Kybernetika 49 no. 5, 765-779, 2013

Efficient application of e–invariants in finite element method for an elastodynamic equation

Martin Balazovjech and Ladislav Halada


We introduce a new efficient way of computation of partial differential equations using a hybrid method composed from FEM in space and FDM in time domain. The overall computational scheme is explicit in time. The key idea of the suggested way is based on a transformation of standard basis functions into new basis functions. The results of this matrix transformation are e-invariants (effective invariants) with such suitable properties which save the number of arithmetical operations needed for a problem solution. The application of this procedure and its effectiveness for 2D problem was the first time published in \cite{halabala}. Now we describe the generalization of this procedure for 3D problem. In order to present the main principle of our process and its advantage, we first explain the main idea of our approach on a simple 1D example and then the application of the e-invariants on an elastodynamics equation using hexahedral elements in 3D is described. Finally, the efficiency of the suggested method in both cases from the point of the required number of arithmetical operations is analyzed. The result of this analysis confirms computational efficiency the suggested method and the usefulness of e-invariants which save only the essential information needed for the computation. Moreover, the method can be used for various types of elements and equations.


finite element method, e-invariants, elastodynamics equation, efficient computation




  1. R. J. Archuleta: Experimental and Numerical Three-Dimensional Simulations of Strike- Slip Earthquakes. PhD. Thesis. University of California, San Diego 1976.   CrossRef
  2. M. Balazovjech and L. Halada: Effective computation of restoring force vector in FEM. Kybernetika 43 (2007), 6, 767-776.   CrossRef
  3. M. Balazovjech: Efektivny vypocet seizmickeho pohybu metodou konecnych elementov. Dissertation, FMPH Bratislava 2008.   CrossRef
  4. T. Belytscho, W. K. Liu and B. Moran: Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons, 2000.   CrossRef
  5. C. A. Felippa: Introduction to Finite Element Methods. Lecture Notes. University of Colorado, Boulder 2005.   CrossRef
  6. G. A. Frazier and C. M. Petersen: 3-D stress wave code for the Illiac IV. Systems. Systems, Science and Software Report SSS-R-74-2103, 1974.   CrossRef
  7. T. J. R. Hughes: The Finite Element Method. Linear Static and Dynamic Finite Element Method Analysis. Prentice Hall, 2000.   CrossRef
  8. P. Moczo, J. Kristek, M. Galis, P. Pazak and M. Balazovjech: The finite-difference and finite-element modeling of seismic wave propagation and earthquake motion. Acta Phys. Slovaca 57 (2007), 2, 177-406.   CrossRef
  9. J. N. Reddy: An Introduction to the Finite Element Method. McGraw-Hill, New York 1993.   CrossRef
  10. G. Strang and G. J. Fix: An Analysis of the Finite Element Method. Wellesley Cambridge Press 1988.   CrossRef
  11. R. Taborda, J. Lopez, H. Karaoglu, J. Urbanic and J. Bielak: Speeding up Finite Element Wave Propagation for Large-scale Earthquake Simulations. Technical Report CMU-PDL-10-109, Carnegie Mellon University, Parallel Data Lab. 2010.   CrossRef