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

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## 数学代写|有限元代写Finite Element Method代考|The Galerkin method

For the choice of weight function $\psi_i$ equal to the approximation function $\phi_i$, the weighted-residual method is known as the Galerkin method. When $A$ is a linear operator, the algebraic equations of the Galerkin approximation are
$$\begin{gathered} \sum_{j=1}^N A_{i j} c_j=F_i \ A_{i j}=\int_{\Omega} \phi_i A\left(\phi_j\right) d x d y, \quad F_i=\int_{\Omega} \phi_i\left[f-A\left(\phi_0\right)\right] d x d y \end{gathered}$$
We note that $A_{i j}$ are not symmetric (i.e., $A_{i j} \neq A_{j i}$ ).
In general, the Galerkin method is not the same as the Ritz method. This should be clear from the fact that the former uses the weighted-integral form whereas the latter uses the weak form to determine the coefficients $c_j$.
Consequently, the approximation functions used in the Galerkin method are required to be of higher order than those in the Ritz method. The method that uses the weak form in which the weight function is the same as the approximation function is sometimes called the weak-form Galerkin method, but it is the same as the Ritz method. The Ritz and Galerkin methods yield the same solutions in two cases: (i) when the specified boundary conditions of the problem are all of the essential type, and therefore the requirements on $\phi_i$ in the two methods become the same and the weighted-integral form reduces to the weak form; and (ii) when the approximation functions of the Galerkin method are used in the Ritz method.

## 数学代写|有限元代写Finite Element Method代考|The least-squares method

In least-squares method, we determine the parameters $c_j$ by minimizing the integral of the square of the residual $R$ in Eq. (2.5.55):
$$\frac{\partial}{\partial c_i} \int_{\Omega} R^2\left(x, y, c_1, c_2, \cdots, c_n\right) d x d y=0$$
Or
$$\int_{\Omega} \frac{\partial R}{\partial c_i} R d x d y=0, i=1,2, \ldots, n$$
Comparison of Eq. (2.5.59) with Eq. (2.5.56) shows that $\psi_i=\partial R / \partial c_i$. If $A$ is a linear operator, we have $\psi_i=\partial R / \partial c_i=A\left(\phi_i\right)$, and Eq. (2.5.59) becomes
$$\begin{gathered} \sum_{j=1}^N\left[\int_{\Omega} A\left(\phi_i\right) A\left(\phi_j\right) d x d y\right] c_j=\int_{\Omega} A\left(\phi_i\right)\left[f-A\left(\phi_0\right)\right] d x d y \ \sum_{j=1}^N A_{i j} c_j=F_i \ A_{i j}=\int_{\Omega} A\left(\phi_i\right) A\left(\phi_j\right) d x, \quad F_i=\int_{\Omega} A\left(\phi_i\right)\left[f-A\left(\phi_0\right)\right] d x \end{gathered}$$
Note that the coefficient matrix $A_{i j}$ is symmetric whenever $A$ is a linear operator, but it involves the same order of differentiation as in the governing differential equation $A(u)-f=0$.

## 数学代写|有限元代写Finite Element Method代考|The Galerkin method

$$\begin{gathered} \sum_{j=1}^N A_{i j} c_j=F_i \ A_{i j}=\int_{\Omega} \phi_i A\left(\phi_j\right) d x d y, \quad F_i=\int_{\Omega} \phi_i\left[f-A\left(\phi_0\right)\right] d x d y \end{gathered}$$

## 数学代写|有限元代写Finite Element Method代考|The least-squares method

$$\frac{\partial}{\partial c_i} \int_{\Omega} R^2\left(x, y, c_1, c_2, \cdots, c_n\right) d x d y=0$$

$$\int_{\Omega} \frac{\partial R}{\partial c_i} R d x d y=0, i=1,2, \ldots, n$$

$$\begin{gathered} \sum_{j=1}^N\left[\int_{\Omega} A\left(\phi_i\right) A\left(\phi_j\right) d x d y\right] c_j=\int_{\Omega} A\left(\phi_i\right)\left[f-A\left(\phi_0\right)\right] d x d y \ \sum_{j=1}^N A_{i j} c_j=F_i \ A_{i j}=\int_{\Omega} A\left(\phi_i\right) A\left(\phi_j\right) d x, \quad F_i=\int_{\Omega} A\left(\phi_i\right)\left[f-A\left(\phi_0\right)\right] d x \end{gathered}$$

## 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。

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

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## 数学代写|有限元代写Finite Element Method代考|Virtual Work

The term configuration means the simultaneous positions of all material points of a body. A body with specific geometric constraints takes different configurations under different loads. The set of configurations that satisfy the geometric constraints (e.g., geometric boundary conditions) of the system is called the set of admissible configurations (i.e., every configuration in the set corresponds to the solution of the problem for a particular set of loads on the system). Of all admissible configurations only one of them corresponds to the equilibrium configuration under a set of applied loads, and it is this configuration that also satisfies Newton’s second law. The admissible configurations for a fixed set of loads can be obtained from infinitesimal variations of the true configuration (i.e., infinitesimal movement of the material points). During such variations, the geometric constraints of the system are not violated, and all applied forces are fixed at their actual equilibrium values. When a mechanical system experiences such variations in its equilibrium configuration, it is said to undergo virtual displacements. These displacements need not have any relationship with the actual displacements. The displacements are called virtual because they are imagined to take place (i.e., hypothetical) with the actual loads acting at their fixed values.

For example, consider a beam fixed at $x=0$ and subjected to any arbitrary loading (e.g., distributed as well as point loads), as shown in Fig. 2.3.6. The possible geometric configurations the beam can take under the loads may be expressed in terms of the transverse deflection $w(x)$ and axial displacement $u(x)$. The support conditions require that
$$w(0)=0, \quad\left(-\frac{d w}{d x}\right)_{x=0}=0, \quad u(0)=0$$
These are called the geometric or displacement boundary conditions. Boundary conditions that involve specifying the forces applied on the beam are called force boundary conditions.

The set of all functions $w(x)$ and $u(x)$ that satisfy the geometric boundary conditions is the set of admissible configurations for this case. This set consists of pairs of elements $\left{\left(u_i, w_i\right)\right}$ of the form
$$\begin{gathered} u_1(x)=a_1 x, \quad w_1(x)=b_1 x^2 \ u_2(x)=a_1 x+a_2 x^2, \quad w_2(x)=b_1 x^2+b_2 x^3 \end{gathered}$$
where $a_i$ and $b_i$ are arbitrary constants. The pair $(u, w)$ that also satisfies, in addition to the geometric boundary conditions, the equilibrium equations and force boundary conditions (which require the precise nature of the applied loads) of the problem is the equilibrium solution. The virtual displacements, $\delta u(x)$ and $\delta w(x)$, must be necessarily of the form
$$\delta u_1=a_1 x, \delta w_1=b_1 x^2 ; \quad \delta u_2=a_1 x+a_2 x^2, \delta w_2=b_1 x^2+b_2 x^3$$
and so on, which satisfy the homogeneous form of the specified geometric boundary conditions:
$$\delta w(0)=0, \quad\left(\frac{d \delta w}{d x}\right)_{x=0}=0, \quad \delta u(0)=0$$

## 数学代写|有限元代写Finite Element Method代考|The Principle of Virtual Displacements

Consider the system of linear algebraic equations
\begin{aligned} & b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \ & b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \ & b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3 \end{aligned}
We see that there are nine coefficients $a_{i j}, i, j=1,2,3$ relating the three coefficients $\left(b_1, b_2, b_3\right)$ to $\left(x_1, x_2, x_3\right)$. The form of these linear equations suggests writing down the coefficients $a_{i j}$ (jth components in the ith equation) in the rectangular array
$$\mathbf{A}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right]$$
This rectangular array $\mathbf{A}$ of numbers $a_{i j}$ is called a matrix, and the quantities $a_{i j}$ are called the elements of matrix $\mathbf{A}$.

If a matrix has $m$ rows and $n$ columns, we will say that is $m$ by $n(m \times n)$, the number of rows always being listed first. The element in the $i$ th row and $j$ th column of a matrix $\mathbf{A}$ is generally denoted by $a_{i j}$, and we will sometimes designate a matrix by $\mathbf{A}=[A]=\left[a_{i j}\right]$. A square matrix is one that has the same number of rows as columns. An $n \times n$ matrix is said to be of order $n$. The elements of a square matrix for which the row number and the column number are the same (that is, $a_{i i}$ for any fixed $i$ ) are called diagonal elements. A square matrix is said to be a diagonal matrix if all of the off-diagonal elements are zero. An identity matrix or its unit matrix, denoted by $\mathbf{I}=[I]$, is a diagonal matrix whose elements are all 1’s. Examples of diagonal and identity matrices are:
$$\left[\begin{array}{rrrr} 6 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & -2 \end{array}\right], \quad \mathbf{I}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
If the matrix has only one row or one column, we will normally use only a single subscript to designate its elements. For example,
$$\mathbf{X}=\left{\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right}, \quad \mathbf{Y}=\left{\begin{array}{lll} y_1 & y_2 & y_3 \end{array}\right}$$
denote a column matrix and a row matrix, respectively. Row and column matrices can be used to denote the components of a vector.

## 数学代写|有限元代写Finite Element Method代考|Virtual Work

$$w(0)=0, \quad\left(-\frac{d w}{d x}\right)_{x=0}=0, \quad u(0)=0$$

$$\begin{gathered} u_1(x)=a_1 x, \quad w_1(x)=b_1 x^2 \ u_2(x)=a_1 x+a_2 x^2, \quad w_2(x)=b_1 x^2+b_2 x^3 \end{gathered}$$

$$\delta u_1=a_1 x, \delta w_1=b_1 x^2 ; \quad \delta u_2=a_1 x+a_2 x^2, \delta w_2=b_1 x^2+b_2 x^3$$

$$\delta w(0)=0, \quad\left(\frac{d \delta w}{d x}\right)_{x=0}=0, \quad \delta u(0)=0$$

## 数学代写|有限元代写Finite Element Method代考|The Principle of Virtual Displacements



& b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \
& b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \
& b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3




\ mathbf{一}=左[开始{数组}{微光}
A_ {11} & A_ {12} & A_ {13} \
A_ {21} & A_ {22} & A_ {23} \
A_ {31} & A_ {32} & A_ {33}





6 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 3 & 0 \
0 & 0 & 0 & -2
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0

## 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。

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

## avatest™帮您通过考试

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## 数学代写|有限元代写Finite Element Method代考|Integration by parts formulae

Let $p, q, u, v$, and $w$ be sufficiently differentiable functions of the coordinate $x$. Then the following integration by parts formulae hold:
$$\int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x=-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}$$

\begin{aligned} \int_a^b v \frac{d^2}{d x^2}\left(q \frac{d^2 w}{d x^2}\right) d x= & \int_a^b q \frac{d^2 v}{d x^2} \frac{d^2 w}{d x^2} d x \ & -v(a)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=a}+v(b)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=b} \ & +\left(\frac{d v}{d x}\right){x=a}\left(q \frac{d^2 w}{d x^2}\right){x=a}-\left(\frac{d v}{d x}\right){x=b}\left(q \frac{d^2 w}{d x^2}\right){x=b} \end{aligned}
These relations can easily be established.
To establish the relation in Eq. (2.2.26), we begin with the identity
$$\frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right)=\frac{d w}{d x} p \frac{d u}{d x}+w \frac{d}{d x}\left(p \frac{d u}{d x}\right)$$
Therefore, we have
\begin{aligned} \int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x & =\int_a^b \frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right) d x-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \ & =-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|Definition of a matrix

Consider the system of linear algebraic equations
\begin{aligned} & b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \ & b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \ & b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3 \end{aligned}
We see that there are nine coefficients $a_{i j}, i, j=1,2,3$ relating the three coefficients $\left(b_1, b_2, b_3\right)$ to $\left(x_1, x_2, x_3\right)$. The form of these linear equations suggests writing down the coefficients $a_{i j}$ (jth components in the ith equation) in the rectangular array
$$\mathbf{A}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right]$$
This rectangular array $\mathbf{A}$ of numbers $a_{i j}$ is called a matrix, and the quantities $a_{i j}$ are called the elements of matrix $\mathbf{A}$.

If a matrix has $m$ rows and $n$ columns, we will say that is $m$ by $n(m \times n)$, the number of rows always being listed first. The element in the $i$ th row and $j$ th column of a matrix $\mathbf{A}$ is generally denoted by $a_{i j}$, and we will sometimes designate a matrix by $\mathbf{A}=[A]=\left[a_{i j}\right]$. A square matrix is one that has the same number of rows as columns. An $n \times n$ matrix is said to be of order $n$. The elements of a square matrix for which the row number and the column number are the same (that is, $a_{i i}$ for any fixed $i$ ) are called diagonal elements. A square matrix is said to be a diagonal matrix if all of the off-diagonal elements are zero. An identity matrix or its unit matrix, denoted by $\mathbf{I}=[I]$, is a diagonal matrix whose elements are all 1’s. Examples of diagonal and identity matrices are:
$$\left[\begin{array}{rrrr} 6 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & -2 \end{array}\right], \quad \mathbf{I}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
If the matrix has only one row or one column, we will normally use only a single subscript to designate its elements. For example,
$$\mathbf{X}=\left{\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right}, \quad \mathbf{Y}=\left{\begin{array}{lll} y_1 & y_2 & y_3 \end{array}\right}$$
denote a column matrix and a row matrix, respectively. Row and column matrices can be used to denote the components of a vector.

## 数学代写|有限元代写Finite Element Method代考|Integration by parts formulae

$$\int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x=-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}$$

\begin{aligned} \int_a^b v \frac{d^2}{d x^2}\left(q \frac{d^2 w}{d x^2}\right) d x= & \int_a^b q \frac{d^2 v}{d x^2} \frac{d^2 w}{d x^2} d x \ & -v(a)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=a}+v(b)\left[\frac{d}{d x}\left(q \frac{d^2 w}{d x^2}\right)\right]{x=b} \ & +\left(\frac{d v}{d x}\right){x=a}\left(q \frac{d^2 w}{d x^2}\right){x=a}-\left(\frac{d v}{d x}\right){x=b}\left(q \frac{d^2 w}{d x^2}\right){x=b} \end{aligned}

$$\frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right)=\frac{d w}{d x} p \frac{d u}{d x}+w \frac{d}{d x}\left(p \frac{d u}{d x}\right)$$

\begin{aligned} \int_a^b w \frac{d}{d x}\left(p \frac{d u}{d x}\right) d x & =\int_a^b \frac{d}{d x}\left(w \cdot p \frac{d u}{d x}\right) d x-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \ & =-w(a)\left(p \frac{d u}{d x}\right){x=a}+w(b)\left(p \frac{d u}{d x}\right){x=b}-\int_a^b p \frac{d w}{d x} \frac{d u}{d x} d x \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|Definition of a matrix

\begin{aligned} & b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \ & b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \ & b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3 \end{aligned}

$$\mathbf{A}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right]$$

$$\left[\begin{array}{rrrr} 6 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & -2 \end{array}\right], \quad \mathbf{I}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$

$$\mathbf{X}=\left{\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right}, \quad \mathbf{Y}=\left{\begin{array}{lll} y_1 & y_2 & y_3 \end{array}\right}$$

## 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。

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## 数学代写|有限元代写Finite Element Method代考|Summation convention

It is useful to abbreviate a summation of terms by understanding that a repeated index means summation over all values of that index. Thus the summation
$$\mathbf{A}=\sum_{i=1}^3 A_i \mathbf{e}i$$ can be shortened to $$\mathbf{A}=A_i \mathbf{e}_i$$ The repeated index is a dummy index and thus can be replaced by any other symbol that has not already been used. Thus we can also write $$\mathbf{A}=A_i \mathbf{e}_i=A_m \mathbf{e}_m$$ and so on. The “dot product” $\hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j$ and “cross product” $\hat{\mathbf{e}}_i \times \hat{\mathbf{e}}_j$ of base vectors in a righthanded system are defined by \begin{aligned} & \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j \equiv \delta{i j}= \begin{cases}0, & \text { if } i \neq j \ 1, & \text { if } i=j\end{cases} \ & \hat{\mathbf{e}}i \times \hat{\mathbf{e}}_j \equiv \varepsilon{i j k} \hat{\mathbf{e}}_k \end{aligned}

where $\delta_{i j}$ is the Kronecker delta and $\varepsilon_{i j k}$ is the alternating symbol or permutation symbol
\varepsilon_{i j k}=\left{\begin{aligned} 1, & \text { if } i, j, k \text { are in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ -1, & \text { if } i, j, k \text { are not in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ 0, & \text { if any of } i, j, k \text { are repeated. } \end{aligned}\right.
Note that in Eq. (2.2.6), $k$ is a dummy index, while $i$ and $j$ are not. The latter are called free indices. A free index can be changed to some other index only when it is changed in every expression of the equation to the same index. Thus, we can write Eq. (2.2.6) as
$$\hat{\mathbf{e}}m \times \hat{\mathbf{e}}_j=\varepsilon{m j k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_m \times \hat{\mathbf{e}}_n=\varepsilon{m n k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_p \times \hat{\mathbf{e}}_q=\varepsilon{p q k} \hat{\mathbf{e}}_k$$

## 数学代写|有限元代写Finite Element Method代考|The del operator

Differentiation of vector functions with respect to the coordinates is common in science and engineering. Most of the operations involve the “del operator”, denoted by $\nabla$. In a rectangular Cartesian system it has the form
$$\nabla \equiv \hat{\mathbf{e}}_x \frac{\partial}{\partial x}+\hat{\mathbf{e}}_y \frac{\partial}{\partial y}+\hat{\mathbf{e}}_z \frac{\partial}{\partial z}$$
It is important to note that the del operator has some of the properties of a vector but it does not have them all, because it is an operator. The operation $\nabla \phi(\mathbf{x})$ is called the gradient of a scalar function $\phi$ whereas $\nabla \times \mathbf{A}(\mathbf{x})$ is called the curl of a vector function $\mathbf{A}$. The operator $\nabla^2 \equiv \nabla \cdot \nabla$ is called the Laplace operator. In a 3-D rectangular Cartesian coordinate system it has the form
$$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$
We have the following relations between the rectangular Cartesian coordinates $(x, y, z)$ and cylindrical coordinates $(r, \theta, z)$ (see Fig. 2.2.2):

$$x=r \cos \theta, y=r \sin \theta, z=z$$
The base vectors in the two coordinate systems are related by
$$\hat{\mathbf{e}}r=\cos \theta \hat{\mathbf{e}}_x+\sin \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}\theta=-\sin \theta \hat{\mathbf{e}}x+\cos \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}_z=\hat{\mathbf{e}}_z$$ Note that the base vectors of the cylindrical coordinate system are not constant; the direction of $\theta$ and $r$-coordinates change as we move around the cylindrical surface. Thus, we have $$\frac{\partial \hat{\mathbf{e}}_r}{\partial \theta}=-\sin \theta \hat{\mathbf{e}}_x+\cos \theta \hat{\mathbf{e}}_y=\hat{\mathbf{e}}\theta, \frac{\partial \hat{\mathbf{e}}\theta}{\partial \theta}=-\cos \theta \hat{\mathbf{e}}_x-\sin \theta \hat{\mathbf{e}}_y=-\hat{\mathbf{e}}_r$$ and all other derivatives of the base vectors are zero. The operators $\nabla$ and $\nabla^2$ in the cylindrical coordinate system are given by (see Reddy $[2,3]$ ) $$\boldsymbol{\nabla}=\hat{\mathrm{e}}_r \frac{\partial}{\partial r}+\frac{1}{r} \hat{\mathrm{e}}\theta \frac{\partial}{\partial \theta}+\hat{\mathrm{e}}_z \frac{\partial}{\partial z}, \quad \nabla^2=\frac{1}{r}\left[\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r} \frac{\partial^2}{\partial \theta^2}+r \frac{\partial^2}{\partial z^2}\right]$$

## 数学代写|有限元代写Finite Element Method代考|Summation convention

$$\mathbf{A}=\sum_{i=1}^3 A_i \mathbf{e}i$$可以缩写为$$\mathbf{A}=A_i \mathbf{e}_i$$重复索引是一个虚拟索引，因此可以用任何其他尚未使用过的符号代替。因此，我们也可以写$$\mathbf{A}=A_i \mathbf{e}_i=A_m \mathbf{e}_m$$等等。右手坐标系中基向量的“点积”$\hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j$和“叉积”$\hat{\mathbf{e}}_i \times \hat{\mathbf{e}}_j$定义为 \begin{aligned} & \hat{\mathbf{e}}_i \cdot \hat{\mathbf{e}}_j \equiv \delta{i j}= \begin{cases}0, & \text { if } i \neq j \ 1, & \text { if } i=j\end{cases} \ & \hat{\mathbf{e}}i \times \hat{\mathbf{e}}_j \equiv \varepsilon{i j k} \hat{\mathbf{e}}_k \end{aligned}

\varepsilon_{i j k}=\left{\begin{aligned} 1, & \text { if } i, j, k \text { are in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ -1, & \text { if } i, j, k \text { are not in cyclic order } \ & \text { and not repeated }(i \neq j \neq k), \ 0, & \text { if any of } i, j, k \text { are repeated. } \end{aligned}\right.

$$\hat{\mathbf{e}}m \times \hat{\mathbf{e}}_j=\varepsilon{m j k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_m \times \hat{\mathbf{e}}_n=\varepsilon{m n k} \hat{\mathbf{e}}k ; \quad \hat{\mathbf{e}}_p \times \hat{\mathbf{e}}_q=\varepsilon{p q k} \hat{\mathbf{e}}_k$$

## 数学代写|有限元代写Finite Element Method代考|The del operator

$$\nabla \equiv \hat{\mathbf{e}}_x \frac{\partial}{\partial x}+\hat{\mathbf{e}}_y \frac{\partial}{\partial y}+\hat{\mathbf{e}}_z \frac{\partial}{\partial z}$$

$$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

$$x=r \cos \theta, y=r \sin \theta, z=z$$

$$\hat{\mathbf{e}}r=\cos \theta \hat{\mathbf{e}}_x+\sin \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}\theta=-\sin \theta \hat{\mathbf{e}}x+\cos \theta \hat{\mathbf{e}}_y, \quad \hat{\mathbf{e}}_z=\hat{\mathbf{e}}_z$$注意圆柱坐标系的基向量不是恒定的;当我们绕圆柱面移动时，$\theta$和$r$坐标的方向会改变。因此，我们有$$\frac{\partial \hat{\mathbf{e}}_r}{\partial \theta}=-\sin \theta \hat{\mathbf{e}}_x+\cos \theta \hat{\mathbf{e}}_y=\hat{\mathbf{e}}\theta, \frac{\partial \hat{\mathbf{e}}\theta}{\partial \theta}=-\cos \theta \hat{\mathbf{e}}_x-\sin \theta \hat{\mathbf{e}}_y=-\hat{\mathbf{e}}_r$$所有基向量的导数都是零。柱坐标系中的算子$\nabla$和$\nabla^2$由(参见Reddy $[2,3]$)给出。 $$\boldsymbol{\nabla}=\hat{\mathrm{e}}_r \frac{\partial}{\partial r}+\frac{1}{r} \hat{\mathrm{e}}\theta \frac{\partial}{\partial \theta}+\hat{\mathrm{e}}_z \frac{\partial}{\partial z}, \quad \nabla^2=\frac{1}{r}\left[\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r} \frac{\partial^2}{\partial \theta^2}+r \frac{\partial^2}{\partial z^2}\right]$$

## 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。

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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。

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## 数学代写|有限元代写Finite Element Method代考|Solution Process

The significance of the information provided in the above input file is very similar to the previous case study. Therefore, this section will highlight the differences that are mainly used for the transient analysis.

The definition of amplitude curve is important here as it enables the load (or boundary condition) to be defined as a function of time here. In this case the load will follow the sinusoidal function defined in the amplitude curve block. The sinusoidal function is defined as a periodic function whereby the formula used is actually the Fourier series. The data lines in the amplitude curve block basically define the angular frequency and the other constants in the Fourier series.

The control card specifies that the analysis is a direct integration, transient analysis. In ABAQUS, Newmarks’s method (Section 3.7.2) together with the Hilber-Hughes-Taylor operator [1978] applied on the equilibrium equations is used as the implicit solver for direct integration analysis. The time increment is specified to be $0.1 \mathrm{~s}$, and the total time of the step is $1.0 \mathrm{~s}$. As mentioned in Chapter 3, implicit methods involve solving of the matrix equation at each individual increment in time, therefore the analysis can be rather computationally expensive. The algorithm used by ABAQUS is quite complex, involving the capabilities of having automatic deduction of the required time increments. Details are beyond the scope of this book.

## 数学代写|有限元代写Finite Element Method代考|Result and Discussion

Upon the analysis of the problem defined by the input file above, the displacement, velocity and acceleration components throughout each individual time increment can be obtained until the final time step specified. Therefore, we have what is known as the displacementtime history, the velocity-time history and the acceleration-time history, as shown in Figures $8.16,8.17$ and 8.18 , respectively. The plots show the displacement, velocity and acceleration histories of nodes 210 and 300 .

A three-dimensional (3D) solid element can be considered to be the most general of all solid finite elements because all the field variables are dependent of $x, y$ and $z$. An example of a 3D solid structure under loading is shown in Figure 9.1. As can be seen, the force vectors here can be in any arbitrary direction in space. A 3D solid can also have any arbitrary shape, material properties and boundary conditions in space. As such, there are altogether six possible stress components, three normal and three shear, that need to be taken into consideration. Typically, a 3D solid element can be a tetrahedron or hexahedron in shape with either flat or curved surfaces. Each node of the element will have three translational degrees of freedom. The element can thus deform in all three directions in space.

## 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。

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## 数学代写|有限元代写Finite Element Method代考|Solution Process

Looking at the mesh in Figure 8.6 , one can see that quadrilateral shell elements are used. Therefore, the equations for a linear, quadrilateral shell element must be formulated by ABAQUS. As before, the formulation of the element matrices would require information from the nodal cards and the element connectivity cards. The element type used here is S4, representing four nodal shell elements. There are other types of shell elements available in the ABAQUS element library.

After the nodal and element cards, next to be considered would be the property and material cards. The properties for the shell element used here must be defined, which in this case includes the material used and the thickness of the shell elements. The material cards are similar to those of the case study in Chapter 7 except that here the density of the material must be included, since we are not carrying out a static analysis as in Chapter 7.
The boundary $(\mathrm{BC})$ cards then define the boundary conditions on the model. In this problem, we would like to obtain only the flexural vibration modes of the motor, hence the components of displacements in the plane of the motor are not actually required. As mentioned, this is very much the characteristic of the plate elements. Therefore, DOFs 1, 2 and 6 corresponding to the $x$ and $y$ displacements, and rotation about the $z$ axis, is constrained. The other boundary condition would be the constraining of the displacements of the nodes at the centre of the motor.

Without the need to define any external loadings, the control cards then define the type of analysis ABAQUS would carry out. ABAQUS uses the sub-space iteration scheme by default to evaluate the eigenvalues of the equation of motion. This method is a very effective method of determining a number of lowest eigenvalues and corresponding eigenvectors for a very large system of several thousand DOFs. The procedure is outlined in the case study in Chapter 5. Finally, the output control cards define the necessary output required by the analyst.

## 数学代写|有限元代写Finite Element Method代考|Result and Discussion

Using the input file above, an eigenvalue extraction is carried out in ABAQUS. The output is extracted from the ABAQUS results file showing the first eight natural frequencies and tabulated in Table 8.1. The table also shows results obtained from using triangular elements as well as a finer mesh of quadrilateral elements. It is interesting to note that for certain modes, the eigenvalues and hence the frequencies are repetitive with the previous one. This is due to the symmetry of the circular rotor structure. For example, modes 1 and 2 have the same frequency, and looking at their corresponding mode shapes in Figures 8.7 and 8.8, respectively, one would notice that they are actually of the same shape but bending at a plane $90^{\circ}$ from each other. As such, many consider this as one single mode. Therefore, though eight eigenmodes are extracted, it is effectively equivalent to only five eigenmodes However, to be consistent with the result file from ABAQUS, all the modes extracted will be shown here. Figure 8.9 to 8.14 show the other mode shapes from this analysis. Remember that, since the in-plane displacements are already constrained, these modes are only the flexural modes of the rotor.

Comparing the natural frequencies obtained using 768 triangular elements with those obtained using the quadrilateral elements, one can see that the frequencies are generally higher using the triangular elements. Note that for the same number of nodes, using the quadrilateral elements requires half the number of elements. The results obtained using 384 quadrilateral elements do not differ much from those that use 1280 elements. This again

## 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。

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## 数学代写|有限元代写Finite Element Method代考|Element Matrices

To obtain the stiffness matrix $\mathbf{k}e$, we substitute Eq. (8.15) into Eq. (8.6), from which we obtain $$\mathbf{k}_e=\int{A_e} \frac{h^3}{12}\left[\mathbf{B}^{\mathrm{I}}\right]^{\mathrm{T}} \mathbf{c} \mathbf{B}^{\mathrm{I}} \mathrm{d} A+\int_{A_e} \kappa h\left[\mathbf{B}^{\mathrm{O}}\right]^{\mathrm{T}} \mathbf{c}s \mathbf{B}^{\mathrm{O}} \mathrm{d} A$$ The first term in the above equation represents the strain energy associated with the in-plane stress and strains. The strain matrix $\mathbf{B}^{\mathrm{I}}$ has the form of $$\mathbf{B}^{\mathrm{I}}=\left[\begin{array}{llll} \mathbf{B}_1^{\mathrm{I}} & \mathbf{B}_2^{\mathrm{I}} & \mathbf{B}_3^{\mathrm{I}} & \mathbf{B}_4^{\mathrm{I}} \end{array}\right]$$ where $$\mathbf{B}{\mathrm{j}}^{\mathrm{I}}=\left[\begin{array}{ccc} 0 & 0 & -\partial N_j / \partial x \ 0 & \partial N_j / \partial y & 0 \ 0 & \partial N_j / \partial x & -\partial N_j / \partial y \end{array}\right]$$
Using the expression for the shape functions in Eq. (8.14), we obtain
\begin{aligned} & \frac{\partial N_j}{\partial x}=\frac{\partial N_j}{\partial \xi} \frac{\partial \xi}{\partial x}=\frac{1}{4 a} \xi_i\left(1+\eta_i \eta\right) \ & \frac{\partial N_j}{\partial y}=\frac{\partial N_j}{\partial \eta} \frac{\partial \eta}{\partial y}=\frac{1}{4 b}\left(1+\xi_i \xi\right) \eta_i \end{aligned}
In deriving Eq. (8.23), the relationship $\xi=x / a, \eta=y / b$ has been employed.

## 数学代写|有限元代写Finite Element Method代考|Higher Order Elements

For an eight-node rectangular thick plate element, the deflection and rotations can be summed as
$$w=\sum_{i=1}^8 N_i w_i, \quad \theta_x=\sum_{i=1}^8 N_i \theta_{x_i}, \quad \theta_y=\sum_{i=1}^8 N_i \theta_{y_i}$$
where the shape function $N_i$ is the same as the eight-node $2 \mathrm{D}$ solid element given by Eq. (7.52). The element constructed will be a conforming element, as $w, \theta_x$ and $\theta_y$ are continuous on the edges between elements. The formulation procedure is the same as for the rectangular plate elements.

## 数学代写|有限元代写Finite Element Method代考|Element Matrices

$$\mathbf{k}e=\int A_e \frac{h^3}{12}\left[\mathbf{B}^{\mathrm{I}}\right]^{\mathrm{T}} \mathbf{c} \mathbf{B}^{\mathrm{I}} \mathrm{d} A+\int{A_e} \kappa h\left[\mathbf{B}^{\mathrm{O}}\right]^{\mathrm{T}} \mathbf{c} s \mathbf{B}^{\mathrm{O}} \mathrm{d} A$$

$$\mathbf{B}^{\mathrm{I}}=\left[\begin{array}{llll} \mathbf{B}_1^{\mathrm{I}} & \mathbf{B}_2^{\mathrm{I}} & \mathbf{B}_3^{\mathrm{I}} & \mathbf{B}_4^{\mathrm{I}} \end{array}\right]$$

$$\frac{\partial N_j}{\partial x}=\frac{\partial N_j}{\partial \xi} \frac{\partial \xi}{\partial x}=\frac{1}{4 a} \xi_i\left(1+\eta_i \eta\right) \quad \frac{\partial N_j}{\partial y}=\frac{\partial N_j}{\partial \eta} \frac{\partial \eta}{\partial y}=\frac{1}{4 b}\left(1+\xi_i \xi\right) \eta_i$$

## 数学代写|有限元代写Finite Element Method代考|Higher Order Elements

$$w=\sum_{i=1}^8 N_i w_i, \quad \theta_x=\sum_{i=1}^8 N_i \theta_{x_i}, \quad \theta_y=\sum_{i=1}^8 N_i \theta_{y_i}$$

## 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。

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## 数学代写|有限元代写Finite Element Method代考|Serendipity type elements

The method used in constructing the Lagrange type of elements is very systematic. However, the Lagrange type of elements is not very widely used, due to the presence of the interior nodes. Serendipity type elements are created by inspective construction methods. We intentionally construct high order elements without interior nodes.

Consider the eight-node element shown in Figure 7.17a. The element has four corner nodes and four mid-side nodes. The shape functions in the natural coordinates for the quadratic rectangular element are given as
$$\begin{array}{ll} N_j=\frac{1}{4}\left(1+\xi_j \xi\right)\left(1+\eta_j \eta\right)\left(\xi_j \xi+\eta_j \eta-1\right) \quad \text { for corner nodes } j=1,2,3,4 \ N_j=\frac{1}{2}\left(1-\xi^2\right)\left(1+\eta_j \eta\right) & \text { for mid-side nodes } j=5,7 \ N_j=\frac{1}{2}\left(1+\xi_j \xi\right)\left(1-\eta^2\right) & \text { for mid-side nodes } j=6,8 \end{array}$$
where $\left(\xi_j, \eta_j\right)$ are the natural coordinates of node $j$. It is very easy to observe that the shape functions possess the delta function property. The shape function is constructed by simple inspections making use of the shape function properties. For example, for the corner node 1 (where $\xi_1=-1, \eta_1=-1$ ), the shape function $N_1$ has to pass the following three lines as shown in Figure 7.18 to ensure its vanishing at remote nodes:
\begin{aligned} 1-\xi=0 \Rightarrow \text { vanishes at nodes } 2,6,3 \ 1-\eta=0 \Rightarrow \text { vanishes at nodes } 3,4,7 \ -\xi-\eta-1=0 \Rightarrow \text { vanishes at nodes } 5,8 \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|ELEMENTS WITH CURVED EDGES

Using high order elements, elements with curved edges can be used in the modelling. Two relatively frequently used higher order elements of curved edges are shown in Figure 7.20(a). In formulating these types of elements, the same mapping technique used for the linear quadrilateral elements (Section 7.4) can be used. In the physical coordinate system, elements with curved edges, as shown in Figure 7.20 (a), are first formed in the problem domain. These elements are then mapped into the natural coordinate system using Eq. (7.67). The elements mapped in the natural coordinate system will have straight edges, as shown in Figure $7.20(\mathrm{~b})$

Higher order elements of curved edges are often used for modelling curved boundaries. Note that elements with excessively curved edges may cause problems in the numerical integration. Therefore, more elements should be used where the curvature of the boundary is large. In addition, it is recommended that in the internal portion of the domain, elements with straight edges should be used whenever possible. More details on modelling issues will be discussed intensively in Chapter 11 .

## 数学代写|有限元代写Finite Element Method代考|Serendipity type elements

$N_j=\frac{1}{4}\left(1+\xi_j \xi\right)\left(1+\eta_j \eta\right)\left(\xi_j \xi+\eta_j \eta-1\right) \quad$ for corner nodes $j=1,2,3,4 N_j=\frac{1}{2}\left(1-\xi^2\right)\left(1+\eta_j \eta\right) \quad$ for mid-side nodes

$1-\xi=0 \Rightarrow$ vanishes at nodes $2,6,31-\eta=0 \Rightarrow$ vanishes at nodes $3,4,7-\xi-\eta-1=0 \Rightarrow$ vanishes at nodes 5,8

## 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。

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## 数学代写|有限元代写Finite Element Method代考|Gauss Integration

Consider first a one-dimensional integral. Using the Gauss integration scheme, the integral is evaluated simply by a summation of the integrand evaluated at $m$ Gauss points multiplied by corresponding weight coefficients as follows:
$$I=\int_{-1}^{+1} f(\xi) \mathrm{d} \xi=\sum_{j=1}^m w_j f\left(\xi_j\right)$$
The locations of the Gauss points and the weight coefficients have been found for different $m$, and are given in Table 7.1. In general, the use of more Gauss points will produce more accurate results for the integration. However, excessive use of Gauss points will increase the computational time and use up more computational resources, and it may not necessarily give better results. The appropriate number of Gauss points to be used depends upon the complexity of the integrand. It has been proven that the use of $m$ Gauss points gives the exact results of a polynomial integrand of up to an order of $n=2 m-1$. For example, if the integrand is a linear function (straight line), we have $2 m-1=1$, which gives $m=1$. This means that for a linear integrand, one Gauss point will be sufficient to give the exact result of the integration. If the integrand is of a polynomial of a third order, we have $2 m-1=3$, which gives $m=2$. This means that for an integrand of a third order polynomial, the use of two Gauss points will be sufficient to give the exact result. The use of more than two points will still give the same results, but takes more computation time. For two-dimensional integrations, the Gauss integration is sampled in two directions, as follows:
$$I=\int_{-1}^{+1} \int_{-1}^{+1} f(\xi, \eta) \mathrm{d} \xi \mathrm{d} \eta=\sum_{i=1}^{n_x} \sum_{j=1}^{n_y} w_i w_j f\left(\xi_i, \eta_j\right)$$
Figure 7.9(b) shows the locations of four Gauss points used for integration in a square region.

## 数学代写|有限元代写Finite Element Method代考|Coordinate Mapping

Figure 7.10 shows a 2D domain with the shape of an airplane wing. As you can imagine, dividing such a domain into rectangular elements of parallel edges is impossible. The job can be easily accomplished by the use of quadrilateral elements with four straight but unparallel edges, as shown in Figure 7.10. In developing the quadrilateral elements, we use the same coordinate mapping that was used for the rectangular elements in the previous section. Due to the slightly increased complexity of the element shape, the mapping will become a little more involved, but the procedure is basically the same.

Consider now a quadrilateral element with four nodes numbered 1, 2, 3 and 4 in a counter-clockwise direction, as shown in Figure 7.11. The coordinates for the four nodes are indicated in Figure 7.11(a) in the physical coordinate system. The physical coordinate system can be the same as the global coordinate system for the entire structure. As there are two DOFs at a node, a linear quadrilateral element has a total of eight DOFs, like the rectangular element. A local natural coordinate system $(\xi, \eta)$ with its origin at the centre of the squared element mapped from the global coordinate system is used to construct the shape functions, and the displacement is interpolated using the equation
$$\mathbf{U}^h(\xi, \eta)=\mathbf{N}(\xi, \eta) \mathbf{d}_e$$

## 数学代写|有限元代写Finite Element Method代考|Gauss Integration

$$I=\int_{-1}^{+1} f(\xi) \mathrm{d} \xi=\sum_{j=1}^m w_j f\left(\xi_j\right)$$

，这使 $m=2$. 这意味着对于三阶多项式的被积函数，使用两个高斯点就足以给出准确的结果。使用两个以上 的点仍会给出相同的结果，但需要更多的计算时间。对于二维积分，高斯积分在两个方向上采样，如下:
$$I=\int_{-1}^{+1} \int_{-1}^{+1} f(\xi, \eta) \mathrm{d} \xi \mathrm{d} \eta=\sum_{i=1}^{n_x} \sum_{j=1}^{n_y} w_i w_j f\left(\xi_i, \eta_j\right)$$

## 数学代写|有限元代写Finite Element Method代考|Coordinate Mapping

$$\mathbf{U}^h(\xi, \eta)=\mathbf{N}(\xi, \eta) \mathbf{d}_e$$

## 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。