CFD作业：采用代数法生成结构网格
将变换关系定义为
{ e m ξ − 1 e m − 1 = x − x m i n x m a x − x m i n e n η − 1 e n − 1 = y − y 1 ( x ) y 2 ( x ) − y 1 ( x ) \left\lbrace \begin{array}{l} \frac{e^{m\xi } -1}{e^m -1}=\frac{x-x_{min} }{x_{max} -x_{min} }\\ \frac{e^{n\eta } -1}{e^n -1}=\frac{y-y_1 \left(x\right)}{y_2 \left(x\right)-y_1 \left(x\right)} \end{array}\right. {em−1emξ−1=xmax−xminx−xminen−1enη−1=y2(x)−y1(x)y−y1(x)
化简有
{ x = x m i n + ( x m a x − x m i n ) e m ξ − 1 e m − 1 y = y 1 ( x ) + ( y 2 ( x ) − y 1 ( x ) ) e n η − 1 e n − 1 \left\lbrace \begin{array}{l} x=x_{min} +\left(x_{max} -x_{min} \right)\frac{e^{m\xi } -1}{e^m -1}\\ y=y_1 \left(x\right)+\left(y_2 \left(x\right)-y_1 \left(x\right)\right)\frac{e^{n\eta } -1}{e^n -1} \end{array}\right. {x=xmin+(xmax−xmin)em−1emξ−1y=y1(x)+(y2(x)−y1(x))en−1enη−1
对于本题 y 1 ( x ) y_1 \left(x\right) y1(x)、 y 2 ( x ) y_2 \left(x\right) y2(x)均为常数， y 1 ( x ) = y m i n y_1 \left(x\right)=y_{min} y1(x)=ymin， y 2 ( x ) = y m a x y_2 \left(x\right)=y_{max} y2(x)=ymax，本题中，取为
{ x m i n = 0 x m a x = 10 y m i n = 0 y m a x = 10 \left\lbrace \begin{array}{l} x_{min} =0\\ x_{max} =10\\ y_{min} =0\\ y_{max} =10 \end{array}\right. ⎩⎪⎪⎨⎪⎪⎧xmin=0xmax=10ymin=0ymax=10
因此只需取值 m m m与 n n n来保证 ( x 1 , j − x 0 , j ) ∈ ( 0.05 , 0.1 ) \left(x_{1,j} -x_{0,j} \right)\in \left(0.05,0.1\right) (x1,j−x0,j)∈(0.05,0.1)， ( y i , 1 − y i , 0 ) ∈ ( 0.05 , 0.1 ) \left(y_{i,1} -y_{i,0} \right)\in \left(0.05,0.1\right) (yi,1−yi,0)∈(0.05,0.1)即可。
已经满足题目要求，即给出答案为
{ a = 10 b = 10 m = 1 n = 1 \left\lbrace \begin{array}{l} a=10\\ b=10\\ m=1\\ n=1 \end{array}\right. ⎩⎪⎪⎨⎪⎪⎧a=10b=10m=1n=1
但我们进一步分析一下这些参数的影响
为了有比对效果，我们先将x、y方向的分区数都提高到20，可以看到，间隔相应减小。
再只调整y方向的参数而保持x方向的参数不变。
调整n的取值，发现随着n的增大，y方向上的不均匀性更强了。
我们再随便画个拉瓦尔喷管试试
{ y 1 ( x ) = x ( 10 − x ) 10 y 2 ( x ) = 10 − x ( 10 − x ) 10 \left\lbrace \begin{array}{l} y_1 \left(x\right)=\frac{x(10-x)}{10}\\ y_2 \left(x\right)=10-\frac{x(10-x)}{10} \end{array}\right. {y1(x)=10x(10−x)y2(x)=10−10x(10−x)
程序还算有效。
%% 图1
%% 程序初始化
close;clear,clc
%% 调整参数
m = 1;
n = 1;
%% 选取分区数
count_x = 10;
count_y = 10;
%% 描述计算区域
x_max = 10;
x_min = 0;
y_1 = @(x) 0;
y_2 = @(x) 10;
%% 计算划分网格
xi = (0:1:count_x)/count_x;
eta = (0:1:count_y)/count_y;
for i = 1:(count_x+1)
for j = 1:(count_y+1)
x(i,j) = x_min + (x_max-x_min)*(exp(m*xi(i))-1)/(exp(m)-1);
y(i,j) = y_1(x(i,j)) + (y_2(x(i,j))-y_1(x(i,j)))*(exp(n*eta(j))-1)/(exp(n)-1);
end
end
%% 绘制划分结果
figure(1);
for i = 1:(count_x)
for j = 1:(count_y)
line([x(i,j) x(i,j+1)],[y(i,j) y(i,j+1)],'color','k','linewidth',2);
line([x(i,j) x(i+1,j)],[y(i,j) y(i+1,j)],'color','k','linewidth',2);
end
end
for i = 1:(count_x)
line([x(i,count_y+1) x(i+1,count_y+1)],[y(i,count_y+1) y(i+1,count_y+1)],'color','k','linewidth',2);
end
for j = 1:(count_y)
line([x(count_x+1,j) x(count_x+1,j+1)],[y(count_x+1,j) y(count_x+1,j+1)],'color','k','linewidth',2);
end
axis equal;
title({['m = ',num2str(m,'%d'),'，n = ',num2str(n,'%d'),'，x、y方向各分',num2str(count_x,'%d'),'、',num2str(count_y,'%d'),'个区域'],...
['x_{1,0} - x_{0,0} = ', num2str((x(2,1)-x(1,1)),'%.3f'),'，y_{0,1} - y_{0,0} = ', num2str((y(1,2)-y(1,1)),'%.3f')]});
xlabel('x');
ylabel('y');
xlim([-2.00 12.00])
ylim([-2.00 12.00])
%% 后面的几个图
%% 程序初始化
clear;
%% 调整参数
m = 1;
n = 1;
%% 选取分区数
count_x = 20;
count_y = 20;
%% 描述计算区域
x_max = 10;
x_min = 0;
y_1 = @(x) 0;
y_2 = @(x) 10;
%% 计算划分网格
xi = (0:1:count_x)/count_x;
eta = (0:1:count_y)/count_y;
for i = 1:(count_x+1)
for j = 1:(count_y+1)
x(i,j) = x_min + (x_max-x_min)*(exp(m*xi(i))-1)/(exp(m)-1);
y(i,j) = y_1(x(i,j)) + (y_2(x(i,j))-y_1(x(i,j)))*(exp(n*eta(j))-1)/(exp(n)-1);
end
end
%% 绘制划分结果
figure(2);
for i = 1:(count_x)
for j = 1:(count_y)
line([x(i,j) x(i,j+1)],[y(i,j) y(i,j+1)],'color','k','linewidth',2);
line([x(i,j) x(i+1,j)],[y(i,j) y(i+1,j)],'color','k','linewidth',2);
end
end
for i = 1:(count_x)
line([x(i,count_y+1) x(i+1,count_y+1)],[y(i,count_y+1) y(i+1,count_y+1)],'color','k','linewidth',2);
end
for j = 1:(count_y)
line([x(count_x+1,j) x(count_x+1,j+1)],[y(count_x+1,j) y(count_x+1,j+1)],'color','k','linewidth',2);
end
axis equal;
title({['m = ',num2str(m,'%d'),'，n = ',num2str(n,'%d'),'，x、y方向各分',num2str(count_x,'%d'),'、',num2str(count_y,'%d'),'个区域'],...
['x_{1,0} - x_{0,0} = ', num2str((x(2,1)-x(1,1)),'%.3f'),'，y_{0,1} - y_{0,0} = ', num2str((y(1,2)-y(1,1)),'%.3f')]});
xlabel('x');
ylabel('y');
xlim([-2.00 12.00])
ylim([-2.00 12.00])
%% 程序初始化
clear;
%% 调整参数
m = 1;
n = 10;
%% 选取分区数
count_x = 20;
count_y = 20;
%% 描述计算区域
x_max = 10;
x_min = 0;
y_1 = @(x) 0;
y_2 = @(x) 10;
%% 计算划分网格
xi = (0:1:count_x)/count_x;
eta = (0:1:count_y)/count_y;
for i = 1:(count_x+1)
for j = 1:(count_y+1)
x(i,j) = x_min + (x_max-x_min)*(exp(m*xi(i))-1)/(exp(m)-1);
y(i,j) = y_1(x(i,j)) + (y_2(x(i,j))-y_1(x(i,j)))*(exp(n*eta(j))-1)/(exp(n)-1);
end
end
%% 绘制划分结果
figure(3);
for i = 1:(count_x)
for j = 1:(count_y)
line([x(i,j) x(i,j+1)],[y(i,j) y(i,j+1)],'color','k','linewidth',2);
line([x(i,j) x(i+1,j)],[y(i,j) y(i+1,j)],'color','k','linewidth',2);
end
end
for i = 1:(count_x)
line([x(i,count_y+1) x(i+1,count_y+1)],[y(i,count_y+1) y(i+1,count_y+1)],'color','k','linewidth',2);
end
for j = 1:(count_y)
line([x(count_x+1,j) x(count_x+1,j+1)],[y(count_x+1,j) y(count_x+1,j+1)],'color','k','linewidth',2);
end
axis equal;
title({['m = ',num2str(m,'%d'),'，n = ',num2str(n,'%d'),'，x、y方向各分',num2str(count_x,'%d'),'、',num2str(count_y,'%d'),'个区域'],...
['x_{1,0} - x_{0,0} = ', num2str((x(2,1)-x(1,1)),'%.3f'),'，y_{0,1} - y_{0,0} = ', num2str((y(1,2)-y(1,1)),'%.3f')]});
xlabel('x');
ylabel('y');
xlim([-2.00 12.00])
ylim([-2.00 12.00])
%% 程序初始化
clear;
%% 调整参数
m = 1;
n = 1;
%% 选取分区数
count_x = 20;
count_y = 20;
%% 描述计算区域
x_max = 10;
x_min = 0;
y_1 = @(x) x*(10-x)/10;
y_2 = @(x) 10 - x*(10-x)/10;
%% 计算划分网格
xi = (0:1:count_x)/count_x;
eta = (0:1:count_y)/count_y;
for i = 1:(count_x+1)
for j = 1:(count_y+1)
x(i,j) = x_min + (x_max-x_min)*(exp(m*xi(i))-1)/(exp(m)-1);
y(i,j) = y_1(x(i,j)) + (y_2(x(i,j))-y_1(x(i,j)))*(exp(n*eta(j))-1)/(exp(n)-1);
end
end
%% 绘制划分结果
figure(4);
for i = 1:(count_x)
for j = 1:(count_y)
line([x(i,j) x(i,j+1)],[y(i,j) y(i,j+1)],'color','k','linewidth',2);
line([x(i,j) x(i+1,j)],[y(i,j) y(i+1,j)],'color','k','linewidth',2);
end
end
for i = 1:(count_x)
line([x(i,count_y+1) x(i+1,count_y+1)],[y(i,count_y+1) y(i+1,count_y+1)],'color','k','linewidth',2);
end
for j = 1:(count_y)
line([x(count_x+1,j) x(count_x+1,j+1)],[y(count_x+1,j) y(count_x+1,j+1)],'color','k','linewidth',2);
end
axis equal;
title({['m = ',num2str(m,'%d'),'，n = ',num2str(n,'%d'),'，x、y方向各分',num2str(count_x,'%d'),'、',num2str(count_y,'%d'),'个区域'],...
['x_{1,0} - x_{0,0} = ', num2str((x(2,1)-x(1,1)),'%.3f'),'，y_{0,1} - y_{0,0} = ', num2str((y(1,2)-y(1,1)),'%.3f')]});
xlabel('x');
ylabel('y');
xlim([-2.00 12.00])
ylim([-2.00 12.00])