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数学代写|有限元代写Finite Element Method代考|Explicit and Implicit Formulations and Mass Lumping

The solution of the fully discretized equations of parabolic equations and hyperbolic equations (after assembly and imposition of boundary and initial conditions) require the inversion of $\hat{\mathbf{K}}$ appearing in Eqs. (7.4.29a) and (7.4.37a) to march forward in time and find the solution at different times. This can be an enormous computational expense, depending on the size of the mesh and the number of time steps. For example, if one needs to solve these equations for 1,000 time steps, the cost is equivalent to solving 1,000 static problems. Thus, it is of practical interest to find ways to reduce the computational cost. It is clear that if the element $\hat{\mathbf{K}}^e$ were a diagonal matrix, then the assembled or global coefficient matrix $\hat{\mathbf{K}}$ would be diagonal, and there is no inversion required to solve for $u_i^{s+1}$ (i.e., simply divide each equation with the diagonal element):
$$U_i^{s+1}=\frac{1}{\hat{K}{(i i)}}\left(\sum{j=1}^{N E Q} \bar{K}_{i j} U_j^s+\bar{F}_i^{s, s+1}\right) \quad(\text { no sum on } i)$$
Formulations that require the inversion of $\hat{\mathbf{K}}$ (because it is not diagonal) are termed implicit formulations and those in which no inversion is required are called explicit formulations.

In the finite element method, no time-approximation scheme results in a diagonal matrix $\hat{\mathbf{K}}$ because matrices $\mathbf{C}$ and/or $\mathbf{M}$ appearing in $\hat{\mathbf{K}}$ are not diagonal matrices. A matrix ( $\mathbf{C}$ or $\mathbf{M}$ ) computed according to the definition is called a consistent (mass) matrix, and it is not diagonal unless the approximation functions $\psi_i$ are orthogonal over the element domain. In realworld problems where hundreds of thousands of degrees of freedom are involved, the cost of computation precludes the inversion of large systems of equations. Thus, one needs to pick a scheme that eliminates $\mathbf{K}$ from $\hat{\mathbf{K}}$

(because, it would be a gross approximation to diagonalize $\mathbf{K}$ ) and then diagonalize $\mathbf{C}$ and/or $\mathbf{M}$ to have an explicit formulation.

For example, the forward difference scheme (i.e., $\alpha=0$ ) results in the following time-marching scheme [see Eq. (7.4.29a)]:
$$\mathbf{C U}^{s+1}=(\mathbf{C}-\Delta t \mathbf{K}) \mathbf{U}^s+\Delta t \mathbf{F}^s$$
If the matrix $\mathbf{C}$ is diagonal then the assembled equations can be solved directly (i.e., without inverting a matrix). Similarly, the central difference scheme for an undamped system (i.e., $\mathbf{C}=0$ ) is [see Eq. (7.4.42)]
$$\mathbf{M U}^{s+1}=(\Delta t)^2 \mathbf{F}^{s+1}+\left(2 \mathbf{M}-(\Delta t)^2 \mathbf{K}\right) \mathbf{U}^s-\mathbf{M} \dot{U}^s$$
which requires diagonalization of $\mathbf{M}$ (and $\mathbf{C}$, in the case of a damped system) in order for the central difference formulation to be explicit. The explicit nature of Eq. (7.4.48) motivated analysts to find rational ways of making $\mathbf{C}$ and/or $\mathbf{M}$ diagonal. There are several ways of constructing diagonal mass matrices by lumping the mass at the nodes, while preserving the total mass. Two such approaches are discussed next.

数学代写|有限元代写Finite Element Method代考|Row-sum lumping

The sum of the elements of each row of the consistent (mass) matrix is used as the diagonal element and setting the off-diagonal elements to zero [(ii) means no sum on $i]$ :
$$M_{(i i)}^e=\sum_{j=1}^n \int_{x_a^e}^{x_b^e} \rho \psi_i^e \psi_j^e d x=\int_{x_a^e}^{x_b^e} \rho \psi_i^e d x$$
where the property $\sum_{j=1}^n \psi_j^e=1$ of the interpolation functions is used.
When $\rho_e$ is element-wise constant, the consistent matrices associated with the linear and quadratic 1-D elements are
$$\mathbf{M}_{\mathrm{C}}^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{ll} 2 & 1 \ 1 & 2 \end{array}\right], \quad \mathbf{M}_C^e=\frac{\rho_e A_e h_e}{30}\left[\begin{array}{rcr} 4 & 2 & -1 \ 2 & 16 & 2 \ -1 & 2 & 4 \end{array}\right]$$
As per Eq. (7.4.51), the associated diagonal matrices for the linear and quadratic elements are
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right], \quad \mathbf{M}_L^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{array}\right]$$
Here subscripts $L$ and $C$ refer to lumped and consistent mass matrices, respectively.
The consistent mass matrix for the Euler-Bernoulli beam is given in Eq. (7.3.57). The row-sum diagonal mass matrix is obtained in two ways: (a) neglecting the terms corresponding to the rotational degrees of freedom in each row and (b) neglecting the terms corresponding to the rotational degrees of freedom in rows 1 and 3 and neglecting the terms associated with the translational degrees of freedom in rows in 2 and 4 :
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{420}\left[\begin{array}{rrrr} 210 & 0 & 0 & 0 \ 0 & h_e^2 & 0 & 0 \ 0 & 0 & 210 & 0 \ 0 & 0 & 0 & h_e^2 \end{array}\right]$$
The consistent mass matrix of the Timoshenko beam theory is given in Eq. (7.3.63b). The lumped mass matrices for the Timoshenko beam are
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & r_e & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & r_e \end{array}\right], \quad r_e=\frac{I_e}{A_e}$$

数学代写|有限元代写Finite Element Method代考|Explicit and Implicit Formulations and Mass Lumping

$$U_i^{s+1}=\frac{1}{\hat{K}{(i i)}}\left(\sum{j=1}^{N E Q} \bar{K}_{i j} U_j^s+\bar{F}_i^{s, s+1}\right) \quad(\text { no sum on } i)$$

(因为，这将是对角化$\mathbf{K}$的粗略近似值)，然后对角化$\mathbf{C}$和/或$\mathbf{M}$以获得显式公式。

$$\mathbf{C U}^{s+1}=(\mathbf{C}-\Delta t \mathbf{K}) \mathbf{U}^s+\Delta t \mathbf{F}^s$$

$$\mathbf{M U}^{s+1}=(\Delta t)^2 \mathbf{F}^{s+1}+\left(2 \mathbf{M}-(\Delta t)^2 \mathbf{K}\right) \mathbf{U}^s-\mathbf{M} \dot{U}^s$$

数学代写|有限元代写Finite Element Method代考|Row-sum lumping

$$M_{(i i)}^e=\sum_{j=1}^n \int_{x_a^e}^{x_b^e} \rho \psi_i^e \psi_j^e d x=\int_{x_a^e}^{x_b^e} \rho \psi_i^e d x$$

$$\mathbf{M}_{\mathrm{C}}^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{ll} 2 & 1 \ 1 & 2 \end{array}\right], \quad \mathbf{M}_C^e=\frac{\rho_e A_e h_e}{30}\left[\begin{array}{rcr} 4 & 2 & -1 \ 2 & 16 & 2 \ -1 & 2 & 4 \end{array}\right]$$

$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array}\right], \quad \mathbf{M}_L^e=\frac{\rho_e A_e h_e}{6}\left[\begin{array}{lll} 1 & 0 & 0 \ 0 & 4 & 0 \ 0 & 0 & 1 \end{array}\right]$$

$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{420}\left[\begin{array}{rrrr} 210 & 0 & 0 & 0 \ 0 & h_e^2 & 0 & 0 \ 0 & 0 & 210 & 0 \ 0 & 0 & 0 & h_e^2 \end{array}\right]$$
Timoshenko梁理论的一致质量矩阵如式(7.3.63b)所示。Timoshenko梁的集总质量矩阵为
$$\mathbf{M}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 \end{array}\right], \quad \hat{\mathbf{M}}_L^e=\frac{\rho_e A_e h_e}{2}\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \ 0 & r_e & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & r_e \end{array}\right], \quad r_e=\frac{I_e}{A_e}$$

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|Time approximations

The time approximation is discussed with the help of a single first-order differential equation for $u_i$, and then we generalize for a vector of unknowns, u. Suppose that we wish to determine $u_i(t)$ for $t>0$ such that $u_i(t)$ satisfies
$$a \frac{d u_i}{d t}+b u_i=f_i(t), 0<t<T \quad \text { and } u_i(0)=u_i^0$$
where $a \neq 0, b$, and $u_i^0$ are constants and $f_i$ is a function of time $t$. The exact solution of the problem consists of two parts: the homogeneous and particular solutions. The homogeneous solution is
$$u_i^h(t)=A e^{-k t}, k=\frac{b}{a}$$
where $A$ is a constant of integration. The particular solution is given by
$$u_i^p(t)=\frac{1}{a} e^{-k t}\left(\int_0^t e^{k \tau} f_i(\tau) d \tau\right)$$
The complete solution is given by
$$u_i(t)=e^{-k t}\left(A+\frac{1}{a} \int_0^t e^{k \tau} f_i(\tau) d \tau\right)$$
Finite difference approximations. The finite difference methods are based on truncated (using the desired degree of accuracy) Taylor’s series expansions. For example, Taylor’s series expansions of function $F(t)$ about $t=t_S$ and $t=$ $t_{s+1}$ are given by
$$F(t)=F\left(t_s\right)+\left(t-t_s\right) \dot{F}\left(t_s\right)+\frac{1}{2 !}\left(t-t_s\right)^2 \ddot{F}\left(t_s\right)+\cdots$$
$$F(t)=F\left(t_{s+1}\right)+\left(t-t_{s+1}\right) \dot{F}\left(t_{s+1}\right)+\frac{1}{2 !}\left(t-t_{s+1}\right)^2 \ddot{F}\left(t_{s+1}\right)+\cdots$$
In particular, for $t=t_{\mathrm{s}+1}$ in Eq. (7.4.12a), we have
$$F\left(t_{s+1}\right)=F\left(t_s\right)+\left(t_{s+1}-t_s\right) \dot{F}\left(t_s\right)+\frac{1}{2 !}\left(t_{s+1}-t_s\right)^2 \ddot{F}\left(t_s\right)+\frac{1}{3 !}\left(t_{s+1}-t_s\right)^3 \ddot{F}\left(t_s\right)+\cdots$$
If we truncate the series after the second term and solve for $\dot{F}\left(t_s\right)$, we obtain
$$\dot{F}\left(t_s\right)=\frac{F\left(t_{s+1}\right)-F\left(t_s\right)}{t_{s+1}-t_s}+\mathrm{O}\left(\Delta t_{s+1}\right)$$

数学代写|有限元代写Finite Element Method代考|Numerical stability

Since the time-marching scheme in Eq. (7.4.17b) uses the previous time step solution $u_i^s$, which itself is an approximate solution, there is a possibility that the error introduced in $u_i^s$ may be amplified when $u_i^{s+1}$ is computed and may grow unboundedly with time. When the error grows without bounds, the underlying scheme is said to be unstable. Here we wish to determine the conditions under which the error remains bounded.
Consider Eq.(7.4.17b) in operator form
$$u_i^{s+1}=B u_i^s+\bar{F}^{s, s+1}$$
where
$$B=\frac{a-(1-\alpha) \Delta t b}{a+\alpha \Delta t b}, \bar{F}{s, s+1}=\Delta t \frac{\alpha f{s+1}+(1-\alpha) f_s}{a+\alpha \Delta t b}$$
If the magnitude of the operator $B$, known as the amplification operator, is greater than $1,|B|>1$, the error will be amplified during each time step. On the other hand, if the magnitude is equal to or less than unity, the error will not grow with time. Therefore, in order for the scheme to be stable it is necessary that $|B| \leq 1$ :
$$|B|=\left|\frac{a-(1-\alpha) \Delta t b}{a+\alpha \Delta t b}\right| \leq 1$$
The above equation places a restriction on the magnitude of the time step for certain values of $\alpha$. When the error remains bounded for any time step [i.e., condition (7.4.24) is trivially satisfied for any value of $\Delta t$ ], it is known as a stable scheme. If the error remains bounded only when the time step remains below certain value [in order to satisfy (7.4.24)], it is said to be conditionally stable scheme. In the finite element method, the coefficients $a$ and $b$ appearing in Eq. (7.4.24) depend on problem material parameters (which cannot be changed) and the element length (which can be selected). Hence, for a given mesh there is a value of $\Delta t$ that makes the scheme conditionally stable (more details on this topic can be found in the book by Surana and Reddy [4]).

数学代写|有限元代写Finite Element Method代考|Time approximations

$$a \frac{d u_i}{d t}+b u_i=f_i(t), 0<t<T \quad \text { and } u_i(0)=u_i^0$$

$$u_i^h(t)=A e^{-k t}, k=\frac{b}{a}$$

$$u_i^p(t)=\frac{1}{a} e^{-k t}\left(\int_0^t e^{k \tau} f_i(\tau) d \tau\right)$$

$$u_i(t)=e^{-k t}\left(A+\frac{1}{a} \int_0^t e^{k \tau} f_i(\tau) d \tau\right)$$

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。

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|Discretization of the Domain

The domain $\Omega=(0, L)$ of the straight beam shown in Fig. 5.2.3(a) is divided into a set of, say, $N$ line elements, a typical element being $\Omega^e=\left(x_a^e, x_b^e\right)$, as indicated in Fig. 5.2.3(b). Although the element is geometrically the same as that used for bars, the number and form of the primary and secondary unknowns at each end point, which constitutes a node, are dictated by the weak formulation of the differential equation, (5.2.10). We isolate a typical element $\Omega^e=\left(x_a^e, x_b^e\right)$ and construct the weak form of Eq. (5.2.10) over the element. The weak form provides the form of the primary and secondary variables of the problem. The primary variables are kinematic quantities that are required to be continuous throughout the domain, while the secondary variables are kinetic entities that are required to satisfy equilibrium conditions. When the secondary variables are physically not meaningful, the integration-by-parts step that yields them should not be carried out.
The weak forms of problems in solid mechanics can be developed either from the principle of virtual work (i.e., the principle of virtual displacements or virtual forces) or from the governing differential equations. Here we start with the given differential equation, Eq. (5.2.10), and using the three-step procedure obtain the weak form. We shall also consider the principle of virtual work in the sequel.
Suppose that $w_h^e$ is the finite element approximation of $w$ and let $v_i^e$ be a weight function over the element $\Omega^e=\left(x_a^e, x_b^e\right)$. Following the three-step procedure illustrated in Example 2.4.2, we write
\begin{aligned} 0= & \int_{x_a^e}^{x_b^e} v_i^e\left[\frac{d^2}{d x^2}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)+k_f^e w_h^e-q_e\right] d x \ = & \int_{x_a^e}^{x_b^e}\left[-\frac{d v_i^e}{d x} \frac{d}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)+k_f^e v_i^e w_h^e-v_i^e q_e\right] d x+\left[v_i^e \frac{d}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)\right]{x_a^e}^{x_b^e} \ = & \int{x_a^e}^{x_b^e}\left(E_e I_e \frac{d^2 v_i^e}{d x^2} \frac{d^2 w_h^e}{d x^2}+k_f^e v_i^e w_h^e-v_i^e q_e\right) d x \ & +\left[v_i^e \frac{d}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)-\frac{d v_i^e}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)\right]_{x_a^e}^{x_b^e} \end{aligned}
where $\left{v_i^e(x)\right}$ is a set of weight functions that are twice differentiable with respect to $x$. Note that, in the present case, the first term of the equation is integrated twice by parts to trade two differentiations to the weight function $v_i^e$, while retaining two derivatives of the dependent variable, $w_l^e$; that is, the differentiation is distributed equally between the weight function $v_i^e$ and the transverse deflection $w_h^e$. Because of the two integrations by parts, there appear two boundary expressions, which are to be evaluated at the two boundary points $x=x_a^e$ and $x=x_b^e$. Examination of the boundary terms indicates that the bending moment $M_h^e=-E_e I_e d^2 w_h^e / d x^2$ and shear force $V_h^e=-(d / d x)\left(E_e I_e d^2 w_h^e / d x^2\right)$ are the secondary variables and $\left(v_i^e \sim\right) w_h^e$ and slope $\left(d v_i^e / d x \sim\right) d w_h^e / d x$ are the primary variables. Thus, the weak form indicates that the boundary conditions for the EBT involve specifying one element of each of the following two pairs:
$$\left(w, V=\frac{d M}{d x}\right), \quad\left(\theta_x \equiv-\frac{d w}{d x}, M\right)$$
Mixed boundary conditions involve specifying a relationship between the variables of each pair:
Vertical spring: $V+k_s w=0 ; \quad$ Torsional spring: $M+\mu_s \theta_x=0$
where $k_{\mathrm{s}}$ and $\mu_{\mathrm{s}}$ are the stiffness coefficients of linear and torsional springs, respectively.

数学代写|有限元代写Finite Element Method代考|Approximation Functions

The weak form in Eq. (5.2.13) requires that the approximation $w_h^e(x)$ of $w(x)$ over a finite element should be such that it is twice-differentiable and satisfies the interpolation properties; that is, satisfies the following geometric “boundary conditions” of the element, as illustrated in Fig. 5.2.4:
$$w_h^e\left(x_a\right) \equiv \Delta_1^e, \quad w_h^e\left(x_b^e\right) \equiv \Delta_3^e, \quad \theta_x^e\left(x_a^e\right) \equiv \Delta_2^e, \quad \theta_x^e\left(x_b^e\right) \equiv \Delta_4^e$$
Note that $x_a^e$ and $x_b^e$ are the global coordinates of nodes 1 and 2, respectively. In satisfying the essential (or geometric) boundary conditions in Eq. (5.2.16), the approximation automatically satisfies the continuity conditions. Hence, we pay attention to the satisfaction of the conditions in Eq. (5.2.16), which forms the basis for the derivation of the interpolation functions of the EulerBernoulli beam element.
Since the approximation functions to be derived are valid over the element domain, it is simpler to derive them in terms of the local coordinate $\bar{x}$ with origin at node $1, \bar{x}=x-x_a^e$ Since there are a total of four conditions in an element (two per node), a four-parameter polynomial must be selected for $w_h^e$ :
$$w(\bar{x}) \approx w_h^e(\bar{x})=c_1^e+c_2^e \bar{x}+c_3^e \bar{x}^2+c_4^e \bar{x}^3$$
Note that the minimum continuity requirement (i.e., the existence of a nonzero second derivative of $w_h^e$ in the element) is automatically met. In addition, the cubic approximation of $w_h$ allows computation of the shear force, which involves the third derivative of $w_h^e$. Next, we express $c_i^e$ in terms of the primary nodal variables
$$\Delta_1^e=w_h^e(0), \quad \Delta_2^e=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\bar{x}=0}, \Delta_3^e=w_h^e\left(h_e\right), \Delta_4^e=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\bar{x}=h_e}$$
such that the conditions (5.2.16) are satisfied:
\begin{aligned} & \Delta_1^e=w_h^e(0) \quad=c_1^e \ & \Delta_2^e=-\left.\frac{d w_h}{d x}\right|{x=x_a}=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\dot{x}=0}=-c_2^e \ & \Delta_3^e=w_h^e\left(h_e\right)=c_1^e+c_2^e h_e+c_3^e h_e^2+c_4^e h_e^3 \ & \Delta_4^e=-\left.\frac{d w_h^e}{d x}\right|{x=x_b}=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\bar{x}=h_e}=-c_2^e-2 c_3^e h_e-3 c_4^e h_e^2 \end{aligned}

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

\begin{aligned} 0= & \int_{x_a^e}^{x_b^e} v_i^e\left[\frac{d^2}{d x^2}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)+k_f^e w_h^e-q_e\right] d x \ = & \int_{x_a^e}^{x_b^e}\left[-\frac{d v_i^e}{d x} \frac{d}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)+k_f^e v_i^e w_h^e-v_i^e q_e\right] d x+\left[v_i^e \frac{d}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)\right]{x_a^e}^{x_b^e} \ = & \int{x_a^e}^{x_b^e}\left(E_e I_e \frac{d^2 v_i^e}{d x^2} \frac{d^2 w_h^e}{d x^2}+k_f^e v_i^e w_h^e-v_i^e q_e\right) d x \ & +\left[v_i^e \frac{d}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)-\frac{d v_i^e}{d x}\left(E_e I_e \frac{d^2 w_h^e}{d x^2}\right)\right]{x_a^e}^{x_b^e} \end{aligned} 其中$\left{v_i^e(x)\right}$是一组对$x$二阶可微的权函数。请注意，在本例中，方程的第一项通过分部积分两次，以对权重函数$v_i^e$进行两次微分，同时保留因变量$w_l^e$的两次导数;即微分在权函数$v_i^e$和横向挠度$w_h^e$之间均匀分布。由于采用了两次分部积分法，出现了两个边界表达式，分别在两个边界点$x=x_a^e$和$x=x_b^e$处求值。对边界项的检验表明，弯矩$M_h^e=-E_e I_e d^2 w_h^e / d x^2$和剪力$V_h^e=-(d / d x)\left(E_e I_e d^2 w_h^e / d x^2\right)$是次变量，$\left(v_i^e \sim\right) w_h^e$和斜率$\left(d v_i^e / d x \sim\right) d w_h^e / d x$是主变量。因此，弱形式表明EBT的边界条件涉及指定以下两对中的每一对的一个元素: $$\left(w, V=\frac{d M}{d x}\right), \quad\left(\theta_x \equiv-\frac{d w}{d x}, M\right)$$ 混合边界条件包括指定每对变量之间的关系: 垂直弹簧:$V+k_s w=0 ; \quad$扭转弹簧:$M+\mu_s \theta_x=0$ 其中$k{\mathrm{s}}$和$\mu_{\mathrm{s}}$分别为线性弹簧和扭转弹簧的刚度系数。

数学代写|有限元代写Finite Element Method代考|Approximation Functions

(5.2.13)式中的弱形式要求$w(x)$在有限元上的近似$w_h^e(x)$是二次可微的，并满足插值性质;即满足单元的几何“边界条件”，如图5.2.4所示:
$$w_h^e\left(x_a\right) \equiv \Delta_1^e, \quad w_h^e\left(x_b^e\right) \equiv \Delta_3^e, \quad \theta_x^e\left(x_a^e\right) \equiv \Delta_2^e, \quad \theta_x^e\left(x_b^e\right) \equiv \Delta_4^e$$

$$w(\bar{x}) \approx w_h^e(\bar{x})=c_1^e+c_2^e \bar{x}+c_3^e \bar{x}^2+c_4^e \bar{x}^3$$

$$\Delta_1^e=w_h^e(0), \quad \Delta_2^e=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\bar{x}=0}, \Delta_3^e=w_h^e\left(h_e\right), \Delta_4^e=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\bar{x}=h_e}$$

\begin{aligned} & \Delta_1^e=w_h^e(0) \quad=c_1^e \ & \Delta_2^e=-\left.\frac{d w_h}{d x}\right|{x=x_a}=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\dot{x}=0}=-c_2^e \ & \Delta_3^e=w_h^e\left(h_e\right)=c_1^e+c_2^e h_e+c_3^e h_e^2+c_4^e h_e^3 \ & \Delta_4^e=-\left.\frac{d w_h^e}{d x}\right|{x=x_b}=-\left.\frac{d w_h^e}{d \bar{x}}\right|{\bar{x}=h_e}=-c_2^e-2 c_3^e h_e-3 c_4^e h_e^2 \end{aligned}

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|Assembly of Element Equations

In deriving the element equations, we isolated a typical element from the mesh and formulated the weak form and developed its finite element model. The finite element model of a typical element contains $n$ equations among $n+$ 2 unknowns, $\left(u_1^e, u_2^e, \ldots, u_n^e\right)$ and $\left(Q_1^e, Q_n^e\right)$. Hence, they cannot be solved without using the equations from other elements. From a physical point of view, this makes sense because one should not be able to solve the element equations without considering the assembled set of equations and the boundary conditions of the total problem.
To obtain the finite element equations of the total problem, we must put the elements back into their original positions. In putting the elements with their nodal degrees of freedom back into their original positions, we must require that the primary variable $u(x)$ is uniquely defined (i.e., $u$ is continuous) and the source terms $Q_i^e$ are “balanced” at the points where elements are connected to each other. Of course, if the variable $u$ is not continuous, we do not impose its continuity; but in all problems studied in this book, unless otherwise stated explicitly (like in the case of an internal hinge in the case of beam bending), the primary variables are required to be continuous. Thus, the assembly of elements is carried out by imposing the following two conditions:

If the end node $i$ of element $\Omega^e$ is connected to the end node $j$ of element $\Omega^f$ and the end node $k$ of element $\Omega^g$, the continuity of the primary variable $u$ requires
$$u_i^{(e)}=u_j^{(f)}=u_k^{(g)}$$
When node $n$ of element $\Omega e$ is connected to node 1 of element $\Omega^{e+1}$ (with $m$ nodes) in series, as shown in Fig. 3.4.10a, the continuity of $u$ requires
$$u_n^{(e)}=u_1^{(e+1)}$$

For the same three elements, the balance of secondary variables at connecting nodes requires
$$Q_i^{(e)}+Q_j^{(f)}+Q_k^{(g)}=Q_I$$
where $I$ is the global node number assigned to the nodal point that is common to the three elements, and $Q_I$ is the value of externally applied source, if any (otherwise zero), at this node (the sign of $Q_I$ must be consistent with the sign of $Q_e$ in Fig. 3.4.4). For the case shown in Fig. 3.4.10, we have
$$Q_n^e+Q_1^{e+1}= \begin{cases}0, & \text { if no external point source is applied } \ Q_l, & \text { if an external point source of magnitude } \ Q_l \text { is applied }\end{cases}$$

数学代写|有限元代写Finite Element Method代考|Postprocessing of the Solution

The solution of the finite element equations in Eq. (3.4.59) gives the values of the primary variables (e.g., displacement, velocity, or temperature) at the global nodes. Once the nodal values of the primary variables are known, we can use the finite element approximation $u_h^e(x)$ to compute the desired quantities. The process of computing desired quantities in numerical form or graphical form from the known finite element solution is termed postprocessing; this phrase is meant to indicate that further computations are made after obtaining the solution of the finite element equations for the nodal values of the primary variables.
Postprocessing of the solution includes one or more of the following tasks:

1. Computation of the primary and secondary variables at points of interest; primary variables are known at nodal points.
2. Interpretation of the results to check whether the solution makes sense (an understanding of the physical process and experience are the guides when other solutions are not available for comparison).
3. Tabular and/or graphical presentation of the results.
To determine the solution $u$ as a continuous function of position $x$, we return to the approximation in Eq. (3.4.28) over each element:
$$u(x) \approx\left{\begin{array}{l} u_h^1(x)=\sum_{j=1}^n u_j^1 \psi_j^1(x) \ u_h^2(x)=\sum_{j=1}^n u_j^2 \psi_j^2(x) \ \vdots \ u_h^N(x)=\sum_{j=1}^n u_j^N \psi_j^N(x) \end{array}\right.$$

数学代写|有限元代写Finite Element Method代考|Assembly of Element Equations

$$u_i^{(e)}=u_j^{(f)}=u_k^{(g)}$$

$$u_n^{(e)}=u_1^{(e+1)}$$

$$Q_i^{(e)}+Q_j^{(f)}+Q_k^{(g)}=Q_I$$

$$Q_n^e+Q_1^{e+1}= \begin{cases}0, & \text { if no external point source is applied } \ Q_l, & \text { if an external point source of magnitude } \ Q_l \text { is applied }\end{cases}$$

数学代写|有限元代写Finite Element Method代考|Postprocessing of the Solution

$$u(x) \approx\left{\begin{array}{l} u_h^1(x)=\sum_{j=1}^n u_j^1 \psi_j^1(x) \ u_h^2(x)=\sum_{j=1}^n u_j^2 \psi_j^2(x) \ \vdots \ u_h^N(x)=\sum_{j=1}^n u_j^N \psi_j^N(x) \end{array}\right.$$

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|One-Dimensional Heat Transfer

The direct approach can also be used to develop finite element models of one-dimensional heat transfer. The relations between temperatures and heats at the ends of a surface insulated solid bar or two surfaces of a plane wall can be developed using the basic principles of heat transfer. We have
\begin{aligned} \text { temperature gradient } & =\text { difference in temperature } / \text { length } \ \text { heat flux, } q & =\text { conductivity } \times(- \text { temperature gradient }) \ \text { heat, } Q & =\text { heat flux } \times \text { area of cross section } \end{aligned}
Then, if there is no internal heat generation and the temperature is assumed to vary linearly between the ends of the bar of length (or plane wall of thickness) $h_e$, cross-sectional area $A_e$, and conductivity $k_e$, the heats at the left and right ends of the bar are
\begin{aligned} & Q_1^e=A_e q_1^e=-A_e k_e \frac{T_2^e-T_1^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_1^e-T_2^e\right) \ & Q_2^e=A_e q_2^e=-A_e k_e \frac{T_1^e-T_2^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_2^e-T_1^e\right) \end{aligned}
Technically, both $Q_1^e$ and $Q_2^e$ are heat inputs. In matrix form, we have
$$\frac{A_e k_e}{h_e}\left[\begin{array}{rr} 1 & -1 \ -1 & 1 \end{array}\right]\left{\begin{array}{l} T_1^e \ T_2^e \end{array}\right}=\left{\begin{array}{l} Q_1^e \ Q_2^e \end{array}\right}$$

数学代写|有限元代写Finite Element Method代考|Model Boundary Value Problem

Consider the problem of finding the function $u(x)$ that satisfies the differential equation
$$-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u-f=0 \text { for } 0<x<L$$
and the boundary conditions
$$u(0)=u_0,\left.\quad\left(a \frac{d u}{d x}\right)\right|_{x=L}=Q_L$$
where $a=a(x), c=c(x), f=f(x), u_0$, and $Q_L$ are known quantities, called the data of the problem. Equation (3.4.1) arises in connection with the analytical description of many physical processes. For example, conduction and convection heat transfer in the bar shown in Fig. 3.4.1(a), flow through channels and pipes, transverse deflection of cables, axial deformation of bars shown in Fig. 3.4.1(b), and many other physical processes are described by Eq. (3.4.1). A sample list of field problems described by Eq. (3.4.1) when $c(x)=0$ is presented in Table 3.4.1. Thus, if we can develop a numerical procedure by which Eq. (3.4.1) can be solved for all possible boundary conditions, the procedure can be used to solve all field problems listed in Table 3.4.1. This fact provides us with the motivation to use Eq. (3.4.1) as the model second-order equation in one dimension.

数学代写|有限元代写Finite Element Method代考|One-Dimensional Heat Transfer

\begin{aligned} \text { temperature gradient } & =\text { difference in temperature } / \text { length } \ \text { heat flux, } q & =\text { conductivity } \times(- \text { temperature gradient }) \ \text { heat, } Q & =\text { heat flux } \times \text { area of cross section } \end{aligned}

\begin{aligned} & Q_1^e=A_e q_1^e=-A_e k_e \frac{T_2^e-T_1^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_1^e-T_2^e\right) \ & Q_2^e=A_e q_2^e=-A_e k_e \frac{T_1^e-T_2^e}{h_e}=\frac{A_e k_e}{h_e}\left(T_2^e-T_1^e\right) \end{aligned}

$$\frac{A_e k_e}{h_e}\left[\begin{array}{rr} 1 & -1 \ -1 & 1 \end{array}\right]\left{\begin{array}{l} T_1^e \ T_2^e \end{array}\right}=\left{\begin{array}{l} Q_1^e \ Q_2^e \end{array}\right}$$

数学代写|有限元代写Finite Element Method代考|Model Boundary Value Problem

$$-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u-f=0 \text { for } 0<x<L$$

$$u(0)=u_0,\left.\quad\left(a \frac{d u}{d x}\right)\right|_{x=L}=Q_L$$

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|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。

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Since this book is concerned with the finite element solutions of differential equations arising mostly in engineering, although the basic developments are also valid for problems in applied sciences, it is useful to review the governing equations, including the boundary conditions (but not the initial conditions) of a continuum occupying a closed bounded region $\Omega$ with boundary $\Gamma$ which serve as references in the coming chapters. Although sufficient background is given every time we consider a differential equation
or a problem for analysis, the summary of equations included in this section serves as a quick reference. The equations are summarized under three subject areas of engineering that will receive considerable attention in this book: heat transfer, fluid mechanics, and solid mechanics. Since the present exposure is only to summarize the equations, the readers may wish to consult books that contain detailed treatment of the subjects.

数学代写|有限元代写Finite Element Method代考|Heat Transfer

The principle of conservation of energy applied to a solid medium yields
$$\rho c_v \frac{\partial T}{\partial t}-\boldsymbol{\nabla} \cdot(k \nabla T)=g$$
where $\nabla$ is the del operator (see Section 2.2.1.3), $T$ is the temperature, $g$ is the rate of internal heat generation per unit volume, $k$ is the conductivity of the (isotropic) solid, $\rho$ is the density, and $c v$ is the specific heat at constant volume. The expanded form of Eq. (2.6.1) (for constant $\mathrm{k}$ ) in rectangular Cartesian system $(x, y, z)$ is given by
$$\rho c_v \frac{\partial T}{\partial t}-k\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)=g$$
In cylindrical coordinate system $(r, \theta, z)$, Eq.(2.6.1) takes the form
$$\rho c_v \frac{\partial T}{\partial t}-k\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}+\frac{\partial^2 T}{\partial z^2}\right]=g$$
The second-order equations in Eqs. (2.6.1)-(2.6.3) are to be solved subjected to suitable boundary conditions. The boundary conditions involve specifying either the value of the temperature $T$ or balancing the heat flux normal to the boundary $q_n=\hat{\mathbf{n}} \cdot \mathbf{q}$ at a boundary point:
$$T=\hat{T} \quad \text { or } \quad q_n+\beta\left(T-T_{\infty}\right)=\hat{q}$$
where $\hat{T}$ and $\hat{q}$ denote the specified temperature and heat flux, respectively. Heat flux vector $\mathbf{q}$ is related to the gradient of temperature by Fourier’s heat conduction law (for the isotropic case)

数学代写|有限元代写Finite Element Method代考|Heat Transfer

$$\rho c_v \frac{\partial T}{\partial t}-\boldsymbol{\nabla} \cdot(k \nabla T)=g$$

$$\rho c_v \frac{\partial T}{\partial t}-k\left(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\right)=g$$

$$\rho c_v \frac{\partial T}{\partial t}-k\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\frac{1}{r^2} \frac{\partial^2 T}{\partial \theta^2}+\frac{\partial^2 T}{\partial z^2}\right]=g$$

$$T=\hat{T} \quad \text { or } \quad q_n+\beta\left(T-T_{\infty}\right)=\hat{q}$$

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。