参数空间边界元法及其在断裂力学中的应用

发布时间：2018-05-12 23:15

本文选题：数值方法 + 边界元法　；参考：《华中科技大学》2014年博士论文

【摘要】：断裂是土木工程中结构破坏的主要形式之一,导致材料发生断裂破坏的因素有很多,而其中最重要的影响因素是材料自身的强度。在实际工程中,由于几何和材料的复杂性,绝大多数的断裂力学问题需要借助于数值分析的方法才能得到解决,只有极少数的简单断裂力学问题存在解析解。由于裂纹尖端附近应力场存在奇异性,传统的数值分析方法在解决裂纹问题时往往效果很差,需要结合断裂力学的特点采取特殊的处理方法,而边界元法中作为权函数的基本解具有奇异性,导致最后形成的代数方程组的系数矩阵中对角线和近对角线元素的值远大于其他元素的值,这一特点使得边界元法特别适用于处理场量变化梯度很大的裂纹问题。另外,边界元法只需在边界上离散,使数值计算的维数降低一维,从而减少了问题的前处理信息量和矩阵规模。本文以边界元法为基础,围绕断裂力学问题,开展了以下三方面工作：(1)如何精确建立裂纹几何模型；(2)如何精确计算裂纹尖端奇异应力场；(3)如何提高裂纹问题的计算精度和效率。首先,提出了弹性力学问题的参数空间边界元法,避免了由于单元离散引起的几何误差,实现了裂纹的精确建模。本文在传统边界元法的基础上,引入了CAD造型中的边界表征(B-rep)数据结构,提出了参数空间边界元法。该方法将B-rep表征模型中的参数曲面看成一个大的等参元,在参数空间中将其继续划分为一系列的边界单元,这些单元只用来进行变量插值和边界积分,不再用来近似求解域的几何形状,几何形状近似是通过参数映射过程实现,从而有效地避免了传统边界元法由于单元离散引起的几何误差。其次,研究了边界元法中近奇异积分的非线性变换方法：距离变换和sinh变换,实现近弱奇异积分和近强奇异积分的精确计算。在计算裂纹尖端奇异应力场时,由于源点非常靠近场点会导致近奇异积分的产生,传统的高斯积分不能有效地计算这类积分。本文在计算近奇异积分时引入距离函数,采用非线性变换的方法,将高斯积分点向奇异点靠拢,得到了很好的计算效果。最后,提出了适用于断裂力学问题的直接应力边界积分方程法,提高了裂纹问题的计算效率和精度。本文在应力边界积分方程的基础上,利用裂纹面上的面力平衡条件和基本解的性质,引入裂纹张开位移作为裂纹边界上的未知量,在裂纹的外部边界和裂纹边界上均采用应力边界积分方程,提出了适用于裂纹问题的直接应力边界积分方程法。本文的研究表明,参数空间边界元法可以建立所求解问题的精确几何模型,在二维和三维弹性力学问题中具有较高的计算精度和收敛性,且适用于复杂几何模型问题,在此基础上提出的适用于断裂力学问题的直接应力边界积分方程法,相比传统的双边界元法,该方法在计算时只需要考虑一个裂纹面,具有较高的计算精度和效率,可以将其推广于实际工程中的断裂力学问题。
[Abstract]:Fracture is one of the main forms of structural damage in civil engineering. There are many factors that cause fracture and failure of the material, and the most important factor is the strength of the material itself. In practical engineering, because of the complexity of geometry and material, the most of the fracture mechanics problems need to be obtained by means of numerical analysis. Only a few simple fracture mechanics problems have an analytical solution. Due to the singularity of the stress field near the crack tip, the traditional numerical analysis method often has a bad effect in solving the crack problem. The special treatment method should be taken in combination with the characteristics of fracture mechanics, and the basic solution of the right function in the boundary element method is odd. The value of diagonal and near diagonal elements in the coefficient matrix of the algebraic equations which result in the final formation is far greater than the value of other elements. This characteristic makes the boundary element method especially suitable for dealing with the crack problem with large variation in the field. In addition, the boundary element method only needs to be discrete in the boundary boundary, so that the dimension of the numerical calculation is reduced one dimension, Thus reducing the pre-processing information volume and matrix size of the problem.
Based on the boundary element method, the following three aspects are carried out on the fracture mechanics problem: (1) how to accurately establish the crack geometry model; (2) how to accurately calculate the singular stress field at the crack tip; (3) how to improve the calculation accuracy and efficiency of the crack problem.
First, the parametric space boundary element method (BEM) for elastic mechanics is proposed, which avoids the geometric error caused by the element dispersion and realizes the accurate modeling of the crack. On the basis of the traditional boundary element method, the boundary representation (B-rep) data structure in the CAD modeling is introduced, and the parameter space boundary element method is proposed. The method is used to characterize the B-rep model. The parametric surface in the type is regarded as a large isoparametric element, and it continues to be divided into a series of boundary elements in the parameter space. These elements are used only for variable interpolation and boundary integral, and are no longer used to approximate the geometry of the domain. The geometric shape is realized by the parametric projection process, thus effectively avoiding the traditional boundary. The geometric error caused by element dispersion in the element method.
Secondly, the nonlinear transformation method of near singular integral in boundary element method is studied. Distance transformation and Sinh transform are used to calculate the near weak singular integral and near strong singular integral. In the calculation of the singular stress field at the crack tip, the traditional Gauss integral can not be effective because the source point is very close to the field point. In this paper, this kind of integral is calculated. In this paper, the distance function is introduced in the calculation of the near singular integral, and the method of nonlinear transformation is used to close the Gauss integral point to the singular point, and a good calculation result is obtained.
Finally, the direct stress boundary integral equation method for fracture mechanics is proposed to improve the calculation efficiency and accuracy of the crack problem. On the basis of the stress boundary integral equation, the crack opening displacement is introduced as the unknown quantity on the crack boundary, and the crack opening displacement is used as the unknown quantity on the basis of the stress boundary integral equation. The stress boundary integral equation is applied to the outer boundary and crack boundary of the stripe. A direct stress boundary integral equation method is proposed for crack problems.
The study of this paper shows that the parametric space boundary element method can establish the exact geometric model of the solved problem. It has high calculation precision and convergence in two-dimensional and three-dimensional elastic mechanics problems, and it is suitable for the problem of complex geometric model. On this basis, the direct stress boundary integral equation method suitable for the fracture mechanics problem is proposed. Compared with the traditional double boundary element method, this method only needs to consider a crack surface, which has high calculation precision and efficiency, and can be extended to the fracture mechanics problem in practical engineering.

【学位授予单位】：华中科技大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TU312.3;O346.1

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