等参圆单元与管单元及其在热传导问题边界元法中的应用

发布时间：2018-03-26 20:39

本文选题：等参圆单元　切入点：等参管单元　出处：《大连理工大学》2015年硕士论文

【摘要】：随着现代工程应用对结构的性能要求越来越高,使得结构呈现出越来越复杂的外貌形状。基于此背景,孔状与管状结构因具有优良的热学与力学性能而得到广‘泛应用。由于孔状与管状结构几何外形的特殊性与复杂性,在采用有限元法求解时,需要划分大量的网格,导致建模与计算工作量很大。边界元法作为继有限元法之后又一重要的数值方法,因其具有降低问题维数、求解精度高等优点而在工程中得到广泛应用。但是,在边界元法中采用常规单元求解此类问题时,为了保证计算精度、减少离散误差,仍需要布置较密的单元来模拟结构的几何外形。这样,边界元法就无法体现自身的优势。为了克服传统边界元法中采用常规单元计算孔状结构时出现的计算节点多、离散误差大的缺点,本文基于Lagrange插值原理,构造了二维边界元法中的等参圆单元。该单元能很好地模拟孔状结构的光滑封闭曲线边界,并能对单元内的物理量进行高阶插值。另外,在二维边界元法中使用等参圆单元时,本文还提出了一种隔离对数奇异项、采用对数高斯积分来计算奇异积分的方法。对热传导问题的算例分析表明,基于等参圆单元的边界元算法在处理孔状结构时具有离散网格少、计算精度高的优点。另外,在等参圆单元的基础上,基于Lagrange插值原理,本文还提出了一种基于三维等参管单元的边界元算法。等参管单元能很好地模拟工程问题中结构的内外管状壁面,并实现物理量的高阶插值。在三维热传导问题中,使用基于等参管单元的边界元法求解时,提出了一种在等参平面内消除积分奇异性的方法。算例分析表明,本文所述方法能计算三维空间中沿任意方向弯曲的管状结构,且具有计算节点少、求解精度高的优点。
[Abstract]:As modern engineering applications become more and more demanding for the performance of structures, the structure presents more and more complex appearance shapes. Porous and tubular structures have been widely used because of their excellent thermal and mechanical properties. Due to the particularity and complexity of the geometrical shapes of porous and tubular structures, a large number of meshes are needed to be solved by finite element method (FEM). Boundary element method (BEM), as an important numerical method after finite element method, is widely used in engineering because of its advantages of reducing the dimension of the problem and high accuracy. In order to ensure the accuracy of the calculation and reduce the discrete error, it is necessary to arrange more dense elements to simulate the geometric shape of the structure when the conventional element is used to solve this kind of problem in the boundary element method. In order to overcome the disadvantages of traditional boundary element method, in which there are many nodes and large discrete errors, this paper is based on the principle of Lagrange interpolation. The isoparametric circular element in the two-dimensional boundary element method is constructed. The element can well simulate the smooth and closed curve boundary of the porous structure, and can interpolate the physical quantities in the element by higher order. In addition, when the isoparametric circular element is used in the two-dimensional boundary element method, In this paper, an isolated logarithmic singular term is proposed, and the method of calculating singular integral with logarithmic Gao Si integral is presented. The boundary element algorithm based on isoparametric circular element has the advantages of less discrete meshes and higher calculation accuracy in dealing with the hole structure. In addition, on the basis of isoparametric circular element, the method is based on the principle of Lagrange interpolation. In this paper, a boundary element algorithm based on 3-D isoparametric element is proposed. The isoparametric element can well simulate the inner and outer wall of the structure in engineering problems, and realize the high-order interpolation of physical quantities. A method to eliminate integral singularity in isoparametric plane is proposed when the boundary element method based on isoparametric element is used. The numerical examples show that the method presented in this paper can be used to calculate the tubular structures bending in any direction in three dimensional space. It has the advantages of less nodes and higher accuracy.
【学位授予单位】：大连理工大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：TK124

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