Large Language Models as Optimization Controllers: Adaptive Continuation for SIMP Topology Optimization

Shaoliang Yang; Jun Wang; Yunsheng Wang

Figure 1: Overall two-level system pipeline. The inner loop executes the SIMP solver for up to

N

main-loop iterations; every

k

th iteration, the LLM agent observes the structured solver state and emits updated solver parameters. The outer meta-optimization loop reflects on completed-run summaries and patches the agent’s own hyperparameters for the next run. A standardized

40

-iteration sharpening tail, applied from the best valid intermediate snapshot, is shared identically across all four continuation controllers, ensuring that all compliance differences at run termination are attributable exclusively to main-loop behavior

Figure 2: Three-field SIMP formulation. Raw design variables

\rho\in[0,1]

pass through a linear density filter of radius

r_{\min}

to produce a smoothed field

\bar{\rho}

, which is then mapped by the Heaviside projection (sharpness

\beta

, threshold

\eta=0.5

) to the physical density

\tilde{\rho}

entering stiffness assembly. The five LLM-controlled parameters (

p

,

\beta

,

r_{\min}

,

\delta

, restart flag) output by the Direct Numeric Control interface are listed in the lower panel with their admissible ranges

Figure 3: LLM agent decision loop at iteration

147

(

49\%

of budget consumed). Left —state observation: the solver’s current state is serialized into thirteen scalar quantities. The compliance

C

measures global structural flexibility (lower is stiffer and better);

C^{*}

is the lowest valid compliance seen so far. The grayness

\mathcal{G}\in[0,1]

quantifies intermediate-density material:

\mathcal{G}=0

is a fully binary (solid/void) design;

\mathcal{G}>0.20

indicates substantial unresolved material that would degrade manufacturability. The stagnation counter counts consecutive iterations without improvement in

C^{*}

; the checkerboard index detects non-physical alternating-density patterns. Centre —reasoning: the model identifies the current budget phase (Stage 2, penalization), notes that the grayness gate is active (

\mathcal{G}=0.224>0.20

), and decides to raise

p

while holding

\beta

to avoid premature binarization. Right —safety rails: the JSON output is clamped to physically valid ranges; the grayness gate hard-caps

\beta\leq 8

while

\mathcal{G}>0.20

;

r_{\min}

is enforced monotonically non-increasing; and restarts are permitted only when a valid best snapshot exists

Stage	Budget	$\boldsymbol{p}$	$\boldsymbol{\beta}$	$\boldsymbol{r_{\min}}$ / $\boldsymbol{\delta}$
1. Exploration	$0$ – $8\%$	$1.0$ – $2.0$	$1.0$	$1.50$ / $0.20$
2. Penalization	$8$ – $50\%$	$2.0$ – $4.5$	$1$ – $4$	$1.35$ / $0.15$
3. Sharpening	$50$ – $75\%$	$4.5$	$4$ – $16$	$1.25$ / $0.08$
4. Converge	$75$ – $100\%$	$4.5$	$32$	$1.20$ / $0.05$
Tail (fixed)	$+40$ iters	$4.5$	$32$	$1.20$ / $0.05$

Table 1: Four-stage advisory parameter schedule embedded in the LLM system prompt. All four continuous parameters are advisory targets, not hard constraints; the agent outputs exact numeric values that may deviate based on the observed solver state. Budget fractions are expressed as a percentage of the total main-loop iteration count

N

Parameter	Symbol	Range	Role
Penalization exp.	$p$	$[1.0,\;5.0]$	SIMP void penalty
Heaviside sharp.	$\beta$	$[1.0,\;64.0]$	Binarization
Filter radius	$r_{\min}$	$[1.1,\;4.0]$	Length scale ( $\downarrow$ )
OC move limit	$\delta$	$[0.03,\;0.40]$	Density step
Restart flag	—	$\{\texttt{T},\texttt{F}\}$	Reload snapshot
Hard gate: $\beta\leq 8.0$ while $\mathcal{G}>0.20$ (not overridable)

Table 2: LLM-controlled solver parameters, admissible ranges, and the hard grayness gate applied by the safety layer before each action is committed to the solver

Figure 4: Meta-optimization outer loop. After each completed comparison run, the meta-optimizer reads the performance summary and submits a second Gemini call that reflects on the relative performance of all five controllers. The model proposes bounded

\Delta

-updates to the primary agent’s own constants—the grayness-gate threshold, API call frequency, and fallback

\beta

-doubling period (Table 3 )—which are bounds-checked and written back to the agent configuration via automated patching. The outer loop repeats for

n_{\text{iters}}

iterations, converging on agent hyperparameters that generalize across problem geometries

Constant	Role	Lower	Upper	Default
GRAYNESS_GATE	$\beta$ -cap threshold	$0.10$	$0.35$	$0.20$
CALL_EVERY	API call period	$3$	$15$	$5$
PENAL_RAMP	Fallback ramp length	$4$	$20$	$12$
BETA_DOUBLE	Fallback $\beta$ period	$5$	$20$	$10$
PHASE.penal	Min iters in Stage 2	$4$	$25$	$22$
PHASE.sharp	Min iters in Stage 3	$4$	$25$	$16$

Table 3: Constants exposed to the meta-optimization outer loop, with hard bounds enforced before write-back. Default values are those obtained after convergence of the joint outer loop across all three 2-D benchmarks; they are used for all primary experiments in Sect. 4

Problem	Preset	Mesh	$\boldsymbol{N}$	$V_{f}$	Primary?
Cantilever	Fast	$60{\times}30$	100	0.40
	Long	$120{\times}60$	300	0.40	$\checkmark$
	Hard	$180{\times}90$	300	0.40
MBB beam	Long	$120{\times}60$	300	0.40	$\checkmark$
L-bracket	Long	$120{\times}60$	300	0.40	$\checkmark$
Cantilever (3-D)	—	$40{\times}20{\times}10$	300	0.40	$\checkmark$

Table 4: Benchmark configurations. The primary results in Sect. 4 use the

120{\times}60

long preset. The fast preset is used during meta-optimization outer-loop iterations. 3-D experiments use AMG-preconditioned conjugate-gradient via pyamg

Problem	Controller	$\bar{C}\pm\sigma$	$\bar{\mathcal{G}}$	vs. Fixed	Time (s)	Best iter
Cantilever	Fixed	$80.978\pm 0.000$	$0.098$	—	384.9	300.0
	Three-field cont.	$76.786\pm 0.000$	$0.000$	$-5.18\%$	337.4	83.0
	Expert heuristic	$76.474\pm 0.014$	$0.000$	$-5.56\%$	348.1	82.4
	Schedule only	$81.330\pm 0.000$	$0.000$	$+0.43\%$	289.0	267.6
	LLM agent	$\mathbf{76.407\pm 0.021}$	$\mathbf{0.000}$	$\mathbf{-5.65\%}$	445.2	167.6
MBB beam	Fixed	$102.281\pm 0.000$	$0.146$	—	425.3	300.0

Table 5: Primary 2-D results: mean

\,\pm\,

std final compliance

C

, grayness

\mathcal{G}

, improvement versus the fixed baseline, wall-clock time, and the mean iteration at which the best valid snapshot entering the sharpening tail was found. Bold : best compliance per problem. All meshes:

120{\times}60

,

V_{f}=0.40

,

N=300

,

n=5

seeds

Figure 5: Final compliance distributions (

n=5

runs) for all five controllers on cantilever (top), MBB beam (middle), and L-bracket (bottom). Each panel plots the mean (filled circle) and

\pm 3\sigma

interval. The LLM agent achieves the lowest mean on every problem with narrow spread. The schedule-only controller underperforms the fixed baseline on cantilever (

+0.43\%

) and MBB beam (

+1.09\%

), confirming that rigid phase-structure adherence without state observation is actively harmful

Figure 6: Final physical density fields

\tilde{\rho}

and element-density histograms for all five controllers (

120{\times}60

,

V_{f}=0.40

, representative run). Top row : cantilever. Middle row : MBB beam. Bottom row : L-bracket. The fixed controller retains substantial gray material (

\mathcal{G}>0

), visible as blurred member boundaries and a non-bimodal histogram. All continuation controllers achieve

\mathcal{G}=0.000

with bimodal histograms concentrated at

\tilde{\rho}\in\{0,1\}

. Column order (left to right): fixed, three-field continuation, expert heuristic, schedule-only, LLM agent

Figure 7: Compliance convergence curves for the cantilever benchmark (

120{\times}60

,

n=5

seeds, shaded band =

\pm 1\sigma

). Left : full 340-iteration trace (300 main loop

+

40 tail). The large compliance spike near iteration

160

in the LLM agent trace (orange) corresponds to aggressive Heaviside sharpening; the subsequent restart recovers compliance well below the heuristic controllers. Right : late-stage zoom (iterations

100

–

350

), confirming that the LLM agent’s final compliance is consistently lower than all alternatives

Figure 8: Grayness convergence

\mathcal{G}(t)

for the cantilever benchmark (

120{\times}60

, representative run). The dashed horizontal line marks the grayness gate threshold (

\mathcal{G}=0.20

). The fixed controller plateaus at

\mathcal{G}\approx 0.10

and never achieves full binarization. The LLM agent crosses the gate threshold around iteration

150

—substantially later than three-field continuation (

\sim 50

) and the expert heuristic (

\sim 60

)—confirming its more patient penalization strategy

Iter.	$p$	$\beta$	$r_{\min}$	$\delta$	Gate	LLM rationale (verbatim)
Stage 1 - Exploration (budget 0–8%)
5	1.2	1.0	1.50	0.20	—	“Stage 1 exploration, slowly increasing penal.”
15	1.6	1.0	1.50	0.20	—	“Increasing penal slightly to encourage feature definition.”
20	1.9	1.0	1.50	0.20	—	“Increasing penal to encourage structural definition.”
Stage 2 - Penalization (budget 8–50%): patient $p$ ramp, $\beta$ held low
25	2.2	1.0	1.50	0.20	—	“Entering Stage 2: increasing penalty to drive grayness reduction.”

Table 6: Representative LLM agent calls during a cantilever run (run 4,

N=300

, final

C=76.381

). restart

=\checkmark

indicates a snapshot reload was requested and executed. The Gate column marks iterations where

\mathcal{G}>0.20

, indicating that the grayness gate was active

Figure 9: Hyperparameter trajectories for all five controllers on the cantilever benchmark (

120{\times}60

, representative run). Four subpanels show

p(t)

,

r_{\min}(t)

,

\delta(t)

, and

\beta(t)

over the full 340-iteration run (300 main

+

40 tail). The LLM agent (orange) maintains

\beta=1

–

4

for the first

\approx\!145

iterations before escalating to

\beta=32

, compared to the schedule-only (purple) and three-field continuation (blue), which reach

\beta\geq 16

by iteration 80

Problem	Controller	$\bar{C}\pm\sigma$	$\bar{\mathcal{G}}$	vs. Fixed	Time (s)	Best iter
Cantilever	Fixed	$4.860\pm 0.000$	$0.213$	—	1044	255
	Three-field cont.	$3.998\pm 0.001$	$0.000$	$-17.74\%$	994	75
	Expert heuristic	$3.999\pm 0.001$	$0.000$	$-17.73\%$	933	94
	Schedule only	$4.030\pm 0.000$	$0.000$	$-17.09\%$	746	68
	LLM agent	$\mathbf{3.981\pm 0.000}$	$\mathbf{0.000}$	$\mathbf{-18.09\%}$	1198	157
MBB beam	Fixed	$6.429\pm 0.000$	$0.260$	—	1014	— $\ast$

Table 7: 3-D results (

40{\times}20{\times}10

,

V_{f}=0.40

,

N=300

,

n=5

seeds). Bold : best compliance per problem. The LLM agent achieves the lowest compliance on both benchmarks. Total improvement from continuation is

\approx\!15

–

18\%

in 3-D versus

5

–

7\%

in 2-D, confirming that problems without an established optimal schedule benefit most from adaptive control

Figure 10: 3-D density projections (X-ray view) for the

40{\times}20{\times}10

cantilever (

n=5

seeds, representative run). Columns: fixed, tail-only, three-field continuation, expert heuristic, schedule-only, LLM agent. Rows:

xy

,

xz

,

yz

projection directions. The fixed controller shows pervasive diffuse shading (

\mathcal{G}=0.213

); all continuation controllers produce binary structures

Figure 11: 3-D density projections (X-ray view) for the

40{\times}20{\times}10

MBB beam (

n=5

seeds, representative run). The LLM agent (rightmost,

C=5.428

) shows a more compact load-carrying core than the heuristic controllers

Problem	Controller	$\bar{C}\pm\sigma$	$\bar{\mathcal{G}}$	vs. Fixed
Cantilever	Fixed	$88.047\pm 0.000$	$0.190$	—
	Tail only	$211.777\pm 0.000$	$0.000$	$+140.5\%$
	Three-field cont.	$79.449\pm 0.000$	$0.000$	$-9.76\%$
	Expert heuristic	$79.486\pm 0.000$	$0.000$	$-9.72\%$
	Schedule only	$83.082\pm 0.000$	$0.000$	$-5.64\%$
	LLM agent	$\mathbf{79.301\pm 0.086}$	$\mathbf{0.000}$	$\mathbf{-9.93\%}$

Table 8: Tail-only ablation (

60{\times}30

,

V_{f}=0.40

,

N=100

,

n=5

seeds). The tail-only controller starts from uniform density (no exploration). Its extreme underperformance confirms that the main-loop exploration is essential. Bold : best compliance per problem

Large Language Models as Optimization Controllers: Adaptive Continuation for SIMP Topology Optimization

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