Abstract In this paper, a new two-probe surface heat flux calibration integral equation method (CIEM) is proposed for resolving nonlinear inverse heat conduction problem possessing temperature-dependent thermophysical properties and an unknown time-varying back surface boundary condition. Compared to the one-probe CIEM, the two-probe CIEM can be applied to a more general class of one-dimensional inverse heat conduction problems that can be used for accurately predicting the front surface heat flux without any restrictions on the back surface boundary condition. Temperature-dependent thermophysical properties and probe locations are not specified a priori but are implicitly accounted through a calibration campaign. The final mathematical formulation involves resolving a Volterra integral equation of the first kind for the unknown surface heat flux. A first kind Chebyshev expansion possessing undetermined coefficients, is used for approximating the introduced property transform function. Next, three well-defined calibration tests are used for estimating the undetermined coefficients associated with the Chebyshev expansion. A time sequential study of the coefficients is shown to yield the optimal truncation of the series expansion. A future-time method is applied for stabilizing the ill-posed first kind Volterra integral equation. Phase plane and cross-correlation analyses are employed for estimating the optimal regularization parameter (i.e., the future-time parameter) from a spectrum of chosen values. The effectiveness of the proposed approach is verified by numerical simulation using stainless steel as the host material. The calibration-based or “parameter free” inverse approaches significantly reduce time and costs in the resolution of the surface condition as there is no need to experimentally determine the thermophysical properties, probe locations, and probe characteristics. These system properties are implicitly included in the proposed calibration procedure. Many applications, especially in aerospace sciences where exotic materials are often used, do not have replicable properties among samples. Conventional inverse methods rely on accurately specifying the input parameters for forming the predictions. These “parameter required” methods are solely based on numerical algorithms and thus their predictions are strongly relied on accurate input of parameters and measured probe data.

中文翻译：

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表面热通量估计非线性逆热传导问题的双探针标定积分方程法

摘要 在本文中，提出了一种新的双探针表面热通量校准积分方程方法（CIEM），用于解决具有温度相关热物理特性和未知时变背面边界条件的非线性逆热传导问题。与单探针 CIEM 相比，双探针 CIEM 可以应用于更一般的一类一维逆热传导问题，可用于准确预测前表面热通量，而对后表面边界条件没有任何限制. 与温度相关的热物理特性和探头位置不是先验指定的，而是通过校准活动隐式计算的。最终的数学公式涉及求解未知表面热通量的第一类 Volterra 积分方程。具有不确定系数的第一类切比雪夫展开式用于逼近引入的属性变换函数。接下来，三个定义明确的校准测试用于估计与切比雪夫展开相关的未确定系数。显示了对系数的时间顺序研究以产生级数展开的最佳截断。应用未来时间方法来稳定不适定的第一类Volterra积分方程。相平面和互相关分析用于从选定值的频谱中估计最佳正则化参数（即，未来时间参数）。通过以不锈钢为主体材料的数值模拟验证了所提出方法的有效性。基于校准或“无参数”逆向方法显着减少了解决表面条件的时间和成本，因为无需通过实验确定热物理特性、探针位置和探针特性。这些系统属性隐含在建议的校准程序中。许多应用，尤其是在经常使用特殊材料的航空航天科学中，在样品之间没有可复制的特性。传统的逆方法依赖于准确指定用于形成预测的输入参数。这些“需要参数”的方法完全基于数值算法，因此它们的预测强烈依赖于参数的准确输入和测量的探针数据。