Unified corner treatment in the boundary element method applied to electrochemical systems.

摘要

A unified framework for the treatment of the non-uniqueness and singular behaviour of fluxes in the vicinity of corners has been derived for the indirect boundary element method (IBEM). The single and double layer loading densities are shown to obey a fractional exponent power-law variation at geometric discontinuities. Proper choice of values for the fractional exponents causes cancellation of the unbounded terms in the singular IBEM integral equations. This results in bounded fluxes that act in a unique direction. Since cancellation of the singular terms is guaranteed, these computationally difficult integrals need not be calculated. Hence, the hypersingular integrals, once thought to be an impediment to the application of the IBEM, are now shown to vanish trivially, and actually contribute to the stability of the flux equation.; The corner methodology is robust, and handles easily an arbitrary number of surfaces, intersecting in any dimensional space. The general analysis formulates the layer densities at vertices as a superposition of the layer densities over the intersecting surfaces. In this way, the layer density at a 3D corner is simply the sum of its 2D edge layer densities. For the simplest case, in 2D, the new formula predicts the same functionality as the well-known 2D result. Excellent agreement is obtained between the flux computed using the new technique and the analytical flux for complex 2D and 3D examples, with linear and higher order boundary conditions.; Practical applications of this work are numerous since geometries containing corners can be cited from virtually every area of applied science and engineering. In the present dissertation, application to electrochemical systems is discussed. Specifically, the simulation of the current and voltage distributions along a cathodically protected infinite pipeline reveals physical, numerical and artificial corners. This example is further complicated by the preponderance of near-singular integrals, which are easily computed using the continuation approach. In particular, continuation equations have been constructed for the convergent, Cauchy and hypersingular integral equations that arise naturally in the IBEM. The final form of these continuation equations can be used directly as an efficient means of quadrature without detailed knowledge of their derivation.