弹性力学和温度场问题的共轭复变量无单元Galerkin方法

发布时间：2018-05-21 11:08

本文选题：无网格方法 + 共轭复变量无单元Galerkin方法　；参考：《湖北工业大学》2017年硕士论文

【摘要】：无网格方法是上个世纪九十年代中期兴起的一种数值方法,由于不需要网格,只需要节点信息,不存在网格移动和网格畸变,因此具有适用范围广和计算精度高等优点,已成为科学和工程计算方法研究的热点,也是科学和工程计算发展的趋势。将共轭复变量移动最小二乘法引入无单元Galerkin方法而形成的共轭复变量无单元Galerkin方法,可有效地解决无单元Galerkin方法存在的配点过多、计算量大等问题。共轭复变量无单元Galerkin方法的优点是采用一维基函数建立二维问题的试函数,使得试函数中所含的待定系数减少,从而提高了计算效率。本文将共轭复变量无单元Galerkin方法应用于弹性力学问题,结合弹性力学问题的Galerkin积分弱形式,采用罚函数法施加本质边界条件,建立了弹性力学问题的共轭复变量无单元Galerkin方法,推导了相应的计算公式,编制了相应的计算程序,对三个弹性力学问题的算例进行了数值分析,并对数值方法中的计算参数进行了分析,确定了合理的参数范围。该方法的优点是具有较高的精度和较好的稳定性。此外,采用外包线图方法,综合考察和评价了相关数值方法的计算精度和计算效率,直观地比较了不同数值方法的优劣。本文将共轭复变量无单元Galerkin方法应用于温度场问题,结合温度场问题的Galerkin积分弱形式,使用罚函数法施加本质边界条件,采用两种场变量表示方法,其中方法I中的场变量采用形函数的实部表示,方法II中的场变量采用试函数的实部或虚部表示,建立了两种温度场问题的共轭复变量无单元Galerkin方法,推导了相应的计算公式,编制了相应的计算程序,对三个温度场问题的算例进行了数值分析,并对数值方法中的计算参数进行了分析,确定了合理的参数范围。两种方法均具有求解精度高、稳定性好等优点,其中方法II相对于方法I具有更高的精度。
[Abstract]:Meshless method is a numerical method developed in the middle of 1990s. Because it does not need mesh, only node information, mesh movement and mesh distortion, it has the advantages of wide application range and high calculation accuracy. It has become a hot spot in the research of scientific and engineering computing methods, and is also the trend of the development of science and engineering computing. The conjugate complex variable moving least square method is introduced into the element free Galerkin method, which can effectively solve the problems of too many collocation points and large computational costs in the element free Galerkin method. The advantage of the element free Galerkin method for conjugate complex variables is that a wiki function is used to establish the trial function of the two-dimensional problem, which reduces the undetermined coefficients in the trial function and improves the computational efficiency. In this paper, the conjugate complex variable element free Galerkin method is applied to the elastic mechanics problem. Combining the weak form of Galerkin integral of the elastic mechanics problem, using penalty function method to apply essential boundary conditions, the conjugate complex variable element free Galerkin method for the elastic mechanics problem is established. The corresponding calculation formula is derived and the corresponding calculation program is worked out. The numerical analysis of three examples of elastic mechanics problems is carried out, and the calculation parameters in the numerical method are analyzed, and the reasonable parameter range is determined. The advantage of this method is that it has higher accuracy and better stability. In addition, the calculation accuracy and efficiency of the relevant numerical methods are comprehensively investigated and evaluated by using the outsourced graph method, and the advantages and disadvantages of different numerical methods are compared intuitively. In this paper, the element free Galerkin method for conjugate complex variables is applied to the temperature field problem. Combining with the weak form of Galerkin integral of the temperature field problem, the penalty function method is used to impose essential boundary conditions, and two kinds of field variable representation methods are adopted. The field variables in method I are represented by the real part of the shape function, and the field variables in method II are represented by the real part or the imaginary part of the trial function. The element free Galerkin method of conjugate complex variables for two kinds of temperature field problems is established, and the corresponding calculation formulas are derived. The corresponding calculation program is compiled and the numerical analysis of three examples of temperature field problems is carried out. The calculation parameters in the numerical method are analyzed and the reasonable range of parameters is determined. Both methods have the advantages of high accuracy and good stability, among which method II has higher accuracy than method I.
【学位授予单位】：湖北工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：TU43

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