Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics
Generalized Differential Quadrature Finite Element Method applied to Advanced Structural Mechanics
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DOI：
10.6092/unibo/amsdottorato/5932
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发表时间：
2013-05
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影响因子：
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通讯作者：
N. Fantuzzi
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文献类型：
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作者：
N. Fantuzzi
近年来，微分正交法（DQ）以其精度高、实现简单、可广泛应用于各种问题而著称。在过去的几年里，一些研究人员经历了重大的发展，他们对这个话题的研究有所增加。DQ本质上是用于数值积分函数的流行的高斯正交（GQ）的推广。GQ将有限内积分近似为问题域中选定点上被积值的加权和，而DQ将光滑函数在某一点上的导数近似为选定节点上函数值的加权和。这种优雅的方法的一个直接应用是求解常微分方程和偏微分方程。此外，近年来在加权系数的计算中对DQ公式进行了推广，使该方法更加灵活和准确。因此，它被称为广义微分正交法（GDQ）。然而，GDQ在其原始形式下的适用性仍然有限。它已被证明对于具有强物质不连续的问题以及涉及奇点和不规则性的问题是失败的。另一方面，众所周知的有限元法可以克服这些问题，因为它将计算域细分为一定数量的单元，在这些单元中计算解。近年来，一些研究人员一直在研究一种既能利用GDQ法的优点又能利用有限元法的优点的数值计算方法。这种方法在各个研究组中有不同的名称，在这里将其称为广义微分正交有限元法（GDQFEM）。
Over the years the Differential Quadrature (DQ) method has distinguished because of its high accuracy, straightforward implementation and general ap- plication to a variety of problems. There has been an increase in this topic by several researchers who experienced significant development in the last years. DQ is essentially a generalization of the popular Gaussian Quadrature (GQ) used for numerical integration functions. GQ approximates a finite in- tegral as a weighted sum of integrand values at selected points in a problem domain whereas DQ approximate the derivatives of a smooth function at a point as a weighted sum of function values at selected nodes. A direct appli- cation of this elegant methodology is to solve ordinary and partial differential equations. Furthermore in recent years the DQ formulation has been gener- alized in the weighting coefficients computations to let the approach to be more flexible and accurate. As a result it has been indicated as Generalized Differential Quadrature (GDQ) method. However the applicability of GDQ in its original form is still limited. It has been proven to fail for problems with strong material discontinuities as well as problems involving singularities and irregularities. On the other hand the very well-known Finite Element (FE) method could overcome these issues because it subdivides the computational domain into a certain number of elements in which the solution is calculated. Recently, some researchers have been studying a numerical technique which could use the advantages of the GDQ method and the advantages of FE method. This methodology has got different names among each research group, it will be indicated here as Generalized Differential Quadrature Finite Element Method (GDQFEM).