## 2 Methods
### Physical and Mathematical Background  
For the simplicity of all formulas, we shall work everywhere with the 1D model reduction. The thermal energy conservation (avoiding other physical processes) can be then read as  
\[\rho c\;\partial t/\partial\tau=\partial q/\partial x \tag{1}\]  
for a certain thermal flux rate \(q\) [W\(\cdot\)m\({}^{-2}\)], which is a function of \(\tau\) in general (observe the formerly introduced temperature \(t\) [K] and time \(\tau\) [s]), and is usually supplied by the empirical linearised Fourier law. \[q=\lambda\;\partial t/\partial x, \tag{2}\]  
Thus, (1) and (2) yield  
\[\rho c\;\partial t/\partial\tau=\partial(\lambda\;\partial t/\partial x)/ \partial x. \tag{3}\]  
Here, the following material characteristics occur: the material density \(\rho\) [kg\(\cdot\)m\({}^{-3}\)], the thermal conductivity \(\lambda\) [W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)], and the thermal capacity \(c\). One can take these characteristics as (nearly) constant and time-independent, thus, the thermal diffusivity \(a=\lambda/(\rho c)\) can be frequently found in engineering considerations. We can introduce \(x\) belonging to a certain finite interval [0, \(d\)]; the natural generalisation will be discussed later. This approach can also be applied to the 2D, 3D, etc., models, with minor modifications of (1), (2), (3): namely \(\partial/\partial x\) must be replaced by the gradient operator, and the material characteristics are allowed to receive a matrix form to handle an anisotropic medium. However, (1), (2), and (3) are able to describe thermal conduction on [0, \(d\)] only; one unknown field \(t\) dependent on both \(x\) and \(\tau\) is present here. Thermal convection can be seen, namely on the exterior surfaces, using the evaluation formula for the surface heat flux rate:  
\[q=h\left(t-t^{\bullet}\right); \tag{4}\]  
Here, \(t^{\bullet}\) refers to the temperature of the outer environment, and \(h\) denotes the thermal transfer factor assigned to the related surface. Thanks to (2) and (4), this can generate the Neumann boundary conditions for all outer building surfaces or their parts. In particular, for the zero-valued \(h\), we come to \(t=t^{\bullet}\), which leads to the Dirichlet boundary conditions; this enables us, i.e., to decompose [0, \(d\)] for (1), (2), (3) to a finite number of subintervals, typically, into constructive and insulation layers in walls, without any change in previous considerations. This is significant for evaluating the total heat passage coefficients \(U\), namely for windows, door(s), etc., required by most technical standards for building structures [69]. Apart from the boundary conditions, the natural Cauchy initial condition, i.e., the prescribed values of \(t\) everywhere on [0, \(d\)], must be implemented. In numerous simplified computations, the total flux rate \(Q\) is also needed instead of the local one, e.g., \(Q=qS\) for (4), where \(S\) means the related surface area, as transferred via gbXML, cf. Section 2.1. Even the surface radiation by the Stefan-Boltzmann law can be incorporated using such a unified approach, replacing both \(t\) and \(t^{\bullet}\) in (4) by their 4 th powers and \(h\) by \(\varepsilon\sigma\); here, \(\sigma=5.670374419\ldots\times 10^{-8}\) W\(\cdot\)m\({}^{-2}\).K\({}^{-4}\) is the famous Stefan-Boltzmann constant, and \(\varepsilon\) denotes the prescribed surface emissivity. The effect of designed ventilation and air exchange loss through imperfect joints and various other gaps can be evaluated as  
\[Q=Vc^{\circ}\left(t-t^{\bullet}\right) \tag{5}\]  
where the temperature \(t\) corresponds to specific rooms, and they are considered constant there (dependent only on \(\tau\)); \(c^{\circ}\) denotes the thermal capacity of air, and the exchanged air volume \(V\) must be estimated in a reasonable way. The simple analogy of (5) can also be formulated for the heat exchange between two different rooms.