柱体内温度分布图 MATLAB

对于下底面和侧面绝热，上底面温度与半径平方成正比的柱体，绘制柱体内温度分布图。

这里给出两种尝试：1、散点图；2、切片云图

1. 散点图仿真

首先使用解析算法求的场解值的解析表达，其次求解Bessel函数零点，带入场解表达。对于空间点阵划分，采用每一定半径划分圆周上等数量点，在总方向复制累计。

下面给出散点图的仿真代码，Nt为划分精度，Nt越大空间划分点越多。

% function y = cal117(R,h,Nt,N)
close all; clear all
R = ;
h = ;
Nt = ;

%% 求bessel函数零点
pi0 = besal_pi0(,Nt);

%% 计算圆柱体分割点坐标，作为u的前三列坐标
m = ;
cot = ;
for i =  :
    [x_,y_,z_] = cylinder((R*i/),m-);
    for k =  : m
        for j =  :
            cot = cot + ;
            u(cot,) = x_(,k);
            u(cot,) = y_(,k);
            u(cot,) = h*j/();
            u(cot,) = ;       %test
        end
    end
end

%% 计算温度值 u第四列为温度值
for i =  : cot
    for j =  : Nt
        ct(j) = *(R^)*(-/(pi0(j)^))/(pi0(j)*besselj(,pi0(j))*sinh(pi0(j)*h/R));
        u(i,) = u(i,)+ ct(j)*sinh(pi0(j)*u(i,)/R)*besselj(,pi0(j)*sqrt(u(i,)^+u(i,)^)/R);
        if(u(i,)>)
            u(i,)=NaN;
        end
    end
end

%% 图像绘制
[x0,y0,z0] = cylinder(R,); %画圆柱体轮廓
z0 = h * z0;
plot3(x0(,:),y0(,:),z0(,:),'k');hold on
plot3(x0(,:),y0(,:),z0(,:),'k');hold on
for i =  :  :
    plot3([x0(,i),x0(,i)],[y0(,i),y0(,i)],[z0(,i),z0(,i)],'k');hold on
end

x = u(:,);
y = u(:,);
z = u(:,);
c = u(:,);

% [X,Y] = meshgrid(linspace(min(x),max(x),)',linspace(min(y),max(y),30)');
% Z = griddata(x,y,z,X,Y,'v4');
% C =griddata(x,y,c,X,Y,'v4');

scatter3(x,y,z,,'filled','k');
axis([-(R+) (R+) -(R+) (R+)  (h+)]);

% scatter3(x,y,z,,c,'filled');
% axis([-(R+) (R+) -(R+) (R+)  (h+)]);
% colorbar;

2. 切片云图绘制

空间点集划分同上，其中将四位数组转化为三维数组，坐标三维即坐标点位置，值即温度值。

可以绘制出纵向切片图、整体剖面图和等温面图三张图：

clear; clc;
pi0 = besal_pi0(,);
R = ;
h = ;
xa = ;
xb = ;
xc = ;
v = zeros(xa,xb,xc);

for i =  : xa
    for j =  : xb
        for k =  : xc
            for t =  :
                ct(t) = *(R^)*(-/(pi0(t)^))/(pi0(t)*besselj(,pi0(t))*sinh(pi0(t)*h/R));
                v(i,j,k)=v(i,j,k)+ ct(t)*sinh(pi0(t)*k/R)*besselj(,pi0(t)*sqrt((i-)^+(j-)^)/R);
            end
%             if(k>)
%                 v(i,j,k) = NaN;
%             end
            if((i-)^+(j-)^>)
                v(i,j,k) = NaN;
            end
        end
    end
end

[x,y,z]=meshgrid(:xb,:xa,:xc);% 坐标值的范围
h=figure();% 第一幅图
% 各大切片
set(h,'name','取单切片')
subplot()% 切片1
slice(x,y,z,v,[],[],[]);
shading interp
% set(gca,'zdir','reverse');
axis equal
grid on
colorbar

subplot()% 切片2
slice(x,y,z,v,[],[],[]);
shading interp
colormap('jet')
% set(gca,'zdir','reverse');
axis equal
grid on
colorbar

subplot()% 切片3
slice(x,y,z,v,[],[],[]);
shading interp
% set(gca,'zdir','reverse');
axis equal
grid on
colorbar

subplot()% 切片4
slice(x,y,z,v,[],[],[]);
shading interp
% set(gca,'zdir','reverse');
axis equal
grid on
colorbar
%%
h2=figure();
figure('menubar','figure');
% rotate 3D
set(h2,'name','全空间切片','MenuBar','none','ToolBar','none')
slice(x,y,z,v,[::xb],[  ],[   ])
shading interp
colorbar
colormap('jet')
% set(gca,'zdir','reverse');
axis equal
grid on
% box on

%%
h3=figure();
figure('menubar','figure');
set(h3,'name','定值包络立体图','MenuBar','none','ToolBar','none')
set(gcf,'InvertHardcopy','off')
fw=;   %%此值为最外层包络面取值
fv=isosurface(x,y,z,v,fw);% 等值面
p=patch(fv);

set(p,'facecolor','b','edgecolor','none');
patch(isocaps(x,y,z,v, fw), 'FaceColor', 'interp', 'EdgeColor', 'none');
colorbar

colormap('jet')
box on
daspect([,,])
view()

camlight
camproj perspective
lighting phong % 光照处理
axis equal
grid on
title(['最外层表面的值为： ' , num2str(fw),'^{o}C']);

最后贴上一些效果图哈：