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UEC Int’l Mini-Conference No.54 33

distance. This process is repeated over genera- objectives throughout the multi-objective opti-
tions to approximate the true Pareto front with mization process.
a set of diverse and well-distributed solutions.

3 Proposed Method
2.3 Topology Optimization

The topology optimization problem considered Building on the theoretical foundations de-
in this study involves optimizing the distribution scribed in the previous section, we now present
of material within a discretized 3D design do- the complete computational framework devel-
main to achieve multiple structural performance oped in this study. The proposed method in-
objectives. The domain is partitioned into vol- tegrates SIMP-based material interpolation, fi-
umetric finite elements, and each element is as- nite element analysis (FEM), and the NSGA-
signed a design variable representing either the II multi-objective evolutionary algorithm into a
presence of material or its relative density. unified and automated optimization pipeline.
By using density as a continuous variable, the As illustrated in Fig. 1, the workflow pro-
resulting optimization problem admits a con- ceeds through a sequence of stages: mesh
tinuous design space, which is favorable for in- initialization, material assignment using the
tegration with both gradient-based and evolu- SIMP method, evaluation of structural re-
tionary algorithms. Structural responses such sponses via FEM, multi-objective optimization
as deformation, stiffness, and vibration charac- using NSGA-II, and post-processing filters to
teristics are computed via finite element anal- ensure manufacturability.
ysis (FEA), and the optimization seeks to find The subsequent subsections describe each
trade-offs among these objectives. In this study, stage in detail, including the formulation of de-
the density-based formulation serves as the foun- sign variables and objective functions, the con-
dation for applying multi-objective evolutionary figuration of the optimization algorithm, and the
optimization. integration of post-processing procedures such
as topology filtering and dynamic analysis.

2.4 SIMP Method for Material Inter-
polation 3.1 SIMP Modeling
The Solid Isotropic Material with Penalization We describe the red segment in Fig. 1. In our
(SIMP) method is a widely used approach in implementation, the SIMP model introduced in
density-based topology optimization for discour- Section 2.4 is applied to a high-resolution tetra-
aging intermediate densities. For each finite el- hedral mesh composed of 1,045 elements. Each
ement, the Young’s modulus E i is interpolated element is assigned a relative density x i ∈ [0, 1],
based on the relative density x i ∈ [0, 1] as fol- which is used to compute a penalized Young’s
lows: modulus via Eq. (2), where E 0 = 210 GPa is the
Young’s modulus of the fully solid material, and
p
E i = x E 0 , (2) the penalization factor is set to p = 3, in ac-
i
cordance with standard practice. The material
3
where E 0 is the Young’s modulus of the fully density is set to ρ = 7,850 kg/m , and the Pois-
solid material, and p is the penalization factor son’s ratio is ν = 0.30. These values correspond
(typically p = 3). This nonlinear relation en- to standard structural steel.
sures that partially solid elements are signifi- The penalized stiffness values E i are assem-
cantly less stiff, promoting binary material dis- bled into the global stiffness matrix K(x), which
tributions in which elements tend to be either is then used to solve the static equilibrium equa-
solid or void. tions and to perform eigenvalue analysis for eval-
In this study, the SIMP interpolation is ap- uating strain energy and the fundamental nat-
plied to a high-resolution tetrahedral mesh and ural frequency. The SIMP model directly in-
serves as the basis for evaluating mechanical fluences two of the three objective functions