Learning rules for finite element mesh design
Contents and links
Introduction to finite element methods
Finite element methods are used extensively by engineers and other modellers
to analyse stresses in physical structures. These structures are represented
quantitatively as finite collections of elements whose deformations can
then be computed using linear algebraic equations.
In order to design a numerical model of a physical structure, the modeller
must decide the appropriate resolution for modelling each component part,
a task requiring considerable expertise. Too fine a mesh will cause unnecessary
computational overheads when running the model, whereas too coarse a mesh
will produce intolerable approximation errors.
We have used ILP to induce, from examples provided by expert modellers,
rules for choosing appropriate resolution values. One advantage of ILP
is that the examples and rules are expressed in predicate logic, so predicates
can be used to describe geometric relations between different elements.
Without such expressiveness, it would be impossible to adequately describe
the structure being modelled.
Learning rules for the number of elements
The resolution of a FE mesh is determined by the number of elements on
each of its edges. The problem of learning rules for determining the resolution
of a FE mesh is therefore, to learn rules that determine the number of
elements on an edge.
The data here is from experiments conducted with Golem as reported in
[Dolsak B. and Muggleton S. (1992)].
The task is to learn rules for the number of elements using the following
information:

The training examples have the form mesh(Edge,Number_of_elements),
where Edge is an edge label (unique for each edge) and Number_of_elements
is the number of elements on the edge denoted by label Edge. The
number of elements on an edge varies from 1 to 17.

The background knowledge describes some of the factors that influence the
resolution of a FE mesh, such as the type of edges, boundary conditions
and loadings, as well as the shape of the structure (relations of neighborhood
etc.).

According to its importance and geometric shape, an edge can belong to
one of the following types: important_long, important,
important_short, not_important, circuit, half_circuit,
quarter_circuit, short_for_hole, long_for_hole,
circuit_hole, half_circuit_hole and quarter_circuit_hole.

With respect to the boundary conditions an edge can be free, one_side_fixed,
two_side_fixed or fixed.

According to the loadings an edge is not_loaded, one_side_loaded,
two_side_loaded or continuously_loaded.

Background knowledge about the shape of a structure includes the symmetric
relations neighbour/2 and opposite/2, as well as the
relation equal/2.
The Golem dataset
The data concerns five structures labelled ``a''  ``e''. The data files
are as used in the original Golem experiments, and are downloadable as
one
compressed TAR file. Within this file, background knowledge files have
a ``.b'' suffix, positive example files have a ``.f'' suffix, and negative
example files have a ``.n'' suffix.
Bibliography
Dolsak B. and Muggleton S. (1992).
The application of Inductive Logic Programming to finite element
mesh design.
In S. Muggleton editor, Inductive Logic
Programming, Academic Press, London.
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