【作者】 黄正；

【导师】 何锃； 江雯；

【作者基本信息】 华中科技大学 ， 工程力学， 2017， 博士

【摘要】 等几何分析是最近提出的一种计算分析方法,主要目标是将计算机辅助设计和有限单元分析整合到一个校型中。等几何分析使用计算机辅助设计中常用的B样条或者非均匀有理B样条基函数描述所分析问题的几何体和未知变量。等几何配点法利用等几何分析中基函数的高阶连续性质和等参的概念,离散强形式的控制偏微分方程,从而得到一种连续体力学的非线性偏微分方程的高效空间离散化方法。在本文中,等几何配点法首次被扩展用于求解具有任意初始曲率和扭率的几何精确的Euler-Bernoulli梁结构的几何非线性问题。基于Euler-Bernoulli假设和几何精确的分析,采用完全客观的、几何的、功共轭的和共旋的Jaumann应变和应力度量,从推广的哈密尔顿变分原理出发,本文推导了三维空间中包含大位移和大转动的几何精确的Euler-Bernoulli梁结构完全的拉格朗日公式。在等几何配点法的框架下,推导得到适合于求解几何非线性静力学和动力学问题的非线性代数方程组,并编写了相应地等几何配点法的计算程序。为了使程序能用于由多个分片组成的梁结构,通过在梁结构参考线的连接点处强制位移和有限转动矩阵旋转变化的连续性以及力和力矩的平衡,本文提出了多个分片连接的刚性耦合模型。此外,拉格朗日乘子法被用来增强刚性耦合的约束关系,从而提高求解精度。在处理三维打限转动时,采川了两个欧拉角参数化旋转向量,并充分利用了位移和基函数的高阶连续性质。本文结合等几何配点法和几何精确的Euler-Bernoulli梁理论,提供了大量的简单结构的几何非线性静力学和动力学实例。在二维静态例子中,考查了本文方法的收敛性。在线性情况下,收敛阶数与样条自由度的奇偶性表现出相关性,这一现象在非线性问题中也存在。在多重分片例子中,不仅验证了本文提出的刚性耦合模型的正确性,同时也表明本文方法可以川于结构的非线性屈曲问题的分析。所有的数值模拟旨在说明基于所提出公式的等几何配点法程序具有良好的准确性和有效性。本文研究发展了等几何配点法的应用领域,也为几何精确梁结构的数值模拟提供了一种高效的替代方法。更多还原

【Abstract】 Isogeometric analysis(IGA)is a recent method of computational analysis with the main objective of integrating Computer Aided Design(CAD)and Finite Element Analysis(FEA)into one model.IGA uses B-Splines or Non-Uniform Rational B-Splines(NURBS)basis functions,which are commonly used in CAD,in order to describe both the geometry and the unknown variables for analysis problems.The isogeometric collocation(IGA-C)method takes advantage of the higher-continuity properties of the basic functions in IGA and the concept of isoparametric to discretize the strong form of the governing partial differential equation,and then an efficient spatial discretization method for nonlinear partial differential equations of continuum mechanics is obtained.In this thesis,the IGA-C approach is extended to solve the geometrically nonlinear problems of the geometrically exact Euler-Bernoulli beam structures with arbitrary initial curvatures and torsion for the first time.Based on Euler-Bernoulli assumptions and the geometrically exact analysis,a total Lagrangian formulation for geometrically exact Euler-Bernoulli beam structures with large displacements and large rotations in three dimensional space is derived from the extended Hamilton principle in this work.And the objective,ge-ometric,work-conjugate and co-rotating Jaumann strains and stresses are adopted.Within the isogeometric framework,a set of nonlinear algebraic equations suitable for the solving geometrically nonlinear statics and dynamics problems is derived.Then the corresponding program of IGA-C is also written.To make the code applicable to beam structures consist-ing of multiple patches,a rigid coupling model is presented by strong enforcement of the continuity of displacements and rotations of the finite rotation matrix as well as the balance of forces and moments at the connection points of the beam reference line.In addition,the Lagrange multiplier method is used to enhance the constraint relations of rigid coupling,so as to improve the accuracy of the solution.In the process of the three-dimensional finite rotation,two Euler angles are used to parameterize the rotation vector,and the higher order continuity of the displacement and the basis functions are fully utilized.Combined with the IGA-C method and the geometrically exact Euler-Bernoulli beam theory,a large number of geometrically nonlinear static and dynamic examples of simple structures are provided in this study.In the examples of 2D statics,the convergence of the proposed method is tested.In the linear case,the order of convergence and the parity of the degree of B-splines or NURBS show correlation,which is also observed in the nonlinear problems.In the case of multiple patches,it not only verifies the correctness of the proposed rigid coupling model,but also shows that the application of present method for nonlinear buckling analysis is feasible.All these numerical simulations are intended to illustrate the good accuracy and effectiveness of the IGA-C program based on the proposed formulation.This study develops the application fields of IGA-C,and it also provides an efficient alter-native to the numerical simulation of geometrically exact beam structures.更多还原