Degenerate scale problem when solving Laplaceʹs equation by BEM and its treatment

In this paper, Laplace problems are solved by using the dual boundary element method (BEM). It is found that a degenerate scale problem occurs if the conventional BEM is used. In this case, the influence matrix is rank deficient and numerical results become unstable. Both the circular and elliptical bars are studied analytically in the continuous system. In the discrete system, the Fredholm alternative theorem in conjunction with the SVD (Singular Value Decomposition) updating documents is employed to sort out the spurious mode which causes the numerical instability. Three regularization techniques, method of adding a rigid body mode, hypersingular formulation and CHEEF (Combined Helmholtz Exterior integral Equation Formulation) concept, are employed to deal with the rankdeficiency problem. The addition of a rigid body term, c, in the fundamental solution is proved to shift the original degenerate scale to a new degenerate scale by a factor e−c. The torsion rigidities are obtained and compared with analytical solutions. Numerical examples including elliptical, square and triangular bars were demonstrated to show the failure of conventional BEM in case of the degenerate scale. After employing the three regularization techniques, the accuracy of the proposed approaches is achieved