A Fast Implicit Integration Scheme to Solve Highly Nonlinear System

  • Saiful Siddiquee
Conference paper
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 146)


Now-a-days researchers are formulating new generation of soil-models based on combined theory. That means researchers are trying to put forward a unified material model, which would predict at least the behaviour of all types of soils under all types of stress and time paths. So the solution techniques so far being used by the nonlinear Finite Element packages no longer can meet the huge demand of computational speed created by those models. It was necessary to develop a new type of solution scheme for the sophisticated models. Usually material nonlinearity makes it difficult to create a robust solution technique. So it is important to develop a solution scheme which will be very robust at the same time. That means the solution scheme should not break-down even for a notoriously complicated unified model. In this paper, we have developed an implicit solution scheme, which solves the resulting nonlinear equations of motion by implicit dynamic relaxation. There are a myriad number of implicit schemes for the use. Here a relatively less used method—called “Houbolt’s integration scheme” has been used. It is very similar to the central difference scheme only difference is the use of the higher-order terms in the definition of velocity and acceleration. In order to make it faster, sparse-matrix solution scheme is used with partial pivoting and reordering of matrix elements to minimize the fill-ins. The combined effect is quite dramatic. It provides the main two traits of a good nonlinear solution technique—i.e., speed and robustness of solution. The solution scheme is applied to trace the full loading path of an elasto-visco-plastically defined material behaviour of a Plane Strain Compression (PSC) test sample. There is a huge gain in speed and robustness compared to the other techniques of solution.


Solution Scheme Dynamic Relaxation Central Difference Scheme Plane Strain Compression Global Stiffness Matrix 
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  1. Houbolt, J. C. (1950), “A Recurrence Matrix Solution for the Dynamic Response of Elastic Aircraft”, Journal of the Aeronautical Sciences, Vol. 17, pp. 540–550.CrossRefMathSciNetGoogle Scholar
  2. Siddiquee, M. S. A. and Tatsuoka, F. (2001), “Modeling time-dependent stress-strain behaviour of stiff geomaterials and its applications”, Proc. 10th International Conference on Computer Methods and Advances in Geomechanics (IACMAG), Tucson, Arizona on January 7–12.Google Scholar
  3. Siddiquee, M. S. A., Tatsuoka, F. and Tanaka, T. (2006), “FEM simulation of the viscous effects on the stress-strain behaviour of sand in plane strain compression”, in press.Google Scholar
  4. Tatsuoka, F., Ishihara, M., Di Benedetto, H. and Kuwano, R. (2002), “Time-dependent shear deformation characteristics of geomaterials and their simulation”, Soils and Foundations, Vol. 42,No.2, pp.103–129.CrossRefGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Saiful Siddiquee
    • 1
  1. 1.Civil Engineering DepartmentBangladesh University of Engineering and Technology (BUET)DhakaBangladesh

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