The relation between incompressibility and divergence-free?

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The discussion centers on the relationship between incompressibility and divergence-free conditions in the context of finite element analysis, specifically regarding mixed displacement and pressure formulations. It is established that both incompressible solids and fluids can be characterized by divergence-free vector fields, with displacement for solids and velocity for fluids. For solids, this relationship holds true primarily under small deformation conditions, while for fluids, the divergence-free condition is equivalent to incompressibility in any scenario due to the Eulerian frame of reference.

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phdggg
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Hi all,

One thing I am really confused is the relation between incompressiblity and divergence-free. Since I am coding a finite element code that use mixed displacement and pressure formulation. From what I got from your book. Both incompressible solid and fluid can be characterized by divergence-free vector field (displacement for solid and velocity for fluid). But is this only for small deformation in the case of solid? Since for large deformation, the determinant of deformation gradient must be one, which seems not equivalent to divergence-free.

What about for fluid, because it uses Eularian frame, the divergence-free should be equivalent to incompressibility for any case. Is this correct? Thanks in advance for your clarification!


Mengda
 
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phdggg said:
What about for fluid, because it uses Eularian frame, the divergence-free should be equivalent to incompressibility for any case. Is this correct? Thanks in advance for your clarification!

You are correct.
 

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