In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM)[1] is an approach to capture essential features of fluid-structure interactions subject to thermal fluctuations while introducing approximations which facilitate analysis and the development of tractable numerical methods. SELM is a hybrid approach utilizing an Eulerian description for the continuum hydrodynamic fields and a Lagrangian description for elastic structures. Thermal fluctuations are introduced through stochastic driving fields. Approaches also are introduced for the stochastic fields of the SPDEs to obtain numerical methods taking into account the numerical discretization artifacts to maintain statistical principles, such as fluctuation-dissipation balance and other properties in statistical mechanics.[1]

The SELM fluid-structure equations typically used are

The pressure p is determined by the incompressibility condition for the fluid

The operators couple the Eulerian and Lagrangian degrees of freedom. The denote the composite vectors of the full set of Lagrangian coordinates for the structures. The is the potential energy for a configuration of the structures. The are stochastic driving fields accounting for thermal fluctuations. The are Lagrange multipliers imposing constraints, such as local rigid body deformations. To ensure that dissipation occurs only through the coupling and not as a consequence of the interconversion by the operators the following adjoint conditions are imposed

Thermal fluctuations are introduced through Gaussian random fields with mean zero and the covariance structure

To obtain simplified descriptions and efficient numerical methods, approximations in various limiting physical regimes have been considered to remove dynamics on small time-scales or inertial degrees of freedom. In different limiting regimes, the SELM framework can be related to the immersed boundary method, accelerated Stokesian dynamics, and arbitrary Lagrangian Eulerian method. The SELM approach has been shown to yield stochastic fluid-structure dynamics that are consistent with statistical mechanics. In particular, the SELM dynamics have been shown to satisfy detailed-balance for the Gibbs–Boltzmann ensemble. Different types of coupling operators have also been introduced allowing for descriptions of structures involving generalized coordinates and additional translational or rotational degrees of freedom. For numerically discretizing the SELM SPDEs, general methods were also introduced for deriving numerical stochastic fields for SPDEs that take discretization artifacts into account to maintain statistical principles, such as fluctuation-dissipation balance and other properties in statistical mechanics.[1]

See also

References

  1. 1 2 3 Atzberger, Paul (2011). "Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations". Journal of Computational Physics. 230 (8): 2821–2837. arXiv:1009.5648. Bibcode:2011JCoPh.230.2821A. doi:10.1016/j.jcp.2010.12.028. S2CID 6067032.
  1. Atzberger, P.J.; Kramer, P.R.; Peskin, C.S. (2007). "A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales". Journal of Computational Physics. 224 (2): 1255–92. arXiv:0910.5748. Bibcode:2007JCoPh.224.1255A. doi:10.1016/j.jcp.2006.11.015. S2CID 17977915.
  2. Peskin, C.S. (2002). "The immersed boundary method". Acta Numerica. 11: 479–517. doi:10.1017/S0962492902000077. S2CID 53517954.

Software : Numerical Codes

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