Accurate modeling of a thermal field is one of the fundamental requirements in engineering thermal management in numerous industries. Existing studies have shown that using differential equations to model a thermal field delivers good performance when the parameters are predetermined through physical or experimental analysis. However, due to variations of the inner medium affected by certain latent factors, the parameters in differential equation models may not be treated as constants while the thermal field is estimated, and this fact poses a new challenge to field estimation by directly solving the differential equation models. In this study, a novel approach to thermal field modeling is developed by considering the parameters as functional variables that vary temporally in partial differential equations (PDEs). This approach provides a new perspective to model the dynamic thermal field by fully using the collected sensor data from the thermal system. Specifically, time-varying parameters can be constructed through a combination of basis functions whose coefficients can be efficiently estimated through the sensor data. A two-level iterative parameter estimation algorithm is also tailored to obtain the parameters in the PDE model. Both simulation and real case studies show that our proposed approach provides satisfactory estimation performance compared with the benchmark method that uses the constant parameter estimation. Note to Practitioners —The proposed method aims to model a thermal field using PDEs with time-varying parameters. To better implement this method in practice, three things are noteworthy: first, the proposed method models a thermal field by fully considering physics-specific engineering knowledge using PDEs and the collected sensor data from thermal systems. Second, because time-varying parameters in PDEs cannot be estimated directly, the proposed model represents the time-varying parameters by a combination of B-spline basis functions in terms of time. Estimating time-varying parameters is converted into estimating the constant coefficients of the basis functions. Because the derivatives of a thermal field might not have an analytical expression, the proposed model represents the thermal field by a combination of B-spline basis functions. Taking the derivatives of the thermal field is converted into taking the derivatives of the corresponding basis functions. Third, the proposed method can not only model a thermal field but can also be applied in other physics-specific engineering cases.

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使用时变参数的偏微分方程对时空温度场进行建模

热场的精确建模是众多行业中工程热管理的基本要求之一。现有研究表明，当通过物理或实验分析预先确定参数时，使用微分方程对温度场进行建模可提供良好的性能。然而，由于内部介质受某些潜在因素的影响，在估计热场时，微分方程模型中的参数可能不会被视为常数，这对直接求解微分方程的场估计提出了新的挑战。楷模。在这项研究中，通过将参数视为在偏微分方程（PDE）中随时间变化的函数变量，开发了一种新颖的热场建模方法。通过充分利用从热系统收集的传感器数据，该方法为动态热场建模提供了新视角。具体地，可以通过基函数的组合来构造时变参数，该基函数的系数可以通过传感器数据被有效地估计。还设计了两级迭代参数估计算法，以获取PDE模型中的参数。仿真和实际案例研究均表明，与使用恒定参数估计的基准方法相比，我们提出的方法可提供令人满意的估计性能。时变参数可以通过基函数的组合来构造，这些基函数的系数可以通过传感器数据进行有效估计。还设计了两级迭代参数估计算法，以获取PDE模型中的参数。仿真和实际案例研究均表明，与使用恒定参数估计的基准方法相比，我们提出的方法可提供令人满意的估计性能。时变参数可以通过基函数的组合来构造，这些基函数的系数可以通过传感器数据进行有效估计。还设计了两级迭代参数估计算法，以获取PDE模型中的参数。仿真和实际案例研究均表明，与使用恒定参数估计的基准方法相比，我们提出的方法可提供令人满意的估计性能。执业者注意 -提出的方法旨在使用具有时变参数的PDE对温度场进行建模。为了在实践中更好地实现此方法，需要注意三件事：首先，所提出的方法通过使用PDE和从热系统收集的传感器数据充分考虑特定于物理的工程知识来对热场建模。其次，由于不能直接估计PDE中的时变参数，因此所提出的模型通过结合B样条基函数在时间上表示时变参数。估计随时间变化的参数将转换为估计基函数的常数系数。由于热场的导数可能没有解析表达式，因此所提出的模型通过结合B样条基函数来表示热场。将热场的导数转换为相应基函数的导数。第三，所提出的方法不仅可以对热场进行建模，还可以应用于其他物理特定的工程案例。