Posted on Categories:Finite Element Method, 数学代写, 有限元

# 数学代写|有限元方法代写finite differences method代考|Derivation of Element Equations: Finite Element Model

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|有限元代写Finite Element Method代考|Derivation of Element Equations: Finite Element Model

In nature, all systems exhibit certain dualities in their behavior or response. For example, a force on a system induces displacement, while heat input to a system elevates its temperature. We call that force and displacements are dual to each other and heat and temperature are dual to each other. This is also referred to as the cause and effect. One element of the pair may be called the primary variable and the other the secondary variable; although the choice of the name given to each variable is arbitrary, the dualities are unique (i.e., if one element is dual to another element, these elements do not appear in other duality pairs again). In this book, we shall call the displacements as the primary variables and the corresponding forces as the secondary variables. Similarly, temperature will be labelled as the primary variable and heat as the secondary variable. Mathematical representations of the relationships between primary and secondary variables are in the form of algebraic, differential, or integral equations, and they are derived with the aid of the laws of physics and constitutive relations. The finite element method is a technique of developing algebraic relations among the nodal values of the primary and secondary variables.
In most cases, the relationships between primary and secondary variables are in the form of differential equations. The objective of any numerical method is to convert these relationships to algebraic form so that one can determine the system response (e.g., force or displacement) associated with a given input to the system. The algebraic relationships for a typical element of a system, called finite element equations or finite element model, can be derived directly (i.e., without going through the differential relationships), in some simple cases, using the underlying physical principles (see Section 3.3). In all continuous systems, the differential equations can be used to derive the algebraic relationships between the primary and secondary variables. In the next section, we shall discuss the derivations of the element equations for discrete systems by a direct or physical approach. Assembly of element equations, imposition of boundary conditions, and solution of algebraic equations for nodal unknowns are presented. In Section 3.4, we systematically develop, starting with a representative differential equation, finite element equations of continuous systems. The reader must have a good background in basic engineering subjects to understand the physical approach and appreciate the application of the general approach presented in Section 3.4 and in the subsequent chapters.

## 数学代写|有限元代写Finite Element Method代考|Linear Elastic Spring

A linear elastic spring is a discrete element (i.e., not a continuum and not governed by a differential equation), as shown in Fig. 3.3.1(a). The loaddisplacement relationship of a linear elastic spring can be expressed as where $F$ is the force $(\mathrm{N})$ in the spring, $\delta$ is the elongation $(\mathrm{m})$ of the spring, and $k$ is a constant, known as the spring constant $(\mathrm{N} / \mathrm{m})$. The spring constant depends on the elastic modulus, area of cross section, and number of turns in the coil of the spring. Often a spring is used to characterize the elastic behavior of complex physical systems.
A relationship between the end forces $\left(F_1^e, F_2^e\right)$ and end displacements $\left(u_1^e, u_2^e\right)$ of a typical spring element $e$ shown in Fig. 3.3.1(b) can be developed with the help of the relation in Eq. (3.3.1). We note that all forces and displacements are taken positive to the right. The force $F_1^e$ at node 1 is (compressive and) equal to the spring constant multiplied by the relative displacement of node 1 with respect to node 2 , that is, $u_1^e-u_2^e$ :
$$F_1^e=k_e\left(u_1^e-u_2^e\right)=k_e u_1^e-k_e u_2^e$$
Similarly, the force at node 2 is (tensile and) equal to elongation $u_2^e-u_1^e$ multiplied by $k_e$ :
$$F_2^e=k_e\left(u_2^e-u_1^e\right)=-k_e u_1^e+k_e u_2^e$$
Note that the force equilibrium, $F_2^1+F_1^2+F_1^3$, is automatically satisfied by the above relations. These equations can be written in matrix form as
$$k_e\left[\begin{array}{rr} 1 & -1 \ -1 & 1 \end{array}\right]\left{\begin{array}{l} u_1^e \ u_2^e \end{array}\right}=\left{\begin{array}{l} F_1^e \ F_2^e \end{array}\right} \text { or } \mathbf{K}^e \mathbf{u}^e=\mathbf{F}^e$$
Equation (3.3.2) is applicable to any spring element whose forcedisplacement relation is linear. Thus a typical spring in a network of springs of different spring constants obeys Eq. (3.3.2). The coefficient matrix $\mathbf{K}^e$ is termed stiffness matrix, $\mathbf{u}^e$ is the vector of displacements, and $\mathbf{F}^e$ is the force vector. We note that Eq. (3.3.2) is valid for any linear elastic spring, and it represents a relationship between point forces and displacements along the length of the spring. The end points are called element nodes and $F_i^e$ and $u_i^e$ are the nodal force and displacement, respectively, of the ith node. We also note that a spring element can only take loads and experience displacements along its length. We consider an example of application of Eq. (3.3.2).

## 数学代写|有限元代写Finite Element Method代考|Linear Elastic Spring

$$F_1^e=k_e\left(u_1^e-u_2^e\right)=k_e u_1^e-k_e u_2^e$$

$$F_2^e=k_e\left(u_2^e-u_1^e\right)=-k_e u_1^e+k_e u_2^e$$

$$k_e\left[\begin{array}{rr} 1 & -1 \ -1 & 1 \end{array}\right]\left{\begin{array}{l} u_1^e \ u_2^e \end{array}\right}=\left{\begin{array}{l} F_1^e \ F_2^e \end{array}\right} \text { or } \mathbf{K}^e \mathbf{u}^e=\mathbf{F}^e$$

## 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。