10 Finite Element Algorithms for Computational Biomechanics of the Brain
255
10.4.1.2 Dynamic Relaxation Algorithm: Estimation of the Minimum
Eigenvalue A0
Estimating the minimum eigenvalue is a challenging task, especially for non-linear
problems, where an adaptive procedure should be used in order to obtain the
optimum convergence parameters. An overview of the procedures proposed by
different authors in the context of DR (including an adaptive one) is presented in
Underwood [39].
The adaptive method proposed in Underwood [39] is based on Rayleigh’s
quotient and the use of a local diagonal stiffness matrix. The elements of this matrix
are computed using finite differences, which can be very difficult to do for degrees
of freedom which have small displacement variation.
In this section we propose a new adaptive method for computing the minimum
eigenvalue, which is also based on Rayleigh’s quotient, but does not have the
shortcomings of the method proposed in [39].
We consider a change of variable:
z n = u n − u k ,
(10.24)
where u k is the point used for linearisation of the nodal forces in Eq. (10.13). The
linearised nodal forces can therefore be expressed as:
F (u n ) = F (u k ) + K k · z n ,
(10.25)
and the linearised equation of motion will become, by replacing Eq. (10.24) and Eq.
(10.25) in Eq. (10.1):
M · ¨
z + K k · z = R − F (u k ) .
(10.26)
We can now rely on Eq. (10.26) to estimate A 0 using Rayleigh’s quotient and the
current value of the displacements:
A 0 ≤
(z n )
T K k z n
(z n )
T Mz n
.
(10.27)
We consider the right hand side of Eq. (10.27) as an estimate of the minimum
eigenvalue. Using Eq. (10.24) and Eq. (10.25), this estimate becomes [42]:
A 0 ≈
(u n − u k )
T (F (u n ) − F (u k ))
(u n − u k )
T M (u n − u k )
,
(10.28)
where u k is a fix point that must be close to u n . We will choose the solution from
a previous iteration as u k , and this point will be updated after a number of steps in
order to keep it close to the current solution u n . No additional information (such
255
10.4.1.2 Dynamic Relaxation Algorithm: Estimation of the Minimum
Eigenvalue A0
Estimating the minimum eigenvalue is a challenging task, especially for non-linear
problems, where an adaptive procedure should be used in order to obtain the
optimum convergence parameters. An overview of the procedures proposed by
different authors in the context of DR (including an adaptive one) is presented in
Underwood [39].
The adaptive method proposed in Underwood [39] is based on Rayleigh’s
quotient and the use of a local diagonal stiffness matrix. The elements of this matrix
are computed using finite differences, which can be very difficult to do for degrees
of freedom which have small displacement variation.
In this section we propose a new adaptive method for computing the minimum
eigenvalue, which is also based on Rayleigh’s quotient, but does not have the
shortcomings of the method proposed in [39].
We consider a change of variable:
z n = u n − u k ,
(10.24)
where u k is the point used for linearisation of the nodal forces in Eq. (10.13). The
linearised nodal forces can therefore be expressed as:
F (u n ) = F (u k ) + K k · z n ,
(10.25)
and the linearised equation of motion will become, by replacing Eq. (10.24) and Eq.
(10.25) in Eq. (10.1):
M · ¨
z + K k · z = R − F (u k ) .
(10.26)
We can now rely on Eq. (10.26) to estimate A 0 using Rayleigh’s quotient and the
current value of the displacements:
A 0 ≤
(z n )
T K k z n
(z n )
T Mz n
.
(10.27)
We consider the right hand side of Eq. (10.27) as an estimate of the minimum
eigenvalue. Using Eq. (10.24) and Eq. (10.25), this estimate becomes [42]:
A 0 ≈
(u n − u k )
T (F (u n ) − F (u k ))
(u n − u k )
T M (u n − u k )
,
(10.28)
where u k is a fix point that must be close to u n . We will choose the solution from
a previous iteration as u k , and this point will be updated after a number of steps in
order to keep it close to the current solution u n . No additional information (such