稳态渗流问题中的FEMOL平面线性样条曲线单元研究

发布时间：2018-12-16 13:16

【摘要】：有限元线法(Finite Element Method of Lines,简称FEMOL)是一种新型的半数值半解析方法,它以常微分方程(Ordinary Differential Equation,简称ODE)求解器为支撑软件。此方法在固体力学中的运用已经较为成熟,在热传导问题中也在不断发展。本文初次将有限元线法引进到二维稳态渗流问题当中,主要研究内容如下:(1)建立二维稳态渗流场的FEMOL平面线性样条曲线单元,基于三次B样条插值基函数和线性Lagrange插值基函数,建立了 FEMOL平面线性样条曲线单元映射。该单元在结线方向采用的是三次B样条插值,端线方向采用的是线性Lagrange插值,将不规则单元映射到[-1,1]局部坐标下的规则单元。(2)利用变分原理,建立了二维稳态渗流场进行了 FEMOL平面线性样条曲线单元半离散泛函,将求解二维稳态渗流问题的偏微分方程的边值问题转化成求解其泛函的极值问题,得到关于二维稳态渗流问题的常微分方程组(ODEs)以及相应的边界条件(BCs)。(3)以FEMOL平面线性样条曲线单元映射以及该单元下二维稳态渗流泛函变分计算为基础,利用FORTRAN 95语言编写了求解二维平面稳态渗流问题的专有程序SSFEMOL1.0,其中调用COLSYS的升级版本——COL90作为常微分方程求解器,使其具有更高的计算效率。(4)运用逆解法,编写二维稳态渗流问题的相关算例,将计算结果与解析解、有限元方法计算结果进行对比,在网格划分、计算精度,方法适用性等几个方面进行分析。
[Abstract]:Finite element linear method (Finite Element Method of Lines,) is a new semi-numerical semi-analytical method, which is supported by (Ordinary Differential Equation, (ODE) solver of ordinary differential equations. The application of this method in solid mechanics is mature, and it is also developing in the field of heat conduction. In this paper, the linear finite element method is first introduced into the two-dimensional steady flow problem. The main research contents are as follows: (1) the FEMOL plane linear spline curve element of the two-dimensional steady flow field is established. Based on cubic B-spline interpolation basis function and linear Lagrange interpolation basis function, the FEMOL plane linear spline curve unit mapping is established. The unit uses cubic B-spline interpolation in the junction direction and linear Lagrange interpolation in the terminal direction. The irregular element is mapped to the regular element in the local coordinate of [-1]. (2) the variational principle is used. In this paper, the semi-discrete functional of two-dimensional steady-state seepage field is established by FEMOL plane linear spline curve element. The boundary value problem of the partial differential equation for two-dimensional steady state seepage is transformed into the extremum problem of solving its functional. It is obtained that the system of ordinary differential equations (ODEs) and the corresponding boundary condition (BCs). (3) for two-dimensional steady state seepage flow are based on the FEMOL plane linear spline curve element mapping and the variational calculation of two-dimensional steady state seepage functional under this element. By using FORTRAN 95 language, the special program SSFEMOL1.0, for solving two-dimensional plane steady seepage problem is written. COL90, which is an upgraded version of COLSYS, is called as an ordinary differential equation solver, which makes it more efficient. (4) the inverse method is used to solve the problem. An example of two-dimensional steady seepage problem is compiled. The results are compared with analytical solution and finite element method. The results are analyzed in the aspects of mesh generation, accuracy and applicability of the method.
【学位授予单位】：北方工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O357.3

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