Boundary Integral Evaluation of Surface Derivatives

Abstract

In boundary element analysis, first order function derivatives, e.g., boundary potential gradient or stress tensor, can be accurately computed by evaluating the hypersingular integral equation for these quantities. However, this approach requires a complete integration over the boundary and is therefore computationally quite expensive. Herein it is shown that this method can be significantly simplified: only local singular integrals need to be evaluated. The procedure is based upon defining the singular integrals as a limit to the boundary and exploiting the ability to use both interior and exterior boundary limits. Test calculations for two-and three-dimensional problems demonstrate the accuracy of the method.

Iterative solution of Hermite boundary integral equations

“…The sparse gradient equations developed previously in [4] reduced to a reasonable level the prohibitive cost of constructing and storing the Hermite system matrix. It has now been shown that this sparsity can also be exploited to reduce the time required to solve the Hermite equations.…”

“…As discussed in detail in [4], the gradient equations exploit the boundary limit formulation of the integral equations. Specifically, the difference between the interior and exterior limits results in…”

“…The gradient equations can now be expressed solely in terms of 'local' singular integrals, avoiding the usual complete boundary integration and, moreover, producing a sparse, rather than dense, set of linear equations [4]. The computation time to assemble these equations is therefore O(N ) rather than O(N 2 ), and if stored in sparse format the storage cost is also O(N ).…”

“…As the gradient equations in [4] are sparse, this suggests, if not demands, an iterative solution of the linear system. Moreover, as will be discussed further below, the principal sub-matrices for the gradient values are well conditioned and the equations for the separate components of the gradient are only weakly coupled.…”

Wave Breaking over Sloping Beaches Using a Coupled Boundary Integral-Level Set Method

“…Note that u and the right hand side of equation (12) are only defined on Γ t (s). In order to solve equation (12) embedded in the whole domain Ω 1 , we need to extend these variables off the front; this is discussed below. In summary, the Level Set model equations, written in a complete Eulerian framework, are u = ∇φ in Ω(t) (13) …”