Extended boundary condition method

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. 

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)... 

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. 

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. 

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... 

Rohde, M., Extending the Lattice-Boltzmann method—novel techniques for local grid refinement and boundary conditions , Ph.D. Thesis, Delft University of Technology, Delft, Netherlands (2004). 

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

In determining the flow in a whole pipe, as above, it is unnecessary to use the infinitesimal shell element the method used in Example 1.8, with an element extending from the centre to a general position r, is preferable because it is simpler. Where the infinitesimal cylindrical shell element is required is for flow in an annulus, for example between r = rx and r = r2. This is necessary because the flow region does not extend to the centre-line so a whole cylindrical element cannot be fitted in. In the case of flow in an annulus, equation 1.56 is valid but the constants of integration must be determined using the boundary conditions that the velocity is zero at both walls. (Note that this specifies a value of vx at two different values of r and therefore provides two boundary conditions as required.)... 

The extended Brusselator [2, 5], Oregonator [5, 10] and other similar systems [4, 7] demonstrate other autowave processes whose distinctive spatial and temporal properties are independent on initial concentrations, boundary conditions and often even on geometrical size of a system. As it was noted by Zhabotinsky [4], Vasiliev, Romanovsky and Yakhno [5], a number of well-documented results obtained in the theory of autowave processes is much less than a number of problems to be solved. In fact, mathematical methods for analytical solution of the autowave equations and for analysis of their stability are practically absent so far. 

The method is later extended to time-varying heat inputs on one face with arbitrary boundary conditions on the back face (C7). Citron also has given a simple method of successive approximations for the finite ablating slab (C6) which is shown to converge rapidly for constant heat input. 

A closely related method is that of Boley (B8), who was concerned with aerodynamic ablation of a one-dimensional solid slab. The domain is extended to some fixed boundary, such as X(0), to which an unknown temperature is applied such that the conditions at the moving boundary are satisfied. This leads to two functional equations for the unknown boundary position and the fictitious boundary temperature, and would, therefore, appear to be more complicated for iterative solution than the Kolodner method. Boley considers two problems, the first of which is the ablation of a slab of finite thickness subjected on both faces to mixed boundary conditions (Newton s law of cooling). The one-dimensional heat equation is once again... 

In order to overcome the limitations of currently available empirical force field param-eterizations, we performed Car-Parrinello (CP) Molecular Dynamic simulations [36]. In the framework of DFT, the Car-Parrinello method is well recognized as a powerful tool to investigate the dynamical behaviour of chemical systems. This method is based on an extended Lagrangian MD scheme, where the potential energy surface is evaluated at the DFT level and both the electronic and nuclear degrees of freedom are propagated as dynamical variables. Moreover, the implementation of such MD scheme with localized basis sets for expanding the electronic wavefunctions has provided the chance to perform effective and reliable simulations of liquid systems with more accurate hybrid density functionals and nonperiodic boundary conditions [37]. Here we present the results of the CPMD/QM/PCM approach for the three nitroxide derivatives sketched above details on computational parameters can be found in specific papers [13]. 

On extending this idea to a quantitative study of the valence state, ionization radius, which is characteristic of each atom, is the important parameter. When using the Hartree-Fock-Slater method to calculate the ionization radii of non-hydrogen atoms the boundary condition is introduced by multiplying... 

To calculate the double-layer force, the nonlinear Poisson-Boltzmann equation was solved for the case of two plane parallel plates, subject to boundary conditions which arise from consideration of the simultaneous dissociation equilibria of multiple ionizable groups on each surface. Deijaguin s approximation is then used to extend these results to calculate the force between a sphere and a plane. Details of the method can be found in Ref. (6). 

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