## 数学代写|有限元方法代写Finite Element Method代考|Fluid Mechanics

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Fluid Mechanics

The equations governing flows of viscous incompressible fluids under isothermal conditions are listed here (listing the conservation principles that give rise to the equations). Further, all nonlinear terms are omitted. In addition to the vector form, only the Cartesian component form is listed, and the summation convention of Section 2.2.1.2 is adopted.
Conservation of mass (Continuity equation)
$$\operatorname{div}(\rho \mathbf{v})=0, \quad \rho \frac{\partial v_i}{\partial x_i}=0$$
Conservation of linear momentum (equations of motion): $\left(\sigma_{i j}=\sigma_{j i}\right)$
$$\boldsymbol{\nabla} \cdot \boldsymbol{\sigma}+\mathbf{f}=\rho \frac{\partial \mathbf{v}}{\partial t}, \quad \frac{\partial \sigma_{j i}}{\partial x_j}+f_i=\rho \frac{\partial v_i}{\partial t}$$
Constitutive relations
$$\sigma=2 \mu \mathbf{D}-P \mathbf{I}, \quad \sigma_{i j}=2 \mu D_{i j}-P \delta_{i j}$$
Kinematic relations
$$\mathbf{D}=\frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{v}+(\boldsymbol{\nabla} \mathbf{v})^{\mathrm{T}}\right], \quad D_{i j}=\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)$$
Here $\mathbf{v}$ is the velocity vector, $\sigma$ is the Cauchy stress tensor, $\mathbf{D}$ is the symmetric part of the velocity gradient tensor, $P$ is the hydrostatic pressure, $\mathbf{f}$ is the body force vector, $\rho$ is the density, and $\mu$ is the viscosity of the fluid. The boundary conditions involve specifying a velocity component $v_i$ or stress vector component $t_i \equiv n_j \sigma_{j i}$ at a boundary point, where $n j$ denote the direction cosines of a unit normal vector on the boundary
$$\mathbf{v}=\hat{\mathbf{v}} \text { or } \hat{\mathbf{n}} \cdot \sigma=\hat{\mathbf{t}} ; \quad v_i=\hat{v}i \text { or } n_j \sigma{j i}=\hat{t}_i$$

## 数学代写|有限元方法代写Finite Element Method代考|Solid Mechanics

Here we summarize the governing equations of a linearized, isotropic, elastic solid.
Momentum equations $\left(\sigma_{j i}=\sigma_{i j}\right)$
$$\boldsymbol{\nabla} \cdot \boldsymbol{\sigma}+\mathbf{f}=\rho \frac{d \mathbf{v}}{d t}, \quad \frac{\partial \sigma_{j i}}{\partial x_j}+f_i=\rho \frac{d \mathbf{v}}{d t}$$
Constitutive relations
$$\sigma=2 \mu \varepsilon+\lambda(\operatorname{tr} \varepsilon) \mathrm{I}, \quad \sigma_{i j}=2 \mu \varepsilon_{i j}+\lambda \varepsilon_{k k} \delta_{i j}$$
Kinematic relations
$$\varepsilon=\frac{1}{2}\left[\nabla \mathbf{u}+(\nabla \mathbf{u})^{\mathrm{T}}\right], \quad \varepsilon_{i j}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$
Here $\mathbf{u}$ is the displacement vector, $\sigma$ is the Cauchy stress tensor, $\varepsilon$ is the symmetric part of the displacement gradient tensor, $\mathbf{f}$ is the body force vector, $\rho$ is the density, and $\mu$ and $\lambda$ are the Lamé (material) parameters. The boundary conditions involve specifying a displacement component $u_i$ or stress vector component $t_i \equiv n_j \sigma_{j i}$ at a boundary point
$$\mathbf{u}=\hat{\mathbf{u}} \text { or } \hat{\mathbf{n}} \cdot \sigma=\hat{\mathbf{t}} ; \quad u_i=\hat{u}i \text { or } n_j \sigma{j i}=\hat{t}_i$$

## 数学代写|有限元方法代写Finite Element Method代考|Fluid Mechanics

$$\operatorname{div}(\rho \mathbf{v})=0, \quad \rho \frac{\partial v_i}{\partial x_i}=0$$

$$\boldsymbol{\nabla} \cdot \boldsymbol{\sigma}+\mathbf{f}=\rho \frac{\partial \mathbf{v}}{\partial t}, \quad \frac{\partial \sigma_{j i}}{\partial x_j}+f_i=\rho \frac{\partial v_i}{\partial t}$$

$$\sigma=2 \mu \mathbf{D}-P \mathbf{I}, \quad \sigma_{i j}=2 \mu D_{i j}-P \delta_{i j}$$

$$\mathbf{D}=\frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{v}+(\boldsymbol{\nabla} \mathbf{v})^{\mathrm{T}}\right], \quad D_{i j}=\frac{1}{2}\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i}\right)$$

$$\mathbf{v}=\hat{\mathbf{v}} \text { or } \hat{\mathbf{n}} \cdot \sigma=\hat{\mathbf{t}} ; \quad v_i=\hat{v}i \text { or } n_j \sigma{j i}=\hat{t}_i$$

## 数学代写|有限元方法代写Finite Element Method代考|Solid Mechanics

$$\boldsymbol{\nabla} \cdot \boldsymbol{\sigma}+\mathbf{f}=\rho \frac{d \mathbf{v}}{d t}, \quad \frac{\partial \sigma_{j i}}{\partial x_j}+f_i=\rho \frac{d \mathbf{v}}{d t}$$

$$\sigma=2 \mu \varepsilon+\lambda(\operatorname{tr} \varepsilon) \mathrm{I}, \quad \sigma_{i j}=2 \mu \varepsilon_{i j}+\lambda \varepsilon_{k k} \delta_{i j}$$

$$\varepsilon=\frac{1}{2}\left[\nabla \mathbf{u}+(\nabla \mathbf{u})^{\mathrm{T}}\right], \quad \varepsilon_{i j}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right)$$

$$\mathbf{u}=\hat{\mathbf{u}} \text { or } \hat{\mathbf{n}} \cdot \sigma=\hat{\mathbf{t}} ; \quad u_i=\hat{u}i \text { or } n_j \sigma{j i}=\hat{t}_i$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|Approximation functions

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Approximation functions

In this section we discuss the properties of the set of approximation functions $\left{\phi_i\right}$ and $\phi_0$ used in the n-parameter Ritz solution in Eq. (2.5.4). First, we note that $u_n$ must satisfy only the specified essential boundary conditions of the problem, since the specified natural boundary conditions are included in the variational problem in Eq. (2.5.1). The particular form of $u_n$ in Eq. (2.5.4) facilitates satisfaction of specified boundary conditions. To see this, suppose that the approximate solution is sought in the form
$$u_n=\sum_{j=1}^n c_j \phi_j(x)$$
and suppose that the specified essential boundary condition is $u\left(x_0\right)=u_0$. Then $u_n$ must also satisfy the condition $u_n\left(x_0\right)=u_0$ at a boundary point $x=x_0$
$$\sum_{j=1}^n c_j \phi_j\left(x_0\right)=u_0$$
Since $c_j$ are unknown parameters to be determined, it is not easy to choose $\phi_j$ (x) such that the above relation holds for all $c_j$. If $u_0=0$, then we can select all $\phi_j$ such that $\phi_j\left(x_0\right)=0$ and satisfy the condition $u_n\left(x_0\right)=0$. By writing the approximate solution $u_n$ in the form Eq. (2.5.4), a sum of a homogeneous part $\sum c_j \phi_j(x)$ and a nonhomogeneous part $\phi_0(x)$, we require $\phi_0(x)$ to satisfy the specified essential boundary conditions while the homogeneous part vanishes at the same boundary point where the essential boundary condition is specified. This follows from
$$\begin{gathered} u_n\left(x_0\right)=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+\phi_0\left(x_0\right) \ u_0=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+u_0 \rightarrow \sum_{j=1}^n c_j \phi_j\left(x_0\right)=0 \end{gathered}$$
which is satisfied, for arbitrary $c_j$, by choosing $\phi_j\left(x_0\right)=0$.

## 数学代写|有限元方法代写Finite Element Method代考|The Method of Weighted Residuals

As noted in Section 2.4.3, one can always write the weighted-integral form of a differential equation, whether the equation is linear or nonlinear (in the dependent variables). The weak form can be developed if the equations are second-order or higher, even if they are nonlinear.

The weighted-residual method is a generalization of the Galerkin method in that the weight functions can be chosen from an independent set of functions, and it requires only the weighted-integral form to determine the parameters. Since the latter form does not include any of the specified boundary conditions of the problem, the approximation functions must be selected such that the approximate solution satisfies all of the specified boundary conditions. In addition, the weight functions can be selected independently of the approximation functions, but are required to be linearly independent so that the resulting algebraic equations are linearly independent.
We discuss the general method of weighted residuals first, and then consider certain special cases that are known by specific names (e.g., the Galerkin method, the collocation method, the least-squares method and so on). Although a limited use of the weighted-residual method is made in this book, it is informative to have a knowledge of this class of methods for use in the formulation of certain nonlinear problems and non-self-adjoint problems.

The method of weighted residuals can be described in its generality by considering the operator equation
$$A(u)=f \text { in } \Omega$$
where $A$ is an operator (linear or nonlinear), often a differential operator, acting on the dependent variable $u$, and $f$ is a known function of the independent variables. Some examples of such operators are given below.
$$A(u)=-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u$$
$$A(u)=\frac{d^2}{d x^2}\left(b \frac{d^2 u}{d x^2}\right)$$
$$A(u)=-\left[\frac{\partial}{\partial x}\left(k_x \frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\left(k_y \frac{\partial u}{\partial y}\right)\right]$$
$$A(u)=-\frac{d}{d x}\left(u \frac{d u}{d x}\right)$$
$$A(u, v)=u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+\frac{\partial^2 u}{\partial x^2}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)$$
For an operator $A$ to be linear in its arguments, it must satisfy the relation
$$A(\alpha u+\beta v)=\alpha A(u)+\beta A(v)$$
for any scalars $\alpha$ and $\beta$ and dependent variables $u$ and $v$. It can be easily verified that all operators in Eq. (2.5.52), except for those in (4) and (5), are linear. When an operator does not satisfy the condition in Eq. (2.5.53), it is said to be nonlinear.

## 数学代写|有限元方法代写Finite Element Method代考|Approximation functions

$$u_n=\sum_{j=1}^n c_j \phi_j(x)$$

$$\sum_{j=1}^n c_j \phi_j\left(x_0\right)=u_0$$

$$\begin{gathered} u_n\left(x_0\right)=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+\phi_0\left(x_0\right) \ u_0=\sum_{j=1}^n c_j \phi_j\left(x_0\right)+u_0 \rightarrow \sum_{j=1}^n c_j \phi_j\left(x_0\right)=0 \end{gathered}$$

## 数学代写|有限元方法代写Finite Element Method代考|The Method of Weighted Residuals

$$A(u)=f \text { in } \Omega$$

$$A(u)=-\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u$$
$$A(u)=\frac{d^2}{d x^2}\left(b \frac{d^2 u}{d x^2}\right)$$
$$A(u)=-\left[\frac{\partial}{\partial x}\left(k_x \frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\left(k_y \frac{\partial u}{\partial y}\right)\right]$$
$$A(u)=-\frac{d}{d x}\left(u \frac{d u}{d x}\right)$$
$$A(u, v)=u \frac{\partial u}{\partial x}+v \frac{\partial u}{\partial y}+\frac{\partial^2 u}{\partial x^2}+\frac{\partial}{\partial y}\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)$$

$$A(\alpha u+\beta v)=\alpha A(u)+\beta A(v)$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|The Principle of Minimum Total Potential Energy

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|The Principle of Minimum Total Potential Energy

The principle of virtual work discussed in the previous section is applicable to any continuous body with arbitrary constitutive behavior (e.g., linear or nonlinear elastic materials). The principle of minimum total potential energy is obtained as a special case from the principle of virtual displacements when the constitutive relations can be obtained from a potential function. Here we restrict our discussion to materials that admit existence of a strain energy potential such that the stress is derivable from it. Such materials are termed hyperelastic.
For elastic bodies (in the absence of temperature variations), there exists a strain energy potential $U_0$ such that [see Eq. (2.3.5)]
$$\sigma_{i j}=\frac{\partial U_0}{\partial \varepsilon_{i j}}$$
The strain energy density $U_0$ is a function of strains at a point and is assumed to be positive definite. The statement of the principle of virtual displacements, $\delta W=0$, can be expressed in terms of the strain energy density $U_0$ as
\begin{aligned} 0=\delta W & =\int_{\Omega} \sigma_{i j} \delta \varepsilon_{i j} d \Omega-\left[\int_{\Omega} \mathbf{f} \cdot \delta \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \delta \mathbf{u} d s\right] \ & =\int_{\Omega} \frac{\partial U_0}{\partial \varepsilon_{i j}} \delta \varepsilon_{i j} d \Omega+\delta V_E \ & =\int_{\Omega} \delta U_0 d \Omega+\delta V_E=\delta\left(U+V_E\right) \equiv \delta \Pi \end{aligned}
where
$$V_E=-\left[\int_{\Omega} \mathbf{f} \cdot \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \mathbf{u} d s\right]$$
is the potential energy due to external loads and $U$ is the strain energy potential
$$U=\int_{\Omega} U_0 d \Omega$$

## 数学代写|有限元方法代写Finite Element Method代考|Residual Function

Consider the problem of solving the differential equation
$$-\frac{d}{d x}\left[a(x) \frac{d u}{d x}\right]+c u=f(x) \text { for } 0<x<L$$
for $u(x)$, which is subject to the boundary conditions
$$u(0)=u_0, \quad\left[a \frac{d u}{d x}+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$$ Here $a(x), c(x)$, and $f(x)$ are known functions of the coordinate $x ; u_0, u{\infty}, \beta$, and $Q_L$ are known values, and $L$ is the size of the one-dimensional domain. When the specified values are nonzero $\left(u_0 \neq 0\right.$ or $\left.Q_L \neq 0\right)$, the boundary conditions are said to be nonhomogeneous; when the specified values are zero the boundary conditions are said to be homogeneous. The homogeneous form of the boundary condition $u(0)=u_0$ is $u(0)=0$, and the homogeneous form of the boundary condition $\left[a(d u / d x)+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$ is $[a(d u / d x)+$ $\left.\beta\left(u-u{\infty}\right)\right]_{x=L}=0$.

Equations of the type in Eq. (2.4.1) arise, for example, in the study of 1-D heat flow in a rod with surface convection (see Example 1.2.2), as shown in Fig. 2.4.1(a). In this case, $a=k A$, with $k$ being the thermal conductivity and $A$ the cross-sectional area, $c=\beta P$, with $\beta$ being the heat transfer coefficient, $P$ the perimeter of the rod, and $L$ the length of the rod; $f$ denotes the heat generation term, $u_0$ is the specified temperature at $x=0, Q_L$ is the specified heat at $x=L$, and $u_{\infty}$ is the temperature of the surrounding medium. Another example where Eqs. (2.4.1) and (2.4.2) arise is provided by the axial deformation of a bar (see Example 1.2.3), as shown in Fig. 2.4.1(b). In this case, $a=E A$, with $E$ being the Young’s modulus and $A$ the cross-sectional area, $c$ is the spring constant associated with the shear resistance offered by the surrounding medium (as discussed in Example 1.2.3), and $L$ is the length of the bar; $f$ denotes the body force term, $u_0$ is the specified displacement at $x$ $=0\left(u_0=0\right), Q_L$ is the specified point load at $x=L$, and $u_{\infty}=0$. Other physical problems are also described by the same equation, but with different meaning of the variables.

## 数学代写|有限元方法代写Finite Element Method代考|The Principle of Minimum Total Potential Energy

$$\sigma_{i j}=\frac{\partial U_0}{\partial \varepsilon_{i j}}$$

\begin{aligned} 0=\delta W & =\int_{\Omega} \sigma_{i j} \delta \varepsilon_{i j} d \Omega-\left[\int_{\Omega} \mathbf{f} \cdot \delta \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \delta \mathbf{u} d s\right] \ & =\int_{\Omega} \frac{\partial U_0}{\partial \varepsilon_{i j}} \delta \varepsilon_{i j} d \Omega+\delta V_E \ & =\int_{\Omega} \delta U_0 d \Omega+\delta V_E=\delta\left(U+V_E\right) \equiv \delta \Pi \end{aligned}

$$V_E=-\left[\int_{\Omega} \mathbf{f} \cdot \mathbf{u} d \Omega+\int_{\Gamma_\sigma} \hat{\mathbf{t}} \cdot \mathbf{u} d s\right]$$

$$U=\int_{\Omega} U_0 d \Omega$$

## 数学代写|有限元方法代写Finite Element Method代考|Residual Function

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

$$u(0)=u_0, \quad\left[a \frac{d u}{d x}+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$$其中$a(x), c(x)$、$f(x)$为坐标$x ; u_0, u{\infty}, \beta$的已知函数，$Q_L$为已知值，$L$为一维域的大小。当指定值为非零$\left(u_0 \neq 0\right.$或$\left.Q_L \neq 0\right)$时，边界条件是非齐次的;当指定值为零时，边界条件称为齐次。边界条件$u(0)=u_0$的齐次形式为$u(0)=0$，边界条件$\left[a(d u / d x)+\beta\left(u-u_{\infty}\right)\right]{x=L}=Q_L$的齐次形式为$[a(d u / d x)+$$\left.\beta\left(u-u{\infty}\right)\right]_{x=L}=0。 例如，在研究具有表面对流的杆内一维热流(见例1.2.2)时，会出现式(2.4.1)式的方程，如图2.4.1(a)所示。在这种情况下，a=k A, k为导热系数，A为截面积，c=\beta P, \beta为传热系数，P为棒的周长，L为棒的长度;f为发热量项，u_0为x=L处的规定温度，x=0, Q_L为处的规定热量，u_{\infty}为周围介质温度。另一个例子是等式。(2.4.1)和(2.4.2)的产生是由杆的轴向变形提供的(见例1.2.3)，如图2.4.1(b)所示。在这种情况下，a=E A, E是杨氏模量，A是截面积，c是与周围介质提供的剪切阻力相关的弹簧常数(如例1.2.3所述)，L是杆的长度;f为体力项，u_0为x处的规定位移，=0\left(u_0=0\right), Q_L为x=L处的规定点荷载，u_{\infty}=0。其他物理问题也可以用相同的方程来描述，但变量的含义不同。 统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。 ## 金融工程代写 金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。 ## 非参数统计代写 非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。 ## 广义线性模型代考 广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。 术语 广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。 ## 有限元方法代写 有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。 有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。 tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。 ## 随机分析代写 随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。 ## 时间序列分析代写 随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。 随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。 ## 回归分析代写 多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。 ## 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。 ## 数学代写|有限元方法代写Finite Element Method代考|Matrix addition and multiplication of a matrix by a scalar 如果你也在 怎样代写有限元方法finite differences method 这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。有限元方法finite differences method在数值分析中，是一类通过用有限差分逼近导数解决微分方程的数值技术。空间域和时间间隔（如果适用）都被离散化，或被分成有限的步骤，通过解决包含有限差分和附近点的数值的代数方程来逼近这些离散点的解的数值。 有限元方法finite differences method有限差分法将可能是非线性的常微分方程（ODE）或偏微分方程（PDE）转换成可以用矩阵代数技术解决的线性方程系统。现代计算机可以有效地进行这些线性代数计算，再加上其相对容易实现，使得FDM在现代数值分析中得到了广泛的应用。今天，FDM与有限元方法一样，是数值解决PDE的最常用方法之一。 statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。 ## 数学代写|有限元方法代写Finite Element Method代考|Matrix addition and multiplication of a matrix by a scalar The sum of two matrices of the same size is defined to be a matrix of the same size obtained by simply adding the corresponding elements. If \mathbf{A} is an m \times n matrix and \mathbf{B} is an m \times n matrix, their sum is an m \times n matrix, \mathbf{C}, with$$ c_{i j}=a_{i j}+b_{i j} \text { for all } i, j $$A constant multiple of a matrix is equal to the matrix obtained by multiplying all of the elements by the constant. That is, the multiple of a matrix \mathbf{A} by a scalar \alpha, \alpha \mathbf{A}, is the matrix obtained by multiplying each of its elements with \alpha :$$ \mathbf{A}=\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1 n} \ a_{21} & a_{22} & \ldots & a_{2 n} \ \vdots & \vdots & \ldots & \vdots \ a_{m 1} & a_{m 2} & \ldots & a_{m n} \end{array}\right], \quad \alpha \mathbf{A}=\left[\begin{array}{cccc} \alpha a_{11} & \alpha a_{12} & \ldots & \alpha a_{1 n} \ \alpha a_{21} & \alpha a_{22} & \ldots & \alpha a_{2 n} \ \vdots & \vdots & & \vdots \ \alpha a_{m 1} & \alpha a_{m 2} & \ldots & \alpha a_{m n} \end{array}\right] $$Matrix addition has the following properties: 1. Addition is commutative: \mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}. 2. Addition is associative: \mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}. 3. There exists a unique matrix \mathbf{0}, such that \mathbf{A}+\mathbf{0}=\mathbf{0}+\mathbf{A}=\mathbf{A}. The matrix \mathbf{0} is called zero matrix; all elements of it are zeros. 4. For each matrix \mathbf{A}, there exists a unique matrix -\mathbf{A} such that \mathbf{A}+(-\mathbf{A}) =\mathbf{0}. 5. Addition is distributive with respect to scalar multiplication: \alpha(\mathbf{A}+\mathbf{B}) =\alpha \mathbf{A}+\alpha \mathbf{B}. ## 数学代写|有限元方法代写Finite Element Method代考|Matrix transpose and symmetric and skew symmetric matrices If \mathbf{A} is an m \times n matrix, then the n \times m matrix obtained by interchanging its rows and columns is called the transpose of \mathbf{A} and is denoted by \mathbf{A}^{\mathrm{T}}. An example of a transpose is provided by$$ \mathbf{A}=\left[\begin{array}{rrr} 2 & -3 & 4 \ 5 & 6 & 8 \ 1 & 5 & 3 \ -2 & 9 & 0 \end{array}\right], \quad \mathbf{A}^{\mathrm{T}}=\left[\begin{array}{rrrr} 2 & 5 & 1 & -2 \ -3 & 6 & 5 & 9 \ 4 & 8 & 3 & 0 \end{array}\right] $$The following basic properties of a transpose should be noted: \left(\mathbf{A}^{\mathrm{T}}\right)^{\mathrm{T}}=\mathbf{A} (\mathbf{A}+\mathbf{B})^{\mathrm{T}}=\mathbf{A}^{\mathrm{T}}+\mathbf{B}^{\mathrm{T}} A square matrix \mathbf{A} of real numbers is said to be symmetric if \mathbf{A}^{\mathrm{T}}=\mathbf{A}. It is said to be skew symmetric or antisymmetric if \mathbf{A}^{\mathrm{T}}=-\mathbf{A}. In terms of the elements of \mathbf{A}, these definitions imply that \mathbf{A} is symmetric if and only if a_{i j}= a_{j i}, and it is skew symmetric if and only if a_{i j}=-a_{j i}. Note that the diagonal elements of a skew symmetric matrix are always zero since a_{i j}=-a_{i j} implies a_{i j}=0 for i=j. Examples of symmetric and skew symmetric matrices, respectively, are$$ \left[\begin{array}{rrrr} 5 & -2 & 11 & 9 \ -2 & 4 & 14 & -3 \ 11 & 14 & 13 & 8 \ 9 & -3 & 8 & 21 \end{array}\right], \quad\left[\begin{array}{rrrr} 0 & -10 & 23 & 3 \ 10 & 0 & 21 & 7 \ -23 & -21 & 0 & 12 \ -3 & -7 & -12 & 0 \end{array}\right] $$## 有限元方法代考 ## 数学代写|有限元方法代写Finite Element Method代考|Matrix addition and multiplication of a matrix by a scalar 两个大小相同的矩阵的和定义为简单地将相应的元素相加得到的大小相同的矩阵。如果\mathbf{A}是一个m \times n矩阵，\mathbf{B}是一个m \times n矩阵，那么它们的和就是一个m \times n矩阵\mathbf{C}$$ c_{i j}=a_{i j}+b_{i j} \text { for all } i, j $$一个常数乘以一个矩阵等于由所有元素乘以这个常数得到的矩阵。即矩阵\mathbf{A}与标量\alpha, \alpha \mathbf{A}的乘积，是矩阵的每个元素与\alpha相乘得到的矩阵:$$ \mathbf{A}=\left[\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1 n} \ a_{21} & a_{22} & \ldots & a_{2 n} \ \vdots & \vdots & \ldots & \vdots \ a_{m 1} & a_{m 2} & \ldots & a_{m n} \end{array}\right], \quad \alpha \mathbf{A}=\left[\begin{array}{cccc} \alpha a_{11} & \alpha a_{12} & \ldots & \alpha a_{1 n} \ \alpha a_{21} & \alpha a_{22} & \ldots & \alpha a_{2 n} \ \vdots & \vdots & & \vdots \ \alpha a_{m 1} & \alpha a_{m 2} & \ldots & \alpha a_{m n} \end{array}\right] $$矩阵加法具有以下性质: 加法是交换的:\mathbf{A}+\mathbf{B}=\mathbf{B}+\mathbf{A}。 加法是结合法:\mathbf{A}+(\mathbf{B}+\mathbf{C})=(\mathbf{A}+\mathbf{B})+\mathbf{C}。 存在一个唯一矩阵\mathbf{0}，使得\mathbf{A}+\mathbf{0}=\mathbf{0}+\mathbf{A}=\mathbf{A}。矩阵\mathbf{0}称为零矩阵;它的所有元素都是0。 对于每个矩阵\mathbf{A}，存在一个唯一的矩阵-\mathbf{A}，使得\mathbf{A}+(-\mathbf{A})$$=\mathbf{0}$。

$$\left[\begin{array}{rrrr} 5 & -2 & 11 & 9 \ -2 & 4 & 14 & -3 \ 11 & 14 & 13 & 8 \ 9 & -3 & 8 & 21 \end{array}\right], \quad\left[\begin{array}{rrrr} 0 & -10 & 23 & 3 \ 10 & 0 & 21 & 7 \ -23 & -21 & 0 & 12 \ -3 & -7 & -12 & 0 \end{array}\right]$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|Boundary Value, Initial Value, and Eigenvalue Problems

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Boundary Value, Initial Value, and Eigenvalue Problems

The objective of most analysis is to determine unknown functions, called dependent variables, that are governed by a set of differential equations posed in a given domain $\Omega$ and some conditions on the boundary $\Gamma$ of the
domain $\Omega$. Often, a domain not including its boundary is called an open domain. A domain $\Omega$ with its boundary $\Gamma$ is called a closed domain and is denoted by $\bar{\Omega}=\Omega \cup \Gamma$.
A function $u$ of several independent variables (or coordinates) $(x, y, \cdots)$ is said to be of class $C^m(\Omega)$ in a domain $\Omega$ if all its partial derivatives with respect to $(x, y, \cdots)$ of order up to and including $m$ exist and are continuous in $\Omega$. Thus, if $u$ is of class $C^0$ in a two-dimensional domain $\Omega$ then $u$ is continuous in $\Omega$ (i.e., $\partial u / \partial x$ and $\partial u / \partial y$ exist but may not be continuous). Similarly, if $u$ is of class $c_1$, then $u, \partial u / \partial x$ and $\partial u / \partial y$ exist and are continuous (i.e., $\partial^2 u / \partial x^2, \partial^2 u / \partial y^2$, and $\partial^2 u / \partial y \partial x$ exist but may not be continuous).

Similarly, if $u$ is of class $c_1$, then $u, \partial u / \partial x$ and $\partial u / \partial y$ exist and are continuous (i.e., $\partial^2 u / \partial x^2, \partial^2 u / \partial y^2$, and $\partial^2 u / \partial y \partial x$ exist but may not be continuous).

When the dependent variables are functions of one independent variable (say, $x$ ), the domain is a line segment (i.e., one-dimensional) and the end points of the domain are called boundary points. When the dependent variables are functions of two independent variables (say, $x$ and $y$ ), the domain is two-dimensional and the boundary is the closed curve enclosing it. In a threedimensional domain, dependent variables are functions of three independent variables (say $x, y$, and $z$ ) and the boundary is a two-dimensional surface.

## 数学代写|有限元方法代写Finite Element Method代考|Boundary value problems

Steady-state heat transfer in a fin and axial deformation of a bar: Find $u(x)$ that satisfies the second-order differential equation and boundary conditions:
$$\begin{gathered} -\frac{d}{d x}\left(a \frac{d u}{d x}\right)+c u=f \text { for } 0<x<L \ u(0)=u_0, \quad\left(a \frac{d u}{d x}\right)_{x=L}=q_0 \end{gathered}$$
The domain and boundary points are identified in Fig. 2.2.3.

Bending of elastic beams under transverse load: Find $w(x)$ that satisfies the fourthorder differential equation and boundary conditions:
$$\begin{gathered} \frac{d^2}{d x^2}\left(E I \frac{d^2 w}{d x^2}\right)+c w=f \quad \text { for } \quad 0<x<L \ w(0)=w_0, \quad\left(-\frac{d w}{d x}\right){x=0}=\theta_0 \ {\left[\frac{d}{d x}\left(E I \frac{d^2 w}{d x^2}\right)\right]{x=L}=V_0, \quad\left(E I \frac{d^2 w}{d x^2}\right)_{x=L}=M_0} \end{gathered}$$
The domain and boundary points for this case are the same as shown in Fig. 2.2.3. However, the physics behind the equations is different, as we shall see shortly.

Steady heat conduction in a two-dimensional region and transverse deflections of a membrane: Find $u(x, y)$ that satisfies the second-order partial differential equation and boundary conditions:
\begin{aligned} & -\left[\frac{\partial}{\partial x}\left(a_{x x} \frac{\partial u}{\partial x}\right)+\frac{\partial}{\partial y}\left(a_{y y} \frac{\partial u}{\partial y}\right)\right]+a_{00} u=f \quad \text { in } \Omega \ & u=u_0 \text { on } \Gamma_u, \quad\left(a_{x x} \frac{\partial u}{\partial x} n_x+a_{y y} \frac{\partial u}{\partial y} n_y\right)=q_0 \text { on } \Gamma_q \end{aligned}
where $\left(n_x, n_y\right)$ are the direction cosines on the unit normal vector $\hat{\mathbf{n}}$ to the boundary $\Gamma_q$. The domain $\Omega$ and two parts of the boundary $\Gamma_u$ and $\Gamma_q$ are shown in Fig. 2.2.4.

## 数学代写|有限元方法代写Finite Element Method代考|Variational Principles and Methods

“直接变分方法”是指利用变分原理，如固体力学和结构力学中的虚功原理和最小总势能原理，来确定问题的近似解的方法(参见Oden和Reddy[1]和Reddy[2])。在经典意义上，变分原理与寻找极值(即，最小值或最大值)或关于问题变量的函数的平稳值有关。泛函包括问题的所有内在特征，如控制方程、边界和/或初始条件以及约束条件(如果有的话)。在固体和结构力学问题中，泛函表示系统的总能量，而在其他问题中，它只是控制方程的积分表示。

## 数学代写|有限元方法代写Finite Element Method代考|Variational Formulations

“变分公式”一词的经典用法是指一个泛函的构造(其含义将很快被阐明)或一个与问题的控制方程等效的变分原理。这个短语的现代用法是指将控制方程转化为等价的加权积分陈述的公式，这些陈述不一定等同于变分原理。甚至那些在经典意义上不承认变分原理的问题(例如，控制粘性或非粘性流体流动的Navier-Stokes方程)现在也可以用加权积分表述出来。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|General Introduction

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

## 数学代写|有限元方法代写Finite Element Method代考|Variational Principles and Methods

This chapter is devoted to a review of mathematical preliminaries that prove to be useful in the sequel and a study of integral formulations and more commonly used variational methods such as the Ritz, Galerkin, collocation, subdomain, and least-squares methods. Since the finite element method can be viewed as an element-wise application of a variational method, it is useful to learn how variational methods work. We begin with a discussion of the general meaning of the phrases “variational methods” and “variational formulations” used in the literature.
The phrase “direct variational methods” refers to methods that make use of variational principles, such as the principles of virtual work and the principle of minimum total potential energy in solid and structural mechanics, to determine approximate solutions of problems (see Oden and Reddy [1] and Reddy [2]). In the classical sense, a variational principle has to do with finding the extremum (i.e., minimum or maximum) or stationary values of a functional with respect to the variables of the problem. The functional includes all the intrinsic features of the problem, such as the governing equations, boundary and/or initial conditions, and constraint conditions, if any. In solid and structural mechanics problems, the functional represents the total energy of the system, and in other problems it is simply an integral representation of the governing equations.
Variational principles have always played an important role in mechanics. First, many problems of mechanics are posed in terms of finding the extremum (i.e., minima or maxima) and thus, by their nature, can be formulated in terms of variational statements. Second, there are problems that can be formulated by other means, such as the conservation laws, but these can also be formulated by means of variational principles. Third, variational formulations form a powerful basis for obtaining approximate solutions to practical problems, many of which are intractable otherwise. The principle of minimum total potential energy, for example, can be regarded as a substitute to the equations of equilibrium of an elastic body, as well as a basis for the development of displacement finite element models that can be used to determine approximate displacement and stress fields in the body. Variational formulations can also serve to unify diverse fields, suggest new theories, and provide a powerful means to study the existence and uniqueness of solutions to problems. Similarly, Hamilton’s principle can be used in lieu of the equations governing dynamical systems, and the variational forms presented by Biot replace certain equations in linear continuum thermodynamics.

## 数学代写|有限元方法代写Finite Element Method代考|Variational Formulations

The classical use of the phrase “variational formulations” refers to the construction of a functional (whose meaning will be made clear shortly) or a variational principle that is equivalent to the governing equations of the problem. The modern use of the phrase refers to the formulation in which the governing equations are translated into equivalent weighted-integral statements that are not necessarily equivalent to a variational principle. Even those problems that do not admit variational principles in the classical sense (e.g., the Navier-Stokes equations governing the flow of viscous or inviscid fluids) can now be formulated using weighted-integral statements.
The importance of variational formulations of physical laws, in the modern or general sense of the phrase, goes far beyond its use as simply an alternative to other formulations (see Oden and Reddy [1]). In fact, variational forms of the laws of continuum physics may be the only natural and rigorously correct way to think of them. While all sufficiently smooth fields lead to meaningful variational forms, the converse is not true: there exist physical phenomena which can be adequately modelled mathematically only in a variational setting; they are nonsensical when viewed locally.
The starting point for the discussion of the finite element method is differential equations governing the physical phenomena under study. As such, we shall first discuss why integral statements of the differential equations are needed.

## 数学代写|有限元方法代写Finite Element Method代考|Variational Principles and Methods

“直接变分方法”是指利用变分原理，如固体力学和结构力学中的虚功原理和最小总势能原理，来确定问题的近似解的方法(参见Oden和Reddy[1]和Reddy[2])。在经典意义上，变分原理与寻找极值(即，最小值或最大值)或关于问题变量的函数的平稳值有关。泛函包括问题的所有内在特征，如控制方程、边界和/或初始条件以及约束条件(如果有的话)。在固体和结构力学问题中，泛函表示系统的总能量，而在其他问题中，它只是控制方程的积分表示。

## 数学代写|有限元方法代写Finite Element Method代考|Variational Formulations

“变分公式”一词的经典用法是指一个泛函的构造(其含义将很快被阐明)或一个与问题的控制方程等效的变分原理。这个短语的现代用法是指将控制方程转化为等价的加权积分陈述的公式，这些陈述不一定等同于变分原理。甚至那些在经典意义上不承认变分原理的问题(例如，控制粘性或非粘性流体流动的Navier-Stokes方程)现在也可以用加权积分表述出来。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|CIVL6840

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|Consistent nodal load vector

Equations $(6.37)(\mathrm{c}, \mathrm{d})$ are used to find statically equivalent forces and moments acting on the nodes due to distributed forces acting along the element. These nodal forces (and moments) are referred to as the consistent nodal forces (and moments). Consider the concentrated and the distributed forces acting on the element shown in Fig. 6.3. By using Eqs. (6.37) and (6.29) the following consistent nodal force vectors are found.
Concentrated force: $-F_y \delta\left(x^{\prime}-a\right)$ acting along the $-y$-axis and located at $x^{\prime}=a$
$$\left{r_q^{\prime}\right}^{(e)}=-F_y^{\prime}\left{\frac{b^2\left(L^{(e)}+2 a\right)}{L^{(e) 3}} \frac{a b^2}{L^{(e) 2}} \frac{a^2\left(L^{(e)}+2 b\right)}{L^{(e) 3}}-\frac{a^2 b}{L^{(e) 2}}\right}^T$$
where $b=L-a$.
Linearly distributed force: $q_{y^{\prime}}=q_{y^{\prime} 1}+\frac{q_{y^{\prime} 2}-q_{y^{\prime} 1}}{L} x^{\prime}$. Note that $q_{y^{\prime}}$ is positive along the $+y^{\prime}$-axis.
$$\left{r_q^{\prime}\right}^{(e)}=\left{\frac{\left(7 q_{y_1}+3 q_{y_2}\right) L^{(c)}}{20} \frac{\left(3 q_{y_1}+2 q_{y_2}\right) L^{(e) 2}}{60} \frac{\left(3 q_{y 1}+7 q_{y_2}\right) L^{(e)}}{20}-\frac{\left(2 q_{y_1 1}+3 q_{y_2 2}\right) L^{(e) 2}}{60}\right}^T$$
Distributed force with constant magnitude: $q_{y^{\prime}}\left(x^{\prime}\right)=q_{y^{\prime} 0}$ which points along the $+y^{\prime}$-axis.
$$\left{r_q{ }^{\prime}\right}^{(e)}=\left{\frac{q_{y_0} L^{(e)}}{2} \frac{q_{y^{\prime} 0} L^{(e) 2}}{12} \frac{q_{y_0 0} L^{(e)}}{2}-\frac{q_{y_0 0} L^{(e) 2}}{12}\right}^T$$
The signs of the forces in these vectors are determined by the sign convention defined in Fig. 6.1.

## 数学代写|有限元方法代写Finite Element Method代考|General beam element with membrane

In many load bearing situations stretching and bending occurs on the same member, simultaneously (Fig. 6.4). The element stiffness matrix that represents the equilibrium of such a member is obtained by using the superposition of membrane and bending responses. Note that here we consider small deflection cases where these two effects can be assumed to be uncoupled from one another. The equilibrium equation for such a generic beam element is represented in matrix form as follows:
$$\left[k_{m b}^{\prime}\right]^{(e)}\left{d_{m b}\right}^{(e)}=\left{r_{m b}\right}^{(e)}$$
This relationship can be obtained by using a superposition of the membrane and beam mechanics, represented by Eqs. (5.22) and (6.38),
$$\left[k_m{ }^{\prime}\right]^{(e)}\left{d_m{ }^{\prime}\right}^{(e)}+\left[k_b^{\prime}\right]^{(e)}\left{d_b{ }^{\prime}\right}^{(e)}=\left{r_m{ }^{\prime}\right}^{(e)}+\left{r_b\right}^{(e)}$$
As the membrane and the bending actions are uncoupled (at least in this presentation) the following variables can be easily identified,
$$\left.\left[k_{m b}\right]^{\prime}\right]^{(e)}=\left[\begin{array}{cccccc} k_m & 0 & 0 & -k_m & 0 & 0 \ 0 & 12 k_b & 6 k_b L^{(e)} & 0 & -12 k_b & 6 k_b L^{(e)} \ 0 & 6 k_b L^{(e)} & 4 k_b L^{(e) 2} & 0 & -6 k_b L^{(e)} & 2 k_b L^{(e) 2} \ -k_m & 0 & 0 & k_m & 0 & 0 \ 0 & -12 k_b & -6 k_b L^{(e)} & 0 & 12 k_b & -6 k_b L^{(e)} \ 0 & 6 k_b L^{(e)} & 2 k_b L^{(e) 2} & 0 & -6 k_b L^{(e)} & 4 k_b L^{(e) 2} \end{array}\right]$$
with $k_m=\frac{E A}{L^{(e)}}$ and $k_b=\frac{E I}{\left.L^{(e)}\right)}$
\begin{aligned} & \left{d_{m b}{ }^{\prime}\right}^{(e)}=\left{\begin{array}{llllll} u_1^{\prime} & v_1^{\prime} & \theta_1^{\prime} & u_2^{\prime} & v_2^{\prime} & \theta_2^{\prime} \end{array}\right}^T \ & \left{r_{m b}\right}^{\prime(e)}=\left[\begin{array}{llllll} F_{x^{\prime} 1} & F_{y^{\prime} 1} & M_1 & F_{x^{\prime} 2} & F_{y^{\prime} 2} & M_2 \end{array}\right}^T \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Consistent nodal load vector

left{ $_r_{-} q^{\wedge}{$ prime $\left.} \backslash r i g h t\right}^{\wedge}{(e)}=-F _y^{\wedge}{\backslash$ prime $} \backslash l e f t\left{\backslash f r a c\left{b^{\wedge} 2 \backslash l e f t\left(L^{\wedge}{(e)}+2\right.\right.\right.$ a right $\left.)\right}{L \wedge{(e) 3}} \backslash f r a c\left{a b^{\wedge} 2\right}\left{L^{\wedge}{(e) 2\right.$

Veft $\left{r_{-} q^{\wedge}{\backslash\right.$ prime $\left.} \backslash r i g h t\right} \wedge{(e)}=\backslash \operatorname{eft}\left{\backslash f r a c\left{\backslash \operatorname{lft}\left(7 q_{-}\left{y_{-} 1\right}+3 q_{_}\left{y_{-} 2\right} \backslash r i g h t\right) L^{\wedge}{(c)}\right}{20} \backslash f r a c\left{\backslash e f t\left(3 q_{-}\left{y_{-} 1\right}+2 q_{-}\left{y_{-}\right.\right.\right.\right.$

Veft{r_q {}$^{\wedge}{\backslash$ prime $\left.} \backslash r i g h t\right}^{\wedge}{(e)}=\bigvee$ left $\left{\right.$ frac $\left{q_{-}\left{y_{-} 0\right} L^{\wedge}{(e)}\right}{2} \backslash$ frac $\left{q_{_}\left{y^{\wedge}{\backslash p r i m e} 0\right} L^{\wedge}{(e) 2}\right}{12} \backslash f r a c\left{q_{-}\left{y_{-} 00\right.\right.$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|ENGR7961

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|Total potential energy of a beam element

Similar to the development presented for the axial bar in Section 5.2.4, the total potential energy of the beam under the effect of external forces is expressed as follows:
\begin{aligned} & \pi_p^{(e)}=U^{(e)}-W^{(e)} \ & \pi_p^{(e)}=\int_A \int_0^{L^{(e)}} \sigma^{\prime} d \varepsilon^{\prime} d A d x^{\prime}-\left(\int_0^{L^{(-)}} v^{\prime}\left(F_b A+q\right) d x^{\prime}+v_1^{\prime} F_{y 1}{ }^{\prime}+v_2^{\prime} F_{y 2}{ }^{\prime}+\theta_1 M_1+\theta_2 M_2\right) \end{aligned}
where the first term represents the strain energy stored in the beam, and the second term represents the work done by the external forces. Note that for a beam we have $d A=b d y^{\prime}$ where $b$ is the breadth of the beam’s cross-section.
The bending strain, is a longitudinal strain $\varepsilon_{x^{\prime}}$, which depends on the normal distance as measured from the neutral axis of the beam (Fig. 6.2) as given in Eq. (2.138),
$$\varepsilon_{x^{\prime}}=-y^{\prime} \frac{d^2 v^{\prime}}{d x^{\prime 2}}$$

The strain energy component of the total potential energy then becomes,
$$U_b^{(e)}=\frac{1}{2} \int_0^{L^{(e)}}\left[\int_A E\left(-y^{\prime} \frac{d^2 v^{\prime}}{d x^2}\right)^2 b d y^{\prime}\right] d x^{\prime}=\frac{1}{2} \int_0^{L^{(e)}} E\left(\frac{d^2 v^{\prime}}{d x^2}\right)^2\left(\int_{-c / 2}^{c / 2} y^2 b d y^{\prime}\right) d x^{\prime}$$
Note that the inner integral over the area represents the definition of the second moment of area
$$I=\int_{-c / 2}^{c / 2} y^{\prime 2} b d y^{\prime}$$
The strain energy due to bending $U_b^{(e)}$ in a beam element is therefore expressed as follows:
$$U_b^{(e)}=\frac{1}{2} \int_0^{L^{(e)}} E I\left(\frac{d^2 v^{\prime}}{d x^{\prime 2}}\right)^2 d x^{\prime}, \text { or } U_b^{(e)}=\frac{1}{2} \int_0^{L^{(e)}} E I\left(\frac{d \theta}{d x^{\prime}}\right)^2 d x^{\prime}$$

## 数学代写|有限元方法代写Finite Element Method代考|Finite element form of the equilibrium equations

The principle of minimum total potential energy states that the equilibrium conditions are found when the variation of the total potential energy functional is zero. For an Euler-Bernoulli beam element, this is expressed as follows:
$$\delta \pi_p^{(\varepsilon)}=\int_0^{L^{(c)}} E I \frac{d \theta}{d x^{\prime}} \frac{d \delta \theta}{d x^{\prime}} d x^{\prime}-\left(\int_0^{L^{(c)}} \delta v^{\prime}\left(F_{B y^{\prime}} A+q_y\right) d x^{\prime}+\delta v_1{ }^{\prime} F_{y_1}+\delta v_2{ }^{\prime} F_{y^{\prime}}+\delta \theta_1 M_1+\delta \theta_2 M_2\right)$$

In order to find an expression for the curvature, $d \theta / d x^{\prime}$ recall that the beam deflection $v^{\prime}$ is approximated as follows:
\begin{aligned} v^{\prime}\left(x^{\prime}\right) & =N_1^{(b)} v_1^{\prime}+N_2^{(b)} \theta_1+N_3^{(b)} v_2{ }^{\prime}+N_4^{(b)} \theta_2 \text { (a) } \ & =\left[N^{(b)}\right]\left{d_b^{\prime}\right}^{(e)} \end{aligned}
where $\left[N^{(b)}\right]$ and $\left{d_b^{\prime}\right}^{(e)}$ are the shape function matrix and the degree of freedom vector, (Eq. (6.10)(a)) respectively,
$$\left[\begin{array}{llll} N^{(b)} \end{array}\right]=\left[\begin{array}{llll} N_1^{(b)} & N_2^{(b)} & N_3^{(b)} & N_4^{(b)} \end{array}\right]$$
$N_i^{(L)}$ are the $\mathrm{C}^1$-continuous shape functions given by Eq. (6.5). Using this approximation, the curvature vector is found as follows:
$$\frac{d \theta}{d x^{\prime}}=\frac{d^2 v^{\prime}}{d x^{\prime 2}}=\frac{d^2}{d x^{\prime 2}}\left(\left[N^{(b)}\right]\left{d_b^{\prime}\right}^{(e)}\right)=\left[B^{(b)}\right]\left{d_b^{\prime}\right}^{(e)}$$
where $\left[\begin{array}{llll}B^{(b)}\end{array}\right]=\left[\begin{array}{llll}B_1^{(b)} & B_2^{(b)} & B_3^{(b)} & B_4^{(b)}\end{array}\right]$ is the curvature-displacement matrix with,
\begin{aligned} & B_1^{(b)}=\frac{d^2 N_1^{(b)}}{d x^2}=\left(-\frac{6}{L^{(e) 2}}+\frac{12 x^{\prime}}{L^{(e) 3}}\right), \quad B_2^{(b)}=\frac{d^2 N_2^{(b)}}{d x^{\prime 2}}=\left(-\frac{4}{L^{(e)}}+\frac{6 x}{L^{(e) 2}}\right), \ & B_3^{(b)}=\frac{d^2 N_3^{(b)}}{d x^{\prime 2}}=\left(\frac{6}{L^{(e) 2}}-\frac{12 x^{\prime}}{L^{(e) 3}}\right), \quad B_4=\frac{d^2 N_4^{(b)}}{d x^{\prime 2}}=\left(-\frac{2}{L^{(e)}}+\frac{6 x^{\prime}}{L^{(e) 2}}\right) \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Total potential energy of a beam element

$$\pi_p^{(e)}=U^{(e)}-W^{(e)} \quad \pi_p^{(e)}=\int_A \int_0^{L^{(e)}} \sigma^{\prime} d \varepsilon^{\prime} d A d x^{\prime}-\left(\int_0^{L^{(-)}} v^{\prime}\left(F_b A+q\right) d x^{\prime}+v_1^{\prime} F_{y 1}^{\prime}+v_2^{\prime}\right.$$

$$\varepsilon_{x^{\prime}}=-y^{\prime} \frac{d^2 v^{\prime}}{d x^{\prime 2}}$$

$$U_b^{(e)}=\frac{1}{2} \int_0^{L^{(e)}}\left[\int_A E\left(-y^{\prime} \frac{d^2 v^{\prime}}{d x^2}\right)^2 b d y^{\prime}\right] d x^{\prime}=\frac{1}{2} \int_0^{L^{(e)}} E\left(\frac{d^2 v^{\prime}}{d x^2}\right)^2\left(\int_{-c / 2}^{c / 2} y^2 b d y^{\prime}\right) d x^{\prime}$$

$$I=\int_{-c / 2}^{c / 2} y^{\prime 2} b d y^{\prime}$$

$$U_b^{(e)}=\frac{1}{2} \int_0^{L^{(e)}} E I\left(\frac{d^2 v^{\prime}}{d x^{\prime 2}}\right)^2 d x^{\prime}, \text { or } U_b^{(e)}=\frac{1}{2} \int_0^{L^{(e)}} E I\left(\frac{d \theta}{d x^{\prime}}\right)^2 d x^{\prime}$$

## 数学代写|有限元方法代写Finite Element Method代考|Finite element form of the equilibrium equations

$$\delta \pi_p^{(\varepsilon)}=\int_0^{L^{(c)}} E I \frac{d \theta}{d x^{\prime}} \frac{d \delta \theta}{d x^{\prime}} d x^{\prime}-\left(\int_0^{L^{(c)}} \delta v^{\prime}\left(F_{B y^{\prime}} A+q_y\right) d x^{\prime}+\delta v_1^{\prime} F_{y_1}+\delta v_2^{\prime} F_{y^{\prime}}+\delta \theta_1 M_1\right.$$

‘begin ${$ aligned $} v^{\wedge}{$ lprime $} \backslash l e f t\left(x^{\wedge}{\backslash\right.$ prime $} \backslash$ \right) $\&=N _1 \wedge{(b)} v_{-} 1^{\wedge}{$ lprime $}+N _2^{\wedge}{(b)} \backslash$ theta_ $1+N_{-} 3^{\wedge}{(b)} v_{-} 2{}^{\wedge}$
$$\left[N^{(b)}\right]=\left[\begin{array}{llll} N_1^{(b)} & N_2^{(b)} & N_3^{(b)} & N_4^{(b)} \end{array}\right]$$
$N_i^{(L)}$ 是 $\mathrm{C}^1$ – 由等式给出的连续形状函数。(6.5)。使用此近似值，曲率向量如下所示:

$$B_1^{(b)}=\frac{d^2 N_1^{(b)}}{d x^2}=\left(-\frac{6}{L^{(e) 2}}+\frac{12 x^{\prime}}{L^{(e) 3}}\right), \quad B_2^{(b)}=\frac{d^2 N_2^{(b)}}{d x^{\prime 2}}=\left(-\frac{4}{L^{(e)}}+\frac{6 x}{L^{(e) 2}}\right), \quad B_3^{(b)}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|EG3001

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|One-dimensional, Lagrange interpolation functions

Shape functions for the domain $x^{\prime} \in\left[0, L^{(e)}\right]$ : In finite element method each small segment is assigned its own local coordinate system. In this case the domain of the element is defined as $x^{\prime} \in\left[0, L^{(e)}\right]$. The shape functions for this domain can be easily obtained by letting,
$x_1=0$ and $x_2=L^{(e)}$ for the linear shape functions, and
$x_1=0, x_2=r L^{(e)}$ and $x_3=L^{(e)}$ for the quadratic shape functions
in Eqs. (4.23) and (4.30), respectively. Note that $r(0<r<1)$ is a coefficient that ensures that node- 2 is located between nodes-1 and-3. This gives,
Linear shape functions:
$$N_1\left(x^{\prime}\right)=1-\frac{x^{\prime}}{L^{(e)}} \text { and } N_2\left(x^{\prime}\right)=\frac{x^{\prime}}{L^{(e)}}$$
for $0<r<1$ :
\begin{aligned} & N_1\left(x^{\prime}\right)=\left(1-\frac{x^{\prime}}{r L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) \ & N_2\left(x^{\prime}\right)=\frac{1}{1-r}\left(\frac{x^{\prime}}{r L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) \ & N_3\left(x^{\prime}\right)=\frac{1}{r-1}\left(\frac{x^{\prime}}{L^{(e)}}\right)\left(r-\frac{x^{\prime}}{L^{(e)}}\right) \end{aligned} for $r=1 / 2$ :
\begin{aligned} & N_1\left(x^{\prime}\right)=\left(1-2 \frac{x^{\prime}}{L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) \ & N_2\left(x^{\prime}\right)=4\left(\frac{x^{\prime}}{L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) \ & N_3\left(x^{\prime}\right)=-\left(\frac{x^{\prime}}{L^{(e)}}\right)\left(1-2 \frac{x^{\prime}}{L^{(e)}}\right) \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|Equilibrium equations in finite element form

The finite element formulation of this boundary value problem described by Eqs. (4.3)-(4.5) will be carried out by using the method of weighted residuals. The weighted residual integral of the boundary value problem is given as follows:
$$\int_0^{L^{(e)}} w\left[\frac{d}{d x^{\prime}}\left(a \frac{d u}{d x^{\prime}}\right)+p u+q\right] d x^{\prime}=0 \text { in } 0<x^{\prime}<L^{(e)}$$
where $w\left(x^{\prime}\right)$ is an arbitrary weight function with the properties described in Section 3.5. Integrating the first term by parts,
\begin{aligned} & w\left(L^{(e)}\right)\left(\left.a_{L^{(e)}} \frac{d u}{d x^{\prime}}\right|{L^{(e)}}\right)-w(0)\left(\left.a_0 \frac{d u}{d x^{\prime}}\right|_0\right) \ & -\int_0^{L(e)} a \frac{d u}{d x^{\prime}} \frac{d w}{d x^{\prime}} d x^{\prime}+\int_0^{L(e)} p w u d x^{\prime}+\int_0^{L(e)} w q d x^{\prime}=0 \end{aligned} and using the natural boundary conditions given in Eqs. (4.4) and (4.5), the weak form of the boundary value problem is found ās föllows: \begin{aligned} & w\left(L^{(e)}\right)\left(a{L^{(e)}}\left(\alpha_{L^{(e)}} u_{L^{(e)}}+\beta_{\left.L^{(e)}\right)}\right)\right)-w(0)\left(-a_0\left(\alpha_0 u_0+\beta_0\right)\right)- \ & \int_0^{L^{(e)}} a \frac{d u}{d x^{\prime}} \frac{d w}{d x^{\prime}} d x^{\prime}+\int_0^{L^{(c)}} p w u d x^{\prime}+\int_0^{L^{(e)}} w q d x^{\prime}=0 \end{aligned}
Note that the weight function has the properties of the variation of $u$. Making the substitution $w \rightarrow \delta u$ and using the nodal notation for variables at the ends of the element, $u_1=u(0)=u_0$ and $u_2=u\left(L^{(e)}\right)=u_{L^{(e)}}$, the weak form is transformed as follows:
\begin{aligned} \delta u_2\left(a_2\left(\alpha_2 u_2+\beta_2\right)\right)-\delta u_1\left(-a_1\left(\alpha u_1+\beta_1\right)\right) \ -\int_0^{L^{(c)}} a \frac{d u}{d x^{\prime}} \frac{d \delta u}{d x^{\prime}} d x^{\prime}+\int_0^{L^{(e)}} p u \delta u d x^{\prime}+\int_0^{L^{(c)}} q \delta u d x^{\prime}=0 \end{aligned}
First, let us consider a two node, $C^{(0)}$ element shown in the Fig. 4.2. Higher order interpolations for $\tilde{u}$ can also be considered.

## 数学代写|有限元方法代写Finite Element Method代考|One-dimensional, Lagrange interpolation functions

$x_1=0$ 和 $x_2=L^{(e)}$ 对于线性形函数，和
$x_1=0, x_2=r L^{(e)}$ 和 $x_3=L^{(e)}$ 对于方程式中的二次形函数
。分别为 (4.23) 和 (4.30)。注意 $r(0<r<1)$ 是确保节点 2 位于节点 1 和节点 3 之间的系数。这给出了 线性形函数:
$$N_1\left(x^{\prime}\right)=1-\frac{x^{\prime}}{L^{(e)}} \text { and } N_2\left(x^{\prime}\right)=\frac{x^{\prime}}{L^{(e)}}$$

$$N_1\left(x^{\prime}\right)=\left(1-\frac{x^{\prime}}{r L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) \quad N_2\left(x^{\prime}\right)=\frac{1}{1-r}\left(\frac{x^{\prime}}{r L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) N_3\left(x^{\prime}\right)=\frac{r}{r}$$

$$N_1\left(x^{\prime}\right)=\left(1-2 \frac{x^{\prime}}{L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) \quad N_2\left(x^{\prime}\right)=4\left(\frac{x^{\prime}}{L^{(e)}}\right)\left(1-\frac{x^{\prime}}{L^{(e)}}\right) N_3\left(x^{\prime}\right)=-\left(\frac{x}{L^{(}}\right.$$

## 数学代写|有限元方法代写Finite Element Method代考|Equilibrium equations in finite element form

$$\int_0^{L^{(e)}} w\left[\frac{d}{d x^{\prime}}\left(a \frac{d u}{d x^{\prime}}\right)+p u+q\right] d x^{\prime}=0 \text { in } 0<x^{\prime}<L^{(e)}$$

$$w\left(L^{(e)}\right)\left(a_{L^{(e)}} \frac{d u}{d x^{\prime}} \mid L^{(e)}\right)-w(0)\left(\left.a_0 \frac{d u}{d x^{\prime}}\right|0\right) \quad-\int_0^{L(e)} a \frac{d u}{d x^{\prime}} \frac{d w}{d x^{\prime}} d x^{\prime}+\int_0^{L(e)} p w u d x^{\prime}+$$ 并使用等式中给出的自然边界条件。(4.4) 和 (4.5) 边值问题的弱形式如下: $$w\left(L^{(e)}\right)\left(a L^{(e)}\left(\alpha{L^{(e)}} u_{L^{(e)}}+\beta_{\left.L^{(e)}\right)}\right)\right)-w(0)\left(-a_0\left(\alpha_0 u_0+\beta_0\right)\right)-\int_0^{L^{(e)}} a \frac{d u}{d x^{\prime}} \frac{d w}{d x^{\prime}} d x^{\prime}$$

$$\delta u_2\left(a_2\left(\alpha_2 u_2+\beta_2\right)\right)-\delta u_1\left(-a_1\left(\alpha u_1+\beta_1\right)\right)-\int_0^{L^{(c)}} a \frac{d u}{d x^{\prime}} \frac{d \delta u}{d x^{\prime}} d x^{\prime}+\int_0^{L^{(e)}} p u \delta u d x^{\prime}+\int_0^{L^{(c)}}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

## 数学代写|有限元方法代写Finite Element Method代考|ENGR7961

statistics-lab™ 为您的留学生涯保驾护航 在代写有限元方法Finite Element Method方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写有限元方法Finite Element Method代写方面经验极为丰富，各种代写有限元方法Finite Element Method相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|有限元方法代写Finite Element Method代考|One-dimensional interpolation for finite element method

We are interested in developing a polynomial approximation $\tilde{u}$ for $u(x)$ in a region $x_1^{\prime} \leq x^{\prime} \leq x_2{ }^{\prime}$ We are going to use an interpolation that passes through $M+1$ points in this region:
$$u\left(x^{\prime}\right) \approx \tilde{u}\left(x^{\prime}\right)=a_0+\sum_{j=1}^M a_j x^j$$
where $a_0$ and $a_j$ are constant coefficients. In this section we transform this polynomial approximation from one whose unknowns are $a_0$ and $a_j$, to one whose unknowns are $u_j$. The polynomial will be required pass through points $\left(x_j^{\prime}, u_j\right)$ wheree $1 \leq j \leq M+1$ with $x_1^{\prime}=0$ and $x_M^{\prime}=L^{(e)}$. The values of $u_j$ aree known at $M+1$ positions in this domain. The approach developed in the next section results in a polynomial approximation of $u(x)$ that is of order $M$, also known as the Lagrange interpolation,
$$\tilde{u}\left(x^{\prime}\right)=\sum_{i=1}^{M+1} N_i\left(x^{\prime}\right) u_i$$
where the shape functions $N_i\left(x^{\prime}\right)$ are defined as follows:
$$N_i\left(x^{\prime}\right)=\frac{\left(x_1^{\prime}-x^{\prime}\right)\left(x_2^{\prime}-x^{\prime}\right) \cdots\left(x_{i-1}^{\prime}-x^{\prime}\right)\left(x_{i+1}^{\prime}-x^{\prime}\right) \cdots\left(x_M^{\prime}-x^{\prime}\right)\left(x_{M+1}^{\prime}-x^{\prime}\right)}{\left(x_1^{\prime}-x_i^{\prime}\right)\left(x_2^{\prime}-x_i^{\prime}\right) \cdots\left(x_{i-1}^{\prime}-x_i^{\prime}\right)\left(x_{i+1}^{\prime}-x_i^{\prime}\right) \cdots\left(x_M^{\prime}-x_i^{\prime}\right)\left(x_{M+1}^{\prime}-x_i^{\prime}\right)}$$
General characteristics of Lagrange shape functions are as follows:
$N_i=1$ at $x^{\prime}=x_i^{\prime}$, but zero at the other positions (nodes)
(C1)
$N_i$ are polynomials of the same degree
(C2)
The sum of all shape functions is equal to one, i.e., $\sum_{i=1}^{M+1} N_i=1$.
(C3)
It is important to keep in mind that the Lagrange interpolation (4.13) gives a polynomial approximation that is exactly of the same polynomial order as Eq. (4.12), but the unknowns are expressed in terms of $u_i$ instead of $a_i$. This will be demonstrated in the following sections.
$\mathrm{C}^0$ Continuity
Note that Lagrange shape functions use only the nodal degrees of freedom $u_i$. Eq. (4.13) ensures the continuity of the degrees of freedom at the nodes. Interpolation functions that only provide the continuity of the primitive variable across element boundaries are said to be $\mathrm{C}^0$ continuous.

Later we will develop interpolation functions that provide higher order continuity, e.g., $u_j$ and $d u_j / d x^{\prime}$.

## 数学代写|有限元方法代写Finite Element Method代考|General form of C0 shape functions

Consider the polynomial approximation,
$$\tilde{u}=a_0+a_1 x^{\prime}+a_2 x^{\prime 2}+a_3 x^{\prime 3}+\ldots a_n x^{\prime M}$$
which can also be represented as follows:
$$\tilde{u}=[X]{a}=\sum_{i=0}^M a_j x^j$$
where,
\begin{aligned} & {[X]=\left[\begin{array}{llllll} 1 & x^{\prime} & x^{\prime 2} & x^{\prime 3} & \ldots & x^{\prime M} \end{array}\right]} \ & {a}=\left{\begin{array}{llllll} a_0 & a_1 & a_2 & a_3 & \ldots & a_M \end{array}\right}^T \end{aligned}
After we evaluate $u$ at $(M+1)$ different $x_i^{\prime}$ locations, we get a system of $(M+1)$ algebraic equations for the unknown $a_i$,
\begin{aligned} & u_1=u\left(x_1^{\prime}\right)=a_0+a_1 x_1^{\prime}+a_2 x^{\prime 2}{ }1+a_3 x^{\prime 3}{ }_1+\ldots a_M x^{\prime M}{ }_1 \ & u_2=u\left(x_2^{\prime}\right)=a_0+a_1 x_2^{\prime}+a_2 x^{\prime 2}{ }_2+a_3 x^{\prime 3}{ }_2+\ldots a_M x^{\prime}{ }_2 \ & \vdots \ & u{M+1}=u\left(x^{\prime}{ }{M+1}\right)=a_0+a_1 x{M+1}^{\prime}+a_2 x^{\prime 2}{ }{M+1}+a_3 x^{\prime 3}{ }{M+1}+\ldots a_M x^M{ }_{M+1} \end{aligned}

## 数学代写|有限元方法代写Finite Element Method代考|One-dimensional interpolation for finite element method

$$u\left(x^{\prime}\right) \approx \tilde{u}\left(x^{\prime}\right)=a_0+\sum_{j=1}^M a_j x^j$$

$$\tilde{u}\left(x^{\prime}\right)=\sum_{i=1}^{M+1} N_i\left(x^{\prime}\right) u_i$$

$$N_i\left(x^{\prime}\right)=\frac{\left(x_1^{\prime}-x^{\prime}\right)\left(x_2^{\prime}-x^{\prime}\right) \cdots\left(x_{i-1}^{\prime}-x^{\prime}\right)\left(x_{i+1}^{\prime}-x^{\prime}\right) \cdots\left(x_M^{\prime}-x^{\prime}\right)\left(x_{M+1}^{\prime}-x^{\prime}\right)}{\left(x_1^{\prime}-x_i^{\prime}\right)\left(x_2^{\prime}-x_i^{\prime}\right) \cdots\left(x_{i-1}^{\prime}-x_i^{\prime}\right)\left(x_{i+1}^{\prime}-x_i^{\prime}\right) \cdots\left(x_M^{\prime}-x_i^{\prime}\right)\left(x_{M+1}^{\prime}-x_i^{\prime}\right)}$$

$N_i=1$ 在 $x^{\prime}=x_i^{\prime}$ ，但在其他位置（节点）为零
(C1)
$N_i$ 是同次多项式
(C2)

(C3)

$\mathrm{C}^0$ 连续性

## 数学代写|有限元方法代写Finite Element Method代考|General form of C0 shape functions

$$\tilde{u}=a_0+a_1 x^{\prime}+a_2 x^{\prime 2}+a_3 x^{\prime 3}+\ldots a_n x^{\prime M}$$

$$\tilde{u}=[X] a=\sum_{i=0}^M a_j x^j$$

Ibegin{aligned $}\left{\left[[X]=\backslash \mid e f t\left[\backslash b e g i n{a r r a y}{\mid I I I I} 1 \& x^{\wedge}{\backslash p r i m e} \& x^{\wedge}{\backslash p r i m e ~ 2} \& x^{\wedge}{\backslash p r i m e ~ 3} \& \mid\right.\right.\right.$ dots\& $x^{\wedge}{\backslash p r i m e ~ M} \backslash e n d{a r r$

$$u_1=u\left(x_1^{\prime}\right)=a_0+a_1 x_1^{\prime}+a_2 x^{\prime 2} 1+a_3 x_1^{\prime 3}+\ldots a_M x_1^{\prime M} \quad u_2=u\left(x_2^{\prime}\right)=a_0+a_1 x_2^{\prime}+a_2$$

## 有限元方法代写

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## MATLAB代写

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