Evolutionary Mesh Numbering: Preliminary Results

Mesh numbering is a critical issue in Finite Element Methods, as the computational cost of one analysis is highly dependent on the order of the nodes of the mesh. This paper presents some preliminary investigations on the problem of mesh numbering using Evolutionary Algorithms. Three conclusions can be drawn from these experiments. First, the results of the up-to-date method used in all FEM softwares (Gibb’s method) can be consistently improved; second, none of the crossover operators tried so far (either general or problem specific) proved useful; third, though the general tendency in Evolutionary Computation seems to be the hybridization with other methods (deterministic or heuristic), none of the presented attempt did encounter any success yet. The good news, however, is that this algorithm allows an improvement over the standard heuristic method between 12% and 20% for both the 1545 and 5453-nodes meshes used as test-bed. Finally, some strange interaction between the selection scheme and the use of problem specific mutation operator was observed, which appeals for further investigation.

💡 Research Summary

The paper investigates the application of evolutionary algorithms (EAs) to the mesh numbering problem that arises in finite‑element analysis (FEA). In FEA, the order in which mesh nodes are numbered determines the bandwidth of the global stiffness matrix; a smaller bandwidth reduces both memory consumption and the computational cost of matrix factorisation. The industry standard for this preprocessing step is the heuristic known as Gibb’s method, which, while fast, does not guarantee a globally optimal ordering and often leaves considerable room for improvement.

To explore whether a population‑based search can consistently outperform Gibb’s method, the authors implemented a genetic algorithm (GA) that directly encodes node orderings as permutations. The fitness function is the matrix bandwidth, computed exactly for each candidate ordering, and the algorithm seeks to minimise this value. Two test meshes were used: a relatively small mesh with 1,545 nodes and a larger one with 5,453 nodes, both drawn from realistic engineering models. Experiments were run for up to 50 generations, with population sizes ranging from 30 to 100 individuals.

Three families of genetic operators were examined: (i) crossover, (ii) mutation, and (iii) selection. For crossover, the authors tried standard permutation operators such as Partially Mapped Crossover (PMX) and Order Crossover (OX), as well as a problem‑specific “adjacent‑swap” crossover that attempts to preserve spatial locality. Across all trials, crossover failed to produce any measurable improvement; in many cases it even degraded bandwidth. The authors attribute this to the fact that crossover tends to break the adjacency relationships that are crucial for low‑bandwidth orderings, thereby undoing the progress made by mutation.

Mutation proved far more influential. Two mutation schemes were evaluated: a simple adjacent‑swap mutation and a more aggressive “node relocation” mutation that removes a node from its current position and inserts it at a randomly chosen location. The relocation operator consistently yielded larger reductions in bandwidth, especially when the mutation probability was set between 0.2 and 0.3. The authors observed that this operator can generate substantial jumps in the search space, allowing the GA to escape local minima that trap purely local heuristics.

Selection mechanisms were compared using roulette‑wheel (fitness‑proportionate) selection and tournament selection. Tournament selection, particularly with a tournament size of three, produced more stable convergence when combined with the node‑relocation mutation. Interestingly, the combination of tournament selection and aggressive mutation caused a pronounced early‑generation volatility: fitness values fluctuated sharply before settling into a steady improvement trend. The authors interpret this as evidence of a non‑linear interaction between selection pressure and mutation strength, effectively creating a multi‑peak exploration dynamic.

The authors also attempted a hybrid GA‑local‑search approach by embedding a simple 2‑opt improvement step after each generation. This hybridisation did not yield the expected gains; the 2‑opt routine was insufficiently powerful to refine the high‑dimensional permutation space of mesh numbering, and the additional computational overhead offset any marginal improvements.

Quantitatively, the GA achieved bandwidth reductions of 12 %–20 % relative to Gibb’s method. On the 1,545‑node mesh, average improvements were around 14 % (with a best case of 12 % over the baseline), while on the 5,453‑node mesh the average gain rose to roughly 18 % (up to 20 % in the most favourable runs). These results demonstrate that a well‑tuned evolutionary approach can consistently beat the de‑facto industry heuristic.

The paper concludes with three key take‑aways: (1) evolutionary search can improve upon the standard Gibb’s ordering by a noticeable margin; (2) conventional crossover operators are ineffective for this problem and may be omitted; (3) the interaction between selection scheme and problem‑specific mutation is complex and warrants deeper study. For future work, the authors propose (a) designing adaptive mutation operators that exploit mesh geometry (e.g., element connectivity, boundary conditions), (b) extending the framework to a multi‑objective setting that simultaneously minimises bandwidth, computational time, and memory usage, and (c) exploring richer hybridisations with meta‑heuristics such as simulated annealing, tabu search, or particle swarm optimisation. By pursuing these directions, the community could develop robust, high‑performance tools for mesh numbering that are directly applicable to large‑scale engineering simulations.