Finite element mesh design (LAI)
The datasets on finite element mesh design come from the
area of mechanical engineering where finite element meshes are used to
analyze the behaviour of structures under different kinds of stress. The
problem addressed here is to determine an appropriate resolution of a finite
element mesh for a given structure so that the corresponding computations
are both accurate and fast.
The data
The resolution of a finite element (FE) mesh is determined
by the number of elements on each of its edges. It depends on the geometry
of the body studied and on the boundary conditions. Given are descriptions
of ten structures for which experts have determined an appropriate mesh
resolution. The positive examples are of the form mesh(Edge,NumberOfElements).
The task is to learn rules that determine an appropriate resolution of
a FE mesh (i.e., an appropriate resolution for each given edge) from the
geometry of the body, the types of edges, boundary conditions and loadings.
The background knowledge can be divided into two parts:
attribute description of the edges and geometric relations between the
edges. The first part consists of unary predicates that have edges as arguments.
These predicates can be grouped in three subgroups:

predicates that describe the type of the edge ( long,
usual, short, circuit,
half_circuit, quarter_circuit, short_for_hole,
long_for_hole,
circuit_hole, half_circuit_hole, quarter_circuit_hole,
not_important),

predicates that describe the supports of the edge (free,
one_side_fixed,
two_side_fixed, fixed),

predicates that describe the loads (not_loaded,
one_side_loaded,
two_side_loaded, cont_loaded).
The second part contains the binary relations neighbour
and opposite, that describe the geometrical relations between
edges. These two relations are nondeterminate. A determinate version also
exists, where each of the two relations is replaced with three relations,
e.g., neighbour_xy, neighbour_yz, and neighbour_zx.
CLAUDIEN experiments
The declarative bias facility of CLAUDIEN was used to specify
the form of classification rules to be induced. Six types of rules were
generated:

rules that use only the properties of a given edge to assign
a proper number of elements to it,

rules that in addition allow the use of properties of one
related (neighboring, opposite or equal) edge,

rules that can use the properties of two edges related to
the given one,

rules that can use the properties of a first edge related
to the given one, and the properties of a second edge, related to the first,

rules that use the properties of a given edge and one additional
related edge, but allow intervals for the number of elements assigned,
and

rules that use only the type of the given edge and allow
intervals for the number of elements assigned.
Each rule has to mention at least one property of each edge
mentioned in the rule. A total of 4462 rules were generated by CLAUDIEN.
These were then postprocessed to eliminate inappropriate rules. A rule
was eliminated if it:

covers less than 3 positive examples,

is a duplicate of another rule,

is subsumed by a more general rule predicting the same class,

covers only additional positive examples (generated taking
into account tolerances for larger number of elements per edge), or

if it is merged with another rule that has the same body,
but specifies a larger interval for the class.
The resulting rule set had 1776 rules. Some rules were added
manually to the above, including recursive rules induced in earlier experiments,
rules that compare the average number of elements for edges of different
types, and rules for determining the appropriate type of finite elements.
The latter encompass some extra expert knowledge.
The new rule set has accuracy of 78% on the entire training
set. Testing on unseen cases was done by leaving out each one of the ten
structures or alternatively, by randomly removing 10% of all edges in all
structures (tenfold crossvalidation). The average accuracies were 59.09%
and 70.16%, respectively. The rules also perform well on a completely new
structure, both in terms of accuracy and in terms of expert evaluation
of the proposed mesh.
References

B. Dolsak, I. Bratko, A. Jezernik. Application of machine
learning in finite element computation. In R.S. Michalski, I. Bratko, and
M. Kubat (eds.) Machine Learning, Data Mining and Knowledge Discovery:
Methods and Applications, John Wiley and Sons, 1997 (in press).
back to index