# Fast Temperature Field Extrapolation Under Non-Periodic Boundary Conditions
## 2 Methodology  
### POD Reduced-Order Method  
As an efficient method of order reduction, the POD technique can use fewer POD modes to describe the original physical field well. The combination of POD and a numerical method can better reduce the degree of freedom of the algebraic equations system of the full-order model to the two-digit level, thus greatly reducing the numerical calculation time and meeting the need for fast calculation or fast prediction. Before using the POD order reduction method, appropriate sample data should be prepared and assembled into a transient matrix. Since this paper mainly studies the solution of unknown time periods in the calculation of an unsteady temperature field, the sample data collected came from discrete data on a known time scale. The temperature \(T(x,t_{i})\) of the unsteady temperature field at a specific time is recorded as a set of temperature snapshots, then the temperature snapshots taken at different times are assembled into a temperature transient image matrix \(T(x,t)\). Firstly, finite orthogonal series expansion is used to approximate the original system, namely:  
\[T(x,t)\approx\sum_{i=1}^{N}\alpha_{i}(t)\varphi_{i}(x)=T^{\prime}(x,t) \tag{1}\]  
In the formula, \(T^{\prime}(x,t)\) represents the POD's approximate solution of the temperature field system, \(\alpha\) represents the coefficient of a linear combination of the POD modes, \(\varphi\) represents the POD modes, and \(N\) represents the number of intercepted POD modes. The POD is used to extract low-order important features from the data in the process of system change, and then a combination of fewer POD modes and coefficients is used to reconstruct the complex physical process. The criterion of POD mode selection is that the reserved modes can achieve the best approximation to the temperature field, namely, the error between the temperature transient matrix and the approximate temperature transient matrix is minimal. The most widely used approach to solve the above problem is the \(L_{2}\)-norm, also known as the least-square estimation, which can be formulated as follows:  
\[T^{\prime}(x,t)=\underset{\varphi_{i}(x)}{\text{argmin}}\bigg{\|}T(x,t)-\sum _{i=1}^{N}\alpha_{i}(t)\varphi_{i}(x)\bigg{\|}_{2}^{2} \tag{2}\]  
The truncation error between the selected POD modes' reconstructed temperature field matrix \(T^{\prime}(x,t)\) and the original temperature field matrix \(T(x,t)\) can be expressed as follows:  
\[error=\sum_{i=N+1}^{\infty}\varphi_{i}^{T}(T^{\prime}T^{\prime T})\varphi_{i} \tag{3}\]  
The Lagrange multiplier \(\pi_{i}\) is introduced to establish the Lagrange function as:  
\[\varphi_{i}^{T}R\varphi_{i}=\pi_{i} \tag{4}\]  
As can be seen from the above formula, the Lagrange multiplier \(\pi_{i}\) is the eigenvalue of the real symmetric matrix \(R\) and \(\varphi_{i}\) is the eigenvector of the real symmetric matrix \(R\). According to the properties of the real symmetric matrix, it can be seen that the eigenvalues of the matrix \(R\) are real numbers and the eigenvectors are orthogonal to each other. Therefore, the truncation error can be expressed as:  
\[error=\sum_{i=N+1}^{\infty}\varphi_{i}^{T}R\varphi_{i}=\sum_{i=N+1}^{M}\pi_{i} \tag{5}\]where \(M\) is the number of matrix eigenvalues. As can be seen from the above formula, the truncation error can be calculated by the sum of the (\(M\)-\(N\)) eigenvalues of the real symmetric matrix \(R\). The eigenvalues of the real symmetric matrix \(R\) are arranged from the largest to the smallest, and we select the first \(N\) largest eigenvalues in order to minimize the sum of the truncated unreserved eigenvalues. Therefore, the eigenvectors corresponding to the retained eigenvalues constitute the optimal orthogonal mode set of the POD.