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1、热传导方程的差分解法PPT课件 热传导方程的差分解法?9.1热传导方程概述考虑三维空间的温度变化情况,设t时刻点(x,y,z)处的温度为u(x,y,z,t),则?t时间内通过横截面积?S传导的热量为(沿n方向):其中:K(x,y,z,t)是介质的热传导系数,为温度梯度的法向量分量.取空间中的一个小区域V,其边界面S为一封闭曲面.则t1到t2时刻通过包面S传入V的热量为:(,)uQ Kxyzt t S?nu?n?Sttdsnut zy xK dtQ),(211由高斯公式:?为哈密顿算子:设介质的比热容为c,密度为?,则V内温度变化消耗的热量:设V内部热源密度为F(x,y,z,t),则内部热源产生
2、的热量为:?211(,)VttKxyzt uQ dVdt?(,)u u uuuxyz ui j kx yz?212VttuctQ dVdt?根据能量守恒原则:Q2=Q1+Q3即:亦即:若F(x,y,z,t)?0,c,?,K,为常数,则:?VttdV F u Ktucdt0)(21?213(,)VttFxyzt QdVdt?),()(t zy xFu Ktuc?uKu Ktuc?)(?其中:?为拉普拉斯算子:所以热传导方程为:其中:?K?c?.222222u u uux yz?222222()u u u ut x yz?9.2一维热传导方程的差分解法一维热传导方程:初值问题初值条件:初边值混合问
3、题初值条件:边值条件:(关于边界点x=0和x=l)第一类.22,0,0u utTt x?(,0)(),|ux x x?(,0)(),0ux x xl?12(0,)()0(,)()u t g ttTult gt?第二类:第三类:其中g1(t),g2(t),?1(t),?2(t)为给定函数,要求?1(t)?,?2(t)?,且不同时为零.12(0,)()0(,)()u tg txt Tul tgtx?1122(0,)()(0,)()0(,)()(,)()u ttutgtxt Tulttultgtx?设空间的步长为h,时间的步长为?.把空间和时间离散化:近似微分:故可定义:对空间一阶向前插商:1,11
4、(,)(,)(,)i k ik i k i k i ki iu u uxt uxt uxtxxxh?00,0,1,2,.;,0,1,2,.i kxxihittkk?,1,i k i k i ku u ux h?对空间一阶向后插商:对空间二阶中心差商:对时间一阶向前插商:,1,i k i ki ku u uxh?,2,1,1,222ik ikik i k ik i ku uu u u u xxx hh?,1,i kiki ku u ut?代入热传导方程:迭代公式:,1,1,1,22ik ik ik ik i ku u u u uh?,11,1,2 (12);ikikikikuuuuh?1,2,.
5、,1;0,1,.,1;l TiNk MNMh?t i-1i i+1x k+1k第一类初边值条件:已知:0,0,1,10,01,01,0,0tM NMNN Nxuuu uuuuu?,0(),0,1,2,.,i iuxi N?0,1(),0,1,2,.,k kugt kM?,2(),0,1,2,.,Nk kugt kM?,11,1,11,1M NMNu uuu?第二类初边值条件:已知即:,0(),0,1,2,.,i iuxi N?1,0,1(),0,1,2,.,k kkuugt k Mh?0,1,1(),0,1,2,.,k k kuuhgt kM?,1,2(),0,1,2,.,Nk Nk kuuh
6、gtk M?,1,2(),0,1,2,.,Nk Nkkuugt kMh?计算过程:0,01,01,0,0tN Nxuuuu?1,11,1Nu u?0,1,1Nu u1,21,2Nu u?0,2,2Nu u第三类初边值条件:已知:即:,0(),0,1,2,.,i iuxi N?1,0,10,1()(),0,1,2,.,k kk k ku utu gtkMh?,1,2,2()(),0,1,2,.,Nk NkkNkkuutu gtkMh?0,1,11()1(),0,1,2,.,kkk kuuhgtht kM?,1,22()1(),0,1,2,.,Nk Nkk kuuhgthtk M?例1:差分方程:
7、初边值条件:22,01,0(,0)4 (1),01(0,)0,(1,)0,0u uxtt xuxxx xututt?,11,1,2 (12);0.1,1,1600ikikikikuuuuhh?,00,10,4 (1),0,1,2,.,100,0,1,2,.ik kuih ih iu uk?function u=rcd(lamda,tao,h,H,T)x=0:h:H;t=0:tao:T;a=tao*lamda/h2;N=length(x);M=length(t);u(:,1)=(4*x.*(1-x);u(1,2:M)=0;u(N,2:M)=0;for k=1:M-1for i=2:N-1u(i,
8、k+1)=a*u(i+1,k)+(1-2*a)*u(i,k)+a*u(i-1,k);end endh1=line(Color,100,Marker,.,MarkerSize,20,EraseMode,xor);for i=1:length(t)set(h1,Xdata,0:0.1:1,Ydata,u(:,i);pause(tao);endX,Y=meshgrid(x,0:0.01:0.2);Z=repmat(u(:,1),size(X,1),1);h2=surface(X,Y,Z);shading interp,axis equal;set(h2,EraseMode,xor);for i=1:
9、length(t)CD=repmat(u(:,i),size(X,1),1);set(h2,Cdata,CD);pause(tao);end?9.3二维热传导方程的差分解法内部无热源均匀介质中二维热传导方程:初值条件:边值条件视具体情况而定.设空间的步长为h,时间的步长为?.设Nh=l,Mh=s,把时间和空间离散化:2222()uuutxy?0,0,0xl ystT?(,0)(,)uxy xy?即:微分近似:000,0,1,2,.;,0,1,2,.;,0,1,2,.iikx xihiNy yjh j Mt tkk?,1,i jki jki jku uut?2,1,1,222ijk i jk i
10、jki jkuuuuxh?2,1,1,222ijk ij k ijkij kuuuuy h?代入热传导方程得:,1,1,1,1,1, (14)()ijk ijkijkijkijkijku uuuuu?2h?k+1ki-1i i+1j+1j j-1例:初值条件:即:恒温边界:绝热边界:即:(,0)(,)0uxy xy?,00,0,1,.,;0,1,.,iju iNj M?,0,0,120,0,1,.,1,.,ikiMkjku ui Nu j M M?,0,120,1,2,.,10,1,.,1,.,1,.,1Njkjkuj MxujM M Mx?,1,0,1,12,1,2,.,1,1,.,1,.,
11、1,.,1Njk Njkjkjkuu j Mu ujMMM?差分公式:,1,1,1,1,1,0,0,0,12,1,0,1, (14)()0,0,1,.,;0,1,.,0,0,1,.,1,.,;1,2,.,1,2,.,1,1,.,i j ki jkijkijkijkij kiji ki Mkj kN jkN jkj kj kuuuuuuu iNjMu ui NujMM kuujMuuj?121,.,1,.,1MMM?function u=rcd2(lamda,tao,h,T,L,S)%二维热传导方程t=0:tao:T;x=0:h:L;y=0:h:S;a=tao*lamda/h2;if a0.25
12、error(lamda*tao/h20.25);end D=length(t);N=length(x);M=length(y);M1=ceil(M/2)-3;M2=ceil(M/2)+3;u=zeros(N,M,D);u(:,:,1)=0;u(:,1,:)=0;u(:,M,:)=0;u(1,M1:M2,2:D)=1;for k=1:D-1for i=2:N-1for j=2:M-1u(i,j,k+1)=(1-4*a)*u(i,j,k)+a*(u(i+1,j,k)+u(i-1,j,k)+u(i,j+1,k)+u(i,j-1,k);u(N,j,k+1)=u(N-1,j,k+1);if(jM2)u(1,j,k+1)=u(2,j,k+1);end end endendX,Y=meshgrid(x,y);Z=u(:,:,1);h2=surface(X,Y,Z);shading interp,axis equal;set(h2,EraseMode,xor);for i=1:length(t)CD=u(:,:,i);set(h2,Cdata,CD);pause(tao);end rcd2(1,0.001,0.1,0.2,2,1);。 内容仅供参考
