Stress analysis for multilayered coating systems using semi-analytical BEM with geometric non-linearities

Abstract

For a long time, most of the current numerical methods, including the finite element method, have not been efficient to analyze stress fields of very thin structures, such as the problems of thin coatings and their interfacial/internal mechanics. In this paper, the boundary element method for 2-D elastostatic problems is studied for the analysis of multicoating systems. The nearly singular integrals, which is the primary obstacle associated with the BEM formulations, are dealt with efficiently by using a semi-analytical algorithm. The proposed semi-analytical integral formulas, compared with current analytical methods in the BEM literature, are suitable for high-order geometry elements when nearly singular integrals need to be calculated. Owing to the employment of the curved surface elements, only a small number of elements need to be divided along the boundary, and high accuracy can be achieved without increasing more computational efforts. For the test problems studied, very promising results are obtained when the thickness of coated layers is in the orders of 10 −6 -10 −9 , which is sufficient for modeling most coated systems in the micro-or nano-scales.

Adaptive Integration Method Based on Sub-Division Technique for Nearly Singular Integrals in Near-Field Acoustics Boundary Element Analysis

“…x,h) is the Jacobian of the transformation from the global three-dimensional coordinate system to the intrinsic twodimensional coordinate system of the surface patch, which can be written as [51,52]: (11) in which, , and are the orthogonal unit basis vectors of the global coordinate axes, and x 1 , x 2 , x 3 are the global coordinate. J(x,h;x¢,h¢) is the Jacobian of the transformation from the original intrinsic coordinate system to new intrinsic coordinate system after subdividing, which can be expressed as:…”

“…Then the analytical integral algorithm is performed for these linear elements. In order to estimate nearly singular integrals occurring on curvilinear geometries, Zhang et al [11,12] proposed a semi-analytical boundary element formulation over curved surface elements. Owing to the employment of the curved surface elements, only a small number of elements need to be divided along the boundary, and high accuracy can be achieved without increasing more computational efforts.…”

A novel semi-analytical algorithm of nearly singular integrals on higher order elements in two dimensional BEM

“…Recently, Fata [15] developed a semi-analytical treatment for the nearly weakly singular surface integrals on the triangular element in the Galerkin BIE. A semianalytical algorithm for the nearly singular integrals on the circular arc element in 2-D BEM has been developed by Zhang et al [16], which can exactly model the circular boundary. After then, the method has been applied to the BE analysis of the thin-walled structures and thermoelastic problems [17,18].…”

Evaluation of Singular and Nearly Singular Integrals in the BEM with Exact Geometrical Representation

“…The direct algorithms are calculating the nearly singular integrals directly. They usually include, but are not limited to, interval subdivision method [12][13], special Gaussian quadrature method [14][15], the exact integration method [16][17][18], and nonlinear transformation method [19][20][21][22][23]. Although great progresses have been achieved for each of the above methods, it should be pointed out that the geometry of the boundary element is often depicted by using linear shape functions when nearly singular integrals need to be calculated [22].…”

“…However, most engineering processes occur mostly in complex geometrical domains, and obviously, higher order geometry elements are expected to be more accurate to solve such practical problems [1][2][3][4]. Recently, two regularized algorithms suitable for calculating the nearly singular integrals occurring on the high-order geometry elements was proposed by the authors of this paper [18,23].…”