摘要
本学位论文针对界面耦合的流固耦振系统,结合目前的几种主要数值方法——有限元法、边界元法及有限元-边界元混合法、半数值半解析方法以及新兴的数值流形方法,开展一些具有创新性的理论及应用研究,主要成果如下:
(1)首先对常规边界元法进行改进,重新构造新的Green函数,使求解半无限域流场中物体附连水质量得到了简化,通过映象法和迭加原理修改常用的Laplace方程基本解,使自由液面边界无需进行边界单元离散,减少了计算量。此外还应用有限元-边界元混合法分析无限大等深度的片状域流场中三维结构的自由耦振问题,借助四面体单元的自动网格剖分技术形成结构网格,并推导了与之配套的混合法计算格式;同样由于采用了合适的流场Green函数,仍只需对流固交界面进行边界元剖分,从而大大减少了方程的自由度数目;在此基础上,实现了与大型有限元分析软件的接口技术,简化了编程,提高了工作效率。
(2)针对二维无限域流场中一般结构的流固耦合问题,提出局部变分原理。用圆形的人工边界将无限域分成两部分,在人工边界内的结构及其附近流场采用有限元法的数值解,人工边界外采用解析解。通过构造泛函,使提出的变分方程和所研究耦振的边值问题完全等价,并推导出杂交元算式,保证了在人工边界上数值解和解析解的场函数及其导数的连续性。文中用不可压缩流场的算例验证了方法的有效性及较高的计算效率。
(3)基于一般旋转薄壳的基本方程首次推导出状态向量的一阶常微分矩阵方程,这为传递矩阵法解决这类壳体的耦振创造了重要条件。同时应用新型的齐次扩容精细积分法进行求解,实现了传递矩阵法简便分析旋转壳的静动力问题。在此基础上进一步研究了等深度流场中旋转壳的流固耦合简谐响应。本文方法扩展了传递矩阵法的应用范围,为旋转壳声弹耦合的半数值半解析分析奠定了基础。
(4)本文开展了高阶数值流形法及其在流固耦合谐振分析中的应用研究。首先推导了基于单纯形积分的高阶流形元公式,并研究了程序自动生成方法,编制了二维和三维流形元线弹性静力分析程序,通过一些算例探讨高阶流形法在连续体静力分析中的计算精度及其适应性;在此基础上,发挥流形法的独特优势将其应用于流固耦合谐振分析中,从而扩展了流形法的应用范围,并提出将近场数值解、远场解析解用覆盖方式联系起来的简便方法,文中算例体现了流形法前处理方便、计算精度高的特点,表明了其在数值解和解析解联合运用上的优势。
(5)本文详细讨论了固定数学网格的流形法相对于现有大变形描述方法的优势,提出了用其解决非线性流固耦合分析的初步思路;提出应力系数反推法,在拉格朗日描述下得到了与理论解符合很好的计算结果;在此基础上初步研究了固定数学网格的流形法,采用固定的矩形数学网格和1阶多项式覆盖函数,实现材料在网格中移动,文中的悬臂梁算例表明了方法的可行性,但现阶段还未解决初应力荷载的准确计算和由此带来的计算稳定性问题。这项研究为将来将固体和流体统一到欧拉描述下进行非线性流固耦合分析打下了基础。
In this dissertation, new theories and applications on solving fluid-solid coupling vibration are studied, concerning some popular numerical approaches -- finite element method (FEM), boundary element method (BEM) and mixed FE-BE method, semi-analytical and semi-numerical method,and numerical manifold method (NMM) which is new developed. The major achievements obtained are summarized as follows:
(1) Ordinary BEM has been improved by modifying or reconstructing the fundamental solution of Laplace equation. A novel approach for solving added mass of structure embedded in semi-infinite liquid field is presented. Using mirror method and superposition method to modify the ordinary Green function, free surfaces of fluid do not need to divide into the boundary elements and then computation time is saved. A mixed FE-BE method is also used to analyze coupling vibration of three dimensional structure in a liquid with constant depth. Tetrahedral meshes of the structure are formed by automatic subdivision, and the corresponding coupling equations are derived. Based on the Green function corresponding to a wave in a liquid with constant depth issuing from any source, only surfaces of the structure need to divide into the boundary elements, and the number of freedoms decreases greatly. Besides, interface of general FEM software is also adopted to improve the efficiency of programming and computation.
(2) A localized variational principle is presented for analyzing fluid-structure interaction problems in two-dimensional infinite liquid field. The infinite liquid field is divided into two parts by an auxiliary circle, inside which the structure and its neighboring liquid are computed by FEM and outside which an analytical solution is used. By means of this variational principle, all governing equations and boundary conditions are satisfied automatically, and the numerical solutions are incorporated with the analytical solution via hybrid element equations derived. Given examples for incompressible liquid case demonstrate validity and high efficiency of the approach.
(3) Based on the general governing equations of revolution thin shells, the 1-order ordinary differential matrix about state parameter vector of revolution shells is firstly derived in this dissertation. This matrix is the important basis for solving interaction problems about revolution shells using the transfer matrix method. By means of the extended homogeneous capacity high precision integration method, static and dynamic solutions are given conveniently, and harmonic responses of revolution shells in a liquid with constant depth are computed. This approach extends the applications of the transfer matrix method, and lays the foundation of semi-analytical and semi-numerical researches on structural vibration and acoustics for revolution shells.
(4) High-order NMM and its applications in fluid-solid interaction harmonic analysis are also studied. A simple method is presented for automatically producing the expressions and writing subroutine codes based on simplex integration. Two and three dimensional programs of high-order NMM are further developed respectively for static analysis. Some examples are given to analyze the precision and the adaptability of NMM. This dissertation also presents two dimensional high-order NMM equations of fluid-solid interaction harmonic analysis based on rectangular mathematical meshes. The given results of computing frequencies and harmonic response prove the validity of the approach, and indicate that high-order NMM has high precision and convenient preprocessing. The approach of using covers of analytical solution to simulate special infinite fluid field is also proposed, and the given example suggests that NMM should be very suitable for combination of numerical solutions and analytical solutions, and more convenient than other approaches.
(5) NMM with fixed mathematical meshes is proposed in this dissertation, which has more advantages than other methods in solving fluid-solid coupling problems considering large deformation of structure and large disturbance of fluid, because it may be convenient to solve the nonlinear coupling problems when treating both fluid field and solid structure in the same background meshes. As the fundamental research, an attempt is made to compute large displacements of structures using fixed rectangular mathematical meshes and 1-order polynomial cover functions, and it is implemented that material particle moves in fixed meshes. The results of large deflection of a cantilever beam show the feasibility of NMM with fixed meshes, and indicate that more research should be further done on the approach of precisely computing initial stresses in the structures.
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