Extended boundary condition

A promising method based on an integral equation formulation of the problem of scattering by an arbitrary particle has come into prominence in recent years. It was developed by Waterman, first for a perfect conductor (1965), later for a particle with less restricted optical properties (1971). More recently it has been applied to various scattering problems under the name Extended Boundary Condition Method, although we shall follow Waterman s preference for the designation T-matrix method. Barber and Yeh (1975) have given an alternative derivation of this method. 

Bayazitoglu, Y., and Tunc, G., (2002) Extended Boundary Condition for Micro-Scale Heat Transfer, AIAA/ASME Thermophysics Conference, in St Eouis, June 2002, Also published in Proceedings of theAlAA J. of Thermophysics and Heat Transfer, Vol.16 no. 3, pp.472-474. 

Since they are first order in accuracy, other extended boundary conditions are proposed by [10]. The accuracy of the equation can be improved in order to increase the Kn range of applicability of the slip-flow regime. For flow over a flat plane, the general form is [11],... 

The simulated domain is 5 x 5 mm square flow field with 100 x 100 grids. Extended boundary condition is applied to the upper gas-liquid interface periodic and bounce-back boundary conditions are chosen, respectively, for the two side walls and solid bottom. The simulated scale is Ax = 5 x 10 m and At = 5 X 10 s. An uniform distributed higher solute concentration is set in the width of 1 mm at the interface at r = 0. During the diffusion process, both Marangoni and Rayleigh convections are simultaneously coupling the former is created at the surface, and the latter is formed perpendicular to the interface. Figure 9.5 shows the simulated results at different times ... 

Doicu, A., Wriedt, T., Formulation of the Extended Boundary Condition Method for Three-dimensional Scattering Using the Method of Discrete Sources,/. Mod Opt, 1998, 45, 199-213. 

F.M. Kahnert, J.J. Stamnes, K. Stamnes, Application of the extended boundary condition method to homogeneous particles with point-group symmetries, Appl. Opt. 40, 3110 (2001)... 

A. Lakhtakia, The extended boundary condition method for scattering by a chiral scatterer in a chiral medium Formulation and analysis. Optik 86, 155 (1991)... 

Note MM-i- is derived from the public domain code developed by Dr. Norm an Allinger, referred to as M.M2( 1977), and distributed by the Quantum Chemistry Program Exchange (QCPE). The code for MM-t is not derived from Dr. Allin ger s present version of code, which IS trademarked MM2 . Specifically. QCMPOlO was used as a starting point Ibr HyperChem MM-t code. The code was extensively modified and extended over several years to include molecular dynamics, switching functuins for cubic stretch terms, periodic boundary conditions, superimposed restraints, a default (additional) parameter scheme, and so on. 

Maxwell obtained equation (4.7) for a single component gas by a momentum transfer argument, which we will now extend essentially unchanged to the case of a multicomponent mixture to obtain a corresponding boundary condition. The flux of gas molecules of species r incident on unit area of a wall bounding a semi-infinite, gas filled region is given by at low pressures, where n is the number of molecules of type r per... 

HyperChem supplements the standard MM2 force field (see References on page 106) by providing additional parameters (force constants) using two alternative schemes (see the second part of this book. Theory and Methods). This extends the range of chemical compounds that MM-t can accommodate. MM-t also provides cutoffs for calculating nonbonded interactions and periodic boundary conditions. 

A switched function extends over the range of inner (Ron) to outer (Roff) radius and a shifted function from zero to outer (Roff) radius. Beyond the outer radius, HyperChem does not calculate non-bonded interactions. The suggested outer radius is approximately 14 Angstroms or, in the case of periodic boundary conditions, less than half the smallest box dimension. The inner radius should be approximately 4 Angstroms less than the outer radius. An inner radius less than 2 Angstroms may introduce artifacts to the structure. 

When q is zero, Eq. (5-18) reduces to the famihar Laplace equation. The analytical solution of Eq. (10-18) as well as of Laplaces equation is possible for only a few boundary conditions and geometric shapes. Carslaw and Jaeger Conduction of Heat in Solids, Clarendon Press, Oxford, 1959) have presented a large number of analytical solutions of differential equations apphcable to heat-conduction problems. Generally, graphical or numerical finite-difference methods are most frequently used. Other numerical and relaxation methods may be found in the general references in the Introduction. The methods may also be extended to three-dimensional problems. 

The space beneath the storage tank. This configuration of extended parallel planes, internally provided with a large number of vertical obstacles (the pylon forest), is an outstanding example of blast-generative boundary conditions. 

The hydraulic oil must provide adequate lubrication in the diverse operating conditions associated with the components of the various systems. It must function over an extended temperature range and sometimes under boundary conditions. It will be expected to provide a long, trouble-free service life its chemical stability must therefore be high. Its wear-resisting properties must be capable of handling the high loads in hydraulic pumps. Additionally, the oil must protect metal surfaces from corrosion and it must both resist emulsification and rapidly release entrained air that, on circulation, would produce foam. 

Studies of double carrier injection and transport in insulators and semiconductors (the so called bipolar current problem) date all the way back to the 1950s. A solution that relates to the operation of OLEDs was provided recently by Scott et al. [142], who extended the work of Parmenter and Ruppel [143] to include Lange-vin recombination. In order to obtain an analytic solution, diffusion was ignored and the electron and hole mobilities were taken to be electric field-independent. The current-voltage relation was derived and expressed in terms of two independent boundary conditions, the relative electron contributions to the current at the anode, jJfVj, and at the cathode, JKplJ. 

Solution An open system extends from —oo to +oo as shown in Figure 9.9. The key to solving this problem is to note that the general solution. Equation (9.18), applies to each of the above regions inlet, reaction zone, and outlet. If k = Q then p=. Each of the equations contains two constants of integration. Thus, a total of six boundary conditions are required. They are... 

To compute each of the n(ct ), one can generalize the methods used to compute ihG- Hence, the most elegant method would be to use basis functions that satisfy the boundary conditions of Eq. (43), if this were practical to implement. A more general method would be to extend the Mead-Truhlar vector-potential approach [6]. This approach would involve carrying out h calculations, each including a... 

The theory developed for tunneling splitting can be easily extended to the decay of the metastable state through multidimensional tunneling, namely, tunneling predissociation of polyatomic molecules. In the case of predissociation, however, the instanton trajectory cannot be fixed at both ends, but one end should be free (see Fig. 17). The boundary conditions are... 

Irrespective of the type of extended system we are interested in we impose periodic boundary conditions in position space - "the large period" BK. Such conditions imply a discretisation of momentum and reciprocal space 27] which means that integrations are replaced by summations ... 

Several boundary conditions have been used to prescribe the outer limit of an individual rhizosphere, (/ = / /,). For low root densities, it has been assumed that each rhizosphere extends over an infinite volume of. soil in the model //, is. set sufficiently large that the soil concentration at r, is never altered by the activity in the rhizosphere. The majority of models assume that the outer limit is approximated by a fixed value that is calculated as a function of the maximum root density found in the simulation, under the assumption that the roots are uniformly distributed in the soil volume. Each root can then extract nutrients only from this finite. soil cylinder. Hoffland (31) recognized that the outer limit would vary as more roots were formed within the simulated soil volume and periodically recalculated / /, from the current root density. This recalculation thus resulted in existing roots having a reduced //,. New roots were assumed to be formed in soil with an initial solute concentration equal to the average concentration present in the cylindrical shells stripped away from the existing roots. The effective boundary equation for all such assumptions is the same ... 

---