Boundary Conditions for an Airy Stress Function

[Hypatio]

I am using Gaussian elimination to solve the airy stress function, but I am having difficulty implementing boundary conditions.
A good synopsis on the problem of identifying boundary conditions is given here (section 5.2.1):

http://solidmechanics.org/text/Chapter5_2/Chapter5_2.htm

Given that I am using a direct numerical method to find the airy stress function, and given that I arbitrarily assign boundary tractions for each boundary node, how do I go about finding the stress function which corresponds with assigned boundary tractions? I assume that you treat the boundary values as known and so you put them on the right side of the equation, but I don't see how that works.I have found that if I simply insert a non-zero constant on the right hand side of the 2D biharmonic equation for boundary nodes, I get results which look much like that known from analysis (eg. flamant/point-contact problem), although the magnitude seems to be off.

What am I missing?

 

[Achy47]

Hello,

Thank you for sharing your experience with using Gaussian elimination to solve the airy stress function. It is not uncommon to encounter difficulties when implementing boundary conditions in numerical methods, but there are some strategies that can help.

First, it is important to carefully consider the boundary conditions that are appropriate for your specific problem. This may involve consulting literature or consulting with colleagues who have experience in the field. It is also important to ensure that the boundary conditions are physically meaningful and consistent with the problem you are trying to solve.

Once you have identified the appropriate boundary conditions, you can then incorporate them into your numerical method. As you mentioned, treating the boundary values as known and including them on the right-hand side of the equation is a common approach. However, it is important to ensure that the boundary values are being applied correctly and consistently throughout the solution process.

In some cases, it may be necessary to apply a non-zero constant on the right-hand side of the equation for boundary nodes, as you have found. This can help to improve the accuracy of the solution, but it is important to carefully consider the magnitude and its impact on the overall solution.

In addition, it may be helpful to check the convergence of your solution and compare it to known analytical solutions or experimental data. This can help to identify any discrepancies and guide further adjustments to your numerical method.

In summary, implementing boundary conditions in numerical methods can be challenging, but it is important to carefully consider the appropriate conditions and ensure that they are being applied correctly and consistently throughout the solution process. I hope this helps and wish you success in your research.