Hybrid nearly singular integration for isogeometric boundary element analysis of coatings and other thin 2D structures

Abstract

We present an isogeometric boundary element method (IGABEM) capable of delivering accurate and efficient solutions in the heat transfer analysis of 2D coated structures such as those commonly found in turbomachinery. Although we consider very thin coatings (of thickness down to 10 −7 m), they are modelled explicitly as BEM zones, and this is made possible by the development of a new integration scheme (sinh +) aimed particularly at the challenging nearly singular integrals that arise. Sinh + is a hybrid of adaptive and sinh transformation approaches, and we make further extensions to the latter to improve its robustness. The scheme is tuned to deliver results of engineering accuracy in an optimal time. The scheme is adaptable, by changing a tolerance, to enable engineers to achieve a different balance between accuracy and computational efficiency as may be required for different applications. A set of numerical examples demonstrates the ability of the scheme to produce accurate temperature distributions efficiently in the presence of very thin coatings.

Hybrid nearly singular integration for three-dimensional isogeometric boundary element analysis of coatings and other thin structures

“…Coatings or other thin structures have been widely used to protect a variety of structural engineering materials from corrosion, wear, and erosion, and to provide lubrication and thermal insulation [1,2]. The great difference in the thermo-mechanical properties of substrate and coatings presents challenges to their use in composite structures subjected to large changes in temperature.…”

“…However, the BEM analysis of solids has classically been restricted to problems without thin sections because such geometries impose a requirement to evaluate a large number of nearly-singular integrals accurately and efficiently. With the development of suitable integration schemes, the BEM has become an attractive method for coated structures and other thin sections, as demonstrated by [33] for the classical (piecewise polynomial) BEM in 2D and extended to isogeometric BEM by Gong et al [2].…”

“…The resulting analysis method is called IGA-BEM (isogeometric analysis boundary element method). The properties of exact geometric description and accurate computation of quantities along the thickness direction inside coatings or thin structures, suggest that IGABEM is a highly promising approach to study the coatings/thin structures [2,20,34]. However, schemes for the efficient and accurate evaluation of the near-singular integrals that arise in IGABEM need to be developed to realise this promise.…”

“…For the computation of near-singular integrals existing in standard BEM, significant progress has been achieved, including analytical methods [35][36][37][38], semi-analytical integration schemes [39,40], cubic transformation [41], radial variable transformation [42], line integral approaches [43,44], implemented method [45], parameterized Gaussian quadrature [46], the particular solution method [47,48], adaptive integration schemes [49,50], regularisation techniques [51,52], domain supplemental approach [53], distance transformation method [54,55], spherical element subdivision method [56], sinh transformation method [57][58][59][60], and the exponential transformation method [61,62]. A more detailed review of these methods can be found in Gong et al [2]. With the adoption of NURBS in IGABEM to replace simple polynomials, some of these methods cannot be applied directly; moreover, the balance between accuracy and efficiency of these methods when applied within a 3D IGABEM context is not yet clear.…”

“…In this work, we develop a scheme for 3D integrals based on the sinh + scheme presented for 2D simulations in [2]; this combines the benefits of the sinh transformation [57][58][59][60] and adaptive integration scheme [49,50]. An essential feature of the method is its ability to produce accurate and efficient integral evaluations for general, industrially relevant IGABEM models of thin sections or components with coatings.…”