# 统计代写|主成分分析代写Principal Component Analysis代考|Title: Evolution of Finite Element Method: From Trial Solutions to Localized Representations

The mathematical foundations of the finite element method (FEM) reach back at least fifty years, though approximate techniques for solving differential equations through trial solutions have an even longer history. Pioneering works by Lord Rayleigh and Ritz involved the use of trial functions (which we now consider as interpolation functions) to estimate solutions to differential equations. Galerkin’s approach also relied on the same concept; however, earlier methodologies differed from the contemporary FEM in that they required trial functions to be valid across the entire problem domain.

It wasn’t until the 1940s when Courant introduced the concept of piecewise-continuous functions within subdomains that the finite element method took its first significant stride towards the modern iteration. This allowed for localized representations of the solution space, a key innovation that distinguishes the FEM from prior approximations.

In the late 1940s, amidst the advent of the jet engine and the need for more complex analysis of aircraft structures under increased loading, engineers began developing matrix-based force analysis methods, collectively known as the flexibility method. This method inverted the traditional formulation by treating forces as unknowns and displacements as knowns. In contrast, the displacement method, widely used in FEM, focuses on finding the system displacements due to applied forces. Throughout this text, the displacement method is exclusively adhered to, where “displacement” is a broad term encompassing not only physical displacement but also other quantities like temperature and fluid velocity.

The term “finite element” was first coined by Clough in 1960 in the context of plane stress analysis, and since then, it has become standard terminology. Over the 1960s and 1970s, the finite element method saw extensive application expansion, being applied to plate bending, shell structures, pressure vessels, and general three-dimensional elasticity problems in structural analysis, as well as to fluid dynamics and heat transfer phenomena.

With advancements during this period, the FEM was extended to handle large deflections and dynamic analyses too. The evolution of the method is chronicled by Noor [16] in an extensive historical account and bibliography.

The FEM’s computational demands necessitate manipulation of large matrices, making it computationally intensive. Initially, calculations were carried out on mainframe computers—powerful by the standards of the day. A landmark development was the creation of the NASTRAN finite element software package in the 1960s in support of the U.S. space exploration program. Since NASTRAN, several other commercial finite element analysis (FEA) software packages have emerged, including ANSYS, ALGOR, and COSMOS/M, which are now capable of handling hundreds of thousands of degrees of freedom on personal desktop computers and engineering workstations.

This text does not focus on promoting any specific software package; instead, it aims to provide a fundamental understanding of finite element analysis so that readers can effectively utilize these powerful tools with an informed grasp of the underlying principles. Today, these packages can tackle large-scale problems in various fields, such as static and dynamic structural analysis, heat transfer, fluid dynamics, electromagnetics, and seismic response analysis.

### 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。