}}} \left( {\mathbf{x}} \right){\mathbf{b}} = \left[ {\begin{array}{*{20}c} {{{\varvec{\Psi}}}^{{\varvec{T}}} \left( {\mathbf{x}} \right)} & {{\mathbf{u}}^{{\varvec{T}}} \left( {\mathbf{x}} \right)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\mathbf{a}} \\ {\mathbf{b}} \\ \end{array} } \right\} $$
(36)
The considered method is very simple for computation of regular and weakly singular domain integrals because all computations are performed in universal Cartesian coordinates, where kernels are defined by irregularly spaced data.
In order to create the RPIM shape functions, we apply the following Gaussian radial basis functions (GRBFs)
$$ \psi_{i} \left( {\mathbf{x}} \right) = {\text{exp}}\left[ { - a_{c} \left( {\frac{{R_{i} }}{{d_{c} }}} \right)^{2} } \right] $$
(37)
where \(\psi_{i}\) are radial basis functions (RBFs), \(n\) is the RBFs number, \(\overline{m}\) is the polynomial basis functions number and \(u_{j} \left( {\mathbf{x}} \right)\), the augmented monomials and \(\alpha_{i}\) and \(b_{j}\) are unknown coefficients which can be evaluated from the following \(n\) linear system of equations.
$$ \mathop \sum \limits_{i = 1}^{n} \alpha_{i} \psi_{i} \left( {{\mathbf{x}}_{{\mathbf{i}}} } \right) + \mathop \sum \limits_{j = 1}^{{\overline{m}}} b_{j} u_{j} \left( {{\mathbf{x}}_{{\mathbf{i}}} } \right) = p\left( {{\mathbf{x}}_{{\mathbf{i}}} } \right), i = 1, 2, \ldots , n $$
(38)
and the following \(\overline{m}\) linear constraints
$$ \mathop \sum \limits_{i = 1}^{n} \alpha_{i} u_{j} \left( {{\mathbf{x}}_{{\mathbf{i}}} } \right) = 0, j = 1,2, \ldots , \overline{m} $$
(39)
From Eqs. (38) and (39), we can write \(\alpha_{i}\) and \(b_{j}\) in the following form
$$ \left\{ {\begin{array}{*{20}c} {\mathbf{a}} \\ {\mathbf{b}} \\ \end{array} } \right\} = {\mathbf{BP}} $$
(40)
Based on [50], and using (40), Eq. (36) may be expressed as follows
$$ p\left( {\mathbf{x}} \right) = \left[ {\begin{array}{*{20}c} {{{\varvec{\uppsi}}}^{T} \left( {\mathbf{x}} \right)} & {{\mathbf{u}}^{T} \left( {\mathbf{x}} \right)} \\ \end{array} } \right]{\mathbf{BP}} = \phi^{{\varvec{T}}} {\mathbf{P}} $$
(41)
where the matrix \({\mathbf{P}}\) is location- and geometry-dependent of boundary nodes and internal nodes.
where \(\phi\) is the RPIM shape functions vector
$$ I = \mathop \sum \limits_{k = 1}^{K} J_{k} \mathop \sum \limits_{i = 1}^{N} w_{i} \mathop \sum \limits_{l = 1}^{L} J_{l} \mathop \sum \limits_{j = 1}^{J} w_{j} \mathop \sum \limits_{r = 1}^{M} p_{r} \phi_{r} \left( {x_{1} \left( {\eta_{j} } \right), x_{2} \left( {\eta_{i} } \right)} \right) $$
(42)
Now, Eq. (42) can be expressed as
$$ I = \mathop \sum \limits_{q = 1}^{M} \gamma_{q} p_{q} = {{\varvec{\upgamma}}}^{{\mathbf{T}}} {\mathbf{p}} $$
(43)
in which \({{\varvec{\upgamma}}}\) is the geometry- and location-dependent weight vector of grid points and \({\mathbf{p}}\) includes the values \(p\) at boundary nodes and internal points.