Boundary elements

Brebbia C A and Walker S 1980 Boundary Element Technique in Engineering (London Newnes-Butterworth)... 

Fulian Q, Fisher A C and Denuault G 1999 Applioations of the boundary element method in eleotroohemistry soanning eleotroohemioal miorosoopy J. Phys. Chem. B 103 4387... 

Ire boundary element method of Kashin is similar in spirit to the polarisable continuum model, lut the surface of the cavity is taken to be the molecular surface of the solute [Kashin and lamboodiri 1987 Kashin 1990]. This cavity surface is divided into small boimdary elements, he solute is modelled as a set of atoms with point polarisabilities. The electric field induces 1 dipole proportional to its polarisability. The electric field at an atom has contributions from lipoles on other atoms in the molecule, from polarisation charges on the boundary, and where appropriate) from the charges of electrolytes in the solution. The charge density is issumed to be constant within each boundary element but is not reduced to a single )oint as in the PCM model. A set of linear equations can be set up to describe the electrostatic nteractions within the system. The solutions to these equations give the boundary element harge distribution and the induced dipoles, from which thermodynamic quantities can be letermined. 

How can Equation (11.79) be solved Before computers were available only simple ihapes could be considered. For example, proteins were modelled as spheres or ellipses Tanford-Kirkwood theory) DNA as a uniformly charged cylinder and membranes as planes (Gouy-Chapman theory). With computers, numerical approaches can be used to solve the Poisson-Boltzmann equation. A variety of numerical methods can be employed, including finite element and boundary element methods, but we will restrict our discussion to the finite difference method first introduced for proteins by Warwicker and Watson [Warwicker and Watson 1982]. Several groups have implemented this method here we concentrate on the work of Honig s group, whose DelPhi program has been widely used. 

Bashin A A 1990. Hydration Phenomena, Classical Electrostatics, and the Boundary Element Methoc Journal of Physical Chemistry 94 1725-1733. 

Mackerle, J., and C. A. Brebbia (eds.). Boundary Element Reference Book, Springer Verlag, Berlin-Heidelberg, New York and Tokyo (1988). 

Subdivision or discretization of the flow domain into cells or elements. There are methods, called boundary element methods, in which the surface of the flow domain, rather than the volume, is discretized, but the vast majority of CFD work uses volume discretization. Discretization produces a set of grid lines or cuives which define a mesh and a set of nodes at which the flow variables are to be calculated. The equations of motion are solved approximately on a domain defined by the grid. Curvilinear or body-fitted coordinate system grids may be used to ensure that the discretized domain accurately represents the true problem domain. 

Boundary conditions are special treatments used for internal and external boundaries. For example, the center line in cylindrical geometry is an internal boundary that is modeled as a plane of symmetry. External boundaries model the world outside the mesh. The outermost row of elements is often used to implement the boundary condition as shown in Fig. 9.13. The mass, stress, velocity, etc., of the boundary elements are defined by the boundary conditions rather than the governing equations. External boundary conditions are typically prescribed through user input. 

The continuum treatment of electrostatics can also model salt effects by generalizing the Poisson equation (12) to the Poisson-Boltzmann equation. The finite difference approach to solving Eq. (12) extends naturally to treating the Poisson-Boltzmann equation [21], and the boundary element method can be extended as well [19]. 

G. Dziuk. A boundary element method for curvature flow. Application to crystal growth. In J. E. Taylor, ed. Computational Crystal Growers Workshop, AMS Selected Lectures in Mathematics. Providence, Rhode Island American Mathematical Society, 1992, p. 34 A. Schmidt. Computation of three dimensional dendrites with finite elements. J Comput Phys 125 293, 1996. 

The solution of the Laplace equation is not trivial even for relatively simple geometries and analytical solutions are usually not possible. Series solutions have been obtained for simple geometries assuming linear polarisation kinetics "" . More complex electrode kinetics and/or geometries have been dealt with by various numerical methods of solution such as finite differencefinite elementand boundary element. ... 

Hoyt, JJ Wolfer, WG, Boundary Element Modeling of Electrokinetically Driven Fluid Flow in Two-Dimensional Microchaimels, Electrophoresis 19, 2432, 1998. 

Wrobel, L. C., The Boundary Element Method, Wiley, New York (2002). 

Demming, F., Jersch, J., Dickmatm, K.and Geshev. P. I. (1998) Calculation of the field enhancement on laser-illuminated scanning probe tips by the boundary element method. Appl. Rhys. B, 66, 593-598. 

To calculate free energies of solvation for several organic molecules, Fortunelli and Tomasi applied the boundary element method for the reaction field in DFT/SCRF framework173. The authors demonstrated that the DFT/SCRF results obtained with the B88 exchange functional and with either the P86 or the LYP correlation functional are significantly closer to the experimental ones than the ones steming from the HF/SCRF calculations. The authors used the same cavity parameters for the HF/SCRF and DFT/SCRF calculations, which makes it possible to attribute the apparent superiority of the DFT/SCRF results to the density functional component of the model. The boundary element method appeared to be very efficient computationally. The DFT/SCRF calculations required only a few percent more CPU time than the corresponding gas-phase SCF calculations. 

In order to calculate eigenvalues of energy use is made of the corresponding Boundary Element Equation (BEE) of this BIE. And for matrix elements of BEE we have ... 

The heat transfer into the boundary surface of a compartment occurs by convection and radiation from the enclosure, and then conduction through the walls. For illustration, a solid boundary element will be represented as a uniform material having thickness, 6, thermal conductivity, k, specific heat, c, and density, p. Its back surface will be considered at a fixed temperature, T0. 

For a fully developed fire, conduction commonly overshadows convection and radiation therefore, a limiting approximation is that h hk, which implies Tw T. This result applies to structural and boundary elements that are insulated, or even to concrete structural elements. This boundary condition is conservative in that it gives the maximum possible compartment temperature. 

The subject of liquid jet and sheet atomization has attracted considerable attention in theoretical studies and numerical modeling due to its practical importance.[527] The models and methods developed range from linear stability models to detailed nonlinear numerical models based on boundary-element methods 528 5291 and Volume-Of-Fluid (VOF) method. 530 ... 

Resistance functions have been evaluated in numerical compu-tations15831 for low Reynolds number flows past spherical particles, droplets and bubbles in cylindrical tubes. The undisturbed fluid may be at rest or subject to a pressure-driven flow. A spectral boundary element method was employed to calculate the resistance force for torque-free bodies in three cases (a) rigid solids, (b) fluid droplets with viscosity ratio of unity, and (c) bubbles with viscosity ratio of zero. A lubrication theory was developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions were derived to accurately represent the numerical data over the entire range of particle positions in a tube for all particle diameters ranging from nearly zero up to almost the tube diameter. The resistance functions formulated are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels. 

Liu YJ, NishimuraN, Otani Y (2005). Large-scale modeling of carbon-nanotube composites by a fast multipole boundary element method. Comput. Mater. Sci. 34 173-187. 

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