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A. Wittek et al.
as estimates of the stiffness matrix) is required and only vector operations are
performed (as M is a diagonal lumped mass matrix).
During the iterative Dynamic Relaxation (DR) procedure, the high frequencies
are damped out, and the system will eventually oscillate at its lowest frequency.
Therefore, Eq. (10.28) will converge towards the minimum eigenvalue. This estimation process, combined with our parameter selection process, leads to an increased
convergence rate, because it always offers an overestimation of the minimum
eigenvalue. The higher the overestimation of the minimum eigenvalue, the higher
the reduction of the high-frequency vibrations (see [40, 42]), and therefore Eq.
(10.28) will convergence faster towards the real minimum eigenvalue.
10.4.1.3 Dynamic Relaxation Algorithm: Termination Criteria
One very important aspect of any finite element (FE) algorithm is the termination
criterion used. If the criterion is too coarse, then the solution might be too inaccurate,
and if the criterion is too tight, then time is lost for unnecessary computations.
The usual criteria used by FE software are based on residual forces, displacements or energy. None of these criteria gives any information about the absolute
error in the solution, and selecting any of these termination criteria is very difficult.
We propose a new termination criterion that gives information about the absolute
error in the solution, particularly suited for our solution method. Because DR
iterations lead to a strong reduction of the high frequencies, the displacement vector
will oscillate around the solution vector with a frequency that converges towards
the smallest oscillation frequency. That implies that the error vector e will converge
towards the eigenvector corresponding to the lowest eigenvalue. Therefore, we can
make the following approximation:
A · e n = A 0 · e n ,
(10.29)
By substituting Eq. (10.29) in Eq. (10.17), and considering relations given in Eq.
(10.19) and Eq. (10.20), we obtain:
u n+1 − u
≈ ρ
u n − u
,
(10.30)
where u* is the solution.
Therefore, after each iteration step, the error is reduced by a ratio equal to ρ. We
can now obtain an approximation of the absolute error in the solution by applying
the infinity norm to Eq. (10.30):
u n+1 − u
∗
∞
≈ ρ ·
u n − u
∗
∞ ≤ ρ ·
u n+1 − u
∗
∞
+ u n+1 − u n ∞
,
(10.31)
u n+1 − u
∗
∞
≤
ρ
1 − ρ
· u n+1 − u n ∞ .
(10.32)
A. Wittek et al.
as estimates of the stiffness matrix) is required and only vector operations are
performed (as M is a diagonal lumped mass matrix).
During the iterative Dynamic Relaxation (DR) procedure, the high frequencies
are damped out, and the system will eventually oscillate at its lowest frequency.
Therefore, Eq. (10.28) will converge towards the minimum eigenvalue. This estimation process, combined with our parameter selection process, leads to an increased
convergence rate, because it always offers an overestimation of the minimum
eigenvalue. The higher the overestimation of the minimum eigenvalue, the higher
the reduction of the high-frequency vibrations (see [40, 42]), and therefore Eq.
(10.28) will convergence faster towards the real minimum eigenvalue.
10.4.1.3 Dynamic Relaxation Algorithm: Termination Criteria
One very important aspect of any finite element (FE) algorithm is the termination
criterion used. If the criterion is too coarse, then the solution might be too inaccurate,
and if the criterion is too tight, then time is lost for unnecessary computations.
The usual criteria used by FE software are based on residual forces, displacements or energy. None of these criteria gives any information about the absolute
error in the solution, and selecting any of these termination criteria is very difficult.
We propose a new termination criterion that gives information about the absolute
error in the solution, particularly suited for our solution method. Because DR
iterations lead to a strong reduction of the high frequencies, the displacement vector
will oscillate around the solution vector with a frequency that converges towards
the smallest oscillation frequency. That implies that the error vector e will converge
towards the eigenvector corresponding to the lowest eigenvalue. Therefore, we can
make the following approximation:
A · e n = A 0 · e n ,
(10.29)
By substituting Eq. (10.29) in Eq. (10.17), and considering relations given in Eq.
(10.19) and Eq. (10.20), we obtain:
u n+1 − u
≈ ρ
u n − u
,
(10.30)
where u* is the solution.
Therefore, after each iteration step, the error is reduced by a ratio equal to ρ. We
can now obtain an approximation of the absolute error in the solution by applying
the infinity norm to Eq. (10.30):
u n+1 − u
∗
∞
≈ ρ ·
u n − u
∗
∞ ≤ ρ ·
u n+1 − u
∗
∞
+ u n+1 − u n ∞
,
(10.31)
u n+1 − u
∗
∞
≤
ρ
1 − ρ
· u n+1 − u n ∞ .
(10.32)