热冲击作用下裂纹扩展数值计算的解析奇异单元

Thermal stress analysis of cracked structures under thermal shock is one of the most concerned projects in engineering applications. However, both of heat flux field and stress field possess singularity characteristics in the vicinity of crack tips, as a result, the relative numerical analysis has been faced with many challenges. More or less, the following shortcomings exist in the existing relative finite element methods including: dense meshes are required around special regions, transition elements are necessary to connect with regular elements, and complex post process has to be conducted to gain relative thermal fracture parameters, etc. This project intents to construct a singular finite element which can be applied to both of the analysis of thermal conduction and thermal stress, in order to propose a numerical method for the analysis of multiple crack propagation problem under thermal shock. Firstly, the transient thermal conduction problem is led into symplectic solving system, and the symplectic eigen-solution and special solution are derived, which are then used to construct a singular finite element for the analysis of thermal conduction problem, to calculate heat flux field around crack tip with highly accuracy. Then, by using the singular finite element constructed using the symplectic eigen-solution and special solution of thermal elastic problem, numerical study of the original problem is continued and finished. The proposed singular finite element fully contains the information of analytical solution of the relative boundary value problem, hence, it can describe the singular fields in the vicinity of crack tip accurately, calculates the relative fracture parameters directly, and possesses satisfactory solving stability. The achievements of this project intent to propose a numerical method with highly solving accuracy and efficiency for the study of multiple crack propagation problem under thermal shock.

热冲击作用下含裂纹结构的热应力分析是实际工程非常关心的课题之一。但因裂纹尖端热流密度场和应力场同时具有奇异性，其数值分析面临一些难点。已有的有限元方法都不同程度存在一些不足，如：局部网格需要加密、需过渡单元连接常规元、需繁琐的后处理计算热断裂参数等。本项目拟构造可同时用于热传导和热应力分析的解析奇异元，用于热冲击作用下多裂纹扩展问题的数值分析。首先，将瞬态热传导问题导入辛体系，推导出辛本征解和特解，并用于构造热传导分析的解析奇异元，从而给出高精度的裂纹尖端热流密度场。然后，再利用辛体系下热弹性裂纹问题的辛本征解和特解所构建的热应力解析奇异元，完成相关问题的数值分析。拟构建的奇异单元由于充分包含相应边值问题的解析信息，因此它能精确地描述裂纹尖端奇异场，直接给出有关的断裂参数值，并且具有非常好的数值稳定性。本项目研究成果将能为热冲击作用下多裂纹扩展问题提供高精度、高效率的数值求解和分析手段。

项目针对实际热能工程中非常关心的研究课题，深入讨论了热冲击下含裂纹结构的传热和热应力分析的相关问题。首先，针对由解析分析指出的裂纹尖端不仅存在应力奇异性也存在热流密度奇异性的特性，构造了传热问题的奇异元，用于瞬态及稳态传热问题的数值分析。同时，还讨论了在复合材料中的该奇异元的构造方式以及相关数值稳定性等问题，相关成果发表在传热传质学领域国际顶尖期刊International Journal of Heat and Mass Transfer上。另一方面，针对线弹性裂纹问题在复合材料领域的应用也具有较大的挑战这一事实，深入研究了双材料直裂纹和斜裂纹两种复合材料中的典型裂纹构型，并构造了相应问题数值分析的解析奇异元。最后，将传热分析与线弹性分析进行结合，并考虑热力耦合的影响，先后在稳态和瞬态传热条件两个方面进行了数值求解格式，并得到了稳定且精确的数值结果。所构建的一系列奇异元中，局部网格加密、过渡单元、繁琐的后处理等常规有限元中普遍存在的问题得到根本解决。所提出的方法可直接给出有关的断裂参数值，并且具有非常好的数值稳定性。本项目研究成果为热冲击作用下多裂纹扩展问题提供了高精度、高效率的数值求解和分析手段。

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