Meshless methods

Complex fluids are fluids which advect a structure. For example, liquid crystals can be modelled as a continuum of microscopic rods immersed in a Newtonian fluid. In order to alter the orientation of these rods, one must consider the velocity field and its spatial gradient. At a point, this data is known as a 1-jet of the velocity and is represented by a finite dimensional group. In [1] we have verified the computational tractability of this group. The resulting integrator satisfies a variant of the circulation theorem, and could be competitive with traditional vortex methods .  This is found in work on particle methods done in [2].  Recent work is allowing these ideas to come to fruition [3], as shown in the movies below.  Finally, the notion of jets can be leveraged to augment the traditional vortex blob method, and permit nontrivial dynamics below the regularization length-scale [4]

[1]  L. Colombo, H.O. Jacobs. Lagrangian mechanics on centered semi-direct products, 13 pages, to appear in “Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden”, Fields Institute Communications Series (2015)

[2] H.O. Jacobs, T.S. Ratiu, M. Desbrun. On the coupling between an ideal fluid and immersed particlesPhysica D, vol. 265, pp. 40–56 (2013) 

[3] C.J. Cotter, D.D. Holm, H.O. Jacobs, D.M. Meier. The jetlet hierarchy of ideal fluid dynamics, Journal of Physics A:47 (2014) (arXiv:1402.0086)

[4] C.J. Cotter, J. Eldering, D.D. Holm, H.O. Jacobs, D.M. Meier. Weak Dual Pairs and Jetlet Methods for Ideal Incompressible Fluid Models. J. Nonlinear Science (2016)

[5] D.D. Holm, H.O. Jacobs. Multiple Vortex Blobs: Symplectic Geometry and Dynamics. J. Nonlinear Science (accepted Dec. 2016) (arXiv:1505.05950)