Three-dimensional singular boundary elements for corner and edge singularities in potential problems

A study of three-dimensional edge and corner problems using the neBEM solver

“…In fact, possibly due to these difficulties, these approaches have remained relatively unpopular and even in very recent papers it is maintained that for evaluating the influence due to source distributed on triangular elements in a general case, one must apply non-analytic procedures [14]. Thus, for solving realistic but difficult problems involving, for example, sharp edges and corners or thin or closely spaced elements, introduction of special formulations (usually involving fairly complicated mathematics, once again) becomes a necessity [8,33,16]. These drawbacks are some of the major reasons behind the relative unpopularity of the BEM despite its significant advantages over domain approaches such as the finite-difference and finite-element methods (FDM and FEM) while solving non-dissipative problems [17,18].…”

“…Later, while discussing the results obtained using our proposed approach, we will return to this issue again. A study on the effect of bias ratio (a ratio of the largest element length to the smallest element length, similar to r-mesh refinement) has led the authors [33] to state that while for normal elements this ratio should be around 4:1 for a given problem, the singular element approach works better with lower bias ratio.…”

“…According to another recent work [33], low-order polynomials used to represent the corners and edges lead to errors in estimation of the derivatives of the potential, and that is the crux of the problem. Despite its advantages (no prior knowledge of singular elements and the order of singularity is necessary) and good convergence characteristics, the mesh refinement approach has been mentioned to be less accurate than two other methods, namely, 1) singular elements, and 2) singular functions.…”

“…Some of the notable two-dimensional (while all the devices are 3D by definition, useful insight is often obtained by performing a 2D analysis) and three-dimensional approaches used to evaluate the singular integrals are discussed in [1,2] and [3,4,5,6,7] and the references in these papers. It is wellunderstood that many of the difficulties in the available BEM solvers stem from the assumption of nodal concentration of singularities which leads to various mathematical difficulties and to the infamous numerical boundary layers [8,9,33] when the source is placed very close to the field point ( [2] and references [4][5][6] therein). While mathematical singularities (that occur when the source and field points coincide) have been shown to be artifacts, several techniques have been used to remove difficulties related to physical or geometrical singularities (that occur when boundaries are degenerate, i.e., geometrically singular, or due to a jump in boundary conditions) such as gaussian quadrature integration, mapping techniques for regularization, bicubic transformation, nonlinear transformation and dual BEM techniques [8].…”

“…Since the singular element approach uses the location information and only the order of the singularity, it is more versatile and popular. Thus, [33] implements the method of singular elements in 3D while attempting to use proper shape function / interpolation function to model correct singular behaviour at corners and edges. It is shown that singular elements are needed to accurately capture the behavior at singular regions, such as sharp corners and edges, where standard elements fail to give an accurate result.…”