问题描述
所以，我是一名 EE 学生，试图创建一个 MATLAB 脚本，以通过有限差分法找到自定义数组不同部分的电压。基本上，我必须遍历每个数组元素，平均它周围的四个元素并将该平均值放在原始位置[所以基本上对于任意点 Vi,j = 1/4(Vi+1,j + Vi-1,j + Vi,j+1 + Vi,j-1)]。但是，当到每个网格点的间隔距离发生变化时，问题会略有变化。该问题成为加权平均值之一。让我们来看看我在 Excel 中为自己制作的一个简单的“图表”。
Click for pictorial representation of a Parallel Plate Grid
这里，整个数组从零开始，除了最后一行用常数 8000 填充。类似地，第 Zeroth（或 MATLAB 中的第一个）行用常数 0 填充。最后一行和第零行不会改变。代码从取平均值开始，经过迭代将收敛到正确的解决方案。可以看出或推断出，沿“X”轴的值不会发生太大变化（也是如此，因为它是等势面）。更改将在“Y”轴上。因此，我基本上可以通过从显示的八 (8) 列更改为仅一 (1) 列来计算网格，并且应该大幅增加行数，以便我可以进行“更精细”/更详细/更高分辨率的分析。这就是我的问题所在。我不知道如何“调整网格大小”。这是我的代码（注释中有一些对前面提供的图形表示的引用）：
% Description: Graphing the potential between
% two parallel plates of set voltages (boundary conditions) using FDM.
% I have considered the area under observation as a grid/array/matrix
% as shown in the accompanying photos. Each element of the matrix gives the
% magnitude of the potential at that very point.
% The matrix starts off zero-filled,and the boundary conditions are set
% (in this case,two parallel edges of the matrix are set to 8 kV and 0 kV
% to represent equipotential surfaces). Since the potential is only
% changing in one direction/axes,the magnitude of each grid element can be
% calculated relatively easily.
% The output graph is a 3D graph. The X and Y axes show the map of the
% region. The Z axis shows the strength of electric potential at every
% given unit grid on the X-Y plane.
% Initial Setup
clc;
clear all;
close all;
% xAxisResolution and yAxisResolution showcase how fine I want to calculate
% the field for. For example,in the attached photo,potential changes
% substantially more drastically over the y axis,but does not change with
% change of x axis. Thus,my yAxisResolution is great by a factor of 10.
xAxisResolution = 5*10^0;
yAxisResolution = 5*10^1;
% Creating the grid I want to calculate my potential strength over and
% zero-filling.
fieldArray = zeros (yAxisResolution,xAxisResolution);
% Setting up the Initial Conditions. Since one of the parallel plates is
% 0kV,and the matrix is already zero-filled,I do not need to do anything.
% For the second plate,the voltage is set to 8 kV.
for column = 1:xAxisResolution
fieldArray(1,column) = 8000;
end
% FDM Calculation
% iterator - Number of times that the whole grid/array is
% refreshed/updated. For best results,use 1:10^5
% To improve accuracy,some boundary condition if statements are added.
% They are described below.
for iterator = 1:10^4
for row = 1:(yAxisResolution) % Traverses each row
for column = 1:(xAxisResolution) % Traverses each column in every row
% Case #1: Check to see if the current row being traversed is
% the user-provided initial/boundary condition. In my case,I
% am testing two parallel plates at two edges. (Colored RED in
% photo) They are 0 kV and 8kV respectively and cannot be
% changed. Thus,when the loop encounters these edges,nothing
% is changed and break condition is encountered.
if (row == 1) || (row == yAxisResolution)
break
% Case #2: Checks to see if the current column is on the edge.
% If so,there will only be 3 adjacent elements,and the
% stencil is changed. This is accounted for with a specific
% formula to account for stencil change (average of only 3
% adjacent elements).
% (Colored YELLOW in photo)
elseif (column == 1)
fieldArray(row,column) = (fieldArray(row+1,column) + fieldArray(row-1,column) + fieldArray(row,column+1))/3;
elseif (column == xAxisResolution)
fieldArray(row,column-1))/3;
% Case #3: Checks to see if the current element is at any other
% position of the matrix/grid that can be changed/overwritten
% that has 4 adjacent elements. These elements are accounted
% for by their specific averaging formula (4 elements).
% (Colored GREEN in photo)
else
fieldArray(row,column+1) + fieldArray(row,column-1))/4;
end
end
end
end
% Produces graph of potential at every point
fieldArray;
surf(fieldArray)
xlabel('X'),ylabel('Y'),zlabel('Potential Strength')
The code output can be seen by clicking this hyperlink.
从这个输出中可以看出，网格被调整为在“Y”轴上有 50 个单位，在“X”轴上有 5 个单位。但是，我想计算恒定长度（例如 5 米 x 1 米区域）上的电位，但想将其划分为不同的网格。基本上，最终目标是拥有一个脚本，可以采用真实世界的物理长度尺寸，将它们分配给“X”或“Y”轴，然后更改计算分辨率。我曾尝试在 Matlab 中使用 linspace 来生成一个向量，但它真的很快就变得很混乱，因为我不知道如何从相同的 2D 矩阵中创建一个 2D 矩阵。我还试图改变我如何处理这些位置。我的意思是，确定矩阵是 50x5，但我可以将其标记为 5x5，并将其纳入我程序的其他区域（加权平均部分）。然而，我的实现根本没有奏效，甚至让我的程序更加混乱。
如果你们有我可以尝试的线索或想法，请告诉我。还有任何问题，请告诉我。
解决方法
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