Some improvements of accuracy and efficiency in three dimensional direct boundary element method

Abstract

SUMMARYNumerical analysis with the Boundary Element Method (BEM) has been used more and more in various engineering fields in recent years. In numerical techniques, however, there are some problems which have not been fully solved even now. The most essential one is the drop in the accuracy of results for internal points near the boundary of the structure, where the singularity of integrands in the boundary integral equation is too strong to be evaluated with the normal numerical method. For the boundary integral equation of stress, this problem became more serious, and the accuracy can be improved only partly, even though very refined boundary elements are used.In this paper, the boundary integral equation is newly formulated using a relative quantity of displacement. In this way, the singularity of boundary integrals is reduced by the order of l/r, and the accuracy of solution is improved significantly. Furthermore, in order to integrate it more accurately, two kinds of numerical integral methods are newly developed. By using these methods, both displacement and stress can be obtained with excellent accuracy at almost any point in the structure without any numerical difficulty, although the discretization may be comparatively coarse. The generality and practicability of the present formulation and integral methods are confirmed through some examples of three dimensional elastic problems.

A new transformation technique for evaluating nearly singular integrals

“…For the onedimensional integrals, results obtained from the proposed transformation with those obtained from the existing variable [17,26,33,34] in which methods such as the modified triangle polar co-ordinate mapping and the Huang-Cruse and the Wu's transformation techniques were employed. The results published in these references are used for comparison.…”

“…A generalized form of this type of transformations is ρ = r m which maps ρ ∈ [0, 1] to r ∈ [0, 1]. Such a transformation has been employed in [26]. Figure 3 shows the ρ − r maps of this transformation corresponding to different m, from which it is clear that this transformation distributes more Gaussian points near the "singular" point, i.e.,ρ = 0.…”

“…This singular behavior becomes more severe when β increases. Various variable transformations were proposed in the past to further regularize the integrand [17,26,33,34]. A common feature of such integrands is the coexistence of two vastly different scales.…”

“…(1) Evaluation point located at (−1.1, −1.1, 0) This example was considered in references [17,26]. In reference [26], a modified triangle-polar coordinate mapping, which is equivalent to the degenerate mapping plus the monomial polynomial transformation mentioned above, was employed.…”

“…For singular integrals, the proposed methods include, but are not limited to, analytical and semi-analytical methods [10,11,28], degenerate mappings [1,24], and various non-linear transformation techniques [20,22,31,32,35]. For nearly singular integrals, which sometimes are more difficult to compute, methods such as domain division [2,4,9,24], singularity subtraction [3], the line integral method for stress analysis of thin structures [27], a method based on continuous integral formulation for linear elasticity [29], and a variety of variable transformation techniques [5,7,[12][13][14][15][16][17][18][19]21,23,25,26,[31][32][33][34] have been proposed. In the variable transformation techniques, nonlinear transformations are employed to smooth out the integrands before conventional Gaussian quadrature is applied.…”