Degenerate scale for multiply connected Laplace problems
Introduction
The non-unique solution in BIE or BEM appears in three types such as (a) a rigid body mode for the Neumann problem, (b) the critical size (degenerate scale) of domain and (c) hypersingular formulation for multiply connected problem with constant Dirichlet boundary condition. The singularity occurs physically and mathematically in the sense that the non-unique solution for the singular matrix includes a rigid body solution for the interior Neumann (traction) problem. The second one is not physically realizable but stems from the zero singular value of influence matrix in BIEs. The numerical instability or failure due to the degenerate scale is only imbedded in plane boundary value problems (BVPs). The influence matrix may be singular for the Dirichlet problem when geometry is special. From the view point of linear algebra, the problem also originates from the rank-deficiency in the influence matrices. For example, the non-unique solution of a circle with a unit radius has been noted by Petrovsky (1991) and by Jaswon and Symm (1977). Jaswon and Symm coined the Γ-contours in their book. Some mathematicians coined it the critical value, transfinite boundary, transfinite radius and logarithmic capacity (Yan and Sloan, 1988) since it is mathematically realizable. As follows from the classical results summarized in the book by Hille (1962), and also from the review of the paper by Yan and Sloan (1988), it is easy to see that the degenerate scale of any bounded and multiply connected domain is equal to the degenerate scale of the outer boundary contour. A simple proof can also be obtained following the steps of the proof of an analogous statement (Proposition 5) given in the paper by Vodička and Mantič, 2004a, Vodička and Mantič, 2004b, in the case of plane elasticity. From the above mentioned classical results in the potential theory it also follows that the degenerate scale of a circle is the inverse of its radius. For the Dirichlet problems, some studies for potential problems (Laplace equation) (Chen et al., 2001), (He et al., 1996) have been done. Also, the degenerate scale of multiply connected problem for the Laplace equation was discussed by Tomlinson et al. (1996). In the recent work, Chen et al. investigated the degenerate scale for the simply connected (circle) and multiply connected problems (annular) (Chen et al., 2002) by using the degenerate kernels and circulant in a discrete system. An annular region has also been considered for the harmonic equation (He et al., 1996) and the possible degenerate scales were studied in both continuous and discrete systems. Regarding to the discrete system, circulant was employed to study the singularity of the influence matrix. However, circulant property fails in the eccentric case. To the authors’ best knowledge, proof in continuous system is well documented in the literature. In potential theory, the problem of the degenerate scales has thoroughly been studied theoretically in the continuous system in the book by Hille (1962). However, only annular case was studied analytically using circulant in the discrete system. Additionally, a lot of numerical studies have been carried out by Christiansen and others. This paper extends the proof of annular problem to eccentric case in the discrete system.
In this paper, we focus on the analytical investigation for the phenomenon of degenerate scale in BIE for multiply connected problems. The eccentric case is addressed to derive the occurring mechanism of the degenerate scale appearance by using degenerate kernels and Fourier series in the null-field integral equation. The addition of rigid body term in the fundamental solution can move the original degenerate scale to a new degenerate scale. Besides, the CHEEF technique is proposed to overcome the non-unique solution in the numerical implementation. The constraint of adding a point outside the domain can promote the rank of the singular matrix. A numerical example is considered to demonstrate the numerical failure in case of degenerate scale. The techniques to avoid the numerical failure or instability are verified. The sensitivity, efficiency and accuracy of the regularization methods are also examined. The main contribution of this paper is that we can prove the existence of degenerate scale for eccentric problems in the discrete system in difference to a large amount of literature in the continuous system.
Section snippets
Boundary integral equations for the Laplace problem
The integral formulation for the domain point of Laplace problem can be derived from Green’s third identitywhere s and x are the source and field points, respectively, B is the boundary and D is the domain of interest, nx is the outward normal vector at the field point x, U(s, x) and T(s, x) are the kernel functions which will be elaborated on later by using the degenerate kernel expansion. The kernel function, U(s, x), is the fundamental solution
Two regularization techniques to solve the non-uniqueness problem
The special scale of outer boundary results in numerical failure and/or instability. A suitable treatment of unstable system is required to solve the rank-deficiency problem and find a unique solution. Two methods, adding a rigid body term and CHEEF concept, are adopted to suppress the occurrence of the degenerate scale.
Illustrative example and discussions
An eccentric case in Fig. 2(a) is examined which has the outer radius of 1 m (a1 = 1.0 m) and the inner radius of 0.4 m (a2 = 0.4 m). The essential boundary conditions on B1 and B2 are u1 = 1 and u2 = 0, respectively. Twenty-one collocation points are both chosen on the outer and inner boundaries. After introducing the two regularization techniques, the non-uniqueness problem due to the critical scale is solved. Fig. 2(b) and (c) show the contour plots of potential after adding a rigid body mode and CHEEF
Concluding remarks
The paper dealt with the Dirichlet problem for Laplace equation solved by the singular BIE in a special case of bounded and multiply connected domains in plane given by circular boundary curves. The contribution of the work is to show in an explicit analytic way, by means of an expansion of the integral kernel functions using degenerate kernels, how this degenerate scale appears if the unknown boundary density is approximated by a Fourier series of trigonometric functions. Also two methods how
Acknowledgement
Financial support form the National Science Council under Grant No. NSC 91-2211-E-019-009 for Taiwan Ocean University is gratefully acknowledged.
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