一种提高平板型壳单元计算精度的改进算法研究

发布时间：2018-01-06 08:25

本文关键词：一种提高平板型壳单元计算精度的改进算法研究　出处：《大连理工大学》2016年博士论文　论文类型：学位论文

【摘要】：壳体结构是工程应用中常见的结构形式之一,广泛应用于土木、水利、汽车、航空航天等各个工程领域中。由于壳体是具有薄膜和弯曲两重受力特性的三维特殊结构,其控制方程是一组形式复杂的偏微分方程,直接解析求解是比较困难的。在电子计算机出现以前只能计算一些结构简单规则的问题,随着以计算机为工具的有限元方法出现以后,经过众多学者的努力,不同类型的壳单元也相继出现。其中,平板型壳单元是最早发展起来的壳单元,它是将壳体离散为一系列折板组成的体系,通过平面膜单元和平板弯曲单元叠加组合来模拟壳体的薄膜和弯曲受力状态。平板型壳单元以其表达格式简单、应用广泛、计算效率高、计算结果可靠等优点在不同类型的壳单元中占有重要的地位,在工程实际中得到了广泛的应用。但是,平板型壳单元是由平面应力单元和平板弯曲单元叠加组合而成的壳单元,采用几何近似性,用折板面代替壳体曲面。因此,在计算不规则壳体结构特别是曲壳结构时不能很好地模拟曲壳形状,从而引起较大的误差,需要将网格合理加密才能达到一定的精度,这会引起较大的计算量。为了进一步提高平板型壳单元的计算性能,本文提出了一种新的单元局部坐标系建立方法。该方法在单元的高斯积分点建立多个局部坐标系,并保证每个局部坐标系都位于单元在高斯点处的切平面上,对每个单元刚度转换矩阵进行及时修正,从而可以有效适应曲面壳体形状。为了能够在新的局部坐标系下求得和单元刚度有关的参数,同时又避免频繁的坐标转换造成的计算误差,本文还引入了计算形函数对局部坐标的导数和积分转换的雅可比行列式的高精度算法。这种高精度算法直接基于三维曲面整体坐标系推导得出,具有较高的计算精度。利用本文提出的改进算法计算平板型壳单元的刚度矩阵以及等效节点载荷列阵,都取得了良好的效果。此外,本文还将一种新的线性方程组求解器应用到平板型壳单元当中,进一步提高其计算性能。本文的主要研究内容可以归纳如下：(1)提出了一种计算平板型壳单元在整体坐标系下的单元刚度矩阵的改进算法,并将其应用到几种不同类型的平板型壳单元当中。通过将几种不同类型的平面膜单元和平板弯曲单元叠加组合形成平板型壳单元,对其计算性能进行讨论,并详细给出了利用本文的改进算法计算单元刚度矩阵的理论公式,从原理上对本文方法在提高平板型壳单元计算性能方面进行分析讨论。(2)对本文所提出的改进算法在平板型壳单元中的应用和实现进行程序开发,给出了详细的编程流程,并通过算例对其计算精度进行比较分析。为了对本文提出的改进算法的计算性能进行验证,将改进算法应用到几种典型的平板型壳单元,并对其进行程序实现,通过具体的算例对其正确性和精确性进行验证分析。(3)将本文提出的改进算法应用扩展到平板型壳单元等效节点载荷的计算上。对平板型壳单元在不同载荷下等效节点载荷的计算进行归纳总结,给出了利用本文方法计算其等效节点载荷的详细的理论公式,并利用程序编制对其应用进行数值实现,通过具体的算例对本文方法在计算等效节点载荷方面的性能进行分析验证。(4)在有限元分析中,引入了一种新的线性方程组直接法求解器——同时消元回代法,提出了新的刚度矩阵组集方法,并将其应用到平板型壳单元问题的求解中。给出了新的刚度矩阵的组集方法和详细计算步骤,并结合同时消元回代法求解器对其求解性能进行分析讨论,给出了程序的编制过程,通过具体算例对其性能进行验证。综上所述,本文提出了一种能够用于提高平板型壳单元计算精度的改进算法,为平板型壳单元计算性能的提高提供了一种新的手段。本文所述方法也可以应用到其他类型的四边形平板型壳单元中,具有广阔的发展前景。
[Abstract]:The shell structure is one of the common structure in engineering application, widely used in civil engineering, water conservancy, automotive, aerospace and other engineering fields. Because the shell is a special three-dimensional structure stress and bending properties of thin film double, the control equation is a partial differential equation of a group of complex forms, direct analytical solution is more difficult. Some simple rules can only be calculated before problems occur in the electronic computer, with finite element method with the computer as a tool, through the efforts of numerous scholars, shell elements of different types have emerged. Among them, flat shell element is the shell element developed early, it is the case for discrete a series of folded plate system, by plane membrane element and plate bending element combination to simulate the shell membrane and bending stress. Flat shell element lattice with its expression Simple, wide application, high computational efficiency, reliable calculation plays an important role in the shell elements of different types, has been widely used in engineering practice. However, flat shell element is shell element from the plane stress element and plate bending element combination and using geometric approximation of the folding surface instead of curved shell. Therefore, in the calculation of irregular shell structure especially the shell structure can well simulate the curved shell shape, which causes the great error, will need to reach a reasonable grid encryption precision, it will cause a large amount of calculation. In order to further improve the computational performance of the flat plate the shell element, the paper puts forward a new method of element local coordinate. The method established a local coordinate system in units of the Gauss integral points, and to ensure that each local coordinate system are located in the unit The tangent plane at Gauss point, timely correction of each element stiffness matrix conversion, which can effectively adapt to the surface of the shell shape. In order to obtain new parameters in local coordinates and unit stiffness, and avoid frequent calculation error caused by coordinate transformation, this paper also introduces high precision algorithm for calculating the shape functions of the local coordinate of the derivative and integral transformation of Jacobi determinant. Derived 3D coordinates the overall accuracy of the algorithm directly based on high precision. Using this improved algorithm to calculate flat shell element stiffness matrix and the equivalent nodal load matrix, have achieved good results. In addition, this paper also will be a new linear equation solver is applied to the flat shell element, further improve its performance. The main research work of this paper Content can be summarized as follows: (1) put forward an improved algorithm for calculating flat shell element in the global coordinate system of the element stiffness matrix, and applied to several different types of flat shell element. The plane membrane element and plate bending element combination of several different types of flat form shell element, the calculation of performance are discussed, and gives the detailed theoretical calculation formula of element stiffness matrix using the improved algorithm, from the principle of this method in improving the performance of flat shell element are discussed. (2) and the improved algorithm of the proposed application in flat shell the unit of program development, detailed programming process is given, and an example of the calculation accuracy were analyzed. In order to calculate the performance of the proposed algorithm is verified, will be changed In the algorithm is applied to several typical flat shell element, and carries on the program, through a numerical example to validate its correctness and accuracy. (3) to expand the application of improved algorithm is proposed in this paper to calculate the flat shell element equivalent node loads. The calculation of flat shell element under different load equivalent node loads are summarized, with the theoretical formula for calculating the equivalent node load is given by using this method, and the application program of the numerical implementation, through specific examples to verify the feasibility of this method in the calculation of equivalent node load performance aspects. (4) in the Co. element analysis, introduced a new direct method of linear equations solver and elimination back generation method, puts forward the stiffness matrix assembly method, and its application to the flat shell element problem In solving the stiffness matrix is given. The new assembly method and the detailed calculation steps, and at the same time combined with the elimination back solver are discussed on the generation method of solving the performance, preparation process are given, through specific examples to verify its performance. In summary, this paper presents a can be used for the improved algorithm can improve the calculation precision flat shell element, which provides a new method for the flat shell element to improve the performance. The method can also be applied to the quadrilateral flat shell element of other types, and has broad prospects for development.

【学位授予单位】：大连理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TP301.6

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