板的弱形式求积元分析
发布时间：2018-12-12 23:13
【摘要】：弱形式求积元法（简称求积元）把数值积分、数值微分与单元形式紧密地结合在一起，形成了一套独具特色的数值计算方法。它针对问题的弱形式描述，，在划分可积域后直接引入数值积分和数值微分来离散问题，得到代数方程组以求解问题。由于采用了数学离散的思想，求积元法不用直接构造形函数，在构造高阶单元等方面展现出其强大的能力。求积元法所构造的高阶板单元的计算效率很高，使用灵活方便，计算结果可靠，后处理精度高，是工程数值计算问题的一个理想选择。本文在课题组多年来对弱形式求积元法的研究基础上研究了薄板等需要满足边界C1类连续的板问题，主要内容包括： 1、薄板的线性分析。利用Gauss-Lobatto积分、微分求积和广义微分求积构造了可处理任意四边形问题的薄板位移型单元，该单元能够实现单元组集、各种形式边界条件的施加，并能够满足解的位移协调性要求。在与有限元计算结果的对比中，该单元显现出了明显的高计算效率。本文尝试了通过单元重构的方法进一步增强求积单元的灵活性，成功地解决了不同单元组集的问题。 2、高阶板的线性分析。在薄板线性单元的基础上构造了针对Reddy三阶板模型以及Kant三阶板模型的求积单元，展示了弱形式求积元法对于高阶板这类位移场较复杂的问题同样能够得到准确可靠的计算结果。通过对厚板的计算进一步明确了高阶板模型的应用范围。 3、薄板的几何非线性分析。对于几何非线性分析这种位移场连续性较好，且计算量大的问题非常适合采用弱形式求积元进行计算。本文利用弱形式求积元法计算了von Kárman薄板大挠度问题，得到可靠的计算结果，计算效率明显强于有限元。求积单元对于非线性屈曲问题也能够得到准确的结果。 4、薄板的弹塑性分析。基于塑性增量理论构造了用于薄板弹塑性分析的高阶求积单元。该单元在计算理想弹塑性材料的薄板问题时表现出强大的能力，甚至可以进行极限分析。 数值算例表明求积元法在处理C1类连续板问题时计算结果可靠、效率与传统有限元相比优势明显，且有着足够的灵活性以满足工程结构的数值计算需要。
[Abstract]:The weak form quadrature element method (abbreviated as quadrature element) combines the numerical integral, numerical differential and the element form closely, and forms a set of unique numerical calculation methods. Aiming at the weak formal description of the problem, the numerical integral and numerical differential are directly introduced to discretize the problem after dividing the integrable domain, and the algebraic equations are obtained to solve the problem. Because of the idea of mathematical discretization, the quadrature element method does not need to construct the shape function directly, and it shows its powerful ability in constructing higher order element and so on. The high order plate element constructed by the quadrature element method is an ideal choice for engineering numerical calculation because of its high efficiency, flexibility and convenience, reliable calculation results and high post-processing accuracy. In this paper, based on the research of weak form quadrature element method in our research group, we study the problems of thin plates and other continuous plates which need to satisfy boundary C1 class. The main contents are as follows: 1. Linear analysis of thin plates. By using Gauss-Lobatto integral, differential quadrature and generalized differential quadrature, a thin plate displacement-type element which can deal with any quadrilateral problem is constructed. This element can realize the application of the set of elements and various forms of boundary conditions. And it can meet the displacement coordination requirement of the solution. Compared with the results of finite element calculation, the element shows obvious high computational efficiency. This paper attempts to further enhance the flexibility of quadrature units by means of cell reconstruction, and successfully solves the problem of different sets of units. 2. Linear analysis of higher order plates. The quadrature element for Reddy third order plate model and Kant third order plate model is constructed on the basis of thin plate linear element. It is shown that the weak form quadrature element method can also obtain accurate and reliable results for the more complicated problems of displacement field such as high order plates. Through the calculation of thick plate, the application scope of high order plate model is further clarified. 3. Geometric nonlinear analysis of thin plates. For the geometric nonlinear analysis, the continuity of the displacement field is good, and the problem of large amount of calculation is very suitable to use the weak form quadrature element to calculate the displacement field. In this paper, the weak form quadrature element method is used to calculate the large deflection problem of von K 谩 rman thin plate. The reliable results are obtained, and the computational efficiency is obviously higher than that of the finite element method. The quadrature element can also obtain accurate results for nonlinear buckling problems. 4. Elastoplastic analysis of thin plates. Based on the theory of plastic increment, a high order quadrature element for elastoplastic analysis of thin plates is constructed. The element can be used to calculate the thin plate problem of ideal elastic-plastic material, and it can even be used for limit analysis. Numerical examples show that the quadrature element method is reliable in dealing with C _ 1 continuous plate problems, has obvious advantages over the traditional finite element method, and has sufficient flexibility to meet the needs of numerical calculation of engineering structures.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：TU311.4
本文编号：2375410
[Abstract]:The weak form quadrature element method (abbreviated as quadrature element) combines the numerical integral, numerical differential and the element form closely, and forms a set of unique numerical calculation methods. Aiming at the weak formal description of the problem, the numerical integral and numerical differential are directly introduced to discretize the problem after dividing the integrable domain, and the algebraic equations are obtained to solve the problem. Because of the idea of mathematical discretization, the quadrature element method does not need to construct the shape function directly, and it shows its powerful ability in constructing higher order element and so on. The high order plate element constructed by the quadrature element method is an ideal choice for engineering numerical calculation because of its high efficiency, flexibility and convenience, reliable calculation results and high post-processing accuracy. In this paper, based on the research of weak form quadrature element method in our research group, we study the problems of thin plates and other continuous plates which need to satisfy boundary C1 class. The main contents are as follows: 1. Linear analysis of thin plates. By using Gauss-Lobatto integral, differential quadrature and generalized differential quadrature, a thin plate displacement-type element which can deal with any quadrilateral problem is constructed. This element can realize the application of the set of elements and various forms of boundary conditions. And it can meet the displacement coordination requirement of the solution. Compared with the results of finite element calculation, the element shows obvious high computational efficiency. This paper attempts to further enhance the flexibility of quadrature units by means of cell reconstruction, and successfully solves the problem of different sets of units. 2. Linear analysis of higher order plates. The quadrature element for Reddy third order plate model and Kant third order plate model is constructed on the basis of thin plate linear element. It is shown that the weak form quadrature element method can also obtain accurate and reliable results for the more complicated problems of displacement field such as high order plates. Through the calculation of thick plate, the application scope of high order plate model is further clarified. 3. Geometric nonlinear analysis of thin plates. For the geometric nonlinear analysis, the continuity of the displacement field is good, and the problem of large amount of calculation is very suitable to use the weak form quadrature element to calculate the displacement field. In this paper, the weak form quadrature element method is used to calculate the large deflection problem of von K 谩 rman thin plate. The reliable results are obtained, and the computational efficiency is obviously higher than that of the finite element method. The quadrature element can also obtain accurate results for nonlinear buckling problems. 4. Elastoplastic analysis of thin plates. Based on the theory of plastic increment, a high order quadrature element for elastoplastic analysis of thin plates is constructed. The element can be used to calculate the thin plate problem of ideal elastic-plastic material, and it can even be used for limit analysis. Numerical examples show that the quadrature element method is reliable in dealing with C _ 1 continuous plate problems, has obvious advantages over the traditional finite element method, and has sufficient flexibility to meet the needs of numerical calculation of engineering structures.
【学位授予单位】：清华大学
【学位级别】：博士
【学位授予年份】：2013
【分类号】：TU311.4
【参考文献】
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本文编号：2375410
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