Volume integral equation method

Peltoniemi, J. I. (1996) Variational volume integral equation method for electromagnetic scattering by irregular grains,/. Quant. Spectrosc. Radiat. Transfer, 55, 627-6A7. 

The volume integral equation method relies on the integral representation... 

Whereas the volume integral equation method deals with the total field, the discrete dipole approximation exploits the concept of the exciting field, each volume element being explicitly modeled as an electric dipole moment. In this regard, we consider the field that excites the volume element Di rn... 

The data vector consists of 780 simulated field components (195 sites for 4 frequencies). The synthetic data set is generated by a full integral equation method and it has been contaminated by 3 percent random noise. The model parameters are the unknown anomalous conductivity values of each cell in the volume over which the inversion is carried out. 

Budko, N. V., Samokhin, A. B., and Samokhin, A. A. [2005] A generalized overrelaxation method for solving singular volume integral equations in low-frequency scattering problems, Diff. Equat, 41,1262-1266. 

K. Sertel and J.L. Volakis, Method of moments solution of volume integral equations using parametric geometry modeling. Radio Sci. 37, 1 (2002)... 

A simpler method arbitrarily picks values for oq and reacts this material in a batch reactor at constant V and T. When the reaction is complete, P is calculated from the molar density of the equilibrium mixture. As an example, set = 22.2 (P=l atm) and react to completion. The long-time results from integrating the constant-volume batch equations are a = 5.53, 5 = c= 16.63, = 38.79mol/m, and y =0.143. The pressure at equili-... 

Experimental determination of excess molar quantities such as excess molar enthalpy and excess molar volume is very important for the discussion of solution properties of binary liquids. Recently, calculation of these thermodynamic quantities becomes possible by computer simulation of molecular dynamics (MD) and Monte Carlo (MC) methods. On the other hand, the integral equation theory has played an essential role in the statistical thermodynamics of solution. The simulation and the integral equation theory may be complementary but the integral equation theory has the great advantage over simulation that it is computationally easier to handle and it permits us to estimate the differential thermodynamic quantities. 

The compressibility factor is defined as Z = FV/RT values of Z and of (9Z/9T)/j therefore come from experimental P VT data, and tire two integrals in Eqs. (6.46) through (6.48) are evaluated by numerical or grapliical methods. Alternatively, tlie two integrals are evaluated analytically when Z is expressed as a function of T and P by a volume-explicit equation of state. Thus, given P VT data or an appropriate equation of state, we can evaluate and S and hence all otlier residual properties. It is this direct comiection with experiment tliat makes residual properties essential to tlie practical application of thermodynamics. 

For unsteady flows, discretization schemes need to be devised to evaluate the integrals with respect to time (refer to Eq. (6.2)). The control volume integration is similar to that in steady flows discussed earlier. The most widely used methods for discretization of time derivatives are two-level methods. In order to facilitate further discussion, let us rewrite the basic governing equation as an ordinary differential equation with respect to time by employing the spatial discretization schemes discussed earlier ... 

Generalized Bom (GB) approach. The most common implicit models used for small molecules are the Conductor-Like Screening Model (COSMO) [77,78], the DPCM [79], the Conductor-Like Modification to the Polarized Continuum Model (CPCM) [80,81], the Integral Equation Formalism Implementation of PCM (IEF-PCM) [82] PB models, and the GB SMx models of Cramer and Truhlar [23,83-86]. The newest Minnesota solvation models are the SMD universal Solvation Model based on solute electron density [26] and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [87-89] with semiempirical terms that account for local electrostatics [90]. Further details on these methods can be found in Chapter 11 of Reference [23]. 

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