# A Dynamic Model Updating Method with Thermal Effects Based on Improved Support Vector Regression
###### Abstract  
The dynamic modeling of structures in a thermal environment has become a new research topic in structural dynamics. The amount of calculation caused by the complexity of the structure and the size of the FEM, which increase the difficulty in modeling the structural dynamic thermal effects are considered. In this study, model updating in thermal temperature environment is proposed based on the hierarchical method and improved SVR, and an iterative procedure is presented. First, the dynamic problem of structure under a thermal environment is classified into a thermal model and a structural dynamic model, and they are both constructed with the FE method. As a result, the model updating process is conducted for both the thermal model and structural dynamic model. Different from the variables in other model updating methods, the updating variables, which are composed of the mechanical characteristics and thermal parameters, in the proposed method are dominated by the temperature distribution of the structure. A surrogate model based on improved SVR is adopted in the hierarchical model updating approach to make the updating process more efficient. Finally, the improved SVR method is validated on a typical nonlinear function, and the proposed method is validated by updating the model of an elastic thin plate and a wing structure in a thermal environment. model updating; improved support vector regression; thermal temperature environment; surrogate model; hierarchical method +
Footnote †: journal: Applied  
12 July 2021  
26 August 2021  
1  
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Considering the existence of the fitting error, the relaxation variables \(\xi\geq 0\) and \(\xi*\geq 0\) are introduced to improve the generalization performance and the mathematical model of SVR, which includes the relaxation variable, is as follows:  
\[ \begin{array}{c}\min\frac{1}{2}\|\mathbf{w}\|^{2}+C\sum\limits_{i=1}^{N}\left( \xi_{i}+\xi_{i}^{*}\right)\\ \text{s.t.}\left\{\begin{array}{l}y_{i}-(<\mathbf{w},\mathbf{x}_{i}>+b)\leq \varepsilon+\xi_{i}^{*}\\ (<\mathbf{w},\mathbf{x}_{i}>+b)-y_{i}\leq\varepsilon+\xi_{i}\\ \xi_{i},\xi_{i}^{*}\geq 0,i=1,\cdots,N\end{array}\right.\end{array} \] (10)  
where \(\mathbf{w}\) is the optimization objectives, \(\varepsilon\) is the parameter of the insensitive error function, which is used to quantify the error between the fitted data and the training sample, \(C\) is penalty functions and a threshold. The SVR problem above is generally solved by dual theory, which is transformed into a quadratic programming problem. The Lagrange equation is established as follows:  
\[ \begin{array}{c}L(\mathbf{w},\xi,\xi*)=\frac{1}{2}\|\mathbf{w}\|^{2}+C\sum \limits_{i=1}^{n}\left(\xi_{i}+\xi_{i}^{*}\right)-\sum\limits_{i=1}^{n}a_{i}( \varepsilon+\xi_{i}+y_{i}-<\mathbf{w},\mathbf{x}_{i}>-b)\\ -\sum\limits_{i=1}^{n}a_{i}^{*}(\varepsilon+\xi_{i}^{*}+y_{i}-<\mathbf{w}, \mathbf{x}_{i}>-b)-\sum\limits_{i=1}^{n}\left(\eta\xi_{i}+\eta*\xi_{i}^{*} \right)\end{array} \] (11)  
The partial derivatives of the above equation for parameters \(\mathbf{w}\), \(b\), \(\xi_{i}\) and \(\xi_{i}^{*}\) are zero, and this condition can be substituted into Equation (10) to obtain the dual optimization problem:  
\[ \begin{array}{c}\min\frac{1}{2}\sum\limits_{i=1}^{N}\left(a_{i}^{*}-a_{i} \right)(a_{j}^{*}-a_{j})<x_{i},x_{j}>+\sum\limits_{i=1}^{N}a_{i}(\varepsilon- y_{i})+\sum\limits_{i=1}^{N}a_{i}^{*}(\varepsilon+y_{i})\\ \text{s.t.}\left\{\begin{array}{l}\sum\limits_{i=1}^{N}\left(a_{i}^{*}-a_{i }\right)=0\\ 0\leq a_{i}^{*}\leq C,0\leq a_{i}\leq C,i=1,\cdots,N\end{array}\right.\end{array} \] (12)  
According to the KKT condition, the optimization problem in the above equation exists the following equations at the optimal solution. \[\left\{ \begin{array}{l}\alpha_{i}(\varepsilon+\xi_{i}+y_{i}-\left(\mathbf{ w}\cdot\mathbf{x}_{i}\right)-b)=0\\ \alpha_{i}^{*}\left(\varepsilon+\xi_{i}^{*}-y_{i}+\left(\mathbf{w}\cdot \mathbf{x}_{i}\right)+b\right)=0\end{array} \right.\] (13)  
In addition:  
\[\left\{