Revisit of the Dual Bem using SVD Updating Technique

Abstract

The boundary element method (BEM) is easier than the finite element method (FEM) on the viewpoint of the discretization of one dimension reduction rather than the domain discretization of finite element method. The disadvantage of BEM is the rank deficiency in the influence matrix, e.g., degenerate boundary, degenerate scale, spurious eigenvalues and fictitious frequencies, which do not occur in the FEM. The conventional BEM can not be straightforward applied to solve a problem which contains a degenerate boundary without decomposing the domain to multi-regions. A hypersingular integral equation is used to ensure a unique solution for the problem containing a degenerate boundary. By combining the singular and hypersingular equations, it's termed the dual BEM due to its dual frame. Following the successful experience on the retrieval of information using the singular value decomposition (SVD) updating term and updating document, this technique is also used to extract out the degenerate-boundary information and the rigid-body information in the dual BEM. It is interesting to find that true information due to a rigid-body mode in physics is found in the right singular vector with respect to the corresponding zero singular value while the degenerate-boundary mode (geometry degeneracy) in mathematics is imbedded in the left singular vector with respect to the corresponding zero singular value. The role of the common right and left singular vectors of SVD for the four influence matrices in the dual BEM is also discussed in this paper. Two examples, a potential flow problem across a cutoff wall and a cracked bar under torsion were demonstrated to see the mathematical SVD structure of four influence matrices in the dual BEM.

Dual null field method for Dirichlet problems of Laplace's equation in circular domains with circular holes

“…This provides a strict analysis of dual techniques for unique solutions and the non-singularity of the dual algorithms, thus to remove the degenerate scales. For algorithm singularity of the dual BEM, an analysis using the SVD is reported in [5]. Eq.…”

“…The algorithm singularity, unique solutions, error bounds, convergence rates, condition numbers, and stability are explored in this paper, while analysis of algorithm singularity of the dual BEM is reported in Chen et al [5] by using the singular value decomposition (SVD).…”

“…In classic algorithms, the rst kind NFM is used for Dirichlet problems; the second kind NFM for Neumann problems. The algorithms using both (2.9) and (2.14) together are called the dual" boundary element method (BEM) in [4,5,20], and the dual" techniques of NFM (simply denoted as the DNFM) in this paper. For the circular domains with circular holes, the second kind of the NFM (simply as the second kind NFM) are developed in [10] for Neumann problems, with the explicit algebraic equations.…”