Three‐dimensional desingularized boundary integral methods for potential problems

SUMMARY The concept of desingularization in three-dimensional boundary integral computations is re-examined. The boundary integral equation is desingularized by moving the singular points away from the boundary and outside the problem domain. We show that the desingularization gives better solutions to several problems. As a result of desingularization, the surface integrals can be evaluated by simpler techniques, speeding up the computation. The effects of the desingularization distance on the solution and the condition of the resulting system of algebraic equations are studied for both direct and indirect versions of the boundary integral method. Computations show that a broad range of desingularization distances gives accurate solutions with significant savings in the computation time. The desingularization distance must be carefully linked to the mesh size to avoid problems with uniqueness and ill-conditioning. As an example, the desingularized indirect approach is tested on unsteady non-linear three-dimensional gravity waves generated by a moving submerged disturbance; minimal computational difficulties are encountered at the truncated boundary. Boundary integral methods provide a powerful technique for the solution of linear, homogeneous boundary value problems. The method employs a fundamental solution, which satisfies the differential equation (and possibly part of the boundary conditions), to reformulate the problem as an integral equation on the boundary. In conventional boundary integral formulations, singularities of the fundamental solution are placed on the domain boundary. This requires special evaluation of singular integrands, which can result in costly numerical calculations. In time-dependent non-linear free surface problems' - a boundary integral problem is solved at each time step. Since most of the computation time is devoted to the boundary integral problem, an effective solution method is critical in the time-marching procedure. When the singularity of the fundamental solution is placed away from the boundary and outside the domain of the problem, a desingularized boundary integral equation is obtained. We will show two advantages to this desingularization: a more accurate solution can be obtained for a given truncation, and a numerical quadrature can be used to reduce the computational time to obtain the algebraic system representing the discretized boundary integral problem. There are two types of non-singular boundary integral formulations: direct and indirect. In the direct method, Green's second identity is used to derive the boundary integral equation, and the solution of the problem is obtained directly by solving the boundary integral equation. In the