# 统计代写|主成分分析代写Principal Component Analysis代考|Finite Element Method: Meshing and Convergence

In the finite element method (FEM), creating a numerical representation of a physical domain involves dividing it into smaller geometric shapes called finite elements, which is known as meshing. The collective assembly of these elements forms the finite element mesh. When dealing with domains that possess curved boundaries, it’s typically not possible to perfectly represent the entire geometry using standard elements with straight sides, as exemplified in Figure 1.2a where a domain with curved edges is meshed with square elements. A finer mesh, as shown in Figure 1.2b, employs a greater number of smaller square elements to more accurately approximate the curved boundary; triangular elements would provide an even better fit.

Finite element solutions converge to the exact solution of a given problem when the interpolation functions meet specific mathematical criteria detailed in Chapter 6. This convergence implies that as the number of elements increases and their size decreases, the FEM solution progressively improves, approaching the exact solution asymptotically.

To illustrate this concept, consider a tapered solid cylinder with one end fixed and the other subjected to a tensile load (depicted in Figure 1.3a). By analyzing the displacement at the loaded end, a progression of finite element models is developed: starting with a one-element approximation treating the cylinder as uniform with an average cross-sectional area (Figure 1.3b), followed by a two-element model dividing the cylinder into segments with areas corresponding to the average half-length areas (Figure 1.3c), and then refining further to a four-element model (Figure 1.3d).

The graph in Figure 1.4a demonstrates the convergence of the finite element solutions towards the exact solution (represented by the dashed line). Moreover, when examining the displacement along the length of the cylinder in Figure 1.4b, the improvement in approximation is evident, although each element uses linear interpolation leading to a linear approximation of the actual nonlinear displacement behavior.

For stress computations in structural problems, stresses are derived from the primary field variable—displacement—using appropriate stress-strain relationships. Strains and stresses are thus considered derived variables. In the tapered cylinder example, Figure 1.5 illustrates how the element stresses converge to the exact solution as the mesh is refined, showing that while the primary variable (displacement) is continuous between elements, there may be discontinuities in the derived variables (stresses and strains). These discontinuities diminish with increased mesh refinement, which is typical of the finite element method.

When the exact solution is unknown, assessing the accuracy of a finite element solution becomes challenging. Even though numerical convergence is observed, it doesn’t guarantee convergence to the correct solution. Users of finite element analysis must thoroughly scrutinize the results through several lenses: numerical convergence, physical reasonableness, adherence to the governing physical laws (e.g., structural equilibrium or energy balance), and the rationale behind any discontinuities in derived variables across element boundaries. Only after satisfactorily answering these and other relevant questions can the results of a finite element analysis be confidently accepted for use in design applications.

### 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。