## 2 Basic Principles and Methodologies
### The Coefficient of Heating Effect
#### 2.3.2 Domain Discretization and the Adaptive Grid Evolution Strategy  
Due to the limited number of acoustic sensors and the sparse valid data of the ill-posed AP problem, the domain discretization assumption was adopted in most of the existing reconstruction method. That is, the ROI is divided into N small grids/cells, and the gas parameter (temperature, sound velocity, etc.) within each cell is considered as a uniform value. Then based on the dependencies of sound velocity and temperature, the grid-averaged temperature is solved by a reconstruction algorithm.  
Theoretically, the length of the sound path traveling through each cell determines its temperature. Tracking the distribution of each path length in the whole domain is crucial. The discretization of the domain affects the accuracy of acoustic pyrometry. Although uniform discretization and square grids are widely used because of their simplicity and reliability, they also restrict the potential for further optimization of domain discretization [47]. [PERSON] [30] used appropriate central nonuniform domain discretization and transceiver collocation to significantly improve the accuracy of acoustic pyrometry. [PERSON] [25] successfully tracked the shift pattern of the hotspot via the spatial moving sample (SMS) method. Specifically, the distribution of uniform hexagonal grids was iteratively shifted to increase the number of acoustic paths in the hotspot grid. Therefore, the SMS method and nonuniform discretization were combined to develop the adaptive grid evolution strategy (AGES) in this paper.  
Figure 2: Flow chart of the adaptive hybrid kernel method.  
Different cell sizes ranging from finer grids to coarser grids are considered, while keeping the number of grids constant. The cell sizes are determined via nonuniform discretization as shown in Figure 3. A hotspot is defined as the local highest temperature point [25; 46]. With \(n=11\), the hotspot grid is located in the HPth division in the x direction. \(f_{b}\) is the bias factor, which represents the degree of inhomogeneity across the mesh, and \(g\) is the growth rate. When \(f_{b}=1\), the AGES undergoes uniform discretization. \(E(i)\) is the length of the \(i\)th division in the x direction, \(L\) is the length of the first element, and SL is the side length of the ROI; that is, the side length of the hotspot grid is \(f_{b}\) times that of the first grid. The lengths of the divisions away from the hotspot grid are in a geometric series. The domain discretization in the y direction is identical to that in the x direction.  
The flow chart of the AGES method is shown in Figure 4. Inspired by the CFD simulation in which refined meshes are employed for modelling complex parts to achieve enhanced results, local mesh refinement for hotspot regions is considered in the adaptive nonuniform domain discretization. Specifically, the SMS method is employed to identify and refine the center position of the hotspot grid, the nonuniform domain discretization is implemented, and then the inverse problem of acoustic pyrometry is solved. In addition, the AGES sets the starting point of the nonuniform discretization to the hotspot grid. This nonuniform discretization is consistent with the intrinsic gradient of the temperature field. The reconstruction errors serve as the evaluation criterion, and the hotspot grid location and nonuniform discretization parameters are optimized through an iterative selection process. In summary, AGES employs a dual-action strategy, which gives local refinement of grids near identified hotspots and sparse meshing across thermally uniform regions without increasing the total grid number.  
Figure 4: Flow chart of the adaptive grid evolution strategy.  
Figure 3: Nonuniform discretization scheme used in the AGES.