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[solids4Foam] ALE formulation for structural displacement equation 

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October 7, 2022, 06:22 
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. 

October 7, 2022, 06:33 

#2 
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Philip Cardiff
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For simulating large deformations in solid mechanics, there are two commonly used approaches:
The following presentation gives a highlevel overview of total Lagrangian and updated Lagrangian methods: https://www.researchgate.net/publica...stoplasticity. 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 timestep, e.g. in metal forming simulations where the Lagrangian mesh gets severely distorted. Philip 

October 7, 2022, 12:44 

#3 
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raynold
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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. 

October 11, 2022, 09:01 

#4  
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Philip Cardiff
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Quote:
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. 

October 12, 2022, 11:12 

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raynold
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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? 

October 12, 2022, 11:36 

#6  
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Philip Cardiff
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Quote:
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: Quote:
Quote:
In updated Lagrangian, the mesh is moved at the end of each timestep 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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