【作者】 谢贵重；

【导师】 张见明；

【作者基本信息】 湖南大学 ， 机械工程， 2014， 博士

【摘要】 CAE分析技术在机械行业发挥了重要的作用。而CAE分析技术的主流数值方法—有限元法却存在一些固有缺陷,而这些缺陷刚好可以使用边界积分方程方法来弥补。在边界积分方程方法的数值实施中,近奇异积分和奇异积分是影响其计算精度的重要因素。因此,本文将重点关注边界积分方程方法中近奇异积分和奇异积分的解决方案及其在薄型结构和断裂力学方面的应用。另外,为了拓宽边界积分方程方法的工程应用,本文也提出了一系列近奇异体积分和奇异体积分的处理方案。以此为核心,本论文完成如下研究：(1)提出了二维和三维问题的近奇异积分变换技术。和传统方法的近奇异积分求解方案不同,本文方法重点分析了近奇异积分的核心问题,即投影点位置和距离函数性质。本文利用泰勒展开得到距离函数,并根据投影点的位置和距离函数的性质将近奇异积分分为三类。从距离函数出发,利用降低被积函数梯度的思想,构造了三种对应的近奇异积分变换。与二维问题不同,在三维问题中,引入新型坐标系,构造出新坐标系下形式比较简单的变换。另外,由于投影点位置的不确定性,引入最近点,开发了一套基于投影点和最近点的近奇异积分子单元划分技术,用以保证积分子单元的良好形状,提高积分精度。数值算例充分证实了本文提出的方法可以成功地应用于薄型结构的求解。(2)提出了全面而系统的奇异积分解决方案。本文直接从柯西主值和哈达玛有限部分积分定义出发,对弱奇异积分、强奇异积分、超奇异积分采取局部坐标近似展开,分析各类型奇异积分的性质以及相应的处理方法。另外根据主值积分和有限部分积分的区间对称性要求,开发一套用于解决三维奇异积分的自适应分块技术,有效地提高了奇异积分的精度。这些方案成功地应用于二维和三维断裂力学问题的求解。(3)实现了二维和三维断裂力学问题的边界积分方程方法求解。针对二维断裂问题,引入裂纹张开位移,利用基本解的性质和裂纹受力平衡的边界条件,改进传统的双边界积分方程,使边界积分方程只用配置在非裂纹边界和裂纹的上表面上,从而减小矩阵规模和计算量。而对于三维问题,则进一步,只用在非裂纹边界和裂纹的上表面上配置面力边界积分方程,这样,既保留了和二维问题一样的优势,又便于奇异积分的模块化编程处理。开发了一种能够捕捉裂纹尖端位移性质的中节点奇异单元,结合裂纹尖端位移的渐近性质,建立了裂纹尖端应力强度因子和裂纹张开位移的线性插值公式。这些方案成功地应用于二维和三维的断裂力学问题求解,并得到了相当好的数值结果。(4)开发了合理的近奇异体积分和奇异体积分技术。为了保证边界积分方程算法的通用性,针对三种常用单元开发了近奇异体积分和奇异体积分技术。对于近奇异体积分,提出以源点到单元的距离和单元尺寸的比例作为控制准则的自适应近奇异体积分方案。对于奇异体积分,首先将积分单元分为四面锥和金字塔子单元,然后对其分别做奇异积分变换技术。这些方案成功地解决了边界积分方程方法中遇到的近奇异体积分和奇异体积分,数值算例证实了本文方法的有效性。更多还原

【Abstract】 CAE analysis technology plays an important role in the machinery industry. The mainstream numerical method of CAE analysis technology is finite element method (FEM). However, there are some inherent flaws with FEM and these flaws can be avoided by the boundary integral equation method (BIEM). In the numerical implementation of the BIEM, nearly singular integrals and singular integrals are important factors which affect the accuracy of BIEM. Thus this work focuses on the nearly singular integrals and singular integrals which arise in the BIEM and their applications in thin structures and fracture mechanics. Moreover, in order to broaden the application of BIEM, this work also includes a series of treatment for nearly singular and singular volume integrals. Treating these core contents, the work of this paper is as follows:(1) This paper presents new transformations for nearly singular integrals in two and three dimensional BIEM. Compared with the traditional methods for nearly singular integrals, the proposed transformations focus on the core issues of nearly singular integrals in nature, namely, the location of the projection point and distance function. In this paper, employing the Taylor expansion, the distance function is obtained. Considering the location of the projection point and the distance function, the nearly singular integrals are divided into three categories. Using the distance function and the idea to lower the gradient of integrands, three different transformations for nearly singular integrals are constructed. And different from that in two dimensional BIEM, a new coordinate system is introduced and new transformation is simpler in the new system. Moreover, due to the uncertainty of the location of the projection point, another nearest point is introduced. An element subdivision technique based on the projection point and the nearest point is developed. With the help of element subdivision technique, subelements of fine shape are obtained, which can improve the integration accuracy. Numerical examples demonstrate that the proposed method can be successfully applied for problems on the thin structures.(2) A comprehensive and systematic analysis of singular integrals in BIEM. Directly from the definition of the Cauchy principal value (C.P.V) and Hadamard finite part integrals (H.F.P), the local coordinate approximate expansion is employed to analyze the nature of weakly singular integrals, strong singular integrals, and hypersingular integrals. And the corresponding treatment is obtained. To meet the symmetry requirement of integration interval for C.P.V and H.F.P, an adaptive subdivision technique is developed to deal with singular integrals in three dimensional BIEM, which can efficiently improve the integration accuracy. These methods are successfully applied for two and three dimensional fracture mechanics.(3) BIEM for two and three dimensional fracture mechanics. For two dimensional fracture mechanics, crack opening displacements (CODs) are introduced. Using the nature of the fundamental solutions and the traction equilibrium boundary conditions of crack faces, the improved dual boundary integral equations can be obtained. The dual boundary integral equations are collocated only on one of the crack faces and the external boundary, thus reducing the matrix size and more efficient. Furthermore, for three dimensional fracture mechanics, only the traction boundary integral equation is needed to collocate on one of the crack faces and the external boundary. Thus the advantages for two dimensional fracture mechanics are retained. Moreover, it is easy for code programming to deal with singular integrals. To capture the nature of the displacement of the crack tips, special crack-front elements are proposed. Combined with the asymptotic properties of the crack tip displacement, crack tip stress intensity factors are obtained by the linear interpolation formula of the crack opening displacements. The proposed method can be successfully applied for two and three dimensional fracture mechanics. Numerical results are in good agreement with existing analytical solutions or numerical results.(4) Reasonable schemes for nearly singular and singular domain integrals. To ensure the universality of BIEM, nearly singular and singular domain integrals techniques are developed for the three common domain cells. An adaptive cell subdivision which is based on the proportion between the distance from the source point to the domain cell and the size of domain cell for nearly singular domain integrals is proposed. For singular integrals, the domain cell is divided into several cone and pyramid subcells. Singular integrals on these subcells are regularized by transformations. The proposed method is successfully applied for the domain integrals. Numerical examples demonstrate the effectiveness of our method.更多还原