傳熱學(xué)MATLAB溫度分布大作業(yè)

1、東南大學(xué)能源與環(huán)境學(xué)院課程作業(yè)報(bào)告作業(yè)名稱：傳熱學(xué)大作業(yè)一一利用 matlab程序解決熱傳導(dǎo)問題院系：能源與環(huán)境學(xué)院專業(yè)：建筑環(huán)境與設(shè)備工程學(xué)號(hào)：姓名：2014年11月9日一、題目及要求1. 原始題目及要求2. 各節(jié)點(diǎn)的離散化的代數(shù)方程3. 源程序4. 不同初值時(shí)的收斂快慢5. 上下邊界的熱流量(入=1W/(m C)6. 計(jì)算結(jié)果的等溫線圖7. 計(jì)算小結(jié)題目：已知條件如下圖所示：10CC絕熱T尸10。t) h=10W/二、各節(jié)點(diǎn)的離散化的代數(shù)方程各溫度節(jié)點(diǎn)的代數(shù)方程ta=(300+b+e)/4 ; tb=(200+a+c+f)/4; tc=(200+b+d+g)/4; td=(2*c+200+

2、h)/4te=(100+a+f+i)/4; tf=(b+e+g+j)/4; tg=(c+f+h+k)/4 ; th=(2*g+d+l)/ 4 ti=(100+e+m+j)/4; tj=(f+i+k+n)/4; tk=(g+j+l+o)/4; tl=(2*k+h+q)/4tm=(2*i+300+n)/24;tn=(2*j+m+p+200)/24;to=(2*k+p+n+200)/24;tp=(l+o+100)/12三、源程序【G-S迭代程序】【方法一】函數(shù)文件為：function y,n=gauseidel(A,b,x0,eps)D=diag(diag(A);L=-tril(A,-1);U=-t

3、riu(A,1);G=(D-L)U;f=(D-L)b;y=G*x0+f;n=1;while norm(y-x0)>=epsx0=y;y=G*x0+f;n=n+1;end命令文件為：A=4,-1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0;-1,4,-1,0,0,-1,0,0,0,0,0,0,0,0,0,0;0,-1,4,-1,0,0,-1,0,0,0,0,0,0,0,0,0;0,0,-2,4,0,0,0,-1,0,0,0,0,0,0,0,0;-1,0,0,0,4,-1,0,0,-1,0,0,0,0,0,0,0;0,-1,0,0,-1,4,-1,0,0,-1,0,0,0,0,

4、0,0;0,0,-1,0,0,-1,4,-1,0,0,-1,0,0,0,0,0;0,0,0,-1,0,0,-2,4,0,0,0,-1,0,0,0,0;0,0,0,0,-1,0,-1,0,4,0,0,0,-1,0,0,0;0,0,0,0,0,-1,0,0,-1,4,-1,0,0,-1,0,0;0,0,0,0,0,0,-1,0,0,-1,4,-1,0,0,-1,0;0,0,0,0,0,0,0,-1,0,0,-2,4,0,0,0,-1;0,0,0,0,0,0,0,0,-2,0,0,0,24,-1,0,0;0,0,0,0,0,0,0,0,0,-2,0,0,-1,24,-1,0;0,0,0,0,0,0,

5、0,0,0,0,-2,0,0,-1,24,-1;0,0,0,0,0,0,0,0,0,0,0,-1,0,0,-1,12;b=300,200,200,200,100,0,0,0,100,0,0,0,300,200,200,100,;x,n=gauseidel(A,b,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,xx=1:1:4;yy=xx;X,Y=meshgrid(xx,yy);Z=reshape(x,4,4);Z=Z'contour(X,Y,Z,30)【方法 2 >> t=zeros(5,5);t(1,1)=100;t(1,2)=100;t(1,3)=10

6、0;t(1,4)=100;t(1,5)=100;t(2,1)=200;t(3,1)=200;t(4,1)=200;t(5,1)=200;for i=1:10t(2,2)=(300+t(3,2)+t(2,3)/ 4 ;t(3,2)=(200+t(2,2)+t(4,2)+t(3,3)/ 4;t(4,2)=(200+t(3,2)+t(5,2)+t(4,3)/ 4;t(5,2)=(2*t(4,2)+200+t(5,3)/ 4;t(2,3)=(100+t(2,2)+t(3,3)+t(2,4)/ 4;t(3,3)=(t(3,2)+t(2,3)+t(4,3)+t(3,4)/ 4;t(4,3)=(t(4,2)

7、+t(3,3)+t(5,3)+t(4,4)/ 4;t(5,3)=(2*t(4,3)+t(5,2)+t(5,4)/ 4;t(2,4)=(100+t(2,3)+t(2,5)+t(3,4)/ 4;t(3,4)=(t(3,3)+t(2,4)+t(4,4)+t(3,5)/ 4;t(4,4)=(t(4,3)+t(4,5)+t(3,4)+t(5,4)/ 4;t(5,4)=(2*t(4,4)+t(5,3)+t(5,5)/ 4;t(2,5)=(2*t(2,4)+300+t(3,5)/ 24;t(3,5)=(2*t(3,4)+t(2,5)+t(4,5)+200)/ 24;t(4,5)=(2*t(4,4)+t(3,

8、5)+t(5,5)+200)/ 24;t(5,5)=(t(5,4)+t(4,5)+100)/12;t'endcontour(t',50);ans =【Jacob迭代程序】函數(shù)文件為：function y,n=jacobi(A,b,x0,eps)D=diag(diag(A);L=-tril(A,-1);U=-triu(A,1);B=D(L+U);f=Db;y=B*x0+f;n=1;while norm(y-x0)>=epsx0=y;y=B*x0+f;n=n+1;end命令文件為:A=4,-1,0,0,-1,0,0,0,0,0,0,0,0,0,0,0;-1,4,-1,0,0,

9、-1,0,0,0,0,0,0,0,0,0,0;0,-1,4,-1,0,0,-1,0,0,0,0,0,0,0,0,0;0,0,-2,4,0,0,0,-1,0,0,0,0,0,0,0,0;-1,0,0,0,4,-1,0,0,-1,0,0,0,0,0,0,0;0,-1,0,0,-1,4,-1,0,0,-1,0,0,0,0,0,0;0,0,-1,0,0,-1,4,-1,0,0,-1,0,0,0,0,0;0,0,0,-1,0,0,-2,4,0,0,0,-1,0,0,0,0;0,0,0,0,-1,0,-1,0,4,0,0,0,-1,0,0,0;0,0,0,0,0,-1,0,0,-1,4,-1,0,0,-1

10、,0,0;0,0,0,0,0,0,-1,0,0,-1,4,-1,0,0,-1,0;0,0,0,0,0,0,0,-1,0,0,-2,4,0,0,0,-1;0,0,0,0,0,0,0,0,-2,0,0,0,24,-1,0,0;0,0,0,0,0,0,0,0,0,-2,0,0,-1,24,-1,0;0,0,0,0,0,0,0,0,0,0,-2,0,0,-1,24,-1;0,0,0,0,0,0,0,0,0,0,0,-1,0,0,-1,12;b=300,200,200,200,100,0,0,0,100,0,0,0,300,200,200,100,;x,n=jacobi(A,b,0,0,0,0,0,0,

11、0,0,0,0,0,0,0,0,0,0',;xx=1:1:4;yy=xx;X,Y=meshgrid(xx,yy);Z=reshape(x,4,4);Z=Z'contour(X,Y,Z,30)n =97Z =四、不同初值時(shí)的收斂快慢1、方法1在Gaus也代和Jacob迭代中，本程序應(yīng)用的收斂條件均為norm(y-x0)>=eps,即使前后所求誤差達(dá)到e的-6次方時(shí)，跳出循環(huán)得出結(jié)果。將誤差改為時(shí)，只需迭代 25次，如下x,n=gauseidel(A,b,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0',運(yùn)行結(jié)果為將誤差改為時(shí)，需迭代20次，可見隨著迭

12、代次數(shù)增加，誤差減小，變化速度也在減小。方法2通過i=1:10判斷收斂，為迭代10次，若改為1:20,則迭代20次。2、在同樣的誤差要求下，誤差控制在e的-6次方內(nèi)，Gaus也代用了 49次達(dá)到要求，而Jacob迭代用了 97次，可見，在迭代中盡量采用最新值，可以大幅度的減少迭代次數(shù)，迭代過程收斂快一些。在Gauss中，初值為100,迭代46次達(dá)到精確度，初值為50時(shí)，迭代47次，初值為0時(shí)，迭代49 次，初值為200時(shí)迭代50次，可見存在一個(gè)最佳初始值，是迭代最快。這一點(diǎn)在jacobi迭代中表現(xiàn)的尤為明顯。五、上下邊界的熱流量：上邊界t=200C,t =10C,所以,200-100* x 2

13、00-ta熱流量1=入*+a200-tbx +=1* (100/2+/2)下邊界200-tcx +t ._t t -1_熱流量2=|入* t一m x +一- yyti-tqx+q*100-10* x tn-t *to-t * tm-th*(* + -* x + -* x + y 2 yyytp-tx +y)|=|1*( / 2)-10*(90/ 2+ -10)+/2)| = |W =六、溫度等值線Gauss:Yacobi:七、計(jì)算小結(jié)導(dǎo)熱問題進(jìn)行有限差分?jǐn)?shù)值計(jì)算的基本思想是把在時(shí)間、空間上連續(xù)的溫度場(chǎng)用有限個(gè)離散點(diǎn)溫度的集合來代替，即有限點(diǎn)代替無限點(diǎn)，通過求解根據(jù)傅里葉定律和能量守恒兩大法則建立關(guān)于控制 面內(nèi)這些節(jié)點(diǎn)溫度值的代數(shù)方程，獲得各個(gè)離散點(diǎn)上的溫度值。要先劃分查分網(wǎng)格，在建立差分代數(shù)方程組，用MATLAB或者其他軟件編程求解。高斯-賽德爾迭代法和雅克比迭代法區(qū)別在于使用新植和舊值進(jìn)行下一次迭代