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34 UEC Int’l Mini-Conference No.54 SIMP Modeling Post processing
mized are defined as follows:
Start
N
X
Total mass: f 1 (x) = x i V i ρ,
Prepare model and population i=1
- Read mesh (3)
- Intit SIMP (x ? 1, p = 3)
- Evaluate initial FEM 1 T
Strain energy: f 2 (x) = 2 u K(x) u,
(4)
g = 1
Fundamental frequency: f 3 (x) = ω 1 (x), (5)
where V i is the volume of element i, ρ is the
g ? G density of the solid material, u is the global dis-
placement vector obtained by solving the equi-
No
librium equations, K(x) is the global stiffness
Yes
matrix dependent on the material distribution
Evaluate objectives (f1,f2, f3)
Post - processing
- solve FEM x, and ω 1 (x) is the first (fundamental) natural
- Connectivity Filter
g ? g+1 - mass calculate - Thickness Filter frequency of the structure for the given material
- energy calculate
- Dynamic Analysis
- calculate_stiffness layout. Note that the third objective function
- Export VTK / PNG
f 3 , the fundamental frequency, is considered op-
SBX crossover and
mutation tional. The constraints are defined as follows:
End N
X
Selection & Update g 1 (x) = x i V i ≤ V max , (6)
i=1
g 2 (x) = U(x) ≤ U max . (7)
Figure 1: Flowchart of the proposed multi- Here, the total volume of the material used,
objective topology optimization framework, in- P N
tegrating the SIMP modeling, the evolution- i=1 x i V i , must not exceed the specified maxi-
mum V max , and the total displacement or com-
ary multi-objective optimization, and the post-
pliance, represented by U(x), must remain be-
processing
low a given threshold U max .
The proposed procedure execute NSGA-II on
in the optimization: structural compliance and the above problem and acquire non-dominated
dynamic performance, the fundamental natural solutions showing the optimal trade-off between
frequency of the structure, which is evaluated the total mass f 1 , the strain energy f 2 , and the
via eigenvalue analysis of the penalized stiffness fundamental frequency f 3 .
matrix. Higher frequencies indicate greater re-
sistance to vibrational deformation. 3.3 Post processing
We describe the green segment in Fig. 1. The
3.2 Evolutionary Multi-Objective post-processing stage converts each Pareto-
Optimization optimal density field into a manufacturable ge-
ometry. First, a connectivity filter removes any
We describe the yellow segment in Fig. 1. As isolated “islands” of material, ensuring that all
the problem formulation, we define the design remaining elements form a single, load-bearing
variable vector as x = (x 1 , x 2 , . . . , x N ). Each structure. Next, a minimum-thickness filter
element x i ranges over [0, 1] and represents the smooths out narrow ligaments by enforcing a
relative density assigned to the i-th tetrahedral user-defined feature-size threshold, eliminating
element, where x i = 1 indicates solid material fragile beams that would otherwise fail during
and x i = 0 indicates void. The full vector x 3D printing. Evolutionary Multi-Objective Optimization
represents the material distribution across all N After filtering, the fundamental natural fre-
elements. The objective functions to be mini- quency is recomputed via eigenvalue analysis to
