*2.1. General Description*

Thermal simulation was performed for the two-dimensional cross-sectional area of the investigated structure in the middle of the resistors' length. To obtain the temperature simulation results, the DPL model, Equation (2), was used. In order to make the analysis more e ffective, the system of Equation (2) was transformed to an equivalent form, presented below, for two-dimensional space [34]:

$$\begin{aligned} \mathbf{c}\_{\upsilon} \cdot \boldsymbol{\tau}\_{q} \cdot \frac{\partial^{2} \mathbf{T}(\mathbf{x}, \boldsymbol{y}, t)}{\partial t^{2}} + \mathbf{c}\_{\upsilon} \cdot \frac{\partial \mathbf{T}(\mathbf{x}, \boldsymbol{y}, t)}{\partial t} - \mathbf{k} \cdot \boldsymbol{\tau}\_{T} \cdot \frac{\partial \boldsymbol{\Lambda} \mathbf{T}(\mathbf{x}, \boldsymbol{y}, t)}{\partial t} - \mathbf{k} \cdot \boldsymbol{\Delta} \mathbf{T}(\mathbf{x}, \boldsymbol{y}, t) &= q \boldsymbol{\nu} \left( \mathbf{x}, \boldsymbol{y}, t \right) \\ \mathbf{x}, \boldsymbol{y} &\in \mathbf{R}, \ t \in \mathbf{R}\_{+} \cup \{0\} \end{aligned} \tag{5}$$

The Laplacian of a temperature function Δ *T* was approximated by using the Finite Di fference Method (FDM) according to the following formulas:

$$
\Delta T(\mathbf{x}, y, t) = \frac{\partial^2 T(\mathbf{x}, y, t)}{\partial \mathbf{x}^2} + \frac{\partial^2 T(\mathbf{x}, y, t)}{\partial y^2}, \qquad \text{or, } y \in \mathbb{R}, \ t \in \mathbb{R}\_+ \cup \{0\} \tag{6}
$$

$$\frac{\partial^2 T(\mathbf{x}, y, t)}{\partial \mathbf{x}^2} \approx \frac{T(\mathbf{x} + \Delta \mathbf{x}, y, t) - 2 \cdot T(\mathbf{x}, y, t) + T(\mathbf{x} - \Delta \mathbf{x}, y, t)}{\left(\Delta \mathbf{x}\right)^2}, \mathbf{x}, y \in \mathcal{R}, \ t \in \mathcal{R}\_+ \cup \{0\} \tag{7}$$

$$\frac{\partial^2 T(\mathbf{x}, y, t)}{\partial y^2} \approx \frac{T(\mathbf{x}, y + \Delta y, t) - 2 \cdot T(\mathbf{x}, y, t) + T(\mathbf{x}, y - \Delta y, t)}{\left(\Delta y\right)^2}, \mathbf{x}, y \in \mathcal{R}, \ t \in \mathcal{R}\_+ \cup \{0\} \tag{8}$$

Thus, considering the same di fference between nodes in both axes, i.e., Δ*x* = Δ*y*, Laplacian Δ *T* can be approximated in the following way:

$$\Delta T(\mathbf{x}, y, t) \approx \frac{T(\mathbf{x} + \Delta \mathbf{x}, y, t) + T(\mathbf{x}, y + \Delta \mathbf{x}, t) - \Phi \cdot T(\mathbf{x}, y, t) + T(\mathbf{x} - \Delta \mathbf{x}, y, t) + T(\mathbf{x}, y - \Delta \mathbf{x}, t)}{\left(\Delta \mathbf{x}\right)^2},\tag{9}$$
 dla  $\mathbf{x}, \mathbf{y} \in \mathbf{R}$ ,  $t \in \mathbf{R}\_+ \cup \{0\}$ 

On the basis of this methodology, the authors' method for structure discretization and FDM matrices generation was proposed. Moreover, taking into consideration the proposed approximation, the DPL equation, in the form of Equation (5), has become an ordinary di fferential equation of a time variable. Finally, the prepared matrix system of equations was solved for di fferent points of time, using a class of Backward Di fferentiation Formulas (BDF) [35–38].

In the following subsections, the structure discretization, as well as the proposed discretization scheme for DPL model describing the temperature distribution in the cross-sectional area of investigated test structure, is characterized in detail.
