Volume preserving

The literature on ergodic theory contains an interesting theorem concerning the spectrum of the Frobenius-Perron operator P. In order to state this result, we have to reformulate P as an operator on the Hilbert space L P) of all square integrable functions on the phase space P. Since and, therefore, / are volume preserving, this operator P L P) —+ L r) is unitary (cf. [20], Thm. 1.25). As a consequence, its spectrum lies on the unit circle. 

Figure 8.5 (A) Schematic representation of the dichotomous branching network. (B) Cross sections at each level. (C) Single tube with continuously increasing radius. (D) Volume-preserving transformation of the varying radius tube to a fixed radius tube. Reprinted from [328] with permission from Springer.

Further, a volume-preserving transformation allows the replacement of the varying radius tube with a tube of fixed radius p0 and fixed area Aq = ttPq (Figure 8.5 D). This is accomplished by replacing z with a new coordinate z with the condition that the constant total flow of the fluid across a section is kept invariant under the transformation ... 

Integrals of flux across the TS can be computed in the normal-form coordinates since the transformation between the original coordinates and the normal-form coordinates is symplectic, hence volume preserving. 

All available, however small a volume (preserved). Enquire if antemortem samples are available. If no urine, submit kidney and bile. 

Hamiltonian systems. Thus, one has to treat this non-volume-preserving piece of the integrator a bit more carefully. To ensure numerical stability, higher order reversible integration schemes in conjunction with multiple time step methods are preferred. The details of implementing this scheme are provided in Ref. 28. 

A vector field v satisfying V v = 0 is called solenoidal. A volume-preserving motion is called isochoric, i.e., a motion for which the density in the neighborhood of any particle remains constant as the particle moves. The flow of an incompressible fluid is necessarily isochoric, but there may also be isochoric flows of compressible fluids [104] (p. 212). 

M. Feingold, L.P. Kadanoff, and O. Piro. Passive scalars, three-dimensional volume preserving maps, and chaos. J. Stat. Phys., 50 529-565, 1988. 

We shall now propose an approach for the evaluation, in the Saharan reservoirs, of the relative importance of the process of compaction and cementation in the reduction of porosity by presenting correlation plots of intergranular volume (VIG) vs. cement (Fig. 4.16). It is generally held that under the conditions of sediment accumulation on surface the VIG of well-graded sandstones is about 40%. This VIG or porosity can only be reduced by compaction, a mechanical process reducing VIG to 30%. Any further reduction by chemical compaction or pressure solution is a specially important process. The intergranular porosity of a sandstone is a function of the volume preserved after compaction and of its (the VIG s) portion filled by cement (Fig. 4.16). The inter-... 

Euler s method is not symplectic for a general Hamiltonian system. Similarly for a general divergence free vector field, Euler s method is not volume preserving. Find conditions on the vector field that imply that Euler s method is volume preserving. Are there special Hamiltonian systems for which Euler s method is a symplectic method ... 

Show that this method is time-reversible, volume-preserving (see Exercise 8), and is a (locally) 0 h ) perturbation of the Takahashi-Imada method. Hint the order can be determined by Taylor-expanding the force evaluations. 

All symplectic methods preserve the volume, but for a system with no first integrals except the energy, it is known from a theorem of Ge and Marsden [400] that a symplectic method cannot preserve the system energy exactly (unless the symplectic method is itself a time-reparameterization of the exact solution). The options for volume preserving non-symplectic discretization methods that conserve energy exactly are limited (standard form schemes that rely on calculation of the vector field at a few points cannot achieve this [380]). 

Technically, Liouville s equation refers to the continuity equation in the setting of a volume preserving flow. Here we use the term liouviUian to refer to the operator whose action on a density gives the right hand side of the general continuity equation. 

Using the volume preservation of Hamiltonian systems, the second term satisfies... 

The deterministic part may however be approximated in various ways. We may solve it by splitting into two (volume preserving) parts, integrating first q = where and p are fixed (fhus a drift along a given direction), followed by solving I = -VU/fjL, which is a similar drift in the auxiliary variable. Composing... 

Feng, K., Shang, Z. Volume-preserving algorithms for source-free dynamical systems. Numer. Math. 71, 451 63 (1995). doi 10.1007/s002110050153... 

Ortiz [106] introduced the isochoric assumption into the kinematics of large deformation by splitting the deformation gradient into volume preserving and nonpreserving parts, as shown in the following ... 

To hold the incompressibihty assumption the volume preserving part of the apphed deformation gradient needs to be utilized in the trial elastic part, with the left and right Cauchy-Green tensors given by [14]... 

The basic idea behind the VOF method is to discretize the equations for conservation of volume in either conservative flux or equivalent form resulting in near-perfect volume conservation except for small overshoot and undershoot. The main disadvantage of the VOF method, however, is that it suffers from the numerical errors typical of Eulerian schemes such as the level set method. The imposition of a volume preservation constraint does not eliminate these errors, but instead changes their symptoms replacing mass loss with inaccurate mass motion leading to small pieces of fluid non-physically being ejected as flotsam or jetsam, artificial surface tension forces that cause parasitic currents, and an inability to calculate accurately geometric information such as normal vector and curvature. Due to this deficiency, most VOF methods are not well suited for surface tension-driven flows unless some improvements are made [19]. 

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