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In COMSOL, convergence issues with nonlinear boundary conditions, like your temperature-dependent heat flux Q(T)=aT3+bT2+cT+100010−5Q(T) = \frac{a T^3 + b T^2 + c T + 1000}{10^{-5}}Q(T)=10−5aT3+bT2+cT+1000​, are common due to the high nonlinearity of the function, especially for high values of TTT. Here are some strategies that might help improve convergence:

1. Scaling the Heat Flux Expression

Since your denominator is on the order of 10−510^{-5}10−5, this makes the flux very large, leading to steep gradients and numerical instability. Try normalizing or scaling the function by a larger constant to reduce the overall magnitude of Q(T)Q(T)Q(T) and see if this helps in achieving convergence. After the problem converges, you can scale the solution back to get the actual heat flux.

2. Starting with Simpler Boundary Conditions (Continuation Method)

Start with a simpler form of your boundary condition and gradually introduce the nonlinear terms. For example:

• Begin with Q(T)=100010−5Q(T) = \frac{1000}{10^{-5}}Q(T)=10−51000​ (i.e., with a=b=c=0a = b = c = 0a=b=c=0, as you did).
• Once that converges, introduce cTc TcT, then bT2b T^2bT2, and finally aT3a T^3aT3.
• This approach, known as the continuation method, helps the solver find an initial solution that it can gradually refine.

3. Use a Transient Study with Initial Temperature Guess

In cases with strong nonlinear dependencies on TTT, a transient solution can sometimes help. This is especially true if the system has large gradients or if the nonlinear boundary condition is very sensitive to TTT. Try running a transient study with a small time step and a reasonable initial guess for the temperature distribution.

• You can use a time-dependent solver with a step function to gradually ramp up the boundary heat source.
• This approach effectively reduces the initial stiffness of the problem and helps the solver find an accurate initial condition, which you can then use for the stationary study.

4. Adding Smoothing Functions

If you observe oscillations or sharp gradients in the boundary heat flux, consider adding a smoothstep function (or another smoothed approximation) to the heat flux expression. Smoothstep functions prevent the solver from having to handle abrupt changes that can disrupt convergence. In COMSOL, you can add smoothing functions like flc2hs(T, smoothing factor) to smooth out temperature gradients.

5. Solver Settings Adjustments

Fine-tuning solver settings can often be essential for these cases. Consider these options:

• Nonlinear Solver Settings: Increase the Nonlinear Method damping factor to help manage the steep nonlinearities.
• Update Frequency: Decrease the Jacobians update frequency to allow for more accurate convergence in each iteration.
• Fine-tune Mesh: Ensure the mesh around the heat flux boundary is fine enough to capture rapid temperature variations without over-refining the entire domain.

6. Stabilization Techniques

COMSOL provides stabilization methods like Automatic stabilization or Isotropic diffusion. These techniques add a small amount of artificial diffusion to the problem, helping it stabilize. Be careful to keep this effect minimal, as it can alter the results slightly, but it often helps the solver find a converged solution in nonlinear problems.

Applying these strategies should enhance your chances of achieving convergence with your nonlinear boundary heat flux. Let me know if you need more specific guidance on implementing any of these in COMSOL!

This should provide your enough guidance to tackle your problem ;)

Best,

Igor

Hi Igor,

Thank you for your reply. I actually started from this too. But couldn't figure out yet. Thanks for sharing your insight.

Best,

Azizul

Hi Md Azizul Hakim,

I suggest making the heat flux independent of temperature. Alternatively, you could try a time-dependent approach or set temperature as the boundary condition and leave out the flux. The convergence issue likely arises because the heat flux, as an independent variable, depends on the temperature (the dependent variable). With each iteration, the previous temperature values affect the heat flux in the next step, leading to a cycle that prevents convergence.

Hence, try to reformulate the problem or try the above solutions given by ChatGPT.

Good Luck.