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SUMMARYFree-surface boundary is not only unknown in a priori but also the boundary conditions are overspecified. In this paper, the Laplace problem with overspecified boundary conditions on the free surface is solved by using the hypersingular equation instead of singular equation used conventionally in boundary element method. The free surface can be determined using an iterative procedure starting from an initial guess. By introducing the hypersingular equation, the convergence rate of free surface can be accelerated. Finally, numerical examples including rectangular dams and canals were demonstrated and were compared with others to show the validity of the present method.

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Following the success of using the null-field integral approach to determine the torsional rigidity of a circular bar with circular inhomogeneities (Chen and Lee in Comput Mech 44(2):221-232, 2009), an extension work to an elliptic bar containing elliptic inhomogeneities is done in this paper. For fully utilizing the elliptic geometry, the fundamental solutions are expanded into the degenerate form by using the elliptic coordinates. The boundary densities are also expanded by using the Fourier series. It is found that a Jacobian term may exist in the degenerate kernel, boundary density or boundary contour integral and cancel out to each other. Null-field points can be exactly collocated on the real boundary free of facing the principal values using the bump contour approach. After matching the boundary condition, a linear algebraic system is constructed to determine the unknown coefficients. An example of an elliptic bar with two inhomogeneities under the torsion is given to demonstrate the validity of the present approach after comparing with available results.