Engineering Analysis with Boundary Elements 95 (2018) 222–237 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Complex Fourier element shape functions for analysis of 2D static and transient dynamic problems using dual reciprocity boundary element method S. Hamzehei-Javaran, N. Khaji ∗ Faculty of Civil and Environmental Engineering, Tarbiat Modares University, P.O. Box 14115-397, Tehran, Iran a r t i c l e i n f o Keywords: Complex Fourier shape functions Complex Fourier radial basis functions Equispaced macroelements Boundary element method Dual reciprocity method 2D elastostatic and transient elastodynamic problems a b s t r a c t In this paper, the boundary element method is reformulated using new complex Fourier shape functions for solv- ing two-dimensional (2D) elastostatic and dynamic problems. For approximating the geometry of boundaries and the state variables (displacements and tractions) of Navier’s differential equation, the dual reciprocity (DR) boundary element method (BEM) is reconsidered by employing complex Fourier shape functions. After enriching a class of radial basis functions (RBFs), called complex Fourier RBFs, the interpolation functions of a complex Fourier boundary element framework are derived. To do so, polynomial terms are added to the functional ex- pansion that only employs complex Fourier RBF in the approximation. In addition to polynomial function fields, the participation of exponential and trigonometric ones has also increased robustness and efficiency in the in- terpolation. Another interesting feature is that no Runge phenomenon happens in equispaced complex Fourier macroelements, unlike equispaced classic Lagrange ones. In the end, several numerical examples are solved to illustrate the efficiency and accuracy of the suggested complex Fourier shape functions and in comparison with the classic Lagrange ones, the proposed shape functions result in much more accurate and stable outcomes. 1. Introduction Elastostatics and dynamics contain an extensive range of phenom- ena in engineering and physical problems including force equilibrium of special structures and analysis of structures subjected to earthquake, vibratory motor, collision and explosion loads. In these cases, the wave propagation is expressed by a governing linear partial differential equa- tion associated with suitable initial and boundary conditions. In gen- eral, obtaining the solution of elastostatic and dynamic problems for the sake of analysis and design can be difficult and laborious when ana- lytical approaches are used. Moreover, it may even become impossible when a little complexity happens in boundary conditions. Therefore, it seems reasonable in most practical engineering cases to solve them numerically. Among the numerical methods considered significantly by researchers, boundary element method (BEM) [1,2] can be mentioned. As it is obvious from its name, only the boundary needs to be discretized in this method, not the domain. Thus, fewer unknown parameters need to be stored and less computational cost and storage space will be spent. For problems such as stress concentration or infinite domains, BEM can be applied to achieve better accuracy in comparison with finite element method (FEM). Many usages of BEM in solving problems related to buck- ∗ Corresponding author. E-mail address: nkhaji@modares.ac.ir (N. Khaji). ling, optimization, crack, wave propagation, and etc. are reported in the literature. [3–11] According to the literature, three formulations known as the Laplace transform, the time domain (TD), and the domain integral techniques exist for solving elastodynamic problems with BEM [1,2]. However, the first two methods are with mathematical complexities and the third one requires domain integration, which are some challenges to be faced in these methods. In the work of Nardini and Brebbia [12–14], the well- known dual reciprocity method (DRM) was introduced to overcome these problems. With the introduction of DRM, a significant develop- ment happened in the BEM analysis of time-dependent problems. One of the advantages of DRM is benefiting from less computational cost than other methods (like TDM) due to its ability in using the simple Green’s function of elastostatics for analyzing elastodynamic problems. Various usages of the DRM for solving a broad range of problems are reported in the literature. In the works of Dehghan et al. [15–19], this method was implemented for solving various equations and problems including stochastic partial differential equations, linear Helmholtz and semi linear Poisson’s equations, and etc. The DRM was applied in the solution of free and force vibration problems by Rashed et al. [20–24]. Hamzehei Javaran et al. [25–30] applied DRM for the analysis of prob- https://doi.org/10.1016/j.enganabound.2018.07.012 Received 3 December 2017; Received in revised form 27 June 2018; Accepted 29 July 2018 0955-7997/© 2018 Elsevier Ltd. All rights reserved.