Multiple scale finite element methods

New temporal and spatial discretization methods are developed for multiple scale structural dynamic problems. The concept of fast and slow time scales is introduced for the temporal discretization. The required time step is shown to be dependent only on the slow time scale, and therefore, large time steps can be used for high frequency problems. To satisfy the spatial counterpart of the requirement on time step constraint, finite-spectral elements and finite wave elements are developed. Finite-spectral element methods combine the usual finite elements with the fast convergent spectral functions to obtain a faster convergence rate; whereas, finite wave elements are developed in parallel to the temporal shifting technique. Therefore, the spatial resolution is increased substantially. These methods are especially applicable to structural acoustics and linear space structures. Numerical examples are presented to illustrate the effectiveness of these methods.

[1]  Thomas J. R. Hughes,et al.  Implicit-Explicit Finite Elements in Transient Analysis: Stability Theory , 1978 .

[2]  Thomas J. R. Hughes,et al.  IMPLICIT-EXPLICIT FINITE ELEMENTS IN TRANSIENT ANALYSIS , 1978 .

[3]  D. Gottlieb,et al.  Numerical analysis of spectral methods : theory and applications , 1977 .

[4]  Wing Kam Liu,et al.  Development of mixed time partition procedures for thermal analysis of structures , 1982 .

[5]  Yi Fei Zhang,et al.  Unconditionally stable implicit-explicit algorithms for coupled thermal stress waves , 1983 .

[7]  Thomas J. R. Hughes,et al.  IMPLICIT-EXPLICIT FINITE ELEMENTS IN TRANSIENT ANALYSIS: IMPLEMENTATION AND NUMERICAL EXAMPLES. , 1978 .

[8]  James M. Kelly,et al.  Seismic analysis of internal equipment and components in structures , 1979 .

[9]  Wing Kam Liu,et al.  Implementation and accuracy of mixed-time implicit-explicit methods for structural dynamics , 1984 .

[10]  Ted Belytschko,et al.  Innovative Methods for Nonlinear Problems , 1984 .

[11]  George M. Fix,et al.  HYBRID FINITE ELEMENT METHODS , 1976 .

[12]  Armen Der Kiureghian,et al.  Dynamic Response of Multiply Supported Secondary Systems , 1985 .

[13]  Ted Belytschko,et al.  Partitioned rational Runge Kutta for parabolic systems , 1984 .

[14]  T. Belytschko,et al.  Stability of explicit‐implicit mesh partitions in time integration , 1978 .

[15]  A. Patera A spectral element method for fluid dynamics: Laminar flow in a channel expansion , 1984 .