### 数学代写|有限元方法代写Finite Element Method代考|Find2022

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• 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|Boundary and initial value problems

Consider a system with dependent variables $u, v$, and $w$ defined over a domain $\Omega$, which itself occupies a subsection of space (Fig. 1.1). In general, each variable can take different values at different points in the domain and these values can also vary in time. Spatial position of a point $P$ in the domain $\Omega$ can be identified with respect to a spatial reference system (e.g., $(x, y, z)$ ). If the position of point $P$ also varies in time, the position of point $P$ is said to be time dependent. Thus, for example, if $u$ is a function of space and time $u=u(x, y, z, t)$. In these notes, we will consider boundary value problems (BVPs) and initial value problems that are formulated by using PDEs. A very general representation of such a problem can be given as follows:
$$\mathcal{L}(u, v, w)=f \text { in } \Omega, \text { for } 0 \leq t \leq \tau$$
where $\mathcal{L}(\cdot)$ is a differential operator of independent spatial variables $x, y, z$ and time $t, f=f(x, y, z, t)$ is typically a function that represents the internal effects that act on the system, and $\tau$ is the duration of interest.

The dependent variables interact with the outside of the domain $\Omega$ through the boundary $\Gamma$ of the domain, and typically experience changes as a result of the external effects that are imposed on the boundary. These external effects are known as the boundary conditions which depend on the physics of the problem.

The Dirichlet boundary condition represents a prescribed value for a dependent variable,
$$u=u_{b}(t) \text { on } \Gamma_{E}$$
Here the variable $u$ of the solution domain is prescribed to $u_{b}$ on a segment of the boundary $\Gamma_{E}$. In general, this prescribed variable can be a function of time $t$. The Dirichlet boundary condition is also known as the essential boundary condition.
The von Neumann boundary condition typically describes the external effects that cause a change in the system. Such effects include external forces, heat flow, etc. As we will demonstrate later in the notes, the von Neumann boundary conditions are typically represented as follows:
$$\mathcal{B}(u, v, w)=g(t) \text { on } \Gamma_{N}$$
where $\mathcal{B}(\cdot)$ is another differential operator, $g$ is a given function, and $\Gamma_{N}$ represents the segment of the boundary over which the von Neumann boundary condition is applied. The von Neumann boundary condition, also known as the natural boundary condition or the nonessential boundary condition, can also vary in time.

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

In some problems, only the steady state of the dependent variables is of interest and the temporal variation is neglected (or negligible). Thus, for example, $u$ becomes only a function of the spatial dimensions $u=u(x, y, z)$. A steady state boundary value problem can be formulated by dropping the time dependence as follows:
$$\mathcal{L}(u, v, w)=f \text { in } \Omega$$
where for a boundary value problem $\mathcal{L}(\cdot)$ is a differential operator of the independent spatial variables $(x, y, z)$ and $f=f(x, y, z)$. A steady state boundary value problem is also subject to the Dirichlet and/or von Neumann conditions on the boundary of the domain.
Example 1.1 Equation of motion of a solid bar
a) Derive the equation of motion of an elastic bar in terms of its deflection $u(x, t)$. Initially, assume that the bar has a variable cross-sectional area $A(x)$ and that it is subjected to distributed axial load $q(x, t)$ and a concentrated force $F$ at its free end as shown in Fig. 1.2. Also assume small deflections, linear elastic material behavior with constant elastic modulus $E$, and constant mass density $\rho$.
b) Obtain the steady state solution for the case of constant cross-section and zero distributed force.

Solution 1.1a: The solution domain $\Omega$ for this problem spans $0<x<L$. The boundaries $\Gamma$ of the solution domain are located at $x=0$ and $x=L$. Internal forces develop in the bar in response to external loading. The internal normal force $N(x)$ at the cross-section $x$ can be defined as follows:
$$N(x)=\bar{\sigma}(x) A(x)$$
where the average normal stress $\bar{\sigma}$ is defined as follows:
$$\bar{\sigma}(x)=\frac{1}{A(x)} \int_{A(x)} \sigma d A$$
and where $\sigma$ is the internal normal stress, $A$ is the cross-sectional area of the bar.

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

$$\mathcal{L}(u, v, w)=f \text { in } \Omega, \text { for } 0 \leq t \leq \tau$$

Dirichlet 边界条件表示因变量的规定值，
$$u=u_{b}(t) \text { on } \Gamma_{E}$$

$$\mathcal{B}(u, v, w)=g(t) \text { on } \Gamma_{N}$$

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

$$\mathcal{L}(u, v, w)=f \text { in } \Omega$$

b) 获得恒定截面和零分布力情况下的稳态解。

$$N(x)=\bar{\sigma}(x) A(x)$$

$$\bar{\sigma}(x)=\frac{1}{A(x)} \int_{A(x)} \sigma d A$$

## 有限元方法代写

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