Multistep Integration Methods Research Articles

Constrained Equations of Motion in Multibody Dynamics as ODEs on Manifolds

A convergence theorem for applying linear multistep numerical integration methods to constrained equations of motion in mechanical systems is presented. Using a differential geometric approach, Euler–Lagrange equations are reduced to ordinary differential equations (ODEs) on a local parameter space of the constraint manifold. The reduced ODEs and the algebraic constraints are discretized by applying numerical integration formulas; the resultant nonlinear equations are then solved using Newton’s method. The order and convergence results of numerical integration methods are proven on the local parameter space. Using constant order and fixed stepsize integration methods, numerical solution of the preliminary examples shows that the theorem is valid.

A dynamic intragranular fission gas behavior model

A model for the non-equilibrium behavior of intragranular fission gas in uranium oxide fuel is developed to study the fundamental phenomena that determine fission gas effects. The dynamic behavior of point defects and the variations in stoichiometry are explicitly represented in the model. The principle of distribution moment invariance is used to allow approximations that significantly reduce computational expense without sacrificing accuracy. A dynamic intragranular gas release and swelling (DIGRAS) computer code, that is based on the non-equilibrium model, was developed for both steady-state and transient applications. The code utilizes implicit multistep numerical integration methods, and is designed to give detailed information on all the physical processes that contribute to fission gas behavior. Simulations of steady-state irradiations indicate that the gas bubble re-solution process is very significant and results in very few large bubbles. The assumptions of equilibrium bubble sizes for normal steady-state irradiations in fast reactors appears to be adequate. On the contrary, a fully dynamic fission gas and point defect treatment was found necessary for transient simulations. The fuel stoichiometry was found to play an important role in determining bubble kinetics. This is mainly due to the strong dependence of point defect populations on stoichiometry. In fast transients, bubbles were found to be highly overpressurized, which suggests that a mechanistic plastic growth model is also needed.

Discretization Error Analysis in Linear DDA Connections

In this paper mathematical models for digital integrators operating with multidigit increments are given, and the discretization error analysis of their linear connections is studied. The integration methods considered are linear multistep integration methods; some examples are studied in detail. Linear digital differential analyzer (DDA) connections are defined, and their use in connection with the numerical solution of the initial value problem is explained. Finally, the discretization error in these linear DDA connections has been analyzed in detail; the theoretical results have also been verified experimentally.