Purpose

This paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.

Design/methodology/approach

The discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.

Findings

The paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.

Originality/value

The paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

中文翻译：

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热导率重构方法在面铣加工中的应用

目的

本文旨在基于具有混合边界条件的偏微分方程（PDE）模型描述的二维逆热传导问题来重建空间变化的正交各向异性传导率。所提出的离散化使用高度精确的技术并且允许简单的实现。此外，作者以在迭代中强制平滑的方式解决了相关的逆问题，在综合示例和移动热源的实际问题中显示了有希望的结果。

设计/方法论/途径

应用于直接问题模型的离散化过程在空间变量中使用伪谱配置策略，在时间相关变量中使用克兰克-尼科尔森方法。然后，通过使用奇异标度矩阵的 Levenberg-Marquardt 方法 (LMM) 的修改版本来解决从温度测量中恢复电导率的相关反问题。由于面铣削操作问题，还考虑了数据可用性有限的问题。数值例子表明了该方法的准确性和效率。

发现

本文提出了偏微分方程模型的离散化，旨在实现简单的实现和数值性能。使用奇异缩放矩阵引入的 LMM 的修改版本显示了在少量迭代下精确恢复数量的能力。数值结果表明，合成噪声数据的精确解和近似解之间具有良好的拟合度，并且当实验数据反转时，逆解解也非常可接受。

原创性/价值

这篇论文之所以重要，是因为伪谱方法以其高精度和易于实现而闻名，并且在 LMM 迭代中使用奇异正则化矩阵，与该方法的经典实现不同，对重建过程产生了积极影响。