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## 数学代写|线性代数代写Linear algebra代考|The Algebra of Projections

If $\rho$ is a projection, then so is $\iota-\rho$, where $\iota$ is the identity operator on $\mathrm{V}$, for we have
$$(\iota-\rho)^2=\iota^2-\iota \rho-\rho \iota+\rho^2=\iota-\rho$$
It is not hard to see that $\operatorname{ker}(\iota-\rho)=i m(\rho)$ and $i m(\iota-\rho)=\operatorname{ker}(\rho)$. Hence, if $\rho$ is projection on $\mathrm{S}$ along $\mathrm{S}^{\mathrm{c}}$, then $\iota-\rho$ is projection on $\mathrm{S}^{\mathrm{c}}$ along $\mathrm{S}$.

Definition Two projections $\rho, \sigma \in \mathcal{L}(\mathrm{V})$ are orthogonal, written $\rho \perp \sigma$, if $\rho \sigma=\sigma \rho=0$.
Note that $\rho \perp \sigma$ if and only if
$$i m(\rho) \subset \operatorname{ker}(\sigma) \text { and } i m(\sigma) \subset \operatorname{ker}(\rho)$$
The following example shows that it is not enough to have $\rho \sigma=0$ in the definition of orthogonality, since it is possible for $\rho \sigma=0$ and yet $\sigma \rho$ may not even be a projection.

## 数学代写|线性代数代写Linear algebra代考|Resolutions of the Identity

If $\rho$ is a projection, then
$$\rho \perp(\iota-\rho) \text { and } \rho+(\iota-\rho)=\iota$$
Let us generalize this to more than two projections.
Definition If $\rho_1, \ldots, \rho_{\mathrm{k}}$ are projections for which
1) $\rho_{\mathrm{i}} \perp \rho_{\mathrm{j}}$ for $\mathrm{i} \neq \mathrm{j}$
2) $\rho_1+\cdots+\rho_{\mathrm{k}}=\iota$
then we refer to the sum in (2) as a resolution of the identity. [
The next theorem displays a correspondence between direct sum decompositions of $\mathrm{V}$ and resolutions of the identity.

Theorem 8.17
1) If $\rho_1+\cdots+\rho_{\mathbf{k}}=\iota$ is a resolution of the identity, then
$$\mathrm{V}=i m\left(\rho_1\right) \oplus \cdots \oplus i m\left(\rho_{\mathbf{k}}\right)$$
2) Conversely, if $\mathrm{V}=\mathrm{S}1 \oplus \cdots \oplus \mathrm{S}{\mathrm{k}}$, and $\rho_{\mathrm{i}}$ is projection on $\mathrm{S}{\mathrm{i}}$ along $\mathrm{S}_1 \oplus \cdots \oplus \widehat{S}{\mathrm{i}} \oplus \cdots \oplus \mathrm{S}{\mathrm{k}}$, where the hat ” means that the corresponding term is missing from the direct sum. Then $$\rho_1+\cdots+\rho{\mathbf{k}}=\iota$$
is a resolution of the identity.

## 数学代写|线性代数代写Linear algebra代考|The Algebra of Projections

$$(\iota-\rho)^2=\iota^2-\iota \rho-\rho \iota+\rho^2=\iota-\rho$$

$$\operatorname{im}(\rho) \subset \operatorname{ker}(\sigma) \text { and } \operatorname{im}(\sigma) \subset \operatorname{ker}(\rho)$$

## 数学代写|线性代数代写Linear algebra代考|Resolutions of the Identity

$\rho \perp(\iota-\rho)$ and $\rho+(\iota-\rho)=\iota$

2) $\rho_1+\cdots+\rho_{\mathrm{k}}=\iota$

1) 如果 $\rho_1+\cdots+\rho_{\mathbf{k}}=\iota$ 是恒等式的解析，那么
$$\mathrm{V}=i m\left(\rho_1\right) \oplus \cdots \oplus i m\left(\rho_{\mathbf{k}}\right)$$
2) 相反，如果 $\mathrm{V}=\mathrm{S} 1 \oplus \cdots \oplus \mathrm{Sk}$ ，和 $\rho_{\mathrm{i}}$ 是投影 $\mathrm{Si}$ 沿着 $\mathrm{S}_1 \oplus \cdots \oplus \widehat{S} \mathrm{i} \oplus \cdots \oplus \mathrm{Sk}$ ，其中“帽子” 表示直和中缺少相应的项。然后
$$\rho_1+\cdots+\rho \mathbf{k}=\iota$$

## 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。

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## 数学代写|有限元代写Finite Element Method代考|Field Variable Interpolation

Consider now a triangular element of thickness $h$. The nodes of the element are numbered 1 , 2 and 3 counter-clockwise, as shown in Figure 7.4. For 2D solid elements, the field variable is the displacement, which has two components $(u$ and $v)$, and hence each node has two Degrees Of Freedom (DOFs). Since a linear triangular element has three nodes, the total number of DOFs of a linear triangular element is six. For the triangular element, the local coordinate of each element can be taken as the same as the global coordinate, since there are no advantages in specifying a different local coordinate system for each element.

Now, let us examine how a triangular element can be formulated. The displacement $\mathbf{U}$ is generally a function of the coordinates $x$ and $y$, and we express the displacement at any point in the element using the displacements at the nodes and shape functions. It is therefore assumed that (see Section 3.4.2)
$$\mathbf{U}^h(x, y)=\mathbf{N}(x, y) \mathbf{d}_e$$
where the superscript $h$ indicates that the displacement is approximated, and $\mathbf{d}_e$ is a vector of the nodal displacements arranged in the order of
$$\mathbf{d}_e=\left{\begin{array}{l} u_1 \ v_1 \ u_2 \ v_2 \ u_3 \ v_3 \end{array}\right} \text { displacements at node } 1$$

and the matrix of shape functions $\mathbf{N}$ is arranged as
in which $N_i(i=1,2,3)$ are three shape functions corresponding to the three nodes of the triangular element. Equation (7.1) can be explicitly expressed as
\begin{aligned} & u^h(x, y)=N_1(x, y) u_1+N_2(x, y) u_2+N_3(x, y) u_3 \ & v^h(x, y)=N_1(x, y) v_1+N_2(x, y) v_2+N_3(x, y) v_3 \end{aligned}
which implies that each of the displacement components at any point in the element is approximated by an interpolation from the nodal displacements using the shape functions. This is because the two displacement components are basically independent from each other. The question now is how can we construct these shape functions for our triangular element that satisfies the sufficient requirements: delta function property; partitions of unity; and linear field reproduction.

## 数学代写|有限元代写Finite Element Method代考|Shape Function Construction

Development of the shape functions is normally the first, and most important, step in developing finite element equations for any type of element. In determining the shape functions $N_i$ ( $\left.i=1,2,3\right)$ for the triangular element, we can of course follow exactly the standard procedure described in Sections 3.4 .3 and 4.2.1, by starting with an assumption of the displacements using polynomial basis functions with unknown constants. These unknown constants are then determined using the nodal displacements at the nodes of the element. This standard procedure works in principle for the development of any type of element, but may not be the most convenient method. We demonstrate here another slightly different approach for constructing shape functions. We start with an assumption of shape functions directly using polynomial basis functions with unknown constants. These unknown constants are then determined using the property of the shape functions. The only difference here is that we assume directly the shape function instead of the displacements. For a linear triangular element, we assume that the shape functions are linear functions of $x$ and $y$. They should, therefore, have the form of
\begin{aligned} & N_1=a_1+b_1 x+c_1 y \ & N_2=a_2+b_2 x+c_2 y \ & N_3=a_3+b_3 x+c_3 y \end{aligned}
where $a_i, b_i$ and $c_i(i=1,2,3)$ are constants to be determined. Equation (7.5) can be written in a concise form,
$$N_i=a_i+b_i x+c_i y, \quad i=1,2,3$$

## 数学代写|有限元代写Finite Element Method代考|Field Variable Interpolation

$$\mathbf{U}^h(x, y)=\mathbf{N}(x, y) \mathbf{d}_e$$

\left 缺少或无法识别的分隔符

$$u^h(x, y)=N_1(x, y) u_1+N_2(x, y) u_2+N_3(x, y) u_3 \quad v^h(x, y)=N_1(x, y) v_1+N_2(x, y) v_2+N_3(x, y) v_3$$

## 数学代写|有限元代写Finite Element Method代考|Shape Function Construction

$$N_1=a_1+b_1 x+c_1 y \quad N_2=a_2+b_2 x+c_2 y N_3=a_3+b_3 x+c_3 y$$

$$N_i=a_i+b_i x+c_i y, \quad i=1,2,3$$

## 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|Equations in Global Coordinate System

Having known the element matrices in the local coordinate system, the next thing to do is to transform the element matrices into the global coordinate system to account for the differences in orientation of all the local coordinate systems that are attached on individual frame members.

Assume that the local nodes 1 and 2 of the element correspond to global nodes $i$ and $j$, respectively. The displacement at a local node should have three translational components in the $x, y$ and $z$ directions, and three rotational components with respect to the $x, y$ and $z$-axes. They are numbered sequentially by $d_1-d_{12}$ corresponding to the physical deformations as defined by Eq. (6.16). The displacement at a global node should also have three translational components in the $X, Y$ and $Z$ directions, and three rotational components with respect to the $X, Y$ and $Z$ axes. They are numbered sequentially by $D_{6 i-5}, D_{6 i-4}, \ldots$, and $D_{6 i}$ for the $i$ th node, as shown in Figure 6.5. The same sign convention applies to node $j$. The coordinate transformation gives the relationship between the displacement vector $\mathbf{d}_e$ based on the local coordinate system and the displacement vector $\mathbf{D}_e$ for the same element but

based on the global coordinate system:
$$\mathbf{d}e=\mathbf{T D}_e$$ where $$\mathbf{D}_e=\left{\begin{array}{c} D{6 i-5} \ D_{6 i-4} \ D_{6 i-3} \ D_{6 i-2} \ D_{6 i-1} \ D_{6 i} \ D_{6 j-5} \ D_{6 j-4} \ D_{6 j-3} \ D_{6 j-2} \ D_{6 j-1} \ D_{6 j} \end{array}\right}$$
and $\mathbf{T}$ is the transformation matrix for the truss element given by
$$\mathbf{T}=\left[\begin{array}{cccc} \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} \ \mathbf{0} & \mathbf{T}_3 & \mathbf{0} & \mathbf{0} \ \mathbf{0} & \mathbf{0} & \mathbf{T}_3 & \mathbf{0} \ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 \end{array}\right]$$

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

In the formulation of the matrices for the frame element in this chapter, the superposition of the truss element and the beam element has been used. This technique assumes that the axial effects are not coupled with the bending effects in the element. What this means simply is that the axial forces applied on the element will not result in any bending deformation, and the bending forces will not result in any axial deformation. Frame elements can also be used for frame structures with curved members. In such cases, the coupling effects can exist even in the elemental level. Therefore, depending on the curvature of the member, the meshing of the structure can be very important. For example, if the curvature is very large resulting in a significant coupling effect, a finer mesh is required to provide the necessary accuracy.
In practical structures, it is very rare to have beam structures subjected to purely transverse loading. Most skeletal structures are either trusses or frames that carry both axial and transverse loads. It can now be seen that the beam element, developed in Chapter 5, as well as the truss element, developed in Chapter 4, are simply specific cases of the frame element. Therefore, in most commercial software packages, including ABAQUS, the frame element is just known generally as the beam element.

The beam element formulated in Chapter 5 , or general beam element formulated in this chapter, is based on so-called Euler-Bernoulli beam theory that is suitable for thin beams with a small thickness to pan ratio $(<1 / 20)$. For thick or deep beams of a large thickness to pan ratio, corresponding beam theories should be used to develop thick beam elements. The procedure of developing thick beams is very similar to that of developing thick plates, to be discussed in Chapter 8. Most commercial software packages also offer thick beam elements, and the use of these elements is much the same as the thin beam elements.

## 数学代写|有限元代写Finite Element Method代考|Equations in Global

Coordinate System 的方向差异。

$$\mathbf{d} e=\mathbf{T D}_e$$

〈left 缺少或无法识别的分隔符

$$\mathbf{T}=\left[\begin{array}{llllllllllllllll} \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{T}_3 \end{array}\right]$$

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

. 对于厚梁比大的厚梁或深梁，应使用相应的梁理论来开发厚梁单元。开发厚梁的过程与开发厚板的过程非常相似，将在第 8 章中讨论。大多数商业软件包也提供厚梁单元，这些单元的使用与薄梁单元非常相似。

## 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|Modelling

The modelling of the bridge is done using one-dimensional beam elements developed in this chapter. The beam is assumed to be clamped at two ends of the beam. The meshing of the structure should not pose any difficulty, but what is important here is the choice of how many elements to use to give sufficient accuracy. Because the exact solution of free vibration modes of the beam is no longer of a polynomial type, the FEM will not be able to produce the exact solution, but an approximated solution. One naturally becomes concerned with whether the results converge and whether they are accurate.

To start, the first analysis will mesh the beam uniformly into ten two-nodal beam elements, as shown in Figure 5.6. This simple mesh will serve to show clearly the steps used in ABAQUS. Refined uniform meshes of 20,40 and 60 elements will then be used to check the accuracy of the results obtained. This is a simplified way of performing what is commonly known as a convergence test. Remember that usually the greater the number of elements, the greater the accuracy. However, we can’t simply use as many elements as possible all the time, since, there is usually a limit to the computer resources available. Hence, convergence tests are carried out to determine the optimum number of elements or nodes to be used for a certain problem. What is meant by ‘optimum’ means the least number of elements or nodes to yield a desired accuracy within the acceptable tolerance.

## 数学代写|有限元代写Finite Element Method代考|ABAQUS Input File

The ABAQUS input file for the above described finite element model is shown below. In the early days, the analyst had to write these cards manually, but now it is generated by the preprocessors of FEM packages. Understanding the input file is very important both for undersanding the FEM and to effectively use the FEM packages. The text boxes to the right of the input file are not part of the input file, but explain what the sections of the file meant.

The input file above shows how a basic ABAQUS input file is set up. Note that all the input file does is provide the information necessary so that the program can utilize them to formulate and solve the finite element equations. It may also be noticed that in the input file, there is no mention of the units of measurement used. This implies that the units must definitely be consistent throughout the input file in all the information provided. For example, if the coordinate values of the nodes are in micrometres, the units for other values like the Young’s modulus, density, forces and so on must also undergo the necessary conversions in order to be consistent, before they are keyed into the preprocessor of ABAQUS. It is noted that in this case study, all the units are converted into micrometres to be consistent with the geometrical dimensions, as can be seen from the values of Young’s modulus and density. This is the case for most finite element software, and many times, errors in analysis occur due to negligence in ensuring the units’ consistency. More details regarding the setting up of an ABAQUS input file will be provided in Chapter 13.

## 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|Properties of the FEM

Using the FEM, one can usually expect only an approximated solution. In Example 4.1, however, we obtained the exact solution. Why? This is because the exact solution of the deformation for the bar is a first order polynomial (see Eq. (4.46)). The shape functions used in our FEM analysis are also first order polynomials that are constructed using complete monomials up to the first order. Therefore, the exact solution of the problem is included in the set of assumed displacements in FEM shape functions. In Chapter 3, we understand that the FEM based on Hamilton’s principle guarantees to choose the best possible solution that can be produced by the shape functions. In Example 4.1, the best possible solution that can be produced by the shape function is the exact solution, due to the reproduction property of the shape functions, and the FEM has indeed reproduced it exactly. We therefore confirmed the reproduction property of the FEM that if the exact solution can be formed by the basis functions used to construct the FEM shape function, the FEM will always produce the exact solution, provided there is no numerical error involved in computation of the FEM solution.
Making use of this property, one may try to add in basis functions that form the exact solution or part of the exact solution, if that is possible, so as to achieve better accuracy in the FEM solution.

## 数学代写|有限元代写Finite Element Method代考|Convergence property of the FEM

For complex problems, the solution cannot be written in the form of a combination of monomials. Therefore, the FEM using polynomial shape functions will not produce the exact solution for such a problem. The question now is, how can one ensure that the FEM can produce a good approximation of the solution of a complex problem? The insurance is given by the convergence property of the FEM, which states that the FEM solution will converge to the exact solution that is continuous at arbitrary accuracy when the element size becomes infinitely small, and as long as the complete linear polynomial basis is included in the basis to form the FEM shape functions. The theoretical background for this convergence feature of the FEM is due to the fact that any continuous function can always be approximated by a first order polynomial with a second order of refinement error. This fact can be revealed by using the local Taylor expansion, based on which a continuous (displacement) function $u(x)$ can always be approximated using the following equation:
$$u=u_i+\left.\frac{\partial u}{\partial x}\right|_i\left(x-x_i\right)+O\left(h^2\right)$$
where $h$ is the characteristic size that relates to $\left(x-x_i\right)$, or the size of the element.

## 数学代写|有限元代写Finite Element Method代考|Convergence property of the FEM

$$u=u_i+\left.\frac{\partial u}{\partial x}\right|_i\left(x-x_i\right)+O\left(h^2\right)$$

## 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|HAMILTON’S PRINCIPLE

Hamilton’s principle is a simple yet powerful tool that can be used to derive discretized dynamic system equations. It states simply that

“Of all the admissible time histories of displacement the most accurate solution makes the Lagrangian functional a minimum.”
An admissible displacement must satisfy the following conditions:
(a) the compatibility equations,
(b) the essential or the kinematic boundary conditions, and
(c) the conditions at initial $\left(t_1\right)$ and final time $\left(t_2\right)$.
Condition (a) ensures that the displacements are compatible (continuous) in the problem domain. As will be seen in Chapter 11, there are situations when incompatibility can occur at the edges between elements. Condition (b) ensures that the displacement constraints are satisfied; and condition (c) requires the displacement history to satisfy the constraints at the initial and final times.
Mathematically, Hamilton’s principle states:
$$\delta \int_{t_1}^{t_2} L \mathrm{~d} t=0$$
The Langrangian functional, $L$, is obtained using a set of admissible time histories of displacements, and it consists of
$$L=T-\Pi+W_f$$
where $T$ is the kinetic energy, $\Pi$ is the potential energy (for our purposes, it is the elastic strain energy), and $W_f$ is the work done by the external forces. The kinetic energy of the entire problem domain is defined in the integral form
$$T=\frac{1}{2} \int_V \rho \dot{\mathbf{U}}^T \dot{\mathbf{U}} \mathrm{d} V$$
where $V$ represents the whole volume of the solid, and $\mathbf{U}$ is the set of admissible time histories of displacements.

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

The solid body is divided into $N_e$ elements. The procedure is often called meshing, which is usually performed using so-called pre-processors. This is especially true for complex geometries. Figure $3.1$ shows an example of a mesh for a two-dimensional solid.

The pre-processor generates unique numbers for all the elements and nodes for the solid or structure in a proper manner. An element is formed by connecting its nodes in a pre-defined consistent fashion to create the connectivity of the element. All the elements together form the entire domain of the problem without any gap or overlapping. It is possible for the domain to consist of different types of elements with different numbers of nodes, as long as they are compatible (no gaps and overlapping; the admissible condition (a) required by Hamilton’s principle) on the boundaries between different elements. The density of the mesh depends upon the accuracy requirement of the analysis and the computational resources available. Generally, a finer mesh will yield results that are more accurate, but will increase the computational cost. As such, the mesh is usually not uniform, with a finer mesh being used in the areas where the displacement gradient is larger or where the accuracy is critical to the analysis. The purpose of the domain discretization is to make it easier in assuming the pattern of the displacement field.

The FEM formulation has to be based on a coordinate system. In formulating FEM equations for elements, it is often convenient to use a local coordinate system that is defined for an element in reference to the global coordination system that is usually defined for the entire structure, as shown in Figure 3.4. Based on the local coordinate system defined on an element, the displacement within the element is now assumed simply by polynomial interpolation using the displacements at its nodes (or nodal displacements) as
$$\mathbf{U}^h(x, y, z)=\sum_{i=1}^{n_d} \mathbf{N}i(x, y, z) \mathbf{d}_i=\mathbf{N}(x, y, z) \mathbf{d}_e$$ where the superscript $h$ stands for approximation, $n_d$ is the number of nodes forming the element, and $\mathbf{d}_i$ is the nodal displacement at the $i$ th node, which is the unknown the analyst wants to compute, and can be expressed in a general form of \mathbf{d}_i=\left{\begin{array}{c} d_1 \ d_2 \ \vdots \ d{n_f} \end{array}\right} \rightarrow \begin{aligned} &\rightarrow \text { displacement component } 1 \ &\rightarrow \text { displacement component } 2 \ &\vdots \ &\text { displacement component } n_f \end{aligned}

## 数学代写|有限元代写Finite Element Method代考|HAMILTON’S PRINCIPLE

“在所有允许的位移时间历史中，最准确的解快方安使拉格朗日函数最小。”

(a) 相容方程,
(b) 基本或运动边界条件，以及
(c) 初始条件 $\left(t_1\right)$ 最后一次 $\left(t_2\right)$.

$$\delta \int_{t_1}^{t_2} L \mathrm{~d} t=0$$

$$L=T-\Pi+W_f$$

$$T=\frac{1}{2} \int_V \rho \dot{\mathbf{U}}^T \dot{\mathbf{U}} \mathrm{d} V$$

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

FEM 公式必须基于坐标系。在为单元制定 FEM 方程时，通常方便地使用为单元定义的局部坐标系，参考通常为整个结构定义的全 局坐标系，如图 $3.4$ 所示。基于单元上定义的局部坐标系，单元内的位移现在通过多项式揷值简单地假设，使用其节点处的位移 (或节点位移) 为
$$\mathbf{U}^h(x, y, z)=\sum_{i=1}^{n_d} \mathbf{N} i(x, y, z) \mathbf{d}_i=\mathbf{N}(x, y, z) \mathbf{d}_e$$

〈left 的分隔符缺失或无法识别

## 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|COMPUTATIONAL MODELLING

The Finite Element Method (FEM) has developed into a key, indispensable technology in the modelling and simulation of advanced engineering systems in various fields like housing, transportation, communications, and so on. In building such advanced engineering systems, engineers and designers go through a sophisticated process of modelling, simulation, visualization, analysis, designing, prototyping, testing, and lastly, fabrication. Note that much work is involved before the fabrication of the final product or system. This is to ensure the workability of the finished product, as well as for cost effectiveness. The process is illustrated as a flowchart in Figure 1.1. This process is often iterative in nature, meaning that some of the procedures are repeated based on the results obtained at a current stage, so as to achieve an optimal performance at the lowest cost for the system to be built. Therefore, techniques related to modelling and simulation in a rapid and effective way play an increasingly important role, resulting in the application of the FEM being multiplied numerous times because of this.

This book deals with topics related mainly to modelling and simulation, which are underlined in Figure 1.1. Under these topics, we shall address the computational aspects, which are also underlined in Figure 1.1. The focus will be on the techniques of physical, mathematical and computational modelling, and various aspects of computational simulation. A good understanding of these techniques plays an important role in building an advanced engineering system in a rapid and cost effective way.

So what is the FEM? The FEM was first used to solve problems of stress analysis, and has since been applied to many other problems like thermal analysis, fluid flow analysis, piezoelectric analysis, and many others. Basically, the analyst seeks to determine the distribution of some field variable like the displacement in stress analysis, the temperature or heat flux in thermal analysis, the electrical charge in electrical analysis, and so on. The FEM is a numerical method seeking an approximated solution of the distribution of field variables in the problem domain that is difficult to obtain analytically. It is done by dividing the problem domain into several elements, as shown in Figures $1.2$ and 1.3. Known physical laws are then applied to each small element, each of which usually has a very simple geometry. Figure $1.4$ shows the finite element approximation for a one-dimensional case schematically. A continuous function of an unknown field variable is approximated using piecewise linear functions in each sub-domain, called an element formed by nodes. The unknowns are then the discrete values of the field variable at the nodes. Next, proper principles are followed to establish equations for the elements, after which the elements are tied’ to one another. This process leads to a set of linear algebraic simultaneous equations for the entire system that can be solved easily to yield the required field variable.

This book aims to bring across the various concepts, methods and principles used in the formulation of FE equations in a simple to understand manner. Worked examples and case studies using the well known commercial software package ABAQUS will be discussed, and effective techniques and procedures will be highlighted.

## 数学代写|有限元代写Finite Element Method代考|PHYSICAL PROBLEMS IN ENGINEERING

There are numerous physical engineering problems in a particular system. As mentioned earlier, although the FEM was initially used for stress analysis, many other physical problems can be solved using the FEM. Mathematical models of the FEM have been formulated for the many physical phenomena in engineering systems. Common physical problems solved using the standard FEM include:

• Mechanics for solids and structures.
• Heat transfer.
• Acoustics.
• Fluid mechanics.
• Others.
This book first focuses on the formulation of finite element equations for the mechanics of solids and structures, since that is what the FEM was initially designed for. FEM formulations for heat transfer problems are then described. The conceptual understanding of the methodology of the FEM is the most important, as the application of the FEM to all other physical problems utilizes similar concepts.

Computer modelling using the FEM consists of the major steps discussed in the next section.

## 数学代写|有限元代写有限元方法代考|工程中的物理问题

• 固体和结构力学。
• 传热

• 音响。
• 流体力学
• 其他这本书首先着重于固体和结构力学的有限元方程的公式，因为这是有限元最初设计的目的。然后描述了传热问题的有限元公式。对有限元方法的概念理解是最重要的，因为对所有其他物理问题的有限元应用都利用了类似的概念

使用有限元的计算机建模包括下一节所讨论的主要步骤

## 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。