# An Efficient Meshless Numerical Method for Heat Conduction Studies in Particle Aggregates
## 2 Materials and Methods  
A truly meshless method based on the MLPG approach is presented below for the solution of heat conduction problems. The success of the MLPG method depends on three important parameters, namely, the approximation of the unknown function and its derivatives (trial function), the weight function of the integration (test function), and the shape of the integration domain. As trial function, the widely used approach of the MLS method was implemented. Also, the DS PSE operator was tested for the first time, to the authors' knowledge, as a trial function of the MLPG method. The solution approximation, \(T^{h}(x)\), is described following the MLS technique [21], as the product of a vector of polynomial base, \(\mathbf{p}(x)\), and a vector of coefficients, \(\mathbf{a}(\mathbf{x})\),  
\[T^{h}(x)=\sum\
olimits_{j}^{m}p_{j}(\mathbf{x})a_{j}(\mathbf{x})\equiv\mathbf{p}^{\mathbf{T}}( \mathbf{x})\mathbf{a}(\mathbf{x}), \tag{1}\]  
\(m\) is a number related with the order of the polynomial base. In order to determine the form of \(\mathbf{a}(\mathbf{x})\), a weighted discrete error norm is constructed and minimized. This process results to the construction of the shape function,  
\[\mathbf{T}^{h}(\mathbf{x})=\mathbf{p}^{\mathbf{T}}(\mathbf{x})\mathbf{A}^{-1}(\mathbf{x})\mathbf{B}(\mathbf{x}) \mathbf{U}_{\mathbf{s}}\equiv\mathbf{\Phi}^{\mathbf{T}}(\mathbf{x})\mathbf{U}_{\mathbf{s}}=\sum\
olimits_{ i=1}^{n}\mathbf{\Phi}_{\mathbf{I}}(\mathbf{x})\mathbf{T}_{\mathbf{I}}, \tag{2}\]  
\(T_{\mathbf{I}}\) is a vector containing the nodal data, and \(\mathbf{A},\;\mathbf{B}\) are matrix functions of the polynomial base and an exponential weight function, and \(n\) is the number of nodes in the support of domain of node \(\mathbf{x}\). Derivatives of the shape function, \(\Phi_{\mathbf{I}}(\mathbf{x})\), can be calculated with the product rule. DC PSE [22; 23] was originally formulated as a correction of the particle strength exchange method (PSE) [24]. Considering a continuous field \(f(\mathbf{x})=f(\mathbf{x},\;y)\), any derivative can be expressed as  
\[D^{m,n}\;f\Big{(}\mathbf{x}_{p}\Big{)}=\left.\frac{\partial^{m+n}}{\partial x^{m} \partial y^{n}}\;f(\mathbf{x},\;y)\right|_{\mathbf{x}=\mathbf{x}_{p},\mathbf{y}=\mathbf{y}_{p}}, \tag{3}\]  
where \(m\) and \(n\) are integers that determine the order of the operator. The DC PSE operator for this derivative is  
\[\mathbf{Q}^{m,n}\;f\Big{(}\mathbf{x}_{p}\Big{)}=\frac{1}{\epsilon\Big{(}\mathbf{x}_{p} \Big{)}^{m+n}}\sum\
olimits_{\mathbf{x}_{q}\in N(\mathbf{x}_{p})}\Big{(}f\Big{(}\mathbf{x }_{q}\Big{)}\pm f\Big{(}\mathbf{x}_{p}\Big{)}\Big{)}\eta\left(\frac{\mathbf{x}_{p}- \mathbf{x}_{q}}{\epsilon\Big{(}\mathbf{x}_{p}\Big{)}}\right), \tag{4}\]  
where \(\epsilon(\mathbf{x})\) is a resolution function, \(N\Big{(}\mathbf{x}_{p}\Big{)}\) are the nodes of the support domain, and \(\eta(\mathbf{x},\epsilon(\mathbf{x}))\) is a kernel function. The weighted integral of the heat conduction equation at the domain \(\Omega_{x}\) around a point \(\mathbf{x}\) is given by  
\[\int_{\Omega_{x}}\overline{\psi}\Big{(}\overline{k\
abla T}\Big{)}vd\Omega=0, \tag{5}\]  
where \(v\) is the weight function (here, the step function), and \(\overline{k}=\overline{k}(\mathbf{x})\) is the thermal conductivity. Reduced quantities are determined based on the physical values of the media, the domain geometry, and the boundary conditions  
\[x=\frac{\overline{x}}{L},\;y=\frac{\overline{y}}{L},\;T=\frac{\overline{T}- \overline{T}_{c}}{\overline{T}_{H}-\overline{T}_{c}},k=\frac{\overline{k}}{k_ {f}}=\left\{ \begin{array}{l}1\;\text{if}\;\mathbf{x}\in\Omega_{f}\\ k_{r}\;\text{if}\;\mathbf{x}\in\Omega_{p}\end{array} \right.. \tag{6}\]\(L\) is the length of the domain, and \(\widetilde{T}_{H}\), \(\widetilde{T}_{c}\) are the values of the temperature imposed at the hot and cold boundary boundaries, respectively. \(\Omega_{f}\) and \(\Omega_{p}\) are the domains of medium \(f\) (exterior) and \(p\) (interior) of the enclosure.