I am trying to solve the transient heat conduction problem using Newton raphson method and backward Euler time integration. Please help me know how to use them together to solve non-linear transient heat conduction problem.
My problem is non-linear due to temperature dependent properties like thermal conductivity, specific heat and density.
I am using 8-noded brick element.
Applied load is heat flux on one surface.
Boundary condition is surface convection on other surfaces.
Initial temperature is known.
Backward Euler method From Wikipedia, the free encyclopedia
In numerical analysis and scientific computing, the backward Euler method (or implicit Euler method) is one of the most basic numerical methods for the solution of ordinary differential equations.
But thermal conduction equation is partial differential equation - not ordinary one.
There is no need for the Newton-Raphson method in the problem you describe. There is also no point using backward Euler. You can easily solve this using the forward (or conventional) Euler method. As Akhan says, this is a partial differential equation. Since you mention a "brick" element, I presume you want a 3D solution. This is easily accomplished with finite differences. You could even do it in an Excel spreadsheet. The nodes on one face are controlled by the heat flux condition. The other exterior nodes are controlled by the convection condition. The interior nodes are calculated from Laplace's equation, which reduces to the average of the surrounding nodes. Set all the nodes to some initial temperature and then step through time. If solution becomes unstable, cut the time step in half. I have attached a spreadsheet with a 1D and 2D steady-state example that crunches until it settles out using Excel's internal iterative solution feature. 3D is a little more complicated, but not much. I also included a Crank-Nicholson example, but it's hardly necessary for your problem.
Dudley J Benton Sir & Akhan Tleoubaev Sir
Thank you a lot for your kind help.
- Badges
- Science topic