等几何法在典型结构力学分析中的有效性研究

发布时间：2018-03-20 06:05

本文选题：等几何分析　切入点：Fortran　出处：《山东大学》2016年博士论文　论文类型：学位论文

【摘要】：等几何分析(IGA,Isogeometric Analysis)于2005年被提出,旨在将设计、分析和优化集成在一起。等几何分析利用准确描述几何形状的基函数(如NURBS, Non Uniform Rational B-spline)作为计算解空间的函数,统一的表达形式使得分析模型和几何模型可以无障碍地交互。随着工程实际中模型的日益复杂化,有限元分析中模型的网格生成耗费了大量的时间,而等几何分析将计算机辅助几何设计(CAD,Computer Aided Design)与计算机辅助工程分析(CAE,Computer Aided Engineering)进行了无缝结合,很好地解决了这一问题。等几何分析已成为当前工程分析发展的趋势,将会对CAD和CAE产生重大影响。现有的有限元环境很多基于Fortran语言,利用Fortran对等几何分析进行编程有希望实现等几何分析与现有有限元分析的集成。作者利用Fortran语言对NURBS等几何分析的理论以及实现方法进行了研究,并将基于Fortran的等几何分析用于不同的力学问题以提高等几何分析的计算效率和准确性。为了提高程序的运行效率,同时也为了使程序能够方便地被不同的算例调用,文中采用Fortran语言中广泛使用的模块化编程。为了使Fortran编程能对不同的算例兼容,作者最大化的将共用子程序,如高斯积分,NURBS基函数及其导数的计算,连接矩阵等,编入同一模块,对Fortran用于等几何分析的研究做了大量的基础性工作。另外,现有的等几何分析软件在解决大型稀疏矩阵问题时,计算效率低且不稳定,作者将一种大型稀疏矩阵的求解器GSS(Grus Sparse Solver)植入Fortran编程中,使得等几何分析的效率大大提高。文中提供可下载的相关算例的Fortran源程序。NURBS基函数通常不满足克罗内克函数的性质,即不具有插值性。控制点不在边界上时,难以直接施加位移边界条件。作者采用罚函数法处理边界条件,但罚因子取值的大小会影响计算结果。作者通过不同的算例探索合适的罚因子取值大小,结果表明,罚因子的取值在高于整体刚度矩阵中绝对值最大值2-3个数量级时,计算结果较为准确。此外,提高模型的网格质量可以减轻罚因子的影响,但高质量的网格势必会导致计算工作量的增加。作者完善了基于罚函数法处理边界条件的Kirchhoff-Love板壳理论。NURBS函数不仅用来描述单元形状和位移场,而且还提供了Kirchhoff-Love理论所需要的高阶连续函数。将基于Fortran的等几何分析用于不同的板壳算例模拟,证明了其准确性和快速收敛性,同时说明了等几何分析即使在粗糙的网格水平上,也能得到准确的计算结果。作者还结合扩展有限元与等几何分析对断裂力学问题进行了研究,证明了基于Fortran的扩展等几何分析模拟不连续问题的有效性,给出了选择强化控制点的方法并对裂纹不连续域和尖端位移场分别利用Heavisde方程和裂纹尖端方程进行了强化。通过与扩展有限元用于模拟相同的裂纹模型作比较,说明了扩展等几何分析仅需更少的单元便可获得准确的结果。在对不同算例位移和应力场的表达上,高阶函数的应用使得其光滑、连续。为了实现局部细化,作者利用多面片技术对带孔平板问题模型进行了分片处理,并在每一片模型上进行了位移和应力计算,得到了准确的结果。在模拟断裂力学问题时,利用线性节点值的插入对裂纹区域进行了局部细化,减少了计算误差。根据不同的算例应用不同程度的细化方法所获得的结果来看,提高网格的细化质量,可以使计算结果更加准确,位移和应力场表达的更加连续。综上,作者开发了基于Fortran的新的等几何分析工具,并将其用于不同的力学问题以验证其准确性和高效性。同时,完善了等几何分析中的相关理论,对等几何分析中出现的问题进行了分析,并提出合理的改进方法。然而由于时间和条件有限,还需对程序进行优化,也需要将Fortran编程应用于更多的等几何分析算法。
[Abstract]:The geometric analysis (IGA, Isogeometric Analysis) was put forward in 2005, aims to design, analysis and optimization are integrated together. The geometric analysis using the accurate description of the geometry of the base function (such as NURBS, Non Uniform Rational B-spline) as a function of solution space, a uniform expression makes the analysis model and the geometric model can be barrier free interaction with the model. In practical engineering becomes more and more complex, grid generation in the finite element analysis model of the spent a lot of time, and the geometric analysis of computer aided geometric design (CAD Computer, Aided Design) analysis and Computer Aided Engineering (CAE Computer, Aided Engineering) for seamless integration, a good solution this problem. The geometric analysis has become the development trend of current engineering analysis, will have a significant impact on CAD and CAE. Many of the existing finite element environment based on F Ortran language, using Fortran programming equivalence geometric analysis to achieve the integration of geometric analysis with existing finite element analysis. The author analysis by Fortran language on the NURBS geometry theory and realization method is studied, and the analysis of computational efficiency and accuracy of the geometric analysis on different mechanical problems in order to improve the Fortran. Based on the geometry. In order to improve the efficiency of the program, but also to make the program can easily be different examples called modular programming Fortran language widely used in this paper. In order to make the Fortran programming can be compatible with the example of different authors, the maximum common subroutines, such as the Gauss integral calculation, NURBS basis function and derivative, connection matrix, into one module of Fortran for the research of geometric analysis foundation has done a lot of work. In addition, the existing etc. Geometric analysis software in solving large sparse matrix problems, the computational efficiency is low and unstable, the author of a large sparse matrix solver GSS (Grus Sparse Solver) with Fortran programming, the efficiency is greatly improved. The geometric analysis provides downloadable relevant examples of Fortran source program based on.NURBS function normally do not meet the properties of Kronecker function in this paper, which has no interpolation. Control points are not on the boundary, it is difficult to directly impose displacement boundary conditions. The author adopts the penalty function method to deal with the boundary conditions, but the penalty factor value will affect the results. The author through different examples to explore the value of penalty factor, right the results show that the value of penalty factor in higher than the maximum absolute value of 2-3 orders of magnitude in the global stiffness matrix, the calculation result is accurate. In addition, to improve the quality of the mesh model can reduce the penalty for The influence of the grid, but high quality will inevitably lead to the increase of computational effort. The author improves the penalty function method to deal with the boundary conditions of the Kirchhoff-Love shell theory based on.NURBS function is not only used to describe the element shape and the displacement field, but also provides a high order continuous function required by the Kirchhoff-Love theory. The Fortran analysis for different geometry etc. the numerical simulation based on the shell, to prove its accuracy and fast convergence, and discusses the geometric analysis even in rough grid level, also can get accurate results. The author also analyzes the fracture mechanics problems of finite element and geometric expansion, proved the validity analysis of simulation of discontinuous the problem of Fortran extension based on geometry, given the choice of strengthening method of control points and the crack tip displacement field and discontinuous domain respectively by Heavisd The e equation and the equation of crack tip has been strengthened. With the extended finite element method is used to simulate the crack model of the same comparison, illustrates the extension of geometric analysis only less unit can obtain accurate results. In different examples of displacement and the expression of stress field on the application of higher order functions makes it smooth in order to realize the continuous, local refinement, the problem with the plate model were divided by multi slice technique, and the displacement and stress were calculated in each model, get accurate results. In the simulation of fracture mechanics problems, using the linear insert node value by local refinement the crack area, reduce the calculation error. According to the different cases considered refinement methods different application results show that the improved grid refinement quality, can make the results more accurate, the displacement and stress field The more continuous. In summary, the author developed based on new geometric analysis tool Fortran, and used different mechanical problems to verify its accuracy and efficiency. At the same time, improve the relevant theory of geometry analysis, the problems of equivalence in geometric analysis are analyzed, and put forward the improvement method is reasonable however, due to the limited time and conditions, but also need to optimize the process, also need to be used in Fortran programming more geometric analysis algorithm.

【学位授予单位】：山东大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O342

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