Transient implicit wave propagation dynamics with the method of finite spheres Ki-Tae Kim, Klaus-Jürgen Bathe ⇑ Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA article info Article history: Received 17 March 2016 Accepted 15 May 2016 Available online 8 June 2016 Keywords: Wave propagation Transient solutions Method of finite spheres Implicit dynamics Bathe method abstract In the paper by Ham et al. (2014), the method of finite spheres enriched for transient wave propagation problems was used and a monotonic convergence of the calculated solutions with decreasing time step size was seen. This is an important property for practical analyses and different from what is seen in tra- ditional finite element solutions. In this paper we explicitly show and study this characteristic through a dispersion analysis of the solutions calculated by the method of finite spheres using an implicit time inte- gration method, the Bathe method. Another important property identified is that in uniform spatial dis- cretizations for the problems solved, the calculated solution accuracy is almost independent of the solution direction considered. Numerical solutions of some wave propagation problems are given to demonstrate these attributes and show that with the schemes discussed very accurate response predic- tions can be obtained. Ó 2016 Elsevier Ltd. All rights reserved. 1. Introduction In transient wave propagation problems, spatial and temporal discretizations introduce dispersion not present in analytical solu- tions, and hence this error is referred to as ’numerical dispersion’. The accuracy of numerical solutions can be bad due to this disper- sion. When applying numerical methods to wave propagation problems, spatial and temporal discretizations should, therefore, be chosen according to the dispersion properties of the methods used and it is difficult to obtain accurate solutions to complex problems. The finite element method with direct time integrations has been widely used to solve wave propagation problems. Consider- able research efforts have been focused on the dispersion analysis of finite element solutions to the wave or Helmholtz equation [2–9]. The finite element discretization with the consistent mass matrix in general results in a faster phase velocity than the exact propagation velocity, while a lumped mass approximation leads to slower phase velocity [2–5,10]. In addition, the finite element solutions using uniform meshes show numerical anisotropy, i.e., the solution error depends on the direction considered although the exact wave propagation is the same in all directions [4–6,8–10]. The effect of temporal discretizations using various time integration methods on solutions has also been studied [2,3,5,9] with the error giving period elongations and amplitude decays [11,12]. Numerous methods have been proposed to reduce the disper- sion error of finite element solutions [5,10,13–19]. However, in general two- and three-dimensional problems, most of these methods suffer from numerical anisotropy and/or are complicated to use. The higher-order finite element method such as the spectral element method [20] can reduce the dispersion error and the numerical anisotropy [21], but it can be difficult to solve two- and three-dimensional problems in practical analyses. Another approach to improve the finite element solutions of wave propagation problems is based on trying to cancel out oppos- ing effects. The finite element discretization with the consistent mass matrix gives an overestimated phase velocity while the use of the trapezoidal rule results in an underestimated phase velocity, and it is found that the combined effect leads to a decrease in dis- persion error [3]. The Bathe method [22,23], an implicit time integration method, was shown to be very effective when used with bi-linear finite ele- ments because the dissipation property of the Bathe method atten- uates undesired high frequency waves. At optimal spatial and temporal discretizations for single types of waves, the solutions are almost non-dispersive but show numerical anisotropy [9]. Since the solutions of practical problems contain multiple types of waves, e.g., longitudinal, transverse and surface waves at differ- ent wave speeds, the optimality for all wave predictions is lost, and all waves cannot be accurately calculated at the same time. http://dx.doi.org/10.1016/j.compstruc.2016.05.016 0045-7949/Ó 2016 Elsevier Ltd. All rights reserved. ⇑ Corresponding author. E-mail address: kjb@mit.edu (K.J. Bathe). Computers and Structures 173 (2016) 50–60 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc