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Lorenzo
Trainelli’s research Algoritmi
d’integrazione numerica per ODE e DAE ODE and DAE numerical integration algorithms
Optimal conditioning of high index DAEs Differential-algebraic equations (DAEs) arise in many engineering
applications, such as multibody dynamics. These equations are often
difficult
to solve for small values of the time step size, basically due to
the effects of finite precision arithmetics. In fact, truncation
errors and non-null convergence tolerances give rise to perturbations in the numerical solution,
with disastrous effects for state variables (displacement and velocities) and
particularly for Lagrange multipliers. Tipically, this results
in accuracy order degradation and difficulties in convergence. Possible solutions are:
A strikingly simple preconditioning
recipe in the context of approach (2) has been found through a novel analysis
of the effects of finite precision arithmetics. By this procedure, we
achieve full
time step independence for the perturbations of both state
variables and Lagrange multipliers. The potential of this
methodology has been demonstrated for a vast class of commonly used numerical
integrators. I contributed in this
work originated by prof. C.L. Bottasso. This methodology is
detailed in the following works:
Consistent index reduction procedures of high index
DAEs Constrained mechanical
systems are naturally formulated by index-2 or index-3 differential-algebraic
equations (DAEs), depending on the type of constraints (nonholonomic or
holonomic). The direct integration of
these equations is prone to severe numerical difficulties, and specialized
integrators have been devised in the last two decades. These integrators (e.g.
BDF) employ strong
dissipation to overcome the instability inherent to the way the governing
equations are formulated and discretized. We developed the 'Embedded
Projection Method' (EPM), a reformulation of the governing
equations in a form that inherently avoids most of the difficulties connected
with high-index DAEs. This procedure can be
shown to be a consistent
index reduction from arbitrarily high values to one, implying that
constraints are simultanously preserved at all relevant
levels (position, velocity and acceleration), yielding improved accuracy and
stabilty. As a consequence, the
need of specialized DAE integrators is by-passed and, in principle, more gentle
(e.g. conservative) integrators can be employed. I contributed in this
work originated by prof. M. Borri. The EPM formulation is
detailed in the following papers:
Robust integration of structural and multibody
dynamics The solution of problems
governed by highly
nonlinear and numerically stiff differential equations,
such as those often encountered in structural and multibody dynamics, imposes
severe
requirements on numerical integration algorithms in terms of their
stability. It is well known that the
integrators that are commonly used in structural dynamics (e.g.
Newmark-family and generalized-a methods) are endowed only with linear
notions of stability, and therefore do not guarantee a good behavior in the
nonlinear regime. In order to overcome
these difficulties, we developed a family of time integration algorithms that
feature nonlinear
unconditional stability together with controlled dissipation of
high-frequency components in the response. Unconditional stability is
obtained by a rigorous energy bound at the discrete level: energy
preserving (EP) methods guarantee the conservation of the total
mechanical energy, while energy decaying (ED) methods enforce a
controlled decay of the same quantity. The following paper focuses
on the fundamental time discretization scheme, also in relation to other
energy-controlling methods appeared later:
In this scheme, the spectral
radius can be varied continuously between 1 (EP) and 0 (asymptotic
annihilation), and accuracy performances improve considerably with respect to the
generalized-a methods with the same value of the spectral radius. Since EP methods are unable
to prevent undue energy transfers from low to high modes, resulting in
possible numerical instabilities, we advocate the correct use of ED methods when
dealing with nonlinear, highly stiff problems. The following papers
describe different versions of the EP and ED algorithms:
These works show some representative
applications of the methodologies described to realistic problems
in modern rotorcraft system dynamics (sponsor: AgustaWestland) and wind turbine generator dynamics (sponsor: Leitner):
Also, we developed a
formulation of the equations of motion and their space-time discretization to
achieve nonconventional geometric integration properties, such as
These results are
obtained by resorting to the fixed-pole formulation of the equations of
motion, together with special parameterization techniques for rigid motion. Properties (1) and (2)
are detailed on the pages devoted to rotation
theory, while property (4) is clearly related to the
unconditional stability formulation described above. To the authors’
knowledge, property (3) has never been achieved before with respect to
arbitrary multibody systems, and contributes to a ‘gentler’ numerical
treatment of highly nonlinear, stiff challenging problems. The following paper deals
particularly with these geometrical integration aspects:
This body of work
originates at the Politecnico di Milano with profs. M. Borri and C.L. Bottasso
in the early ‘90s, and considerable interaction with prof. O.A. Bauchau
at GaTech must be mentioned. |
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Last updated 19/09/2006. |
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