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

Abstract

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.