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32 UEC Int’l Mini-Conference No.54

true Pareto-optimal trade-offs. In response to 2 Methodology
these limitations, the early 2000s saw the in-
tegration of multi-objective evolutionary algo- 2.1 Multi-objective Optimization
rithms into topology optimization, with NSGA- A multi-objective optimization problem is de-
II standing out for its non-dominated sorting fined as follows:
and crowding-distance mechanisms that pre-

serve solution diversity [5–7]. Researchers have Minimize f i (x), i = 1, 2, . . . , m,
since extended these methods to incorporate Subject to g j (x) ≤ 0, j = 1, 2, . . . , k,
dynamic constraints—either by embedding fre- (1)
quency terms directly into the objective func-
tions [11] or by adopting robust formulations where x = (x 1 , x 2 , . . . , x N ) is the design vari-
that account for material and loading uncertain- able vector with N elements; m is the number of
ties [12]. However, most existing methods are objectives and f i is the i-th objective function;
limited to bi-objective formulations and are of- k is the number of constraints and g j is the j-th
ten confined to 2D domains or regular 3D grid constraint function. The constraint violation is
P k
structures. defined as V (x) = j=1 max {0, g j (x)}. A solu-
tion x with V (x) = 0 is called feasible, whereas
one with V (x) > 0 is called infeasible. A feasible
solution x 1 is said to dominate another feasible
solution x 2 if x 1 is no worse than x 2 in all ob-
jectives and strictly better in at least one. A
feasible solution is considered Pareto optimal if
it is not dominated by any other feasible solu-
tion. The set of objective vectors corresponding
to all Pareto optimal solutions forms the Pareto
In contrast, this study proposes a fully front. The goal of multi-objective optimization
integrated topology optimization framework is to approximate the Pareto front with a repre-
that combines SIMP-based material interpola- sentative set of solutions.
tion with NSGA-II on an unstructured three-
dimensional tetrahedral mesh [3]. The method 2.2 Multi-objective Evolutionary Al-
simultaneously optimizes structural mass, strain gorithms
energy, and fundamental natural frequency,
thereby capturing trade-offs between static and Evolutionary optimization is a promising ap-
dynamic performance in a single run using proach for the target problems addressed in this
NSGA-II [5]. Furthermore, the framework in- study, as it can be applied even when both
cludes post-processing filters to ensure manufac- the objective and constraint functions are black-
turability [8–10] (e.g., enforcing minimum fea- box, and it enables the acquisition of an approx-
ture size and structural connectivity), and au- imate set of solutions representing the Pareto
tomated dynamic validation to assess vibration front in a single run [5–7].
performance [11, 12]. This end-to-end pipeline NSGA-II [5] is one of the most widely used
enables lightweight, stiff, and dynamically ro- algorithms in this field. It has been success-
bust structural design with a degree of general- fully applied to a wide range of real-world multi-
ity and automation not achieved in prior work. objective optimization problems [?]. The al-
The effectiveness of the proposed method is ver- gorithm employs fast non-dominated sorting to
ified through a benchmark experiment with a classify individuals in the population into differ-
detailed Pareto-front analysis and 3D visualiza- ent Pareto fronts. A crowding-distance metric
tions to confirm the efficacy of our approach, is used to maintain diversity among solutions
paving the way for the automated design of within each front. Parent and offspring popula-
lightweight, stiff, and dynamically robust com- tions are merged, and the best individuals are
ponents. selected based on their front rank and crowding