2. CONCEPT OF FUZZY BASED
PARAMETER ESTIMATION
The present concept assumes that spatially distributed
models represented by PDEs (e.g. Navier-Stokes
equations) can be numerically solved by finite element
method. Parameters that can not be calculated
by analytic models (like static or ordinary differential
equations (ODEs)) can often be approximated
by means of expert knowledge that is incorporated
by if-then-rules. With an inference scheme
of Fuzzy Logic [11] the expert knowledge can be
transformed into physical crisp values. This approach
allows the maximal use of expert knowledge
and reduces limitations due to missing information
and parameter values. Figure 1 shows the structure
of a coupled Fuzzy and FEM model where the output
variables of the FEM model is returned to the
Fuzzy model as input variables. This model structure
in general yields a nonlinear FEM model.
The calculation of the Fuzzy model consists in
three steps [11]: 1. Fuzzyfication of input parameters,
2. inference of if-then-rules and 3. defuzzyfi-
cation. Within step one the considered input/output
variables are transformed to linguistic variables, e.g.
small, medium, large and maximum ranges of the
considered variables are defined. In the inference
unit (step two) the actual processing of the linguistic
variables is performed whereby if-then-rules have
to be formulated by experts that describe the interdependence
of the input parameters to the output
parameter(s). The defuzzyfication (step three)
generates crisp physical output values of the Fuzzy
model.
The concept of coupled Fuzzy and FEM models
is demonstrated by the modelling and simulation
of algae growth in flat water bodies and related
eutrophication effects. The proposed approach is
very suited for this problem as the hydrodynamics is
Fig. 1. Structure of coupling spatially distributed
FEM-models and Fuzzy models.
well understood whereas the biochemical processes
are still a topic of research and the developed models
are very complex and require a big number of
parameters [2]. Since in general the required parameters
(e.g. growth rates) are not measurable for
the entire model domain the idea raised to estimate
them by taking the main variables (e.g. temperature,
fluid velocity, nutrients concentration) into account
and estimate the algae growth rate by Fuzzy
model. The next section summarises the main re