A self-regularized method for rank-deficiency systems in BEM and FEM

Abstract

It is well known that a rank-deficiency system appears in the degenerate scale once BEM is used for the Dirichelet problem. For the Neuman problem, either FEM or BEM yields a rank-deficient matrix. Fredholm alternative theorem plays an important role in the linear algebra when the matrix is singular. Based on the singular value decomposition (SVD) for the matrix, range deficiency can be easily and systematically understood. By introducing a slack variable, we obtain a bordered matrix by adding one column vector from the left unitary vector and one row vector from the right unitary vector with respect to the zero singular value. It is interesting to find that an original singular matrix is regularized to a non-singular one. The value of the slack variable indicates the infinite solution (zero) or no solution (non-zero) for the linear algebraic system. To demonstrate this finding, one triangular-domain problem with a degenerate scale and a rigid body mode is solved. Although influence matrices are singular in the BIE formulation for different problems (degenerate scale in the Dirichlet problem and rigid body mode in the Neumann problem), the corresponding unique solution (Dirichlet problem) and infinite solutions containing a constant potential (Neumann problem) can be obtained by using the bordered matrix and SVD technique. In addition, a singular stiffness matrix using the FEM for free-free structure is also regularized to find a reasonable solution.