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In this paper, the boundary element method is reformulated using new complex Fourier shape functions for solving 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 expansion 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 interpolation. 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. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, boundary element analysis for two-dimensional potential problems is investigated. In this study, the boundary element method (BEM) is reconsidered by proposing new shape functions to approximate the potentials and fluxes. These new shape functions, called complex Fourier shape function, are derived from complex Fourier radial basis function (RBF) in the form of exp(iωr). The proposed shape functions may easily satisfy various functions such as trigonometric, exponential, and polynomial functions. In order to illustrate the validity and accuracy of the present study, several numerical examples are examined and compared to the results of analytical and with those obtained by classic real Lagrange shape functions. Compared to the classic real Lagrange shape functions, the proposed complex Fourier shape functions show much more accurate results. [Copyright &y& Elsevier]

Abstract: A new semi-analytical method is proposed for solving boundary value problems of two-dimensional elastic solids, in this paper. To this end, the boundary of the problem domain is discretized by specific non-isoparametric elements that are proposed for the first time in this research. These new elements employ higher-order Chebyshev mapping functions and new special shape functions. For these shape functions, Kronecker Delta property is satisfied for displacement function and its derivatives. Furthermore, the first derivatives of shape functions are assigned to zero at any given control point. Eventually, implementing a weak form of weighted residual method and using Clenshaw–Curtis quadrature, coefficient matrices of equations system become diagonal, which results in a set of decoupled governing equations to be used for solving the whole system. In other words, the governing equation for each degree of freedom (DOF) is independent from other DOFs of the problem. Validity and accuracy of the present method are fully demonstrated through four benchmark problems which are successfully modeled using a few numbers of DOFs. The numerical results agree very well with the analytical solutions and the results from other numerical methods. [Copyright &y& Elsevier]

Abstract: In this paper, a new semi-analytical method is proposed for solving boundary value problems of two-dimensional (2D) potential problems. In this new method, the boundary of the problem domain is discretized by a set of special non-isoparametric elements that are introduced for the first time in this paper. In these new elements, higher-order Chebyshev mapping functions and new special shape functions are used. The shape functions are formulated to provide Kronecker Delta property for the potential function and its derivative. In addition, the first derivative of shape functions are assigned to zero at any given control point. Finally, using weighted residual method and implementing Clenshaw–Curtis quadrature, the coefficient matrices of equations system become diagonal, which results in a set of decoupled governing equations for the whole system. This means that the governing equation for each degree of freedom (DOF) is independent from other DOFs of the domain. Validity and accuracy of the present method are fully demonstrated through four benchmark problems. [Copyright &y& Elsevier]

Abstract: In this paper, a new boundary element analysis for two-dimensional (2D) transient elastodynamic problems is proposed. The dual reciprocity method (DRM) is reconsidered by employing new radial basis functions (RBFs) to approximate the domain inertia terms. These new RBFs, which are in the form of ζ+κ sin(ωr+α), are called Fourier RBFs hereafter. Using the method of variation of parameters, the particular solution kernels of Fourier RBFs corresponding to displacement and traction, whose a few terms are singular, has been explicitly derived. Therefore, a new simple smoothing trick has been employed to resolve the singularity problem. Moreover, the limiting values of the particular solution kernels have been evaluated. In order to find the unknown parameters of Fourier RBFs, an optimization problem seeking for the optimum value of the Houbolt scheme parameter β that minimizes the mean squared error (MSE) function of the problem is established. Since the MSE function of the proposed RBFs is a function of five unknown parameters (i.e., ζ, κ, ω, α, and β), the genetic algorithm (GA) has been used to solve the necessary optimization problem. In order to illustrate the validity, accuracy, and superiority of the present study, several numerical examples are examined and compared to the results of analytical and other RBFs reported in the literature. Compared to other RBFs, Fourier RBFs show more accurate and stable results. Moreover, these results are obtained using less degree of freedom without any additional internal points that are commonly used to improve the accuracy of the results. [Copyright &y& Elsevier]