Boundary integral methods

Pratt, L. R., Tawa, G.J., Hummer, G., Garcia, A. E., and Corcelli, S. A. (1997). Boundary integral methods for the poisson equation of continuum dielectric solvation models. Int.J. Quant. Chem. 64, 121-141. 

Abstract. A method for the computation of eigenvalues of quantum billiard is offered. This method is based on combining of boundary integral method and thermofield dynamics formalism. 

In this work we use thermofield dynamics formalism (Takahashi et.al., 1996 Das, 1997) and boundary integral method (Li et.al., 1995) to get temperature dependence in the billiard energy level spectrum. Instead of the zero-temperature Green s function we use finite-temperature one which is obtained within the TFD formalism. 

Before proceeding to finite-temperature treatment we briefly describe the boundary integral method for zero-temperature case, which was introduced to solve this type of problem (Berry et al, 1984 Tiago et al, 1997). 

In the case of finite temperature a similar approach can be used based on the boundary integral method, where instead of the zero temperature Green s function, finite-temperature Green s function derived within TFD formalism is used. Introducing finite-temperature within the thermofield dynamics formalism is based on two steps, doubling of the Hilbert space and Bogolyubov transformations (Takahashi et.ah, 1996 Ademir, 2005). 

Using in-boundary integral method for this Green s function gives us finite-temperature analog of the matrix given in Eq. (5),... 

B.A. Davis. Investigation of non-linear flows in polymer mixing using the boundary integral method. PhD thesis, University of Wisconsin-Madison, Madison, 1995. 

H. Power and L. C. Wrobel. Boundary Integral Methods in Fluid Mechanics. Computational Mechanics Publications, 1995. 

FIG. 2 A topographical plot of the electrostatic potential adjacent to and outside of a layer of charged hemicylindrical micelles adsorbed to a solid interface. The results were obtained using the boundary integral method (or boundary element method, BEM). All values shown are dimensionless. The half-width of a unit cell was taken to be b = 5 k -1, while the maximum extent into the electrolyte calculated was L = 10k-1. The radius of the cylinder s circular cross-section was assumed to be K —1. The potential is given in dimensionless units, y = efhp. 

Dungan and Hatton [12] solved Eq. (6) together with Eq. (48) for the problem depicted in Fig. 6, where a spherical particle is interacting with an oppositely charged deformable interface. To obtain their solution they used a boundary-integral method, in which the surfaces of the interface and sphere are discretized and assigned constant surface charge density boundary con-... 

In general, the problem just defined is nonlinear, in spite of the fact that the governing, creeping-flow equations are linear. This is because the drop shape is unknown and dependent on the pressure and stresses, which in turn, depend on the flow. Thus n and F are also unknown functions of the flow field, and the boundary conditions (2-112), (2-122), (2-141), and (8-58) are therefore nonlinear. Thus, for arbitrary Ca, for which the deformation may be quite significant, the problem can be solved only numerically. Later in this chapter, we briefly discuss a method, known as the boundary Integral method, that may be used to carry out such numerical calculations. Here, however, we consider the limiting case... 

The formulation (8-198) was used by Youngren and Acrivos16 to calculate the force on solid particles of different shapes translating through an unbounded stationary fluid, u, (xv ) = 0, in what was likely the first application of the boundary-integral method to creeping-flow problems. Many subsequent investigators have used it to calculate forces on bodies of complicated shape, in a variety of undisturbed flows.17... 

The value of the boundary-integral method is particularly evident if we consider problems in which one or more of the boundaries is a fluid interface. Here, for simplicity, we consider the generic problem of a drop in an unbounded fluid that is undergoing some mean motion that causes the drop to deform in shape. This type of problem is particularly difficult because the shape of the interface is unknown and is often changing with time. We shall see that the boundary-integral formulation provides a powerful basis to attack this class of problems, and in fact, is largely responsible for much of the considerable theoretical progress that has... 

In addition to the recent book by Pozrikidis (Ref. 7), a good general reference to the boundary-integral method is S. Weinbaum, P. Ganatos, and Z. Y. Yan, Numerical multipole and boundary integral equation techniques in stokes flow, Annu. Rev. Fluid Mech. 22, 275-316 (1990). 

D. Jiao, A. A. Ergin, B. Shanker, E. Michielssen, and J.-M. Jin, A fast higher-order time-domain finite element boundary integral method for 3-D electromagnetic scattering analysis, IEEE Trans. Antennas Propag., vol. 50, pp. 1192—1202, Sep. 2002. doi 10.1109/TAP.2002.801375... 

M.A. Jawson and G.T. Symm. Integral Equation Methods in Potential Theory and Elastostatics. Academic Press, London, 1977. An introduction to the boundary integral method as applied to potential theory and elastostatics. 

Abstract This chapter reviews atomization modeling works that utilize boundary element methods (BEMs) to compute the transient surface evolution in capillary flows. The BEM, or boundary integral method, represents a class of schemes that incorporate a mesh that is only located on the boundaries of the domain and hence are attractive for free surface problems. Because both primary and secondary atomization phenomena are considered in many free surface problems, BEM is suitable to describe their physical processes and fundamental instabilities. Basic formulations of the BEM are outlined and their application to both low- and highspeed plain jets is presented. Other applications include the aerodynamic breakup of a drop, the pinch-off of an electrified jet, and the breakup of a drop colliding into a wall. 

Tong, R. P. A new approach to modelling an unsteady free surface in boundary integral methods with applications to bubble-structure interactions, Math. Comput. Simul. 44, 415 26 (1997). 

Zhang, Y. L., K. S. Yeo, B. C. Khoo, W. K. Chong Simulation of three-dimensional bubles using desingularized boundary integral method, bit J. Num. Methods Fluids 31, 1311-1320... 

Plastic shear on the (100) [001] system in a lamella can also be initiated in a more general case by the nucleation of a screw-dislocation half loop from the narrow XA face of the lamella of Fig. 9.17. This process has been considered rigorously using the variational boundary-integral method developed by Xu and Ortiz (1993). The problem of interest here has also been solved by Xu and Zhang (2003), giving a stress dependence of the activation free energy of this mode shown in Fig. 9.19 as the upper curve. It is of the form... 

Xu, G., and Ortiz, M. (1993) A variational boundary integral method for the analysis of 3-D cracks of arbitrary geometry modeled as eontinuous distribution of dislocation loops, Int. J. Numer. Methods Eng., 36, 3675-3701. 

The development of fundamental models on droplet dynamics through microchannels has led to a number of analytical and numerical studies reported in the literature in the recent past. Scheeizer and Bonnecaze [8] employed the boundary-integral method to numerically simulate the displacement of a two-dimensional droplet attached to a solid surface when the inertial and gravitational forces are negligible. These authors showed that as the capillary number was increased, the deformation of the droplet increased until a critical value was reached. 

Reynolds number approximation of the Navier-Stokes equations (also known as Stokes equations) is an acceptable model for a number of interfacial flow problems. For instance, the typical example of drop coalescence also belongs to this case. A boundary integral method [3] arises from a reformulation of the Stokes equations in terms of boundary integral expressions and the subsequent numerical solution of the integral equations. 

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