Boundary element methods

In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical
theory of strongly elliptic operators and generalize the above stability results for a large class of such operators.

Marine pipeline cathodic protection systems for asymmetrical situation were systematically investigated by means of a newly proposed approach and Boundary Element Method (BEM). Potential attenuation profiles from BEM modeling indicate that far-field cathode potentials of different pipe sections approach identical values under different coating resistance and different electrolyte resistivity conditions provided anodes are separated by at least 10m and metallic resistance is negligible. A series of equations based on the Slope Parameter Method (SPM) has been modified for more extensive applicability. Several design examples have been analyzed and the results verified by BEM. Cathode potential and current demands projected by the new method are consistent with those of BEM. The inclusive equation for even anode spacing CP has been modified to include the cable parameters by combining cable resistance and the anode resistance. Current demand for existing pipelines can be determined by either of two methods. The first utilizes the inclusive equation and involves solving this for current demand based upon a known potential profile. The other is based on SPM.

A three-dimensional nonlinear time-dependent boundary-integral algorithm is developed to compute wave forces on an underwater vehicle. The effect of viscosity is neglected and the cases for which the effects could be important are discussed. The present algorithm is however an efficient tool to determine wave forces on a submerged body and can also be integrated into a viscous flow algorithm. A numerical wave tank is constructed for the simulation. A damping layer is introduced to minimize spurious reflection of scattered waves at the open boundary. A sinusoidal progressive pressure patch is used to generate incident waves. Wave forces are determined using four different methods: viz., (1) Froude-Krylov volume integration method, (2) Froude-Krylov surface pressure integration method, (3) Linear diffraction analysis and (4) Nonlinear diffraction analysis for a range of parameters including incident wavelength and wave height. Results are compared to quantify effects of nonlinearity and diffraction effect of the body.

Boundary integral algorithms are developed to analyze three-dimensional inviscid fluid-body interactions, including the nonlinear free-surface effects. Hydrodynamic coefficients are computed for various body geometries, some corresponding to that of small underwater vehicles, in deep waters and near the free surface. The fully nonlinear unsteady wave-radiation problem corresponding to forced submerged-body oscillations and forward translation are solved using the mixed Eulerian-Lagrangian formulation (Longuet-Higgins and Cokelet, 1976). By implementing the leading-order free-surface conditions on the calm surface, linear time-domain solutions are also obtained. The nonlinear and linear results are compared to quantify the nonlinear free-surface effects. Linear frequency-domain analysis of the wave-body interactions is also carried out using a boundary-integral method based on the simple-source distribution (Yeung, 1974). The linear time-domain and the latter frequency-domain results are also compared for a validation of the algorithms.