Stochastic Eulerian Lagrangian method

The Stochastic Eulerian Lagrangian Method (SELM) 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. The Stochastic Eulerian Lagrangian Method is related to the Immersed Boundary Method, Brownian-Stokesian Dynamics Method, and Arbitrary Lagrangian Eulerian Method. The SELM fluid-structure equations are

$\rho {\frac {d{u}({x},t)}{d{t}}}=\mu \Delta u(x,t)-\nabla p+\Lambda [\Upsilon (V-\Gamma {u})]+\lambda +f_{thm}(x,t)$

$m{\frac {d{V}}{d{t}}}=-\Upsilon (V-\Gamma {u})-\nabla \Phi [X]+\xi +F_{thm}$

${\frac {d{X}}{d{t}}}=V$

The $\Gamma$ and $\Lambda$ are operators that couple the Eulerian and Lagrangian degrees of freedom. The $X,V$ should be considered as the composite vectors of the full set of coordinates for the elastic structures. The $f_{thm},F_{thm}$ are stochastic driving fields introducing thermal fluctuations. The $\lambda ,\xi$ are Lagrange multipliers imposing constraints, such as local rigid body deformations. The pressure p is determined by the incompressibility condition for the fluid

$\nabla \cdot u=0.\,$

To ensure that dissipation occurs only through the $\Upsilon$ drag and not as a consequence of the interconversion by the operators $\Gamma ,\Lambda$ the following adjoint conditions are imposed

$\Gamma =\Lambda ^{T}.$

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

$\langle f_{thm}(s)f_{thm}^{T}(t)\rangle =-\left(2k_{B}{T}\right)\left(\mu \Delta -\Lambda \Upsilon \Gamma \right)\delta (t-s).$

$\langle F_{thm}(s)F_{thm}^{T}(t)\rangle =2k_{B}{T}\Upsilon \delta (t-s).$

$\langle f_{thm}(s)F_{thm}^{T}(t)\rangle =-2k_{B}{T}\Lambda \Upsilon \delta (t-s).$

This SELM framework has been shown to yield fluid-structure dynamics that are consistent with statistical mechanics. In particular, satisfying the fluctuating-dissipation principle and detailed-balance for the Gibbs-Boltzmann ensemble. To obtain simplified descriptions and efficient numerical methods, approximations in various limiting physical regimes have been considered to remove small time-scale dynamics or inertial degrees of freedom. Coupling operators have also been introduced allowing for descriptions of the structures involving generalized coordinates and additional translational or rotational degrees of freedom, see references.

See also

• Immersed boundary method
• Stokesian dynamics
• Volume of fluid method
• Level set method
• Marker-and-cell method

References

1. P.J. Atzberger, Stochastic Eulerian Lagrangian Methods for Fluid Structure Interactions with Thermal Fluctuations, Journal of Computational Physics, 230, pp. 2821--2837, (2011)[DOI] .
2. P. J. Atzberger, P. R. Kramer, and C. S. Peskin, A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Journal of Computational Physics, vol. 224, Issue 2, 2007. [DOI] .
3. C. S. Peskin, The immersed boundary method, Acta Numerica, 11, pp. 1–39, 2002.