Three-dimensional Coons macroelements: application to eigenvalue and scalar wave propagation problems

This paper discusses the matter of using higher order approximations in three-dimensional problems through Coons macroelements. Recently, we have proposed a global functional set based on ‘Coons interpolation formula’ for the construction of large two-dimensional macroelements with degrees of freedom appearing at the boundaries only of the domain. After successive application in many engineering problems, this paper extends the methodology to large three-dimensional hexahedral macroelements with the degrees of freedom appearing at the 12 edges of the entire domain in case of smooth box-like structures. Closed-form expressions of the global shape functions are presented for the first time. It is shown that these global shape functions can be automatically constructed in a systematic way by arbitrarily choosing univariate approximations such as piecewise-linear, cubic B-splines, Lagrange polynomials, etc., along the control lines. Moreover, the mechanism of adding facial and internal nodes is presented. Relationships with higher order p-methods are discussed. Following to excellent results previously derived for the solution of the Laplace equation as well as static and eigenvalue extraction analysis of structures, the paper investigates the performance of Coons macroelements in 3-D eigenvalue and scalar wave propagation problems by implementing standard time-integration schemes.