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

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes™为您提供可以保分的包课服务

couryes-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|Weak Form

The weak form of Eq. (5.4.8) over a typical element $\Omega^e=\left(r_a^e, r_b^e\right)$ is obtained using the three-step procedure. Let $w_h^e$ be the approximation of $w$. Then the weightedresidual statement (step 1) and subsequent steps are carried out as for the Euler-Bernoulli beams:
\begin{aligned} 0= & \int_{r_a^e}^{r_b^e} v_i^e\left{-\frac{1}{r} \frac{d}{d r}\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]+k_f^e w_h^e-q\right} r d r \quad \text { (Step 1) } \ = & \int_{r_a^e}^{r_b^e}\left{\frac{1}{r} \frac{d v_i^e}{d r}\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]+k_f^e v_i^e w_h^e-v_i^e q\right} r d r \ & -\left{v_i\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]\right}_{r_a^e}^{r_b^e} \ = & \int_{r_a^e}^{r_b^e}\left[-\frac{d^2 v_i^e}{d r^2} M_{r r}^h-\frac{1}{r} \frac{d v_i^e}{d r} M_{\theta \theta}^h+k_f^e v_i^e w_h^e-v_i^e q\right] r d r \ & -\left{v_i^e\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]\right}_{r_a^e}^{r_b^e}-\left[-\frac{d v_i^e}{d r} r M_{r r}^h\right]{r_a^e}^{r_b^e} \quad \text { (Step 2) } \end{aligned} (Step 2) where $\left{v_i^e\right}$ is the set of weight functions, and $M{r r}^h$ and $M_{\theta \theta}^h$ are the bending moments derived from $w_h$ according to Eq. (5.4.4). From the last expression, it is clear that
Primary variables: $w_h^e,-\frac{d w_h^e}{d r}$
Secondary variables: $\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right] \equiv r V_r^h, r M_{r r}^h$

Thus, one element of each of the pairs $\left(w_h^e, r V_r^h\right)$ and $\left(-d w_h^e / d r, r M_{r r}^h\right)$ must be known at any boundary point.
With the notation in Eq. (5.4.10), the final weak form is given by
\begin{aligned} 0= & \int_{r_a^e}^{r_b^c}\left[-\frac{d^2 v_i^e}{d r^2} M_{r r}^h-\frac{1}{r} \frac{d v_i^e}{d r} M_{\theta \theta}^b+k_f^e v_i^e w_h^e-v_i^c q\right] r d r-v_i^e\left(r_a^r\right) Q_1^e-v_i\left(r_b^e\right) Q_3^e \ & -\left(-\frac{d v_i^e}{d r}\right){r_a^t} Q_2^c-\left(-\frac{d v_i^e}{d r}\right){r_b^r} Q_4^e \text { (Step 3) } \end{aligned}
where

$$\begin{array}{ll} Q_1^e=-\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]{r_a^e}, & Q_2^e=\left[-r M{r r}^h\right]{r_a^e} \ Q_3^e=\left[\frac{d}{d r}\left(r M{r r}^h\right)-M_{\theta \theta}^h\right]{r_b^e}, & Q_4^e=\left[r M{r r}^h\right]{r_b^e} \end{array}$$ Clearly, $Q_1^e$ and $Q_3^e$ are the shear forces and $Q_2^e$ and $Q_4^e$ are the bending moments. In order to express the weak form in Eq. (5.4.11) in terms of the displacement $w_h$, the bending moments $M{r r}^h$ and $M_{\theta \theta}^h$ appearing in Eq.
(5.4.11) should be expressed in terms of $w_h$ by means of Eq. (5.4.4). We have
\begin{aligned} 0= & \int_{r_a^e}^{r_b^e}\left[D \frac{d^2 v_i}{d r^2}\left(\frac{d^2 w_h}{d r^2}+\frac{v}{r} \frac{d w_h^e}{d r}\right)+\frac{D}{r} \frac{d v_i^e}{d r}\left(v \frac{d^2 w_h^e}{d r^2}+\frac{1}{r} \frac{d w_h^e}{d r}\right)\right. \ & \left.+k_f^e v_i^e w_h^e-v_i^e q_c\right] r d r-v_i^e\left(r_a^e\right) Q_1^e-v_i^e\left(r_b^e\right) Q_3^e \ & -\left(-\frac{d v_i^e}{d r}\right){r_a^e} Q_2^e-\left(-\frac{d v_i^e}{d r}\right){r_b^e} Q_4^e \end{aligned}

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

The weak-form Galerkin (or Ritz) finite element model of axisymmetric bending of circular plates is obtained by assuming Hermite cubic approximation of $w(r)$ over a typical element $\Omega^e=\left(r_a^e, r_b^e\right)$ in the form
$$w(r) \approx w_h^e(r)=\sum_{j=1}^4 \Delta_j^e \phi_j^e(r)$$
where $\phi_j^e(r)$ are the Hermite cubic polynomials given in Eq. (5.2.20) with $x$ replaced by $r$ (i.e., $\bar{r}=\bar{x}$ and $\left.\bar{r}=r-r_a^2\right)$ and $\Delta_j$ are the nodal values $(\theta=$ $-d w / d r)$
$$\Delta_1^e=w\left(r_a^e\right), \Delta_3^e=w\left(r_b^e\right), \Delta_2^e=\theta\left(r_a^e\right), \Delta_4^e=\theta\left(r_b^e\right)$$
Substituting Eq. (5.4.14) for $w_h^e$ and $v_i^e=\phi_i^e$, we obtain
$$\begin{gathered} \mathbf{K}^e \Delta^e=\mathbf{f}^e+\mathbf{Q}^e \ K_{i j}^e=\int_{r_s^e}^{r_b^r}\left[D \frac{d^2 \phi_i^e}{d r^2}\left(\frac{d^2 \phi_j^e}{d r^2}+\frac{\nu}{r} \frac{d \phi_j^e}{d r}\right)+\frac{D}{r} \frac{d \phi_i^e}{d r}\left(\nu \frac{d^2 \phi_j^e}{d r^2}+\frac{1}{r} \frac{d \phi_j^e}{d r}\right)+k_f^e \phi_i^e \phi_j^e\right] r d r \ f_i^e=\int_{r_a^e}^{r_b^r} q_c \phi_i^e r d r \ Q_i^e=\phi_i^e\left(r_a^e\right) Q_1^e+\phi_i^e\left(r_b\right) Q_3^e+\left(-\frac{d \phi_i^e}{d r}\right){r_a^e} Q_2^e+\left(-\frac{d \phi_i}{d r}\right){r_b^c} Q_4^e \end{gathered}$$
Clearly, the stiffness matrix $\mathbf{K}^e$ is symmetric and is of the order $4 \times 4$.

# 有限元方法代考

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

(5.4.8)式在典型元素$\Omega^e=\left(r_a^e, r_b^e\right)$上的弱形式是用三步程序得到的。设$w_h^e$为$w$的近似。然后对Euler-Bernoulli梁进行加权独立表述(步骤1)及后续步骤:
\begin{aligned} 0= & \int_{r_a^e}^{r_b^e} v_i^e\left{-\frac{1}{r} \frac{d}{d r}\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]+k_f^e w_h^e-q\right} r d r \quad \text { (Step 1) } \ = & \int_{r_a^e}^{r_b^e}\left{\frac{1}{r} \frac{d v_i^e}{d r}\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]+k_f^e v_i^e w_h^e-v_i^e q\right} r d r \ & -\left{v_i\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]\right}{r_a^e}^{r_b^e} \ = & \int{r_a^e}^{r_b^e}\left[-\frac{d^2 v_i^e}{d r^2} M_{r r}^h-\frac{1}{r} \frac{d v_i^e}{d r} M_{\theta \theta}^h+k_f^e v_i^e w_h^e-v_i^e q\right] r d r \ & -\left{v_i^e\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]\right}{r_a^e}^{r_b^e}-\left[-\frac{d v_i^e}{d r} r M{r r}^h\right]{r_a^e}^{r_b^e} \quad \text { (Step 2) } \end{aligned}(步骤2)，其中$\left{v_i^e\right}$为权函数集合，$M{r r}^h$和$M_{\theta \theta}^h$为根据式(5.4.4)由$w_h$导出的弯矩。从最后一个表达式可以清楚地看出

\begin{aligned} 0= & \int_{r_a^e}^{r_b^c}\left[-\frac{d^2 v_i^e}{d r^2} M_{r r}^h-\frac{1}{r} \frac{d v_i^e}{d r} M_{\theta \theta}^b+k_f^e v_i^e w_h^e-v_i^c q\right] r d r-v_i^e\left(r_a^r\right) Q_1^e-v_i\left(r_b^e\right) Q_3^e \ & -\left(-\frac{d v_i^e}{d r}\right){r_a^t} Q_2^c-\left(-\frac{d v_i^e}{d r}\right){r_b^r} Q_4^e \text { (Step 3) } \end{aligned}

$$\begin{array}{ll} Q_1^e=-\left[\frac{d}{d r}\left(r M_{r r}^h\right)-M_{\theta \theta}^h\right]{r_a^e}, & Q_2^e=\left[-r M{r r}^h\right]{r_a^e} \ Q_3^e=\left[\frac{d}{d r}\left(r M{r r}^h\right)-M_{\theta \theta}^h\right]{r_b^e}, & Q_4^e=\left[r M{r r}^h\right]{r_b^e} \end{array}$$ 显然，$Q_1^e$和$Q_3^e$是剪力，$Q_2^e$和$Q_4^e$是弯矩。为了将式(5.4.11)中的弱形式用位移$w_h$表示，式(5.4.11)中出现的弯矩$M{r r}^h$和$M_{\theta \theta}^h$
(5.4.11)应用式(5.4.4)表示为$w_h$。我们有
\begin{aligned} 0= & \int_{r_a^e}^{r_b^e}\left[D \frac{d^2 v_i}{d r^2}\left(\frac{d^2 w_h}{d r^2}+\frac{v}{r} \frac{d w_h^e}{d r}\right)+\frac{D}{r} \frac{d v_i^e}{d r}\left(v \frac{d^2 w_h^e}{d r^2}+\frac{1}{r} \frac{d w_h^e}{d r}\right)\right. \ & \left.+k_f^e v_i^e w_h^e-v_i^e q_c\right] r d r-v_i^e\left(r_a^e\right) Q_1^e-v_i^e\left(r_b^e\right) Q_3^e \ & -\left(-\frac{d v_i^e}{d r}\right){r_a^e} Q_2^e-\left(-\frac{d v_i^e}{d r}\right){r_b^e} Q_4^e \end{aligned}

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

$$w(r) \approx w_h^e(r)=\sum_{j=1}^4 \Delta_j^e \phi_j^e(r)$$

## 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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)