# 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.
 Application domain: Finite element mesh design Further specification: Complete data set Pointers: http://www.gmd.de/ml-archive/general/data/mesh_design Data complexity: 642 positive examples + 3804 background facts Data format: Prolog facts

## 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),

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 post-processed 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 (ten-fold cross-validation). 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

1. 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).

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