Computing the new mesh in Lagrangian co-ordinates
 

Suppose I am working in a 2D compressible elasticity. I am using Lagrangian co-ordinates , and I want to compute the new shape. How would I go about it? Naively I would solve the equations:



I did that in 1D but I got errors, but I'm not sure about the parameters I used.

what do you mean? in 1D you have only one equation but you should be aware about the global character of your problem.

For example, the Burgers equation in lagrangian form is Du/Dt=0 but you can create the evolution of the regular lagrangian grid only if the characteristic lines dx/dt=u do not intersect.

In other words until the u is regular.

I use the amount of mass as my "length" co-ordinate, and as such I have this magical property that the mass in the cell is constant for all time. I compute the density at each time-step, and mass=density*length, to find the new length, you just need to compute length=mass of cell/new density.

I don't have a hyperbolic system, so I don't have characteristics to compute with.

Could you better detail the problem you see during the computation?

I have a 2D elastic system with diffusive terms that cause the shape to be deformed because I have an external heat source that is causing it to shrink. I believe I have the correct equations describing the process but I don't know how to compute the shape as it deforms as this changes the grid.

I think you can not use a fully Lagrangian grid. At some time you need to remapping it and restart.
Deformation will be high to have a stable computation. I don’use lagariangian grid for several issues.

It's the remapping aspect that I need to understand. I haven't seen any references for it.

Are you referring to ALE at all?

ALE is a possible way. But you could use also the way presented by Peskin in his immersed boundary method.

Which book/paper is this in?

Charles Peskin is a famous scientist, you can find his home page and all his papers.
However, have a look to my references here

https://www.researchgate.net/publica...ructured_grids