Galerkin boundary integral analysis for the axisymmetric Laplace equation

Abstract

SUMMARYThe boundary integral equation for the axisymmetric Laplace equation is solved by employing modified Galerkin weight functions. The alternative weights smooth out the singularity of the Green's function at the symmetry axis, and restore symmetry to the formulation. As a consequence, special treatment of the axis equations is avoided, and a symmetric-Galerkin formulation would be possible. For the singular integration, the integrals containing a logarithmic singularity are converted to a non-singular form and evaluated partially analytically and partially numerically. The modified weight functions, together with a boundary limit definition, also result in a simple algorithm for the post-processing of the surface gradient.

Numerical simulation of non‐viscous liquid pinch off using a coupled level set boundary integral method

“…The integral formulation of equation (2) is solved using a BEM with an axisymmetric Galerkin approximation of the BI [5], with linear and quadratic shape functions [4], and an interface velocity obtained by postprocessing the BI solution ∂φ ∂n on the interface using a Galerkin technique. For equations (3) and (4), first or second order upwind finite differences in space and first order in time are used.…”

Simulation of the droplet-to-bubble transition in a two-fluid system

“…The free surface equations (7) and (8) have now been embedded into the higher dimension equations (15) and (16) and it can be shown that system (13)- (16) is equivalent to system (5)- (8). In fact, this enriches the kinematics of the system, in the sense that it can incorporate topological changes of the free surface, and as well the evolution of the associated potential function within this boundary; see [8,14].…”

“…It is also worth noting that there are other significant differences with the boundary integral approximations employed in [6,7]. Herein, a linear element Galerkin method [16] is employed, in contrast to the high-order collocation approximation in [6,7], and the methods for computing the critical surface gradient are completely different. With the linear element discretization and N p free boundary nodes, the boundary integral equations for the exterior and interior fluids, Eqs.…”

“…Moreover, with the Galerkin scheme described in [16] the second subsystem is symmetric. Consequently, the construction of the two-fluid boundary integral matrices is no more expensive than for a single fluid.…”

“…Once the exterior boundary functions have been obtained from Eq. (22), the interior functions are known as well, and the algorithm discussed in [16] can be applied in each fluid to obtain the needed boundary gradients.…”