等几何分析方法的本质边界条件处理研究

发布时间：2017-12-30 17:43

本文关键词：等几何分析方法的本质边界条件处理研究　出处：《西北工业大学》2016年博士论文　论文类型：学位论文

【摘要】：等几何分析是近年来备受关注的一种新型数值分析方法。随着问题规模越来越大、几何模型越来越复杂,繁琐的网格划分成为制约有限元发展的瓶颈。等几何分析采用CAD中精确描述几何形状的样条函数进行力学分析,设计模型和分析模型采用同一种几何表示,解决CAD和CAE的模型异构问题。论文选择本质边界条件处理作为切入点,以固体介质传热、线弹性结构,不可压缩流动和曲面上的偏微分方程为应用对象,研究等几何分析在应用中遇到的问题。由于样条函数缺乏插值性,等几何分析很难直接处理本质边界条件。如果像有限元一样施加在单元结点上,会造成计算精度降低和收敛退化。针对这一问题,论文提出两类本质边界条件处理方法。并在曲面偏微分方程几何精确有限元求解的基础上,提出参数曲面水平集方程及其数值计算。论文的主要研究内容包括:(1)提出基于样条拟合的“强处理”方法“强处理”法是在有限元空间施加边界约束的一类方法。基于样条拟合的思想,论文提出三种“强处理”方法,包括:边界配点法、最小二乘扩展以及构造等效插值基函数的转换矩阵法。边界配点法是在边界上引入一组精心设计的插值点,将本质边界条件处理转化为样条拟合问题。最小二乘扩展是在配点法的基础上,构造边界条件误差残余的最小二乘泛函,在求解精度、数值稳定性和适用性上都要优于配点法。论文第三章以固体介质传热为例,与有限元的“直接施加法”对比,提出的方法在精度和收敛性上都有显著提升。论文第五章从NURBS曲面提取对应的有理Bézier单元,基于等几何分析求解定义在参数曲面的Laplace-Beltrami方程。针对Bernstein多项式缺乏插值性的问题,提出一种等效插值多项式的矩阵转化算法。转换后的多项式满足插值属性,避免了本质边界条件处理的困难。由于有理Bézier单元可以精确描述球面、柱面等二次曲面表示的工业产品外形,在求解曲面偏微分方程时比有限元的精度更高。(2)提出基于Nitsche变分法的“弱处理”方法“弱处理”法是对等效积分形式进行修正的一类方法。第三章将Nitsche法应用到线弹性问题的位移边界条件处理,通过拉格朗日乘子识别,构造对应的无约束泛函。并针对线弹性问题,证明了Nitsche变分法的条件稳定性,给出稳定系数的计算公式。与同类方法相比,Nitsche法具有自由度少、数值稳定、惩罚项容易控制的优势,第四章将其推广到两类重要的流体力学问题。对于速度场-压强形式的二维Stokes流动,采用混合有限元求解需要构造满足inf-sup稳定条件的单元格式。针对这一问题,引入流函数将其转化四阶偏微分方程,采用高阶连续的NURBS基函数求解。最后混合采用Nitsche法和最小二乘扩展法对该问题的两组本质边界条件进行处理。对于不可压缩Navier-Stokes方程,论文推导出包含切向和法向全部速度Dirichlet边界条件的新型等效积分公式。新公式更为一般化,可以“弱形式”处理无穿透壁面条件。对于生成的非线性方程组,给出切线刚度矩阵,采用Newton-Raphson法求解。(3)研究参数曲面水平集方程及其数值计算在传统欧氏空间水平集的基础上,论文提出参数曲面上的水平集方程,可应用于曲面上的动态界面追踪问题。研究了参数曲面水平集方程的几何性质,推导出约束在参数曲面上、隐式表示的空间曲线的切面法矢和曲率计算公式。最后采用样条配点法数值求解参数曲面水平集方程。
[Abstract]:Geometric analysis is a new numerical analysis method has attracted much attention in recent years. With the increasingly large scale, the geometric model is more and more complex, cumbersome mesh has become a bottleneck restricting the development of finite element. The geometric analysis using a precise description of geometry in CAD splineto mechanics analysis, the design model and analysis model using the same geometry model solve the heterogeneous problem of CAD and CAE. The treatment of essential boundary conditions as a starting point, with solid medium heat transfer, linear elastic structure, incompressible flow and partial differential equations on the surface as the application object, on the geometric analysis of the problems in the application of the spline function. The lack of interpolation, geometric analysis is very difficult to directly deal with the essential boundary conditions. If a finite element like force on the element node, will cause the calculation accuracy and reduce the convergence deterioration. To solve this problem, this paper puts forward two kinds of essential boundary condition processing method. And based on the precise surface partial differential equation of finite element solution geometry, calculating equation and numerical parametric level set. The main research contents of this thesis include: (1) based on the spline fitting of the "strong" process method "processing method is applied in the finite element space boundary constraints for a class of spline fitting method. Based on the idea, the paper proposes three" strong processing methods, including the boundary collocation method, least squares expansion and conversion structure equivalent polynomial matrix method. The boundary collocation method is introducing a set of carefully designed the interpolation points on the boundary, the treatment of essential boundary conditions into spline fitting problem. The least square collocation method is extended based on the least squares functional structure boundary conditions of residual error, the accuracy of numerical solution. Stability and applicability is better than the collocation method. The third chapter in the solid medium heat transfer as an example, and directly applying the finite element method, proposed method has significantly improved the accuracy and convergence. The fifth chapter extracts the corresponding rational B zier unit from NURBS based on the surface. The geometric analysis solution is defined in the Laplace-Beltrami equation for Bernstein polynomial parametric surfaces. The lack of interpolation problems, put forward a matrix equivalent polynomial interpolation algorithm. The polynomial transformation after conversion satisfies the interpolation property, to avoid the difficulties in the treatment of essential boundary conditions. The rational B zier unit can accurately describe the sphere, cylinder two the surface shapes of industrial products, in solving surface partial differential equations than finite element accuracy. (2) based on Nitsche variational method of "weak" process method "weak processing method" Is a kind of modified method of equivalent integral form. The third chapter will deal with the displacement boundary condition Nitsche method is applied to the linear elastic problem, the Lagrange multiplier recognition, unconstrained functional structure. According to the corresponding linear problem, the stability conditions of Nitsche's variational method is proved, calculation formula of stability coefficient is given. Compared with other methods, the Nitsche method has less degree of freedom, numerical stability, the penalty is easy to control the advantage, the fourth chapter will be extended to the two important problems in fluid mechanics. For velocity field pressure in the form of two-dimensional Stokes flow, using mixed finite element solution need to be constructed to meet the stability conditions of inf-sup unit in format. This problem, introducing the stream function to convert the four order partial differential equation, by solving NURBS basis function continuous high order. Finally, by combining Nitsche method and least square method to expand the problem The two group of essential boundary conditions for processing. For the incompressible Navier-Stokes equation, this paper derives the new equivalent integral formula contains tangential and normal velocity Dirichlet all boundary conditions. The new formula is more general, can be "weak form" impenetrable wall conditions. For the nonlinear equations generated by the given tangent the stiffness matrix is calculated by Newton-Raphson method. (3) based on parametric level set equation and its numerical calculation in the traditional Euclidean space level set, the parameters on the surface level set equation, dynamic tracking interface can be applied to the surface of the problem. The study of geometrical parameters of surface level set equation, push derived constraints in parametric surface, calculation formula of plane normal vector and curvature of space curve is represented implicitly. Finally using the spline collocation method for numerical parametric level set equation.

【学位授予单位】：西北工业大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241

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