[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/stochastic-eulerian-lagrangian-method-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/stochastic-eulerian-lagrangian-method-wikipedia\/","headline":"Stochastic Eulerian Lagrangian method – Wikipedia","name":"Stochastic Eulerian Lagrangian method – Wikipedia","description":"before-content-x4 From Wikipedia, the free encyclopedia after-content-x4 In computational fluid dynamics, the Stochastic Eulerian Lagrangian Method (SELM)[1] is an approach","datePublished":"2016-12-27","dateModified":"2016-12-27","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4717cde69581c6ffa8b6c9462e534744270cdae","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b4717cde69581c6ffa8b6c9462e534744270cdae","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/stochastic-eulerian-lagrangian-method-wikipedia\/","about":["Wiki"],"wordCount":3200,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4From Wikipedia, the free encyclopedia       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In 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       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u03c1dudt=\u03bc\u0394u\u2212\u2207p+\u039b[\u03a5(V\u2212\u0393u)]+\u03bb+fthm(x,t){displaystyle rho {frac {d{u}}{d{t}}}=mu ,Delta u-nabla p+Lambda [Upsilon (V-Gamma {u})]+lambda +f_{mathrm {thm} }(x,t)}mdVdt=\u2212\u03a5(V\u2212\u0393u)\u2212\u2207\u03a6[X]+\u03be+Fthm{displaystyle m{frac {d{V}}{d{t}}}=-Upsilon (V-Gamma {u})-nabla Phi [X]+xi +F_{mathrm {thm} }}dXdt=V.{displaystyle {frac {d{X}}{d{t}}}=V.}The pressure p is determined by the incompressibility condition for the fluid       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u2207\u22c5u=0.{displaystyle nabla cdot u=0.,}The \u0393,\u039b{displaystyle Gamma ,Lambda } operators couple the Eulerian and Lagrangian degrees of freedom.  The X,V{displaystyle X,V} denote the composite vectors of the full set of Lagrangian coordinates for the structures.  The \u03a6{displaystyle Phi } is the potential energy for a configuration of the structures.  The fthm,Fthm{displaystyle f_{mathrm {thm} },F_{mathrm {thm} }} are stochastic driving fields accounting for thermal fluctuations.  The \u03bb,\u03be{displaystyle lambda ,xi } are Lagrange multipliers imposing constraints, such as local rigid body deformations.  To ensure that dissipation occurs only through the \u03a5{displaystyle Upsilon } coupling and not as a consequence of the interconversion by the operators \u0393,\u039b{displaystyle Gamma ,Lambda } the following adjoint conditions are imposed\u0393=\u039bT.{displaystyle Gamma =Lambda ^{T}.}Thermal fluctuations are introduced through Gaussian random fields with mean zero and the covariance structure\u27e8fthm(s)fthmT(t)\u27e9=\u2212(2kBT)(\u03bc\u0394\u2212\u039b\u03a5\u0393)\u03b4(t\u2212s).{displaystyle langle f_{mathrm {thm} }(s)f_{mathrm {thm} }^{T}(t)rangle =-left(2k_{B}{T}right)left(mu Delta -Lambda Upsilon Gamma right)delta (t-s).}\u27e8Fthm(s)FthmT(t)\u27e9=2kBT\u03a5\u03b4(t\u2212s).{displaystyle langle F_{mathrm {thm} }(s)F_{mathrm {thm} }^{T}(t)rangle =2k_{B}{T}Upsilon delta (t-s).}\u27e8fthm(s)FthmT(t)\u27e9=\u22122kBT\u039b\u03a5\u03b4(t\u2212s).{displaystyle langle f_{mathrm {thm} }(s)F_{mathrm {thm} }^{T}(t)rangle =-2k_{B}{T}Lambda Upsilon delta (t-s).}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\u2013Boltzmann 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[edit]References[edit]Software\u00a0: Numerical Codes[edit]       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/stochastic-eulerian-lagrangian-method-wikipedia\/#breadcrumbitem","name":"Stochastic Eulerian Lagrangian method – Wikipedia"}}]}]