Modified Bessel function second kind

Hi everyone,

I have been working on the analytical solution on keyhole welding. And the first paper in this topic is A.H.F Kaplan 1994. He formulated heat flow inside keyhole in terms of 2nd kind modified Bessel function. However, in my implementation when r approaches zero (r=0), the ratio between Bessel function K1(0)/K0(0) returns NaN. Do you have any idea how to handle this singularity problem? I want to calulate qfront from -r to +r. Related paper is

Kaplan, Alexander. "A model of deep penetration laser welding based on calculation of the keyhole profile." Journal of Physics D: Applied Physics 27.9 (1994): 1805.

My implementation is

if true
  model.Material.Density=7860*1e-9; % density (kg/mm^3)
  model.Material.Cp=465; % specific heat capacity (J/(kg*K))
  model.Material.Lamda=43/1000; % thermal conductivity (W/(mm*K))
  model.Material.K=model.Material.Lamda/(model.Material.Density * model.Material.Cp); % thermal diffusivity (mm^2/s)

    model.Process.Power=600; % W
    model.Process.Speed=50; % mm/s

    model.Material.Ta=300; % ambient temperature (K)
    model.Material.Tm=1893; % melting temperature (K)
    model.Material.Tv=3123; % vaporisation temperature (K)

    % function q=getHeatFlowKeyHole(model, abs(r), phi)
    % read constants

    Ta=model.Material.Ta; % ambient temperature (K)
    Tv=model.Material.Tv; % vaporisation temperature (K)
    lamba=model.Material.Lamda;
    s=model.Process.Speed;
    k=model.Material.K;
    Pnumber=s/(2*k);

    % Modified Bessel Function Second Kind
    K0=besselk(0,Pnumber*r);
    K1=besselk(1,Pnumber*r);

    qfront=(Tv-Ta) * lamba * Pnumber * (1 + K1 / K0);

end

Thank you Erkan

[David Goodmanson]

For small z,

 K1(z) --> 1/z and K0(z) --> -ln(z)

which means that

 K1(z)/K0(z) --> inf.

But since Matlab evaluates not the ratio but K0 and K1 separately, the result is inf/inf --> NaN.

[Walter Roberson]

@David Goodmanson it's more complicated than that...

syms z
K0 = besselk(sym(0),z)

K0 = 

K1 = besselk(sym(1),z)

K1 = 

subs(K0, z, sym(0))

ans = 

NaN

subs(K1, z, sym(0))

ans = 

NaN

but

besselk(sym(0), sym(0))

ans = 

∞

besselk(sym(1), sym(0))

ans = 

∞

I don't understand at the moment how they can give two different results depending on whether you subs() or not.

[David Goodmanson]

Hello Walter,

I believe that my comment is correct when the bessel functions are calculated numerically, although that is switching gears from what the OP was asking in terms of syms. The limits

K1(z) --> 1/z     and      K0(z) --> -ln(z)  

both work when z --> 0 from any direction in the complex plane. However, the K0 expression gets to the limit quite slowly. A better approximation along the way is

K0(z) --> -ln(z/2)-eulerC             % eulerC = 0.577215664901533...

As to syms considerations, of course there are many questions on this site asking for a symbolic solution to equations so difficult that no such solution is possible. Can't blame syms for those. Other times though, the questions are more along the lines of your observation here, trying to determine why syms is doing something according to its own arbitrary, obscure symsian rules.

Answer 1

[John D'Errico]

1. Are you sure this generates something meaningful for negative r? Those bessel functions return complex results for negative r.

help besselk
 besselk Modified Bessel function of the second kind.
  K = besselk(NU,Z) is the modified Bessel function of the second kind,
  K_nu(Z).  The order NU need not be an integer, but must be real.
  The argument Z can be complex.  The result is real where Z is positive.

If I had to guess, the function is symmetric across zero, since when r is negative, I see a comment in your code about abs( r ), but then you never bother to use abs() in the code.

2. The limit of K1/K0 as r --> 0 from above appears to be +inf. If you try to evaluate those functions exactly at r==0, you have a 0/0 condition, which will be NaN.

[Walter Roberson]

Looking at the definitions of BesselK, it looks like it is definitely undefined at 0, with a (0)^(-1)/0 term