微积分网课代修|偏微分方程代写Partial Differential Equation代考|MATH054 Geometry Function for a Circle

• 单变量微积分
• 多变量微积分
• 傅里叶级数
• 黎曼积分
• ODE
• 微分学

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Geometry Function for a Circle

This example shows how to use a geometry function to create a circular region. Of course, you could just as easily use a circle basic shape.
You can parametrize a circle with radius 1 centered at the origin $(0,0)$ as follows:
\begin{aligned} &x=\cos (t) \ &y=\sin (t) \ &0 \leq t \leq 2 \pi . \end{aligned}
A geometry function needs to have at least two segments. So break up the circle into four segments: $0 \leq t \leq \Pi / 2, \Pi / 2 \leq t \leq \Pi, \Pi \leq t \leq 3 \Pi / 2$, and $3 \Pi / 2 \leq t \leq 2 \Pi$.
Now that you have a parametrization, write the geometry function. Save this function file as circlefunction.m on your MATLAB path.

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Arc Length Calculations for a Geometry Function

This figure shows how the direction of parameter increase relates to label numbering. The arrows in the following figure show the directions of increasing parameter values. The black dots indicate curve beginning and end points. The red numbers indicate region labels. The red 0 in the center of the figure indicates that the center square is a hole.

• The arrows by curves 1 and 2 show region 1 to the left and region 0 to the right.
• The arrows by curves 3 and 4 show region 0 to the left and region 1 to the right.
• The arrows by curves 5 and 6 show region 0 to the left and region 1 to the right.
• The arrows by curves 7 and 8 show region 1 to the left and region 0 to the right.
• Use a parametrization that is not proportional to arc length plus a constant. This approach is simplest, but can yield a distorted mesh that does not give the most accurate solution to your PDE problem.

微积分网课代修|偏微分方程代写Partial Differential Equation代 考 Geometry Function for a Circle

$$x=\cos (t) \quad y=\sin (t) 0 \leq t \leq 2 \pi .$$

$0 \leq t \leq \Pi / 2, \Pi / 2 \leq t \leq \Pi, \Pi \leq t \leq 3 \Pi / 2$ ， 和 $3 \Pi / 2 \leq t \leq 2 \Pi$.

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Arc Length Calculations for a Geometry Function

• 曲线 1 和 2 的箭头表示左侧的区域 1 和右侧的区域 0 。
• 曲线 3 和 4 的箭头表示左侧的区域 0 和右侧的区域 1 。
• 曲线 5 和 6 的箭头表示左侧的区域 0 和右侧的区域 1 。
• 曲线 7 和 8 的箭头表示左侧的区域 1 和右侧的区域 0 。
• 使用与弧长加一个常数不成比例的参数化。这种方法最简单，但会产生扭曲的网 格，无法为您的 PDE 问题提供最准确的解决方案。