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一种求解瞬态热传导方程的无条件稳定方法

季奕 邢誉峰

季奕, 邢誉峰. 一种求解瞬态热传导方程的无条件稳定方法. 力学学报, 2021, 53(7): 1859-1869 doi: 10.6052/0459-1879-21-140
引用本文: 季奕, 邢誉峰. 一种求解瞬态热传导方程的无条件稳定方法. 力学学报, 2021, 53(7): 1859-1869 doi: 10.6052/0459-1879-21-140
Ji Yi, Xing Yufeng. An unconditionally stable method for transient heat conduction. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(7): 1859-1869 doi: 10.6052/0459-1879-21-140
Citation: Ji Yi, Xing Yufeng. An unconditionally stable method for transient heat conduction. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(7): 1859-1869 doi: 10.6052/0459-1879-21-140

一种求解瞬态热传导方程的无条件稳定方法

doi: 10.6052/0459-1879-21-140
基金项目: 国家自然科学基金资助项目(11872090)
详细信息
    作者简介:

    邢誉峰, 教授, 主要研究方向: 计算固体力学, 结构动力学和复合材料力学等. E-mail: xingyf@buaa.edu.cn

  • 中图分类号: O302

AN UNCONDITIONALLY STABLE METHOD FOR TRANSIENT HEAT CONDUCTION

  • 摘要: 瞬态热传导问题普遍存在于航空航天、土木和冶金等领域中, 对这类问题精确、高效的数值求解方法一直备受关注. 为此, 本文提出了一种无条件稳定的单步时间积分方法. 在所提出的方法中, 拉格朗日插值函数被用来近似真实的温度场及其一次导数场, 之后, 加权残量法被用来建立二者之间的关系. 通过对算法参数的巧妙设计, 本文提出的方法具有二阶精度和L型数值耗散, 其中, L型耗散使得本文方法能够快速过滤掉虚假的高频振荡. 多数现有时间积分方法仅对线性系统具有无条件稳定性, 对非线性系统则是条件稳定的. 为此, 本文改进了Hughes针对一阶非线性热传导问题提出的时间积分方法稳定性评估理论, 并将改进的理论用于方法的参数设计中. 理论分析的结果表明本文方法对线性和非线性热传导系统均是无条件稳定的. 即使对于著名的Crank-Nicolson方法失稳的非线性热传导问题, 本文方法仍能给出稳定且精确的预测. 数值测试结果显示, 所提出的方法相较于当前流行的方法具有明显的精度、耗散和稳定性优势.

     

  • 图  1  传递因子

    Figure  1.  Amplification factor

    图  2  收敛率

    Figure  2.  Convergence rate

    图  3  算例1的几何尺寸和边界条件

    Figure  3.  Geometry and boundary conditions for Ex.1

    图  4  A和点B的温度−时间曲线

    Figure  4.  Temperatures of point A and point B versus time

    图  5  温度误差分布(t = 0.1)

    Figure  5.  Errors of temperature at t = 0.1

    图  6  温度−时间曲线(γ = 0, β = η = 1)

    Figure  6.  Temperature-time history

    图  7  温度−时间曲线(β = 100, γ = η = 1)

    Figure  7.  Temperature-time history

    图  8  算例3的几何尺寸和边界条件

    Figure  8.  Geometry and boundary conditions for Ex.3

    图  9  A处温度和其一阶导数与时间的关系曲线

    Figure  9.  Temperature and its derivative -time histories of point A

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出版历程
  • 收稿日期:  2021-04-16
  • 录用日期:  2021-06-03
  • 网络出版日期:  2021-06-03

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