I suggest you simply move the source in time within a fixed grid. No adaptive grid has ever been shown to be desirable or effective in the field of reservoir simulation. Maintaining effective species and energy conservation is virtually impossible with adaptive gridding.

Dear Brian Coats ,

Thank you for your answer. I am not focusing more on the result but on programming an adaptive moving grid. I want to understand how it works at the algorithmic level.

Best regards,

Using an adaptive moving mesh method is one way to solve a 3D heat conduction equation involving a moving heat source, such as a laser. The main idea is to adapt the mesh to the changing heat source position so that the elements around the laser spot have a finer resolution.

using Model-based prediction, If the laser's motion is known, it can be predicted in advance using a mathematical model. The mesh can then be updated based on the predicted motion of the laser.

Hi, you may be helped by one of my recent publications,

"A p-refinement Method Based on a Library of Transition Elements for 3D Finite Element Applications". Link:

https://www.dropbox.com/scl/fo/xf0vyalm1z4hkia9x7yck/h?rlkey=kcho4xadjjmlqsi7t65sy5cdi&dl=0

Here, the heat can be implemented at the center of a fourth-order element that transitions from order 4 to 1. I have implemented the refinement procedure to a Matlab app you can use readily. Please reach me if you have any questions.

App link: https://github.com/AdnanShahriar64/P-refinery

YouTube tutorial video link: https://www.youtube.com/watch?v=81O3n6KFZmg