Solution Of Elastodynamic Problems Research Articles

The solution of elastodynamic problems using modified radial integration boundary element method (MRIBEM)

In this paper, the modified radial integration boundary elements method (MRIBEM) is applied to analyze two-dimensional (2D) elastodynamic problems. The boundary and domain variables are approximated by Lagrangian shape functions and global radial basis functions (RBFs), respectively. As a key idea of this study, the domain integrals that result from implying the inertia force effect are computed by means of a modified radial integration method (MRIM). With the MRIBEM, we are able to deal with problems with concave shapes when the domain variables are approximated by global RBFs by defining an auxiliary point or, in case of complex domain shapes, two or more auxiliary points as the origin for radial integration. Two-time marching schemes, i.e. Newmark method and the Houbolt method, are implemented for the solution of governing differential equations along time. The performance of the proposed method is examined in the solution of four numerical examples.

A new LBIE method for solving elastodynamic problems

A new local boundary integral equation (LBIE) method for the solution of elastodynamic problems in both frequency and time domain is proposed. Non-uniformly distributed points covering the analyzed domain are used for the interpolation of the involved fields. The key-point of the proposed methodology is that the support domain of each point is divided into parts with the aid of cells formed by connecting the point of interest with the nearby points. Then an efficient radial basis functions (RBF) interpolation scheme is exploited for the representation of displacements in each cell, while on the intersections between the local domains and the global boundary, tractions are treated as independent variables via conventional boundary elements. For each point the corresponding LBIE is written in terms of displacements only, since on the boundary of support domains tractions are eliminated with the aid of the elastostatic companion solution. The integration in support domains is performed easily and with high accuracy, while due to cells the extension of the method to three dimensions is straightforward. Transient solutions are obtained after inversion of frequency domain results with the inverse fast Fourier transform (FFT). Two representative numerical examples that demonstrate the accuracy of the proposed methodology are provided.

Vibration analysis of plane elasticity problems by the [formula omitted]-continuous time stepping finite element method

This paper proposes a C 0 -continuous time stepping finite element method to solve vibration problems of plane elasticity. In the time direction, unlike the existing methods [F. Costanzo, H. Huang, Proof of unconditional stability for a single-field discontinuous Galerkin finite element formulation for linear elasto-dynamics, Comput. Methods Appl. Mech. Engrg. 194 (2005) 2059–2076; D.A. French, A space–time finite element method for the wave equation, Comput. Methods Appl. Mech. Engrg. 107 (1993) 145–157; H. Huang, F. Costanzo, On the use of space–time finite elements in the solution of elasto-dynamic problems with strain discontinuities, Comput. Methods Appl. Mech. Engrg. 191 (2002) 5315–5343; T.J.R. Hughes, G. Hulbert, Space–time finite element methods for elastodynamics: Formulations and error estimates, Comput. Methods Appl. Mech. Engrg. 66 (1988) 339–363; G. Hulbert, T.J.R. Hughes, Space–time finite element methods for second-order hyperbolic equations, Comput. Methods Appl. Mech. Engrg. 84 (1990) 327–348; C. Johnson, Discontinuous Galerkin finite element methods for second order hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 107 (1993) 117–129; X.D. Li, N.E. Wiberg, Structural dynamic analysis by a time-discontinous Galerkin finite element method, Int. J. Numer. Methods Engrg. 39 (1996) 2131–2152; X.D. Li, N.E. Wiberg, Implementation and adaptivity of a space–time finite element method for structural dynamics, Comput. Methods Appl. Mech. Engrg. 156 (1998) 211–229], this method does not use the discontinuous Galerkin (DG) method to simultaneously discretize the displacement and velocity fields, but only use the C 0 -continuous Galerkin method to discretize the displacement field instead. This greatly reduces the size of the linear system to be solved at each time step. The finite element in the space directions is taken as the usual P r − 1 -conforming element with r ⩾ 2 . It is proved that the error of the method in the energy norm is O ( h r − 1 + k 3 ) , where h and k denote the mesh sizes of the subdivisions in the space and time directions, respectively. Some numerical tests are included to show the computational performance of the method.