To account for thermal ﬂuctuations, a random force density fthm is introduced in the ﬂuid equations and Fthm in the structure equations.
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
The thermal ﬂuctuations are taken into account by two stochastic ﬁelds gthm and Fthm .
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
One particularly convenient feature of this reformulation is that the stochastic driving ﬁeld Fthm and gthm become independent.
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
In this expression, we let fthm = gthm − Λ[Fthm ] for the particular realized values of the ﬁelds gthm and Fthm .
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
The identity fthm = gthm − Λ[Fthm ] could be used to eﬃciently generate the required stochastic driving ﬁelds in the equations for u.
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
As a result, the ﬁeld fthm can be generated using the Fast Fourier Transform (FFT) with at most O(N log(N )) computational steps.
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
The covariance operator G is associated with the Gaussian ﬁeld g = [gthm , Fthm , 0]T by (cid:104)g(s)gT (t)(cid:105) = Gδ(t − s).
Stochastic Eulerian Lagrangian Methods for Fluid-Structure Interactions with Thermal Fluctuations
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