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# 数学代写|有限元方法代写finite differences method代考|A Brief Review of the History of the Finite Element Method

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## 数学代写|有限元代写Finite Element Method代考|A Brief Review of the History of the Finite Element Method

The idea of representing a given domain as a collection of discrete parts is not unique to the finite element method. It was recorded that ancient mathematicians estimated the value of $\pi$ by noting that the perimeter of a polygon inscribed in a circle approximates the circumference of the latter. They predicted the value of $\pi$ to accuracy of almost 40 significant digits by representing the circle as a polygon of a finitely large number of sides (see Reddy $[5,6])$. In modern times, the idea first found a home in structural analysis, where, for example, wings and fuselages are treated as assemblages of stringers, skins, and shear panels. In 1941, Hrenikoff [7] introduced the socalled framework method, in which a plane elastic medium was represented as a collection of bars and beams. The use of piecewise continuous functions defined over a subdomain to approximate an unknown function can be found in the work of Courant [8], who used an assemblage of triangular elements and the principle of minimum total potential energy to study the St. Venant torsion problem. Although certain key features of the finite element method can be found in the works of Hrenikoff [7] and Courant [8], its formal presentation is attributed to Argyris and Kelsey [9] and Turner et. al. [10]. The term “finite element” was first used by Clough [11]. Since its inception, the literature on finite element applications has grown exponentially, and today there are numerous books and journals that are primarily devoted to the theory and application of the method. Additional information on the history of the finite element method can be found in [12-16].

In recent years, extensions and modifications of the finite element method have been proposed. These include the partition of unity method (PUM) of Melenk and Babuska [17], the $h-p$ cloud method of Duarte and Oden [18], meshless methods advanced by Belytschko and his colleagues [19], and generalized finite element method (GFEM) detailed by Babuska and Strouboulis [20]. All of these methods and numerous other methods not named here are very closely related to the original idea.

## 数学代写|有限元代写Finite Element Method代考|The Present Study

This book deals with an introduction to the finite element method and its application to linear problems in engineering and applied sciences. Most introductory finite element textbooks written for use in engineering schools are intended for students of solid and structural mechanics, and these introduce the method as an offspring of matrix methods of structural analysis. A few texts that treat the method as a variationally based technique leave the variational formulations and the associated methods of approximation to an appendix. The approach taken in this book is one in which the finite element method is introduced as a numerical technique of solving classes of problems, each class having a common mathematical structure in the form of governing differential equations. This approach makes the reader understand the generality of the finite element method, irrespective of the reader’s subject background. It also enables the reader to see the mathematical structure common to various physical problems, and thereby to gain additional insights into various engineering problems. Review of engineering problems that are governed by each class of equations will receive significant attention because the review helps the reader to understand the connection between the continuum problem and its discrete model.

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。