基于微分转换边界元法和布谷鸟搜索算法的瞬态热传导反问题研究

The research on inverse transient heat conduction problems has important theoretical significance and engineering application value in aerospace and petrochemical industry. The project proposes an improved cuckoo search algorithm to identify boundary conditions and geometry shapes for inverse transient heat conduction problems. The direct problems are analyzed by the differential transformation boundary element method(BEM) and the radial integral method. Firstly, differential transformations are implemented to treat relative physical variables so the time-independent and recursive governing equations are obtained. At the same time, the boundary integral equations are established by the weighted residual method to solve direct transient heat conduction problems, while the domain integrals are transformed into boundary integrals by the radial integral method. Secondly, the inverse problems of boundary conditions or geometry shapes are transformed into the undetermined coefficient problems by using the expansion of time-independent basis functions. After that, the objective function of transient inverse heat conduction problems is presented. The cuckoo search algorithm is applied to obtain numerical results. Finally, the improved schemes of cuckoo search algorithm are put forward, in order to increase its local search capability and convergence speed. The improved cuckoo search algorithm is compared with the conventional gradient method. Furthermore, the influence rules on inverse results are investigated by considering different initial values, position of measure points, number of measure points and measurement errors. Thus the validity and stability of the improved cuckoo search algorithm are tested and verified. The research can expand the application of the differential transformation boundary element method and provide a new theory and method to solve inverse heat conduction problems.

研究瞬态热传导反问题在航空航天、石油化工等领域具有重要的理论意义和应用价值。本项目提出改进的布谷鸟搜索算法反演瞬态热传导问题的边界条件和几何形状。正问题采用微分转换边界元法和径向积分法进行分析。首先，将物理量进行微分转换，推导与时间无关且具有递推格式的控制方程，并通过加权余量法建立边界积分方程，对域积分采用径向积分法处理，实现瞬态热传导正问题的求解。其次，利用与时间无关的基函数将反演边界条件和几何形状问题转化为求解待定系数问题，建立瞬态热传导反演问题的目标函数，采用布谷鸟搜索算法求解。最后，为提高布谷鸟算法的局部搜索能力和收敛速度提出改进方案，且对比分析布谷鸟算法相对常规梯度算法的优势。探讨初值、测点位置、测点数量、测量误差对反演结果的影响规律，验证改进的布谷鸟搜索算法的有效性和稳定性。该研究可拓展微分转换边界元法的应用，为瞬态热传导反问题提供新的理论和求解方法。

瞬态热传导反问题的研究在航空航天、石油化工等领域具有重要的理论意义和应用价值。反问题的求解方法包含梯度类算法和元启发式优化算法。梯度类算法具有局部搜索能力强，收敛速度快，计算精度高等优点，但是对计算初值较为敏感，且容易陷入局部最优解。元启发式优化算法如布谷鸟搜索算法具有全局搜索能力强，不需要梯度信息，待求参数少等优点，但是收敛速度较慢，计算精度较低。本项目结合梯度类算法和元启发式算法的优点，提出了BFGS、共轭梯度法等改进的布谷鸟搜索算法，反演了稳态和瞬态热传导问题的边界温度和热流条件、热物性参数和边界几何形状。.热传导正问题的求解采用边界元方法。针对功能梯度材料等非线性材料或瞬态问题中的域积分，采用径向积分法或双重互易法处理。对于瞬态热传导问题中出现的时间导数项积分，采用有限差分法或微分转换法计算。建立了微分转换-双重互易边界元法。微分转换法受时间步长影响小，微分转换双重互易边界元法计算结果精度高。.针对边界条件重构反问题，提出与空间或时空相关的多项式函数来近似未知的温度边界条件，使用BFGS改进布谷鸟算法寻找最优的未知多项式系数，进一步可求得待求的边界温度和热流。也提出了改进萤火虫算法求解。针对热物性参数识别反问题，采用多项式函数近似，将未知热物性系数识别问题转换为多项式系数识别问题。利用改进布谷鸟算法、LM算法等获得反演结果。针对边界几何形状识别反问题，利用布谷鸟算法选出一组最优的未知边界，通过正问题求解测点的计算温度，使用测量温度和计算温度差值的平方和作为目标函数，应用BFGS或共轭梯度法等对目标函数进行优化。识别了管道内壁为圆形、肾形和正方形等不同的边界形状。.本项目建立了微分转换—双重互易边界元法，提出了改进的布谷鸟搜索算法和改进萤火虫算法，丰富了瞬态热传导反问题的数值分析方法，拓展了瞬态热传导反问题的工程应用前景。