我正在尝试模拟一圈铜线的热传导。我相信pdepe函数是适当的，因为问题是圆柱极坐标中的强制一维热方程。
按照本论坛其他地方的建议，我将整个复合域（绝缘-铜-绝缘）建模为一个在空间中具有可变属性的单个域。
pdepe语法中的等式为（1 / alpha（r）） du / dt =（1 / r）（r * du / dr）+ q（r）/ k（r）
温度u（r，t），扩散率alpha（r），体积生热率q（r），电导率k（r）
内表面具有外部边界条件du / dr = 0（绝缘），外表面具有k（r） du / dr + h （u-u0）= 0（对流）表面。
任何地方的初始条件都是零。
我将输出t-> inf与解析稳态解决方案（附带）进行了比较，后者完全不同。在稳态下，温升应为120摄氏度左右。
但是，代码的输出（以下是相关摘录）均为O（1）。如果有人能说明我做错了什么，或者pdepe在这里不合适，我将不胜感激。预先表示感谢。
function [sol,systInfo]=ThreeLayerHeat
%uses pdepe()
%% geometry
rInner=0.05;
rOuter=0.1;
tIns=50e-6; %insulation thickness
r2=rInner+tIns; %internal boundaries
r3=rOuter-tIns;
tDomain=rOuter-rInner;
tCond=tDomain-tIns; %one conductor layer
m = 1; %CPCs
%% material properties
%copper
k_cond=394;
rho_cond=8960;
cp_cond=390;
%mica
k_ins=0.5;
rho_ins=2800;
cp_ins=880;
alpha_cond=k_cond/(rho_cond*cp_cond);
alpha_ins=k_ins/(rho_ins*cp_ins);
%%Internal generation
Jlimit=4e6; %4A/mm^2
ACfactor=1.2; %CF design of electrical machines
rho_conductor=1.75e-8; %electrical resistivity Table 9.3 293K %at reference level
%%boundary details
Tinf=0; %fluid temp on outer boundary
h2=100; %HTC on outer boundary
%%Characteristic (Fo) times
tau_chic_cond=tCond^2/alpha_cond;
tau_chic_ins=tIns^2/alpha_ins;
tau_chic_limiting=max(tau_chic_cond,tau_chic_ins);
%%discretisation
%spatial
npoints=50;
x=linspace(rInner,rOuter,npoints);
%ensure mesh points exist at internal boundaries r2 and r3
x=unique(sort(horzcat(x,r2,r3)));
%temporal
nt=10;
nPeriods=50; %how many times limiting (longest) tau (>1 to reach equilibrium)
%linspace
t0=0;
tf=t0+tau_chic_limiting*nPeriods;
t=linspace(t0,tf,nt);
if t0~=0
t=horzcat(0,t); %add 0 padding if doesn't start at 0 to allow initialisation
end
%logspace - uncomment this to use log time
% logi=-2; %initial point
% logf=log10(tf);
% t=logspace(logi,logf,nt);
% t=horzcat(0,t);
%%solve the system
sol= pdepe(m,@pdefun,@pdeic,@pdebc,x,t);
% Extract the first solution component as u.
u = sol(:,:,1);
% A surface plot
surf(x./rOuter,t./tau_chic_limiting,u-Tinf)
title(horzcat('Numerical solution computed with ',num2str(npoints), ' mesh points.'))
xlabel('Nondimensional radial distance r/r_O_u_t_e_r')
ylabel('Limiting Fourier number Fo')
zlabel('Temperature rise T')
set(gca, 'YScale','log') %if logspace used
% A limiting solution profile.
figure
plot(x,u(end,:))
hold on
plot([r2 r2],[0 max(u(end,:))],'--')
plot(x,0,'.')
plot([r3 r3],[0 max(u(end,:))],'--')
hold off
title('Final timestep (Fo \rightarrow \infty) solution')
xlabel('Distance r')
ylabel('T_\infty')
% --------------------------------------------------------------
function [c,f,s] = pdefun(x,t,u,DuDx) %computes cfs
%geometry and material info passed through nesting
%material distribution of alpha, k
if (x>=r2 && x<=r3)
alpha1=alpha_cond;
k=k_cond;
else
alpha1=alpha_ins;
k=k_ins;
end
%define for pdepe
c = 1/alpha1;
%definition of flux term
f = DuDx;
%heating distribution (currently constant with t)
rho_cond_max=rho_conductor; %for invariable resistivity
sigma_cond=1/rho_cond_max; %definition of conductivity = 1/resistivity
qvdot=ACfactor*(Jlimit^2)/sigma_cond; %volumetric heating rate (of conductor)
%source terms: heating only in conductor
s_cond=qvdot/k_cond;
s_ins=0/k_ins;
if (x>=r2 && x<=r3)
s=s_cond;
else
s=s_ins;
end
end
% --------------------------------------------------------------
function u0 = pdeic(x) %ICs
u0 = 0.*x; %starts at zero relative temp
end
% --------------------------------------------------------------
function [pl,ql,pr,qr] = pdebc(xl,ul,xr,ur,t) %BCs
%exterior BCs only
%constants passed through nesting
%nb f is set as du/dx
% %insulated LHS (inner)
pl = 0;
ql = -1*k_ins;
%convective RHS (outer)
pr = -1*h2*(ur-Tinf);
qr = -1*k_ins;
end
%end of nesting
end