Suppose a thermoelastic rod is fixed to a wall at one end and may expand to make contact with a wall at the other end. J. R. Barber [1] proposed a boundary condition that models a physically realistic transition between thermal insulation, when far from contact, and perfect thermal contact.

Linear stability analysis suggests a change from stable to unstable behavior as the temperature difference between the walls increases. The eigenvalue problem governing the stability of the perturbation $\phi(x)$ is nondimensionally

$$\phi''(x) = \lambda \phi(x),\qquad 0 < x < 1,$$

$$\phi(0) = 0,~~ \phi'(1) + \phi(1) = 4 \delta\int_0^1\phi(x) dx ,$$

where the value of $\delta$ is a function of the thermal gradient. The transition from stable to unstable happens at $\delta=1$. The presence of the integral of $\phi$ in the boundary condition makes the problem unusual from a classical standpoint, but from the Chebfun point of view it's just another linear boundary condition.

First, we solve the eigenvalue problem in a stable case.

N = chebop( @(x,u) diff(u,2), [0 1] );    % operator on 0<x<1
N.lbc = 0;              % fixed end
delta = 0.96;           % stable choice
N.bc = @(x,u) feval(diff(u),1) + u(1) - 4*delta*sum(u);  % Barber condition
[Vs,Ls] = eigs(N,4,0);  % eigenmodes closest to zero

The eigenvalues are all negative, indicating stability:

format long
diag(Ls)
ans =
1.0e+02 *
-1.234915472724630
-0.626486098335608
-0.251462532662759
-0.001601435706946


Here is what happens in a slightly unstable case:

delta = 1.02;  % unstable choice
N.bc = @(x,u) feval(diff(u),1) + u(1) - 4*delta*sum(u);  % Barber condition
[Vu,Lu] = eigs(N,4,0);
diag(Lu)
ans =
1.0e+02 *
-1.235278901227600
-0.625884455974818
-0.252000055363520
0.000799646105231


Here we see the perturbation which is least stable in the first case, or unstable in the second case.

LW = 'linewidth'; MS = 'markersize';
subplot(1,2,1)
plot(Vs(:,4),LW,1.6)
title(sprintf('Stable, \\lambda = %.3f',Ls(4,4)))
subplot(1,2,2)
plot(Vu(:,4),LW,1.6)
title(sprintf('Unstable, \\lambda = %.3f',Lu(4,4)))

The solutions above look linear, but they do have significant Chebyshev coefficients out to degree 8.

Without knowing the transition value $\delta=1$ in advance, we could locate it through a simple Chebfun rootfinding search. First, we parameterize the boundary conditions and the maximum real eigenvalue.

BC = @(delta) @(x,u) [u(0); feval(diff(u),1) + u(1) - 4*delta*sum(u)];
maxlam = @(delta) eigs( chebop(@(x,u)diff(u,2),[0 1],BC(delta)), 1, 0 );

Then, we construct a chebfun for the maximum $\lambda$. A polynomial of degree 10 captures the behavior of the maximum eigenvalue to about 11 digits.

stability = chebfun(maxlam,[0.5,2],'eps',1e-11,'vectorize')
stability =
chebfun column (1 smooth piece)
interval       length   endpoint values
[     0.5,       2]       11        -2      3.9
Epslevel = 1.000000e-11.  Vscale = 3.928224e+00.


Finally, the transition in stability occurs when the eigenvalue passes through zero.

dstar = find(stability==0)
clf, plot(stability,LW,1.6), hold on, plot(dstar,0,'ro',MS,16)
xlabel('\delta'), ylabel('max \lambda'), grid on
dstar =
1.000000000023135


References

1. J. R. Barber, "Contact problems involving a cooled punch," Journal of Elasticity, 8 (1978), 409-423.

2. J. A. Pelesko, "Nonlinear stability, thermoelastic contact, and the Barber condition", Journal of Applied Mechanics, 68 (2001), 28-33.