# 统计代写|主成分分析代写Principal Component Analysis代考|”Limitations of Traditional Finite Elements and Introduction of a New Quadrilateral Element”

The passage discusses the limitations of using traditional two- and three-dimensional finite elements such as triangles, rectangles, tetrahedra, and bricks to accurately represent complex, irregular geometries. When creating a mesh for the plane area in Figure 6.20a, triangular elements can closely approximate a curved boundary, but they can lead to high aspect ratios and computational inefficiencies as the mesh gets finer towards the center of the domain. Rectangular elements, on the other hand, might exclude significant portions of the domain when dealing with curved boundaries unless the elements are very small.

To address these issues, the text suggests combining different element types, though this can create inconsistencies in the polynomial representations of the field variable across element boundaries. It then introduces a new element type—a general two-dimensional quadrilateral element with four nodes—that can effectively mesh with rectangular elements while also better approximating curved boundaries.

This quadrilateral element is derived from a parent four-node rectangular element using isoparametric mapping, where the same interpolation functions describe both the geometry and the field variable variation within the element. The geometry of the quadrilateral element is mathematically represented by Equations 6.77 and 6.78, with the interpolation functions satisfying the condition of unity at the associated node and zero at others.

When computing element characteristic matrices and solving for derivatives of the interpolation functions with respect to global coordinates, there’s an added level of complexity because these are initially expressed in natural coordinates. To overcome this, one uses the Jacobian matrix, which relates changes in global and natural coordinates, and its inverse to transform the partial derivatives of the interpolation functions accordingly.

The passage further illustrates how to show that the mapping described by Equation 6.80 correctly captures the line connecting nodes 2 and 3 in a quadrilateral element and calculates the global (x, y) coordinates corresponding to the natural coordinates (r, s) = (1, 0.5). By using the interpolation functions and geometric properties of the element, it confirms the consistency between the mapped geometry and the actual geometry of the element edge.

### MATLAB代写

MATLAB 是一款高性能的技术计算语言，集成了计算、可视化和编程环境于一体，以熟悉的数学符号表达问题和解决方案。MATLAB 的基本数据元素是一个不需要维度的数组，使得能够快速解决带有矩阵和向量公式的多种技术计算问题，相比使用 C 或 Fortran 等标量非交互式语言编写的程序，效率大大提高。MATLAB 名称源自“矩阵实验室”（Matrix Laboratory）。最初开发 MATLAB 的目标是为了提供对 LINPACK 和 EISPACK 项目的矩阵软件的便捷访问，这两个项目代表了当时矩阵计算软件的先进技术。经过长期发展和众多用户的贡献，MATLAB 已成为数学、工程和科学入门及高级课程的标准教学工具，在工业界，MATLAB 是高效研究、开发和分析的理想选择。MATLAB 提供了一系列名为工具箱的特定应用解决方案集，这对广大 MATLAB 用户至关重要，因为它们极大地扩展了 MATLAB 环境，使其能够解决特定类别问题。工具箱包含了针对特定应用领域的 MATLAB 函数（M 文件），涵盖信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等诸多领域。