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首页> 外文期刊>American journal of applied sciences >Weighted Integration Route to Stiffness Matrix of Quadrilaterals for Speed, Accuracy and Functionally Graded Material Application
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Weighted Integration Route to Stiffness Matrix of Quadrilaterals for Speed, Accuracy and Functionally Graded Material Application

机译：四边形刚度矩阵的加权积分路径，用于速度，精度和功能渐变材料的应用

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摘要

A weighted integration route with robust one-point integration (hourglass-controlled) is proposed as efficient, time saving alternative to Gauss quadrature for stiffness matrix of bilinear quadrilaterals. One-point rule relies on sampling at the center of the element to linearize the geometric transformation and average the material property over it. This enables, for a given element, explicit integration of stiffness matrix yielding a first approximation. For a second and better approximation, this procedure is applied independently to each of the four sub-squares of the mapped 2-square of the element and the matrices are assembled. A weighted addition of the two approximations produces a stiffness matrix as accurate as from 3-point Gauss-quadrature (G9P). Whereas, due to explicit integrations, obtaining stiffness matrix in this way demands less than a third of the time needed for 2-point Gauss-quadrature (G4P). On both counts (speed and accuracy) this approach outperforms Gauss-quadrature. Sampling (material and geometry) at 5-points makes this element superior to G4P for Functionally Graded Material (FGM) applications. Bench mark examples by this approach are validated with Gauss quadrature and analytical solutions.
机译：对于双线性四边形的刚度矩阵，提出了一种具有鲁棒的单点积分（沙漏控制）的加权积分路线，作为高斯积分的一种高效，省时的替代方案。一点法则依赖于元素中心的采样来线性化几何变换并平均其上的材料属性。对于给定的元素，这可以实现刚度矩阵的显式积分，从而得出第一近似值。为了获得第二个更好的近似值，此过程将独立地应用于元素的映射2平方的四个子平方中的每一个，并且将矩阵组装在一起。两个近似值的加权加法产生的刚度矩阵与三点高斯正交（G9P）一样准确。而由于显式积分，以这种方式获得刚度矩阵所需的时间少于2点高斯正交（G4P）所需时间的三分之一。在速度和准确性两方面，这种方法都优于高斯正交。 5点采样（材料和几何形状）使该元素优于功能分级材料（FGM）应用程序的G4P。这种方法的基准示例已通过高斯正交和分析解决方案进行了验证。

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