Dear colleagues,
I am currently working on a multi-material arbitrary Lagrangian-Eulerian (ALE) method for applications in geomechanics and geotechnical engineering. In most multi-material ALE methods, a typical calculational cycle is divided into (i) a Lagrangian step where the motions of the material and the computational mesh coincide, and (ii) a remap step in which the mesh is rezoned (relaxed) and the variables are transferred to the rezoned mesh; see reviews in [1, 2]. In multi-material Eulerian formulations the mesh after the Lagrangian step is moved to its original configuration, so that it appears to be fixed throughout the calculation.
Depending on how the mesh is rezoned, so-called multi-material elements may arise which contain a mixture of materials separated by interfaces. The mixture is treated as an effective single-phase material on the element level by using some kind of mixing rules (mixture theory). Free surfaces are modeled by multi-material elements containing "void material" with (nearly) zero stiffness and mass density. Interfaces resp. free surfaces are typically reconstructed and propagated element by element using Volume of Fluid (VOF) or related techniques.
Almost all ALE or Eulerian methods use explicit time integration schemes in both the Lagrangian and remap steps. The range of application of such "hydrocodes" is therefore limited to dynamic problems having a relatively short-time duration. Reference [3] is the only one I know which deals with implicit integration in the Lagrangian step. Can somebody provide other references? Thanks in advance.
Regards,
Daniel
[1] D. J. Benson. Computational methods in Lagrangian and Eulerian hydrocodes. Computer Methods in Applied Mechanics and Engineering, 99:235-394, 1992
[2] H. U. Mair. Review: Hydrocodes for structural response to underwater explosions. Shock and Vibration, 6:81-96, 1999.
[3] D. J. Benson. An implicit multi-material Eulerian formulation. International Journal for Numerical Methods in Engineering, 48:475-499, 2000
Perhaps this PhD Thesis might be helpful:
On Single-and-Multi Material ALE Approaches with Application to Micromechanical Problems at Finite Deformations, D. Frebmann, University of Hannover, 2004.
Thank you, Dr. Nazem. I know this reference very well and, indeed, the proposed ALE methods use implicit time integration of the finite element equations in the Lagrangian step.
Unfortunately, there is little information about the implementation into the multi-material ALE framework. Moreover, in the numerical examples concerned with multi-material configurations the mechancal behavior of the different materials is quite similar. Therefore, there is no large difference between a pure and a mixed (resp. multi-material) element. No free surface problems or heterogeneous multi-material configurations are analyzed.
Regards
I am not sure if can help you in your work but you can give a look to the papers of C.Peskin about the Immersed Boundary Method. Many papers illustrate the fluid-(elastic) structure interactions.
Thank you, Dr. Denaro. These papers of Prof. Peskin you mentioned are indeed groundbreaking. However, it is a technique for fluid-structure-interaction, with a Lagrangian description for the structure and a Eulerian description for the fluid. I am instead interested in a particulr variant of a one-field numerical method for fluid-fluid resp. fluid-solid interaction.
Regards
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