微积分网课代修|偏微分方程代写Partial Differential Equation代考|MATH480 Finite Element Method (FEM) Basics

微积分网课代修|偏微分方程代写Partial Differential Equation代考|MATH480 Finite Element Method (FEM) Basics

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微积分网课代修|偏微分方程代写Partial Differential Equation代考|MATH480 Finite Element Method (FEM) Basics

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Finite Element Method (FEM) Basics

The core Partial Differential Equation Toolbox algorithm is a PDE solver that uses the Finite Element Method (FEM) for problems defined on bounded domains in the plane.
The solutions of simple PDEs on complicated geometries can rarely be expressed in terms of elementary functions. You are confronted with two problems: First you need to describe a complicated geometry and generate a mesh on it. Then you need to discretize your PDE on the mesh and build an equation for the discrete approximation of the solution. The PDE app provides you with easy-to-use graphical tools to describe complicated domains and generate triangular meshes. It also discretizes PDEs, finds discrete solutions and plots results. You can access the mesh structures and the discretization functions directly from the command line (or from a file) and incorporate them into specialized applications.
Here is an overview of the Finite Element Method (FEM). The purpose of this presentation is to get you acquainted with the elementary FEM notions. Here you find the precise equations that are solved and the nature of the discrete solution. Different extensions of the basic equation implemented in Partial Differential Equation Toolbox software are presented. A more detailed description can be found in “Elliptic Equations” on page 5-2, with variants for specific types in “Systems of PDEs” on page 5-13, “Parabolic Equations” on page 5-17, “Hyperbolic Equations” on page 5-20, “Eigenvalue Equations” on page 5-22, and “Nonlinear Equations” on page 5-26.
You start by approximating the computational domain $\Omega$ with a union of simple geometric objects, in this case triangles (2-D geometry) or tetrahedra (3-D geometry). (This discussion applies to both triangles and tetrahedra, but speaks of triangles.) The triangles form a mesh and each vertex is called a node. You are in the situation of an architect designing a dome. The architect has to strike a balance between the ideal rounded forms of the original sketch and the limitations of the simple building-blocks, triangles or quadrilaterals. If the result does not look close enough to a perfect dome, the architect can always improve the result by using smaller blocks.
Next you say that your solution should be simple on each triangle. Polynomials are a good choice: they are easy to evaluate and have good approximation properties on small domains. You can ask that the solutions in neighboring triangles connect to each other continuously across the edges. You can still decide how complicated the polynomials can be. Just like an architect, you want them as simple as possible. Constants are the simplest choice but you cannot match values on neighboring triangles. Linear functions come next. This is like using flat tiles to build a waterproof dome, which is perfectly possible.

Now you use the basic elliptic equation (expressed in $\Omega$ )
$$
-\nabla \cdot(c \nabla u)+a u=f .
$$
If $u_{h}$ is the piecewise linear approximation to $u$, it is not clear what the second derivative term means. Inside each triangle, $\nabla u_{h}$ is a constant (because $u_{h}$ is flat) and thus the second-order term vanishes. At the edges of the triangles, $c \nabla u_{h}$ is in general discontinuous and a further derivative makes no sense.

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Open the PDE App

For basic information on 2-D geometry construction, see “Create 2-D Geometry” on page $2-17$

Partial Differential Equation Toolbox software includes the PDE app, which covers all aspects of the PDE solution process. You start it by typing
pdetool
at the MATLAB command line. It may take a while the first time you launch the PDE app during a MATLAB session. The following figure shows the PDE app as it looks when you start it.

At the top, the PDE app has a pull-down menu bar that you use to control the modeling. Below the menu bar, a toolbar with icon buttons provide quick and easy access to some of the most important functions.

To the right of the toolbar is a pop-up menu that indicates the current application mode. You can also use it to change the application mode. The upper right part of the PDE app also provides the $x$ – and $y$-coordinates of the current cursor position. This position is updated when you move the cursor inside the main axes area in the middle of the PDE app.
The edit box for the set formula contains the active set formula.
In the main axes you draw the 2-D geometry, display the mesh, and plot the solution.
At the bottom of the PDE app, an information line provides information about the current activity. It can also display help information about the toolbar buttons.

微积分网课代修|偏微分方程代写Partial Differential Equation代考|MATH480 Finite Element Method (FEM) Basics

微积分网课代修|偏微分方程代写Partial Differential Equation代 考|Finite Element Method (FEM) Basics


核心偏微分方程工具箱算法是一种 PDE 求解器,它使用有限元法 (FEM) 来解决在平面 中的有界域上定义的问题。
复杂几何上的简单偏微分方程的解很少可以用初等函数来表示。您面临两个问题: 首 先,您需要描述一个复杂的几何图形并在其上生成一个网格。然后,您需要在网格上离 散化您的 PDE,并为解的离散近似建立一个方程。PDE 应用程序为您提供易于使用的 图形工具来描述复杂的域并生成三角形网格。它还离散化 PDE,找到离散解并纭制结 果。您可以直接从命令行 (或从文件) 访问网格结构和离散化函数,并将它们合并到专 门的应用程序中。
以下是有限元法 (FEM) 的概述。本演示文稿的目的是让您熟悉基本的 FEM 概念。在这 里,您可以找到求解的精确方程以及离散解的性质。介绍了在偏微分方程工具箱软件中 实现的基本方程的不同扩展。更详细的描述可以在第 5-2 页的“椭圆方程”中找到,特定 类型的变体在第 5-13 页的“偏微分方程系统”、第 5-17 页的“抛物线方程”和第 5-17页 的”双曲方程”中第 5-20页、第 5-22 页的“特征值方程”和第 5-26页的“非线性方程”。 你从近似计算域开始 $\Omega$ 与简单几何对象的联合,在本例中为三角形 (2-D 几何) 或四面 体 (3-D 几何)。(这个讨论适用于三角形和四面体,但谈到三角形。) 三角形形成一 个网格,每个顶点称为一个节点。您处于建筑师设计圆顶的情况。建筑师必须在原始草 图的理想圆形形式和简单的积木、三角形或四边形的限制之间取得平衡。如果结果看起 来不够接近完美的圆顶,建筑师总是可以通过使用更小的块来改善结果。
接下来你说你的解决方案应该在每个三角形上都很简单。多项式是一个不错的选择: 它 们易于评估并且在小域上具有良好的近似特性。您可以要求相邻三角形中的解在边上连 续地相互连接。您仍然可以决定多项式的复杂程度。就像建筑师一样,您希望它们尽可 能简单。常数是最简单的选择,但不能匹配相邻三角形的值。接下来是线性函数。这就 像使用平地砖建造防水窀顶一样,这是完全可能的。
现在您使用基本的椭圆方程 (表示为 $\Omega$ )
$$
-\nabla \cdot(c \nabla u)+a u=f .
$$
如果 $u_{h}$ 是分段线性逼近 $u$ ,不清楚二阶导数是什么意思。在每个三角形内部, $\nabla u_{h}$ 是 一个常数(因为 $u_{h}$ 是平坦的),因此二阶项消失。在三角形的边缘, $c \nabla u_{h}$ 通常是不连 续的,并且进一步的导数没有意义。


微积分网课代修|偏微分方程代写Partial Differential Equation代 考|Open the PDE App


有关 2-D 几何构造的基本信息,请参阅第 1 页的“创建 2-D 几何”2-17
偏微分方程工具箱软件包括 PDE 应用程序,它涵盖了 PDE 求解过程的所有方面。您可 以通过在 MATLAB 命令行中键入
pdetool来启动它。
在 MATLAB 会话期间首次启动 PDE 应用程序可能需要一段时间。下图显示了 PDE 应 用程序在您启动时的样子。
在顶部,PDE 应用程序有一个下拉菜单栏,您可以使用它来控制建模。在菜单栏下方, 带有图标按钮的工具栏可让您快速轻松地访问一些最重要的功能。
工具栏右侧是一个弹出菜单,指示当前应用程序模式。您还可以使用它来更改应用程序 模式。PDE 应用程序的右上角还提供 $x$ – 和 $y$ – 当前光标位置的坐标。当您在 PDE 应用 程序中间的主轴区域内移动光标时,此位置会更新。
设置公式的编辑框包含激活的设置公式。
在主轴上绘制二维几何图形,显示网格并绘制解。
在 PDE 应用程序的底部,一条信息行提供有关当前活动的信息。它还可以显示有关工
具栏按钮的帮助信息。

微积分网课代修|偏微分方程代写Partial Differential Equation代考|MATH480 Finite Element Method (FEM) Basics
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