【作者】 王宇；

【导师】 钟宏志；

【作者基本信息】 清华大学 ， 土木工程， 2010， 硕士

【摘要】 弱形式求积元法是基于变分原理及微分求积思想的新数值方法。自被提出以来,该方法在结构分析中得到应用,并展示了其处理复杂几何形状,复杂荷载,非均匀材料,强非线性问题相比有限元的优势。弱形式求积元法核心思想是:将复杂的物理域划分为一些较大的连续子区域,将每个不规则的子区域应用自然-笛卡尔几何变换到标准区域内建立问题的控制泛函。在每个标准域内,将问题的控制泛函用数值积分近似表示。再引入微分求积法,将积分点处函数的导数值用积分点的函数值表示。最终,将泛函表示成各个积分点函数值的函数。引入泛函驻值条件得到关于待求函数的代数方程组,求解代数方程组即得到问题的解答。功能梯度材料是一种新型的材料,它是通过将两种或多种材料按照一定的规律组合起来而得到的一种混合材料。由于出色的性能,功能梯度材料已经在航天,机械等领域有了广泛的应用,发挥了巨大的作用,在不远的未来也很有可能会应用到土木工程中来。功能梯度材料构件的计算分析成为了研究人员关注的热点。高阶梁、板理论为了满足对功能梯度材料构件计算分析的需要提供了有效途径。本论文中将弱形式求积元法应用于高阶梁、板的分析中。研究了功能梯度梁、板的弯曲和振动。同时,通过对功能梯度梁、板计算分析的研究,讨论了功能梯度梁、板计算的中性轴问题,并提出了可行的修正方法。通过数值算例研究,与已有文献和有限元软件ANSYS计算结果进行对比,充分地展示了弱形式求积元法的高效性、稳定性和准确性。特别是应用于功能梯度构件的计算时,弱形式求积元法能够准确描述材料在空间中的连续变化,展示了弱形式求积元法应用于功能梯度材料计算中的广阔前景。更多还原

【Abstract】 The weak form quadrature element method (QEM) is a new method based on variational principles and the conventional differential quadrature method. Since its inception, it has found applications in structural analysis. The QEM has demostrated its superiority over the finite element method in handling complex geometry, complicated boundary conditions, inhomogeneous materials and strong nonlinearity. Implementation of the QEM consists of following essential procedures. Firstly, the variational description of a problem is established. Secondly, the physical domain is divided into a number of quadrature elements, each being mapped onto a standard computational domain. Thirdly, numerical integrations and the differential quadrature method are introduced to characterize the problem in terms of quantities at quadrature points. A set of algebraic equations result eventualy after introduction of the stationary condition. The weak form QEM inherits the high computational efficiency of the differential quadrature method and enjoys flexibility and wide applicability of weak formulation. Functionally Graded Materials (FGMs) are a new kind of material. FGM are composed of two different materials and there is continuous gradient of composition and structure between the two materials, without any explicit interfaces. Because of its excellent properties, FGMs have been playing more and more important roles in the areas of aerospace and mechanical engineering. It will hopefully find application in civil engineering in the future.High-order beam and plate theories have been adopted in analysis of FGM components which necessiates an efficient and accurate numerical method.The purpose of this thesis is to extend the weak form QEM to high-order beam and plate analysis, and study functionally graded beams and plates. The issue of neutral-axis for FGM components is addressed. Numerical examples are presented and comparison with results in the literature and those of the commecial software ANSYS is made to exhibit the efficacy , accuracy and superiority of the QEM. The present work exploits the advantages of the QEM and performs preliminary analysis of FGM components. The achievement will serve as the stepping-stone for its application in analysis and design of FGMs.更多还原