Degenerate scales and boundary element analysis of two dimensional potential and elasticity problems

SUMMARYFor a plane elasticity problem, the boundary integral equation approach has been shown to yield a non-unique solution when geometry size is equal to a degenerate scale. In this paper, the degenerate scale problem in the boundary element method (BEM) is analytically studied using the method of stress function. For the elliptic domain problem, the numerical di culty of the degenerate scale can be solved by using the hypersingular formulation instead of using the singular formulation in the dual BEM. A simple example is shown to demonstrate the failure using the singular integral equations of dual BEM. It is found that the degenerate scale also depends on the Poisson's ratio. By employing the hypersingular formulation in the dual BEM, no degenerate scale occurs since a zero eigenvalue is not imbedded in the in uence matrix for any case.

Section: Introductionmentioning

confidence: 99%

Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity

Chen

Kuo

Lin³

2002

“…In this case, when a/b = 1 two critical values must merge into one size a = 1 = 2 = 1.32168 (see Table I). This tendency can be found from Figure 5 and Table I (refer to three Also, some researchers suggested an equation for the critical value [8]. After the 'a' value (for the annular region) in Equation (16) it follows…”

Section: Numerical Examination For the Critical Value For Degenerate mentioning

confidence: 92%

“…Without losing generality, it is assumed that the ratios b/a and c/a are given beforehand. It is preferable to write the kernel in Equation (8) in the form of U * i j ( , x, a). The relevant homogeneous equation to Equation (8) is introduced…”

Section: Formulation Of the Degenerate Scale Problem For The Ellipse-mentioning

confidence: 99%

Numerical examination for degenerate scale problem for ellipse‐shaped ring region in BIE

Chen

Wang

Lin

2007

SUMMARYThis paper investigates the degenerate scale problem for an ellipse-shaped ring region in boundary integral equation (BIE). A homogenous integral equation is introduced. The integral equation is reduced to an algebraic equation after discretization. The critical value for the degenerate scale can be obtained from the vanishing condition of a determinant. It is proved that there are two critical values for the degenerate scale, rather than one. This finding is first proposed in the paper. Two particular problems with known solutions are examined numerically. The loadings applied on the exterior boundary may result in a resultant force in the x-direction or in the y-direction. The improper numerical solutions have been found once the real size approaches the critical value. Two techniques for avoiding the improper solutions are suggested. The techniques depend on the appropriate choice of the used size or adding a constant in a kernel of the integral equation. It is proved that both techniques will give accurate numerical results. Numerical examinations for the problem are emphasized in the paper.

“…Such matrices are not symmetric, however exactly satisfy equilibrium and rigid body conditions, hence, an efficient and robust BEM-FEM coupling algorithm can be achieved if an iterative coupling procedure is employed [6][7][8]. For more recent publications where the topic 'BE formulations satisfying the self-equilibrium condition' is addressed, the reader is referred to References [9][10][11][12][13][14]. For some other publications concerning related topics, the reader is referred to References [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Boundary elements with equilibrium satisfaction—a consistent formulation for dynamic problems considering non‐linear effects

Soares

Telles

Mansur

2005

SUMMARYThis paper presents a boundary element (BE) formulation with complete dynamic equilibrium satisfaction with respect to the co-ordinate axis directions and moments, including inertial forces. The new procedure is quite general and very easy to implement into BE existing codes. All the required expressions for both static and dynamic formulations are shown in the text and two dynamic examples, which take into account linear and non-linear material behaviour, are presented at the end of the paper, showing the potentialities of the proposed methodology.