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Calculation of singular field near the tip of a multi-material notch under thermal load with an over-determined method

DOI：

 作者 单位 邮编 姚善龙* 广西大学 土木建筑工程学院 530004 赵光鹏 广西大学 土木建筑工程学院 张　建 广西大学 土木建筑工程学院 张志梅 合浦县中等职业技术学校

本文建立一种分析多复合材料平面切口尖端热流密度奇异性的有效方法,再将奇性特征分析结果与有限元法相结合,确定多复合材料切口尖端热学奇异物理场。首先,基于切口尖端物理场的Williams级数展开式,推导多复合材料切口的热传导奇性特征方程,数值求解此特征方程后获得切口尖端物理场渐近展开式的奇性指数、特征角函数及其各阶导数。其次,利用稀疏有限元网格初步计算切口尖端附近的温变场,将有限元计算结果与奇性特征分析结果相结合,构建超定方程组,计算切口尖端邻域的物理场渐近展开式的幅值系数,进而确定切口尖端邻域的温变场和奇异热流密度。本文方法基于稀疏网格有限元计算结果,获得多复合材料切口尖端较高精度的奇异物理场,摆脱常规有限元法在计算切口尖端奇异场时对稠密网格的依赖,提高计算效率。

This study presents an effective approach for analyzing the singularity of heat flux at the tip of a plane notch in multi-composite materials. By integrating the findings of singularity characteristics analysis with the finite element method, the thermally singular physical field at the notch tip is determined. Firstly, based on the Williams series expansion of the physical field near the notch tip, the thermal conduction singularity characteristic equation for the multi-composite material notch is derived. The singularity orders, characteristic angular functions, and their respective derivatives for the asymptotic expansion of the physical field at the notch tip are yielded by solving the characteristic equation numerically. Secondly, using a sparse finite element mesh, the temperature variation field near the notch tip are conducted. The finite element results are then integrated with the singularity characteristics analysis results for constructing an over-determined system of equations. This system allows for the computation of the amplitude coefficients of the asymptotic expansion of the physical field near the notch tip, enabling the determination of the temperature variation field and the singular heat flux in the vicinity of the notch tip. The proposed method leverages the finite element computations with a sparse mesh, whereas high-precision computational results are obtained. The present method overcomes the reliance on dense meshes in conventional finite element methods for computing singular fields at notch tips, thereby enhancing computational efficiency.
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