.
.
,
and
load vector 
as given by equation (3.19) and (3.24).
from:
from:
.
Stability of the Generalized VIP Representation
Although obvious, the stability of the generalized VIP representation is illustrated based on a spectral analysis of the amplification matrix.
The conditions for spectral stability of a numerical algorithm are (see [4]):
1. The spectral radius of the amplification matrix must be strictly less than one:
.
2. Eigenvalues of of multiplicity greater than one are strictly less than one in modulus.
The eigenvalues of the amplification matrix given by (3.19), are obtained as:
where 
.
Since the system under consideration is assumed to be under-damped ( 
), or undamped ( 
), the spectral radius of the amplification matrix has the obvious property:
where the equality is valid only for undamped systems ( ).
The plot of the spectral radius versus 
for different values of the damping ratio (see Figure 17) shows that the spectral radius is less than one for all values of the frequency and decreasing with the increase in frequency.
This ensures good damping properties for the higher frequency modes, which is a very desirable property.
Figure 17: Spectral radius of the generalized VIP representation for different values of the damping coefficient
Accuracy of the Generalized VIP Representation
The accuracy characteristics of an algorithm are measured by the truncation error, numerical dissipation and dispersion. As observed, the generalized VIP representation yields the exact solution for a free (undamped) system.
The accuracy of the generalized VIP representation applied to linear systems with proportional damping is presented next (and those involving non-proportional damping are addressed in a later section).
Truncation error
The most widely used measure of the accuracy of a numerical algorithm is the evaluation of the local truncation error, 
, generally defined as:
Following the standard procedures, the modulus of the local truncation error vector can be shown to satisfy the inequality:
in which:
and
Since the constant C does not depend on the time step 
, the generalized VIP representation is shown to be consistent and second-order accurate.
Convergence is automatically established.
Numerical Dissipation and Dispersion
The numerical dissipation and dispersion are evaluated by the algorithmic damping ratio and the relative period error. These are obtained as:
and
and also show in Figures 18(a) and 18(b).
Figure 18: Plot for numerical dissipation and dispersion
The methodology outlined here represents a complete generalization of the original VIP time integral representation described in [88, 81, 8], in which, the Green's function of the free undamped system is used via a similar formulation.
In contrast to the present representation, the original VIP time integral representation (described subsequently) requires one extra approximation, namely assuming a linear interpolation for the velocity, since the virtual or weighted time field was selected based on an undamped system.
Remarks
1. In an attempt to capitalize on the advantages and explain the basis of both modal type methods and time integration techniques for dynamic field problems, the methodology outlined above leading to the development of the `exact integral operator' and its consequences are initially described in [87, 89, 88, 8, 81], in a series of papers for dynamic analysis.
The issues relevant to the exact integral operator described in Case 1 also follow Case 2. Since a virtual or weighted time field is employed for the time discretization as the fundamental solution to the adjoint operator set equal to zero based on the initial state with 
representative of a Dirac pulse type behavior, the methodology is termed as the VIrtual-Pulse (VIP) time integral methodology.
2. An `approximate integral operator' is constructed by making an approximation to the forcing function, 
. For dynamic field problems, this leads to an explicit self starting time integral methodology of computation with several computationally attractive, and obviously excellent algorithmic stability and accuracy attributes. It is of n-th order accuracy for (n-1)th order approximation of the load .
For example, for a linear interpolation of the loads, the approximate integral operator is of second-order accuracy.
Furthermore, the time integral operator naturally inherits features of unconditional stability and does not suffer the drawbacks of conditional consistency. For dynamic analysis, obviously the `approximate integral operator' has the following algorithmic characteristics: the algorithmic damping, 
, and the relative period error, 
for damped dynamic systems.
These quantities are significantly lower than those of the commonly advocated `integration operators'.
As initial efforts, the `approximate integral operator', however, based on the original VIP methodology (see section to follow) of computation has been extended to a class of nonlinear inertial structural dynamics considering elasto-plastic material behavior described by 
, with the need to compute the eigenproblem only once based on the initial state, and with the argument that with the selection of the virtual or weighted time field being arbitrary, the original semi-discretized nonlinear dynamic system is not disturbed.
For nonlinear situations, the approximate integral operator is, however, explicit with iterations.
For linear situations, it readily reduces to the linear time integral operator.
The views and opinions expressed in this page are strictly those of the page author.
The contents of this page have not been reviewed or approved by the University of Minnesota.
