## Graded Finite(分级有限)研究综述

Graded Finite 分级有限 - Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2.^{[1]}We study Betti numbers of graded finitely generated modules over a quadratic complete intersection.

^{[2]}

令$k$ 是一个代数闭域，$A$ 是一个$\mathbb{Z}$-分级有限生成的简单$k$-代数，它是 Gelfand-Kirillov 维数为 2 的域。

^{[1]}我们研究了二次完全交点上分级有限生成模块的 Betti 数。

^{[2]}

## Brick Graded Finite 砖分级有限

The three-dimensional (3-D) brick graded finite element is programmed and incorporated into the code via the user-defined material subroutine UMAT.^{[1]}Using the finite element code ABAQUS and the user-defined material utilities UMAT and UMATHT, a solid brick graded finite element is developed for three-dimensional (3D) modeling of free vibrations of thermally loaded functionally gradient material (FGM) sandwich plates.

^{[2]}

通过用户定义的材料子程序 UMAT 对三维 (3-D) 砖分级有限元进行编程并合并到代码中。

^{[1]}使用有限元代码 ABAQUS 和用户定义的材料实用程序 UMAT 和 UMATHT，开发了实心砖分级有限元，用于热加载功能梯度材料 (FGM) 夹层板的自由振动的三维 (3D) 建模。

^{[2]}

## graded finite element 分级有限元

The three-dimensional (3-D) brick graded finite element is programmed and incorporated into the code via the user-defined material subroutine UMAT.^{[1]}To perform accurate and efficient finite element analysis for heat transfer and transient thermal stress analyses in two-dimensional functionally graded materials, incompatible graded finite elements are developed and verified.

^{[2]}Using the finite element code ABAQUS and the user-defined material utilities UMAT and UMATHT, a solid brick graded finite element is developed for three-dimensional (3D) modeling of free vibrations of thermally loaded functionally gradient material (FGM) sandwich plates.

^{[3]}In order to investigate the nonlinear elastoplastic response of the FGM, the nonlinear graded finite element method is developed from three-dimensional continuum concepts that admit arbitrarily large displacements and rotations.

^{[4]}On the contrary, in the conventional homogeneous finite element formulation the isoparametric graded finite element formulation is adopted on the level of Gaussian integration points to realize the gradation in material properties.

^{[5]}

通过用户定义的材料子程序 UMAT 对三维 (3-D) 砖分级有限元进行编程并合并到代码中。

^{[1]}为了对二维功能梯度材料中的传热和瞬态热应力分析执行准确有效的有限元分析，开发和验证了不兼容的梯度有限元。

^{[2]}使用有限元代码 ABAQUS 和用户定义的材料实用程序 UMAT 和 UMATHT，开发了实心砖分级有限元，用于热加载功能梯度材料 (FGM) 夹层板的自由振动的三维 (3D) 建模。

^{[3]}为了研究 FGM 的非线性弹塑性响应，非线性梯度有限元方法是从允许任意大位移和旋转的三维连续统概念发展而来的。

^{[4]}相反，在传统的齐次有限元公式中，在高斯积分点的水平上采用等参梯度有限元公式来实现材料性质的梯度。

^{[5]}