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A. Wittek et al.
where A 0 and A m are the minimum and maximum eigenvalues of matrix A and
therefore ω 0 and ω max are the lowest and highest circular frequencies of the
undamped equation of motion [39].
The effect of eigenvalue estimation accuracy on the convergence of the method is
presented in Joldes et al. [40]. To ensure convergence, it is critical that the maximum
eigenvalue A m is overestimated, even if this will lead to a decreased convergence
speed. If, at the same time, the minimum eigenvalue A 0 is underestimated, uniform
convergence will be obtained for all eigenvalues, with a further decrease in the convergence speed. If A 0 is overestimated, then it is possible to increase the convergence
speed for all eigenvalues except a very narrow range of small eigenvalues.
10.4.1.1 Dynamic Relaxation Algorithm: Maximum Eigenvalue A m
and Mass Matrix
Using Gershgorin’s theorem [25, 41], it has been demonstrated that the maximum
eigenvalue of an assembled finite element mesh is bounded by the maximum
eigenvalue of any of the elements in the mesh [4]:
A m ≤ max
e
λ
e
max
.
(10.23)
Therefore, an estimation of the maximum eigenvalue can be obtained by
estimating the maximum eigenvalue of each element in the mesh. Such estimations
for different element types are presented in Hughes [26].
In the case of a non-linear problem, the maximum eigenvalue changes during
the simulation as the geometry of the elements changes, and therefore it must be
estimated after every iteration step.
Because the mass matrix has no influence on the steady state (as the time
derivatives in Eq. 10.9 become zero), a fictitious mass matrix that improves the
convergence rate can be used. The mass matrix can be chosen such that it reduces
the condition number of matrix A, leading to a decrease in the spectral radius ρ (see
Eq. 10.20).
In order to reduce the condition number, we propose to align the maximum
eigenvalue of all elements in the mesh to the same value by changing the density of
each element. By doing this we can still use Eq. (10.23) for estimating the maximum
eigenvalue, and the condition number is at least preserved and generally decreased,
as shown in Underwood [39]. This process guarantees that the selected maximum
eigenvalue A m is an overestimation of the actual maximum eigenvalue during the
simulation, therefore ensuring the convergence.
A. Wittek et al.
where A 0 and A m are the minimum and maximum eigenvalues of matrix A and
therefore ω 0 and ω max are the lowest and highest circular frequencies of the
undamped equation of motion [39].
The effect of eigenvalue estimation accuracy on the convergence of the method is
presented in Joldes et al. [40]. To ensure convergence, it is critical that the maximum
eigenvalue A m is overestimated, even if this will lead to a decreased convergence
speed. If, at the same time, the minimum eigenvalue A 0 is underestimated, uniform
convergence will be obtained for all eigenvalues, with a further decrease in the convergence speed. If A 0 is overestimated, then it is possible to increase the convergence
speed for all eigenvalues except a very narrow range of small eigenvalues.
10.4.1.1 Dynamic Relaxation Algorithm: Maximum Eigenvalue A m
and Mass Matrix
Using Gershgorin’s theorem [25, 41], it has been demonstrated that the maximum
eigenvalue of an assembled finite element mesh is bounded by the maximum
eigenvalue of any of the elements in the mesh [4]:
A m ≤ max
e
λ
e
max
.
(10.23)
Therefore, an estimation of the maximum eigenvalue can be obtained by
estimating the maximum eigenvalue of each element in the mesh. Such estimations
for different element types are presented in Hughes [26].
In the case of a non-linear problem, the maximum eigenvalue changes during
the simulation as the geometry of the elements changes, and therefore it must be
estimated after every iteration step.
Because the mass matrix has no influence on the steady state (as the time
derivatives in Eq. 10.9 become zero), a fictitious mass matrix that improves the
convergence rate can be used. The mass matrix can be chosen such that it reduces
the condition number of matrix A, leading to a decrease in the spectral radius ρ (see
Eq. 10.20).
In order to reduce the condition number, we propose to align the maximum
eigenvalue of all elements in the mesh to the same value by changing the density of
each element. By doing this we can still use Eq. (10.23) for estimating the maximum
eigenvalue, and the condition number is at least preserved and generally decreased,
as shown in Underwood [39]. This process guarantees that the selected maximum
eigenvalue A m is an overestimation of the actual maximum eigenvalue during the
simulation, therefore ensuring the convergence.