Lagrangian coordinates

Lagrangian coordinates

[lə′grän·jē·ən ko′ȯrd·ən·əts]
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The RBCs are considered as elastic boundaries immersed in the fluid flow and are represented in Lagrangian coordinates. The coupling method between the fluid and solid membranes is based on the IBM, which uses a uniform and fixed Eulerian mesh and removes the burden of expensive mesh updating in the traditional Arbitrary Lagrangian Eulerian approach.
In the Immersed Boundary method the elastic membrane is modeled in Lagrangian coordinate but the flow field is discretized using a uniform and fixed Eulerian grid.
In this way, the structure is reduced to a classical holonomic system, with 2 n degrees of freedom, in particular, n vertical displacements [v.sub.1,i], for inner tube, and n vertical displacements [v.sub.2,i], for outer tube, at the cells abscissae will be conveniently assumed as Lagrangian coordinates and will be organized into the 2n-dimensional vector v.
The strain energies should be expressed as functions of the Lagrangian coordinates as follows: