Boundary integral equation

D. B. Ingham and M. A. Kelmanson. Boundary Integral Equation Analysis of Singular, Potential and Biharmonic Problems. Heidelberg Springer Verlag, 1994. 

Atkinson, K.E., 1997, The numerical solution of boundary integral equations, Cambridge University Press, Cambridge. 

J.P. Hernandez-Ortiz. Boundary integral equations for viscous flows non-Newtonian behavior and solid inclusions. PhD thesis, University of Wisconsin-Madison, Madison, 2004. 

A.M. Linkov Boundary Integral Equations in Elasticity Theory. 2002... 

For the given scheme of partition of the machining zone boundary on the elements, the discretization of the boundary integral equation and the boundary conditions is performed. The set of nonlinear equations, which is obtained by discretization, is solved by Newton s method. As a result, the distribution of the current density over the WP surface is obtained (Fig. 10c). 

Recently Horibe [6] in 1990 proposed a new iterative solution of the Berger equation. The solution is derived by utilizing both the idea of the Kantrovich method and the boundary integral equation method. 

The solution of the integral equations must be done numerically in virtually all cases. In this book, we do not discuss the strictly numerical issues of solving these equations. There have been excellent books and review papers written that cover this topic at a level of depth that could not be emulated here (see, for example, the excellent book of Pozrikidis7). Instead, the focus is on explaining the principles of using (8-117) or (8-119) to formulate the boundary-integral equations. 

The key to using (8-117) or (8-119) as the basis for derivation of the boundary-integral equations is that they must be applied at the boundaries of the flow domain where boundary conditions are specified. For this purpose, we need to understand the behavior of the single- and double-layer terms as we approach a boundary, 3D. The single-layer terms are continuous in the entire fluid domain, including boundaries, provided only that the latter are sufficiently smooth. However, the double-layer terms are not continuous at 3D, but suffer a jump.11 In particular, let us define a function W(x)... 

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). 

The dimensionless shape factor for the isothermal rectangular annulus is derived from the correlation equation of Schneider [89], who obtained accurate numerical values of the thermal constriction resistance of doubly connected rectangular contact areas by means of the boundary integral equation method ... 

Some Theoretical Aspects of Boundary Integral Equations". 

Regular Boundary Integral Equations for Stress Analysis". 

Ligguet, J. A., P. L. F. Liu The boundary integral equation method for porous media flow, Allen Unwin, London (1983). 

Kelmanson, M. A. Boundary integral equation solution of viscous flows with free surfaces,... 

Nishimura, N., Yoshida, K.-i., and Kobayashi, S., A fast multipole boundary integral equation method for crack problems in 3D. Engineering Analysis with Boundary Elements, 23, 97-105 (1999). 

The initial-boundary value problem represented by Eq. 7.1 can be transmuted into a boundary integral equation by several different methods. Brebbia and Walker (1980) and Curran et al. (1980) approximated the time derivative in the equation in a finite difference form, thus changing the original parabolic partial differential equation to an elliptical partial differential equation, for which the standard boundary integral equation may be established. 

The type of Eq. 8.75 is elliptical and the boundary integral equation for the formulation can be derived to give... 

In the coupled BIE/FD formulation mentioned above, the requirement for the evaluation of the domain integral is certainly a costly feature. Zheng and Phan-Thien (1992), who considered a boundary integral equation for the following problem of heat conduction with a heat source, have employed an alternative approach that transforms the domain integral to a boundary integral. 

Equation 8.89 is the desired boundary integral equation. The domain integral has been eliminated and a boundary-only formulation is obtained. After solving the equation for 7 (we call it the homogeneous solution), the total solution can be obtained byT = 7 + t. Although the particular solution T is not unique, the total solution T obtained in this way is unique, since the equation for 7 satisfies the above modified boundary conditions. 

The Laplace-transform based boundary integral equation was proposed by Rizzo and Shippy (1970) to solve transient heat conduction problems. 

Equation 8.93 is an elliptical partial differential equation, and it is similar in form to Eq. 8.75, so that its fundamental solutions have the same form as Eq. 8.79, with At replaced by 1/s. The boundary integral equation for Eq. 8.93 is... 

Another alternative BIE that can be applied to mold cooling analysis is to use time-dependent fundamental solutions (e.g., Qiao 2005). The boundary integral equation can be written as... 

As has been mentioned in Sect. 7.3, the continuous injection-molding operation results in a cyclic heat transfer behavior in the mold, after a short transient period. The cycle-averaged temperature can be represented by a steady state heat conduction equation, i.e., Eq. 7.10. The mold cooling analysis can be greatly simplihed by solving the steady state problem. The boundary integral equation of ( 7.10) is... 

Modified Boundary Integral Equations for Closely Spaced Surfaces... 

Associated with the midsurface, each variable has two different variables at top and bottom defined by a normal vector n to the mid-surface, one can use superscript + and on the variables (e.g., T and T ) to indicate values on the different sides. An extra formula is thus derived for the additional degree of freedom. For point ro on the midsurface, the following two boundary integral equations are solved ... 

Having defined the boundary integral equations, next step is to discretize the integral equations and find solutions. Let us use Eq. 7.40 as an example. After substituting the boundary conditions on the mold cavity surface (Fp), the cooling channel surface (F ) and the external surface (F ), respectively, the equation can... 

A detailed formulation of the employed 3-D BEM is too extensive and beyond the scope of this paper and can be found in O Brien and Rizos (2005), Rizos (1993), Rizos (2000), Rizos and Karabalis (1994) and Rizos and Loya (2002). The BEM uses the time domain 4th order B-Spline fundamental solutions of the 3-D full space along with higher order spatial discretization of the boundary. The Boundary Integral Equation associated to the Navier-Cauchy governing equations of motion is expressed in a discrete form yielding a system of algebraic equations at step N relating displacements u to forces f at discrete boundary nodes in the BEM model and at discrete time instants tj and Ty, as... 

An equation of motion for the sloshing height of the liquid surface and the hydrodynamic pressure of the liquid on the wall of the cylindrical tank are obtained using the boundary-elements. The boundary integral equation for the inviscid incompressible liquid (Fig. 3) can be expressed using the velocity potential function cp. 

The integral equation for the elastic boundary tractions and displacements is solved by numerical methods. The boundary is divided Into N finite length elements. In this paper the surface tractions and displacements are assumed to change linearly over each of the boundary elements. Figure 2 shows a typical boundary the surface tractions are prescribed on part of the boundary and the displacements are prescribed on the remaining part of the boundary. At each node point on the boundary there are two components of traction and two components of displacement. Thus, for N elements and N nodes there are 2N unknowns in the discretized system. The boundary Integral equation for the elasticity problem Is rewritten as below ... 

Fairweather, G., Rizzo, F., Shlppy, D., Wu, Y. (1979) "On The Numerical Solution of Two-Dimensional Potential Problems by an Improved Boundary Integral Equation Method" Journal of Computational Physics, No. 31, pp. 96-112. 

Kelmanson, M. A. (1984) "A Boundary Integral Equation Method for the Study of Slow Flow in Bearings with Arbitrary Geometries", Journal of Tribology, Vol. 106, pp. 260-264. 

Cahan BD, Scherson D (1988) I-BIEM. An iterative boundary integral equation method for computer solutions of current distribution problems with complex boundaries - a new algorithm. I. Theoretical. J Electrochem Soc 135 285-293... 

IRSCHIK, H. A Boundary integral equation method for bending of orthotropic plates. Int. J. Solids, Structures 20 (1984),... 

KITAHARA, M. Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin Plates. Studies in Applied Mechanics 10, Amsterdam-Oxford-New York-Tokyo Elsevier, 1985. 

To construct the interaction problem, the pressure distribution on F2 is needed. Hence, v/e use the associated boundary integral equation for an efficient calculation with the real valued Green s function of the half-space, see e.g. /I 4/... 

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