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[solids4Foam] ALE formulation for structural displacement equation

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Old   October 7, 2022, 06:22
Default ALE formulation for structural displacement equation
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Philip Cardiff
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Question copied from an email thread:

I wanted to ask you a question regarding the use of ALE formulation on the structural displacement equation itself. Is there already such an implementation in solids4Foam? I was wondering about the case of FSI problems where the solid model is undergoing large strain?

So in this case, the ALE formulation is applied to both the solid and fluid equations.
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Old   October 7, 2022, 06:33
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For simulating large deformations in solid mechanics, there are two commonly used approaches:
  • Total Lagrangian
  • Updated Lagrangian

The following presentation gives a high-level overview of total Lagrangian and updated Lagrangian methods: https://www.researchgate.net/publica...sto-plasticity.

Both total Lagrangian and updated Lagrangian approaches are implemented in solids4foam as solidModels, e.g. nonLinGeomTotalLagSolid, nonLinGeomUpdatedLagSolid.

Both of these are Lagrangian approaches (no mass enters or leaves each cell), in contrast to ALE approaches, which are commonly used in fluids. ALE approaches are possible for solids, but are typically implemented as a mesh smoothing step at the end of a time-step, e.g. in metal forming simulations where the Lagrangian mesh gets severely distorted.

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Old   October 7, 2022, 12:44
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Hi Philip,

Thank you for your reply and the provision of these slides. It seems that from a numerical implementation perspective, the total langrangian approach is easier to implement and should be quicker as well since one need not recompute the update surface and volume information at each time step, the original surface and volume information can be used for all time steps. I was wondering that in the case of a FSI simulation where the forces acting on the structures are spatially and temporally varying as well, it makes more sense to use an updated langrangian approach because it is more accurate? Do we expect that both methods should give more or less the same results in a FSI simulation.
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Old   October 11, 2022, 09:01
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Quote:
Originally Posted by rnoldt View Post
Hi Philip,

Thank you for your reply and the provision of these slides. It seems that from a numerical implementation perspective, the total langrangian approach is easier to implement and should be quicker as well since one need not recompute the update surface and volume information at each time step, the original surface and volume information can be used for all time steps. I was wondering that in the case of a FSI simulation where the forces acting on the structures are spatially and temporally varying as well, it makes more sense to use an updated langrangian approach because it is more accurate? Do we expect that both methods should give more or less the same results in a FSI simulation.
The total and updated Lagrangian approaches should produce the same results. The difference between them is their convergence, robustness and efficiency for a given problem.

Both approach will calculate the same deformed solid domain, where the difference is just how this calculation is done. For FSI, the FSI procedure will use the deformed configuration and so it doesn't care if the solid is total or updated Lagrangian.
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Old   October 12, 2022, 11:12
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Hi Philip,

I was wondering if you could be more specific on the differences between the two approaches? As in what kind of problem should we use total langragian or updated langragian? How should we decide?

Also, I wanted to ask, in this Langragian (total and updated) approach, are we solving the problem on a dynamic Mesh?
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Old   October 12, 2022, 11:36
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Originally Posted by rnoldt View Post
I was wondering if you could be more specific on the differences between the two approaches? As in what kind of problem should we use total langragian or updated langragian? How should we decide?
The short answer is try both approaches and see which one works better for your problem.

For a more general answer, you may like to read about total and updated Lagrangian approaches in standard finite element text books for nonlinear solid mechanics. For example, in Belytschko, Liu, Moran (2000), Nonlinear finite elements for continua and structures, they say:
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Although the total and updated Lagrangian formulations are superficially quite different, it will be shown that the underlying mechanics of the two formulations are identical; furthermore, expressions in the total Lagrangian formulation can be transformed to updated Lagrangian expressions and vice versa. The major difference between the two formulations is in the point of view: in the total Lagrangian formulation variables are described in the original configuration, in the updated Lagrangian formulation in the current configuration. Different stress and deformation measures are typically used in these two formulations. For example, the total Lagrangian formulation customarily uses a total measure of strain, whereas the updated Lagrangian formulation often uses a rate measure of strain. However, these are not inherent characteristics of the formulations, for it is possible to use total measures of strain in updated Lagrangian formulations, and rate measures in total Lagrangian formulations. These attributes of the two Lagrangian formulations are discussed further in Chapter 4.
While, if I remember correctly, in the Bathe text book, he says that one formulation may be more efficient in some scenarios and less efficient in others.

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Originally Posted by rnoldt View Post
Also, I wanted to ask, in this Langragian (total and updated) approach, are we solving the problem on a dynamic Mesh?
In total Lagrangian, the mesh is not moved; however, remember that the correct deformed geometry is calculated and so the deformed mesh could easily be visualised as a post-processing step (e.g. use Warp by vector in ParaView with the displacement field).

In updated Lagrangian, the mesh is moved at the end of each time-step to the deformed position. So "Warp by vector" is not needed in ParaView. However, this will give the same answer as the total Lagrangian approach (albeit there could be small discretisation differences that dissapear as the mesh is refined).
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