微积分网课代修|偏微分方程代写Partial Differential Equation代考|AMATH569 Poisson’s Equation with Complex 2-D Geometry

微积分网课代修|偏微分方程代写Partial Differential Equation代考|AMATH569 Poisson’s Equation with Complex 2-D Geometry

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微积分网课代修|偏微分方程代写Partial Differential Equation代考|AMATH569 Poisson’s Equation with Complex 2-D Geometry

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Poisson’s Equation with Complex 2-D Geometry

This example shows how to solve the Poisson’s equation, $-\Delta u=f$ using the PDE app. This problem requires configuring a 2-D geometry with Dirichlet and Neumann boundary conditions.

To start the PDE app, type the command pdetool at the MATLAB prompt. The PDE app looks similar to the following figure, with exception of the grid. Turn on the grid by selecting Grid from the Options menu. Also, enable the “snap-to-grid” feature by selecting Snap from the Options menu. The “snap-to-grid” feature simplifies aligning the solid objects.

The first step is to draw the geometry on which you want to solve the PDE. The PDE app provides four basic types of solid objects: polygons, rectangles, circles, and ellipses. The objects are used to create a Constructive Solid Geometry model (CSG model). Each solid object is assigned a unique label, and by the use of set algebra, the resulting geometry can be made up of a combination of unions, intersections, and set differences. By default, the resulting CSG model is the union of all solid objects.
To select a solid object, either click the button with an icon depicting the solid object that you want to use, or select the object by using the Draw pull-down menu. In this case, rectangle/square objects are selected. To draw a rectangle or a square starting at a corner, click the rectangle button without a $+$ sign in the middle. The button with the $+$ sign is used when you want to draw starting at the center. Then, put the cursor at the desired corner, and click-and-drag using the left mouse button to create a rectangle with the desired side lengths. (Use the right mouse button to create a square.) Click and drag from $(-1, .2)$ to $(1,-.2)$. Notice how the “snap-to-grid” feature forces the rectangle to line up with the grid. When you release the mouse, the CSG model is updated and redrawn. At this stage, all you have is a rectangle. It is assigned the label R1. If you want to move or resize the rectangle, you can easily do so. Click-and-drag an object to move it, and doubleclick an object to open a dialog box, where you can enter exact location coordinates. From the dialog box, you can also alter the label. If you are not satisfied and want to restart, you can delete the rectangle by clicking the Delete key or by selecting Clear from the Edit menu.

Next, draw a circle by clicking the button with the ellipse icon with the $+$ sign, and then click-and-drag in a similar way, starting near the point $(-.5,0)$ with radius $.4$, using the right mouse button, starting at the circle center.

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

PDE app toolbar provide quick access to key operations that are also available in the menus.

The toolbar consists of three different parts: the five leftmost buttons for draw mode functions, the next six buttons for different boundary, mesh, solution, and plot functions, and the rightmost button for activating the zoom feature.

Five buttons on the left let you draw the geometry. Double-click a button makes it “stick,” and you can then continue to draw solid objects of the selected type until you single-click the button to “release” it.
In draw mode, you can create the 2 -D geometry using the constructive solid geometry (CSG) model paradigm. A set of solid objects (rectangle, circle, ellipse, and polygon) is provided. These objects can be combined using set formulas in a flexible way.

微积分网课代修|偏微分方程代写Partial Differential Equation代考|AMATH569 Poisson’s Equation with Complex 2-D Geometry

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Poisson’s Equation with Complex 2-D Geometry

这个例子展示了如何求解泊松方程,−D在=F使用 PDE 应用程序。这个问题需要用 Dirichlet 和 Neumann 边界条件配置二维几何。

要启动 PDE 应用程序,请在 MATLAB 提示符下键入命令 pdetool。PDE 应用程序与下图类似,但网格除外。通过从选项菜单中选择网格来打开网格。此外,通过从“选项”菜单中选择“对齐”来启用“对齐网格”功能。“对齐网格”功能简化了实体对象的对齐。

第一步是绘制要在其上求解 PDE 的几何图形。PDE 应用程序提供四种基本类型的实体对象:多边形、矩形、圆形和椭圆。这些对象用于创建构造实体几何模型(CSG 模型)。每个实体对象都被分配了一个唯一的标签,并且通过使用集合代数,生成的几何可以由联合、交叉和集合差异的组合组成。默认情况下,生成的 CSG 模型是所有实体对象的并集。
要选择实体对象,请单击带有描述您要使用的实体对象的图标的按钮,或使用“绘图”下拉菜单选择对象。在这种情况下,矩形/正方形对象被选中。要从一个角开始绘制一个矩形或正方形,请单击没有+在中间签到。按钮与+当您想从中心开始绘制时使用符号。然后,将光标放在所需的角上,并使用鼠标左键单击并拖动以创建具有所需边长的矩形。(使用鼠标右键创建一个正方形。)单击并拖动(−1,.2)至(1,−.2). 请注意“对齐网格”功能如何强制矩形与网格对齐。当您释放鼠标时,CSG 模型会更新并重新绘制。在这个阶段,你所拥有的只是一个矩形。它被分配了标签 R1。如果您想移动或调整矩形的大小,您可以轻松地做到这一点。单击并拖动对象以移动它,然后双击对象以打开一个对话框,您可以在其中输入准确的位置坐标。从对话框中,您还可以更改标签。如果您不满意并想重新开始,您可以通过单击 Delete 键或从 Edit 菜单中选择 Clear 来删除该矩形。

接下来,通过单击带有椭圆图标的按钮和+签名,然后以类似的方式单击并拖动,从该点附近开始(−.5,0)带半径.4,使用鼠标右键,从圆心开始。

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

PDE 应用程序工具栏提供对菜单中也可用的关键操作的快速访问。

工具栏由三个不同的部分组成:最左边的五个按钮用于绘制模式功能,接下来的六个按钮用于不同的边界、网格、解决方案和绘图功能,最右边的按钮用于激活缩放功能。

左侧的五个按钮可让您绘制几何图形。双击一个按钮使其“粘住”,然后您可以继续绘制所选类型的实体对象,直到您单击该按钮以“释放”它。
在绘图模式下,您可以使用构造实体几何 (CSG) 模型范例创建二维几何。提供了一组实体对象(矩形、圆形、椭圆和多边形)。这些对象可以使用设置公式以灵活的方式组合。

微积分网课代修|偏微分方程代写Partial Differential Equation代考|AMATH569 Poisson’s Equation with Complex 2-D Geometry
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