功能梯度材料瞬态传热问题的等几何边界元法及其反问题非迭代法研究

The transient heat transfer direct and inverse problems of functional graded materials (FGM) exist widely in aerospace, nuclear engineering fields and so on. The study of direct problem and the inverse of boundary conditions and geometry shapes can provide the reference and guide to design and operate safely for the equipment. Based on the fundamental solution of potential problem and the weighted residual method, the isogeometric boundary integral equation is established, where the dual reciprocity method and precise integration method are used to transform domain integrals into boundary integrals and deal with the time-domain, respectively. On this basis, the non-iterative inverse method is proposed in this project by means of the matrix transformation scheme. Also, the error function is set by seeking the relation between measurement points and boundary points to prepare inverse. To realize the inverse boundary conditions, the extremum of error function is solved based on the least square method. Meanwhile, the inverse boundary condition problems are transformed by the basis function. By introducing the virtual boundary, the inverse geometry shape problem is transformed to inverse the boundary condition of virtual boundary. Furthermore, the contour line or surface is searched to achieve the identification of geometry shape in all over the domain. To verify and improve the non-iterative inverse method, some factors will be discussed such as the measured error, the position and number of measurement points and so on, and comparing with other inversed methods. This study will provide a new method for solving the direct transient heat transfer problems of FGM and establish the new theory simultaneously to quickly inverse transient heat transfer problems.

功能梯度材料瞬态传热正、反问题广泛存在于航空航天、核工程等领域，正问题及反演边界条件和几何形状问题的研究可为设备的设计、安全运行提供参考依据和指导。本项目采用位势问题基本解，利用加权余量法，建立功能梯度材料瞬态传热的等几何边界积分方程，其中双重互易和精细积分法被分别用来进行域积分的转化和时域的处理。基于此，本项目提出非迭代反演方法，借助矩阵变换技术，探寻测点与待演边界点之间的关系，建立误差函数。同时利用基函数将反演边界条件问题进行转化，基于最小二乘法对误差函数求极值，实现边界条件的反演。引入虚边界将反几何形状问题转化为反虚边界条件问题，在全域内搜索等值线、面，进而实现几何形状的反演。考虑测量误差、测点的数量和位置等信息对反演结果的影响，并与其他反演方法进行对比，验证和改进非迭代反演方法。本项目研究将为分析功能梯度材料瞬态传热正问题提供一种新方法的同时，可为快速反演瞬态传热问题建立新的理论。

功能梯度材料作为一种即古老而又充满生命力的非均质材料，已被广泛用于航空航天、汽车等制造业。本项目重点关注功能梯度材料瞬态传热的等几何边界元法和反载荷及几何问题的非迭代法研究，等几何边界元发展至今已有近10年的历史，但基本都关注在求解稳态问题，其根本原因就是无法避免域积分的求解。本项目采用位势问题基本解推导了瞬态二维和三维传热边界积分方程，建立了等几何边界元基本理论，采用双重互易法将域积分转换为边界积分，保留了边界元法仅离散边界单元的优点。分析了单片三维螺旋桨、多片复杂战斗机等模型，耦合精细积分法取得了较为稳定准确的结果。基于此，采用最小二乘法建立了反几何问题的非迭代求解理论，引入了虚拟边界实现了二维及三维几何的重构。同时，为了实现复杂几何的准确反演，项目负责人与合作者提出了逐步域推进法和基于Gappy POD的直接反演计划，并取得了令人满意的识别结果。基于本项目执行期所取得的成果，项目负责人进一步拓展研究了考虑不确定性非傅里叶瞬态传热的等几何边界元法和利用拓扑优化方法进行缺陷识别的研究。项目执行期所取得的研究成果可为结构设计提供正问题分析方法，亦可为结构安全运行提供有效指导。四年间项目负责人共指导毕业研究生5人，在国内外重要期刊共发表高水平学术论文16篇，其中等几何双重互易边界元法一文为高被引论文。本项目的实施为等几何边界元法分析瞬态问题提供了一种新思路，同时进一步拓展了非迭代法的研究范畴。