Abstract The present work conducts a systematic and in-depth algorithm investigation for transient nonlinear heat conduction problems solved by using the proper orthogonal decomposition (POD) method. Part 1 of this two-part articles presents the process and characteristics of basic algorithms, including POD explicit and implicit time-marching methods. The accuracy and efficiency are verified by several numerical examples with various boundary conditions and element types. The results show that the POD-based reduced order model (ROM) can provide high quality temperature prediction of the transient nonlinear heat conduction problems when using implicit method. However, the computational time of implicit method is much longer than that of explicit one. The acceleration effect of POD-based ROM on the calculation of the transient nonlinear heat conduction problems is one order of magnitude lower than that of the corresponding transient linear heat conduction problems. The improvement of computational efficiency is not pronounced. Further studies of the more efficient advanced algorithms to deal with POD-based ROM for transient nonlinear heat conduction problems will be presented in Part 2. Additionally, an approximate POD-based ROM for transient nonlinear heat conduction problem is proposed, which can be constructed quickly by using POD modes obtained from the corresponding transient linear heat conduction system. It is confirmed to be feasible for allowing nonlinear behavior to be modeled at an acceptable level of accuracy. It has significant application potential of solving practical engineering problems.

中文翻译：

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一种适用于瞬态非线性热传导问题的正交分解分析方法。第 1 部分：基本算法

摘要 目前的工作对通过使用适当的正交分解（POD）方法解决的瞬态非线性热传导问题进行了系统和深入的算法研究。这篇由两部分组成的文章的第 1 部分介绍了基本算法的过程和特征，包括 POD 显式和隐式时间推进方法。通过具有各种边界条件和单元类型的几个数值例子验证了精度和效率。结果表明，当使用隐式方法时，基于 POD 的降阶模型 (ROM) 可以提供瞬态非线性热传导问题的高质量温度预测。然而，隐式方法的计算时间比显式方法长得多。基于POD的ROM对瞬态非线性热传导问题计算的加速效果比对应的瞬态线性热传导问题低一个数量级。计算效率的提升并不明显。第 2 部分将进一步研究处理基于 POD 的瞬态非线性热传导问题的 ROM 的更有效的高级算法。此外，还提出了一个近似的基于 POD 的瞬态非线性热传导问题的 ROM，它可以快速构建通过使用从相应的瞬态线性热传导系统获得的 POD 模式。已确认允许以可接受的精度水平对非线性行为进行建模是可行的。