In this paper, a new closed-form formulation with universal matrices strongly recommends that Weighted Richardson Extrapolation (WRE) with robust and hourglass controlled one-point quadrature can absolutely replace the conventional Gauss Quadrature in terms of efficiency, accuracy, and speed to find element stiffness matrices of quadrilateral and hexahedral elements. Linearizing the geometric transformation and averaging the material property over an element, using sampling point at the origin (0,0) of a standard mapped 2-square $(\xi ,\eta )$ plane for two-dimensional analysis and origin (0,0,0) of a standard mapped 2-cube $(\xi ,\eta ,\zeta )$ system in the hexahedral element (mid-point rule), helps to obtain element stiffness matrix quickly and explicitly through constant universal matrices. This technique is used independently to each of the eight sub-cubes of the mapped 2-cube $(\xi ,\eta ,\zeta )$ of the element for three-dimensional problems and four sub-squares of the mapped 2-square of the element for two-dimensional problems. These matrices are assembled appropriately for a second and better approximation. A weighted addition of the two approximations produces a stiffness matrix as accurate as from conventional Gauss Quadrature. However, finding the stiffness matrix in this way, due to explicit integrations, demands only a third of the time needed for Gauss Quadrature.

在本文中，采用通用矩阵的新闭式公式强烈建议具有鲁棒性且由沙漏控制的单点积分的加权Richardson外推法（WRE）在效率，准确性和查找元素的速度方面可以绝对替代传统的高斯正交。四边形和六面体单元的刚度矩阵。使用标准映射2平方的原点（0,0）处的采样点来线性化几何变换并平均元素上的材料属性$（\xi ，\eta ）$ 用于二维分析的平面和标准映射的2多维数据集的原点（0,0,0） $（\xi ，\eta ，\zeta ）$六面体单元中点法（中点法则）有助于通过恒定的通用矩阵快速而明确地获得单元刚度矩阵。此技术独立用于映射的2多维数据集的八个子多维数据集中的每个子多维数据集$（\xi ，\eta ，\zeta ）$三维问题的元素的平方和二维问题的元素的映射2平方的四个子平方。适当地组装这些矩阵，以获得第二个更好的近似值。两个近似值的加权加法产生的刚度矩阵与常规高斯正交算法一样准确。但是，由于显式积分，以这种方式找到刚度矩阵仅需要高斯求积所需时间的三分之一。