Biomedical Engineering Reference
In-Depth Information
1. This generates a two-time-level, fully dis-
crete system that is expressible as matrix equation
where
t
t
n
1
t
n
and 0
AU
n
1
BU
n
C
n
(15.6)
where the entries matrices A and B and column vector C are constructed from
spatial integrations of known functions as defined in references 19 and 27 for
Cartesian coordinates. Provided tissue material property coefficients do not
vary in time, matrices A and B are stationary and can be constructed once.
Because the weighting functions,
i
, and, correspondingly, the solution basis
functions have local support, A and B are extremely sparse relative to their
overall rank (four times the number of sample positions or nodes used to rep-
resent the brain geometry) which lends them to sparse storage methods and
iterative solution schemes that become essential in 3D problems. Hence, the
displacement vector and pressure field evolve by one time increment
through iterative matrix solution to the sparse linear system described sym-
bolically in Equation (15.6).
15.5
Model Validation
The process of model validation involves answering two questions: (1) does the
discrete model represent the continuum mathematics as posed, and (2) does the
discrete model emulate physical reality? The first question is relatively
straightforward to answer, and a number of analysis tools and techniques are
available. One of the keys to this form of validation is to exercise the model
under a number of different conditions where known solutions exist. For
example, benchmark problems of increasing complexity ranging from a rela-
tively simple 1D column consolidation problem to a 3D concentric sphere
case intended to correspond to infusion-induced brain swelling have been
considered in Paulsen et al.
19
There, computational accuracies of 1 to 2% in
both displacement and pressure fields have been readily achieved with mod-
erate levels of finite element discretization.
Another aspect of examining the mathematical integrity of computed
results involves investigation of the propagation of errors during time evolu-
tion of the solution, since numerical convergence on vanishingly small ele-
ment sizes requires numerical stability. Miga et al. have conducted a Fourier
analysis of the spectrum of modes which are sustainable by the discrete FE
equations of consolidation on an infinite mesh having uniform node-to-node
sample spacing.
28
This so-called Von Neumann stability analysis shows that
two dimensionless groups along with the time integration weighting used in
Equation (15.5) control the stability of error propagation for changes in phys-
ical property and mesh discretization parameters. The results indicate that the
