【作者】 孙海涛；

【导师】 王元汉；

【作者基本信息】 华中科技大学 ， 结构工程， 2006， 博士

【摘要】 无网格法的初衷是为了追踪物理量的局部高梯度变化,其显著特征是近似场函数构造不依赖于网格,并且数值积分网格与场函数构造方式无关。无网格法在弹塑性大变形、动态裂纹发展、冲击碰撞、流体力学等领域取得了很大的成功,受到众多研究人员重视,是计算力学的又一重要发展。研究表明,即使在近似场函数有较高精度的情况下,近似场函数的导数也会产生局部数值震荡,极值点会发生相移。近似场函数导数的局部震荡和相移会对追踪局部物理量变化产生不利影响,导致数值计算结果的可靠性降低。另外,无网格法计算工作量大,计算效率低下,不利于大规模数值问题求解。神经网络具有结构自适应确定、输出与初始权值无关及非线性收敛特征,径向基神经网络逼近原理实质是无网格法思想。将基于优化思想的神经网络算法用于构造近似场函数,可得到具有亚插值特性的近似场函数。论文分析了产生近似导数数值局部震荡和相移的原因,讨论了影响无网格法计算量的因素。提出通过改进近似场函数的构造方式,提高近似场函数导数的精度,消除近似导数数值震荡和相移。利用基本解降低计算维数,赋予形函数Kroneckerδ函数性质,方便施加边界条件,减少数值计算工作量。文中提出了两种消除近似导数数值局部震荡和相移的方法。方法之一是将近似场函数和近似场函数的导数分别看作是独立变量,按照相同的方式分别构造近似场函数和近似场函数的导数,得到具有相同构造精度的函数。通过张量积的方式产生二维激励函数,构造直接径向基神经网络,依据混合变分原理建立系统控制方程,形成了基于张量积的径向基神经网络无网格法。适当选择激励函数,这种类型的无网格法不需要作数值积分,只要存储少量的计算基矩阵,就能在规则域上获得高精度的数值解。对不规则域,适当的数值处理也可求得满意的结果。方法之二是利用误差在数学运算过程中的传播规律,构造积分形式的激励函数,得到高精度的近似导数。函数误差在微分过程中将进一步劣化,而在积分过程中光顺钝化。首先构造函数的高阶导数,通过不定积分得到带有权参数的新激励函数,用积分形式的激励函数构造间接径向基神经网络,显著地提高了场函数导数和场函数的近似精度,有效地消除了近似场函数导数出现的局部震荡和相移现象。文中采取两种方法减少计算工作量。第一种方法是利用间接边界型数值方法思想,提出了无网格虚边界法。边界型数值方法通过引入基本解,降低待求问题复杂程度,减少数值计算工作量。在无网格虚边界法中,虚边界始终不与真实边界相交,消除了其它边界型方法中存在的奇异积分和边界效应。虚边界几何形态简单灵活,与实边界的间距在较大范围内变化时,保持数值结果稳定收敛。无网格虚边界法计算不需要真实边界的外法线方向导数,不存在其它边界型方法中必须处理的角点问题。二维弹性力学的应用表明,在无论是简单域,还是复连通域,无网格虚边界法都能获得快速稳定的收敛结果。在物理量梯度变化较大的点,无网格虚边界法能高精度地给出结果,准确地反映出物理量的变化。第二种方法是将边界节点邻近的激励函数进行线性组合,得到便于施加边界条件的扩展激励函数,消除了控制方程中的边界处理项,从而减少了计算工作量。为满足无网格法场函数构造和数值积分计算的需要,文中还分析讨论了节点应满足的分布条件,编制了几何处理程序。程序能按几何变化特征,确定数值积分域内的数值积分点,给出较高精度稳定的数值结果。数值计算结果证明了基于径向基神经网络无网格法的正确和有效。方法在计算精度、收敛速度、计算工作量上的优良表现,为其进一步研究发展奠定了基础。更多还原

【Abstract】 The meshless methods are primarily motivated to trace large gradient variation of the variables in local domain. It is distinctive in that approached field functions can be constructed independently of the grids, and that the numerical integral cells are not concerned with the manner the approached field functions are constructed by. As a class of newly developed methods with great appeal, the meshless methods have made significant successes in such fields as large deformations in elasticity and plasticity, dynamic crack tracing, crash problems, flow mechanics, and etc.However, it has been revealed that the derivative value may exhibit numerical oscillation in local domains and the peak values may be drifted even though the approached field functions are of high-level accuracy. Such local numerical oscillation and peak value drifts might have negative effects on tracing large gradient variation, making the calculation results less reliable. In addition, computation burden and low efficiency of the meshless methods is also an obstacle to solving large-scale numerical problems.Neural networks have many excellent properties such as self-adapted determinant, output insensitive to the initial weighted values and non-linear convergence. The procedure of the radial basis neural networks (RBNN) approaches agrees with the idea of the meshless methods. It is expected that a quasi-interpolated function can be obtained by introducing radial basis neural network algorithms based on the optimum ideas into the meshless methods for constructing approached field functions.The causes of local oscillations of the derivatives and drift of the peak values, as well as those factors that might have influence on the computation cost of the meshless methods, are analyzed in this dissertation. It is suggested that the local derivatives oscillations and the peak values drift should be eliminated by modifying the construction methods of the approached field functions with improved accuracy of the derivatives. On the other hand, in order to make less calculation, the fundamental solutions would be used for declining the dimensions by one, and the shape functions with Kroneckerδfunction properties be constructed so that the boundary conditions can be imposed easily.Two methods for eliminating local oscillations of the derivatives and drift of the peak values are proposed. One is to consider both approached field functions and their derivatives as independent variables respectively. Both are constructed with the same procedure to make the approached fields functions and their derivatives have the same accuracy. Applying two-dimensional prompted functions that are generated by tensor product method to the radial basis neural networks, a novel RBNN meshless method based on the tensor product is then developed, using the mixed variational principles to establish the governing equations of the system. In this meshless method, if the prompted functions are chosen properly, no more numerical integral is needed. Moreover, high accuracy results can be obtained in the normal region rapidly with only a few basis matrices required for completing numerical evaluation. A satisfactory result is also available if a moderate procedure is implemented in an irregular region.Another method proposed here for eliminating the local oscillations of the derivatives and drift of the peak values is to create new prompted functions with the integral operations by taking advantages of the error propagation laws. Generally, the error generated in the approached functions would get worse during the differential process. Inversely, little difference occurs while the integral calculi are executed. By constructing high order derivatives first and establishing indirect radial basis neural networks with prompted functions generated by indeterminate integral operations, the accuracy of the field functions and their derivatives are improved significantly. The local oscillations of the derivatives and drift of the peak values vanish evidently.So far as the calculation burden of meshless methods is concerned, two approaches are proposed to reduce the amount of computation.Firstly, meshless virtual boundary method is developed by taking the profits of the ideas of the indirect boundary numerical procedures. It might be able to decline the complexity and reduce the calculations by introducing the fundamental solutions into the boundary methods. In this proposed method, singular integral and boundary effect associated with other boundary type methods disappear because no intersection between the virtual boundaries and true boundaries exists. The shapes of the virtual boundaries are simple and can be chosen easily. The numerical results are kept stable and convergent when the intervals between the virtual boundaries and true boundaries vary in large range. In addition, the corner problem associated with the other boundary methods is nonexistent as the directional cosines of the outer normal along the virtual boundaries are no more required in the meshless virtual boundary method. The results of the two-dimensional elastic problems illustrate that stabilized solutions are available either in the simple regions or in the complex regions, and the variable values varied in large gradient can be obtained exactly.Secondly, a linear combination of the prompted functions neighboring the boundary neural units is made for generating new expanded prompted functions. Since the expanded prompted functions are subjected to the boundary conditions simply, the terms for imposing boundary conditions in system equations are eliminated. Consequently, the amount of calculation decreases.To meet the requirement of the numerical integral and approached field functions construction in the methods proposed here, the conditions that scattered nodes must be subjected to have also been discussed. A code for determining the nodes in the regions with different geometric characteristics is developed. The code is capable of determining the integral points in each integral cell according to the shape of each integral domain. As a result, accurate numerical integral results can be obtained conveniently.The numerical results have demonstrated the validity and effectiveness of the proposed RBNN meshless method. Such excellent performance the method exhibits as little calculation, high accuracy and rapid convergence, lays a foundation for the future progress.更多还原