1.1Introduction
The widespread use of digital computers and simulation has had a profound effect on engineering and science in the current 4th Industrial Revolution mainly due to digital transformation (Kemp, 2016). With advances in big data (Zikopoulos and Eaton, 2011), cloud computing (Armbrust et al., 2009; Mell and Grance, 2011; Zhang et al., 2010), simulation-based engineering and sciences (Jain, 1990; Oden et al., 2006), computer hardware, and appropriate software, we can model and analyze complex physical systems and problems. However, efficient and accurate use of numerical results obtained from computer programs requires considerable background and advanced working knowledge to avoid blunders and the blind acceptance of computer results.
The solution of an engineering problem generally requires the description of the response of a material body (computer chips, machine part, structural element, or mechanical system) to a given excitation (such as force). In an engineering sense, this description is usually required in numerical form, the objective being to assure the designer or engineer that the response of the system will not violate design requirements. These requirements may include the consideration of deterministic and probabilistic concepts (Thoft-Christensen and Baker, 2012; Wen, 1984; Yao, 1985) and system engineering (Hazelrigg, 1996). In a broad sense, the numerical results are predictions as to whether the system will perform as desired. For example, the solution to the elasticity problem may be obtained by a direct numerical process (numerical stress analysis) or in the form of a general solution (which ordinarily requires further numerical evaluation).
The successful solution of a complex engineering problem begins with an accurate physical model of the problem. In turn, this physical model is transformed into a mathematical model. The solution of the mathematical model is usually obtained by numerical methods that are by definition approximate. Thus, the accuracy, convergence, data set such as material properties, software, model, and other aspects need to be verified and validated (Kleijnen, 1995; Oden et al., 2006; Pardo, 2016). The success of these numerical techniques rests in turn upon high-speed digital computers and quality of the software. Other engineering aspects such as durability and sustainability also need to be considered (Chong et al., 2013; Monteiro et al., 2001).
The finite difference method (FDM) (Mitchell and Griffiths, 1980; Pletcher et al., 2012) and the finite element method (FEM) (Bathe, 2006; Cook, 2007; Hughes, 2012; Zienkiewicz and Taylor, 2005) are widely used numerical techniques. These methods are classified as domain methods, in that the engineering system is analyzed either in terms of discretized finite grids (FDMs) or finite elements (FEMs) throughout the entire region of the system (body). Another method that has emerged as a powerful tool is the boundary element method (BEM) (Brebbia, 1984; Brebbia and Connor, 1989; Cruse, 1988; Rizzo, 1967). In certain problems, this method has some distinct advantages over FDM and FEM, for several reasons. In particular, a discretization of only the boundary of the domain of interest is necessary for BEM—hence the name boundary element method.
All three of the methods noted previously, and a variety of other specialized techniques, provide powerful means of treating complex boundary value problems of engineering. In a particular case, depending on requirements, one of these methods may be more efficient than the others in generating a solution. For example, it may be more advantageous to use BEM for certain classes of linear problems characterized by infinite or semi-infinite domains, stress concentrations, three-dimensional structural effects, and so on (Beskos, 1989; Brebbia and Connor, 1...
