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

Shaoliang Yang; Jun Wang; Yunsheng Wang

cs.CEMar 26, 2026

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

Authors: Shaoliang Yang, Jun Wang, Yunsheng Wang

Organizations: Department of Mechanical Engineering, Santa Clara University, Santa Clara, CA 95053, USA

Abstract

We present a framework in which a large language model (LLM) acts as an online adaptive controller for SIMP topology optimization, replacing conventional fixed-schedule continuation with real-time, state-conditioned parameter decisions. At every $k$ -th iteration, the LLM receives a structured observation $-$ current compliance, grayness index, stagnation counter, checkerboard measure, volume fraction, and budget consumption $-$ and outputs numerical values for the penalization exponent $p$ , projection sharpness $β$ , filter radius $r_{\min}$ , and move limit $δ$ via a Direct Numeric Control interface. A hard grayness gate prevents premature binarization, and a meta-optimization loop uses a second LLM pass to tune the agent's call frequency and gate threshold across runs. We benchmark the agent against four baselines $-$ fixed (no-continuation), standard three-field continuation, an expert heuristic, and a schedule-only ablation $-$ on three 2-D problems (cantilever, MBB beam, L-bracket) at $120\!\times\!60$ resolution and two 3-D problems (cantilever, MBB beam) at $40\!\times\!20\!\times\!10$ resolution, all run for 300 iterations. A standardized 40-iteration sharpening tail is applied from the best valid snapshot so that compliance differences reflect only the exploration phase. The LLM agent achieves the lowest final compliance on every benchmark: $-5.7\%$ to $-18.1\%$ relative to the fixed baseline, with all solutions fully binary. The schedule-only ablation underperforms the fixed baseline on two of three problems, confirming that the LLM's real-time intervention $-$ not the schedule geometry $-$ drives the gain. Code and reproduction scripts will be released upon publication.

Figures & tables

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

Panel of Figure 5 — 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

Panel of Figure 6 — 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

Topology Optimization

Topology optimization finds material layouts that meet performance and manufacturing constraints across engineering systems.

See all 33 papers

Mar 27, 2026

AutoSiMP: Autonomous Topology Optimization from Natural Language via LLM-Driven Problem Configuration and Adaptive Solver Control

We present AutoSiMP, an autonomous pipeline that transforms a natural-language structural problem description into a validated, binary topology without manual configuration. The pipeline comprises five modules: (1) an LLM-based configurator that parses a plain-English prompt into a validated specification of geometry, supports, loads, passive regions, and mesh parameters; (2) a boundary-condition generator producing solver-ready DOF arrays, force vectors, and passive-element masks; (3) a three-field SIMP solver with Heaviside projection and pluggable continuation control; (4) an eight-check structural evaluator (connectivity, compliance, grayness, volume fraction, convergence, plus three informational quality metrics); and (5) a closed-loop retry mechanism. We evaluate on three axes. Configuration accuracy: across 10 diverse problems the configurator produces valid specifications on all cases with a median compliance penalty of

+0.3\%

versus expert ground truth. Controller comparison: on 17 benchmarks with six controllers sharing an identical sharpening tail, the LLM controller achieves the lowest median compliance but

76.5\%

pass rate, while the deterministic schedule achieves

100\%

pass rate at only

+1.5\%

higher compliance. End-to-end reliability: with the schedule controller, all LLM-configured problems pass every quality check on the first attempt

-

no retries needed. Among the systems surveyed in this work (Table 1), AutoSiMP is the first to close the full loop from natural-language problem description to validated structural topology. The complete codebase, all specifications, and an interactive web demo will be released upon journal acceptance.

Shaoliang Yang, Jun Wang, Yunsheng Wang

Santa Clara University, Santa Clara, CA

Topology Optimization Natural Language to Optimization Modeling Structural Engineering Structural Mechanics Language Model-Based Control

Ask PDF

May 22, 2026

Self-Refining Topology Optimization via an LLM-Based Multi-Agent Framework

Topology optimization is a widely used design method that produces optimized material distributions for prescribed objectives and constraints through well-established numerical algorithms. Throughout the workflow, engineers make a series of decisions ranging from setting and adjusting numerical parameters to assessing whether the converged design meets considerations beyond those explicitly included in the optimization problem, such as physical feasibility. These decisions, which draw on domain expertise, interfere with the autonomous design process. To address this difficulty, this study presents TopOptAgents, a multi-agent system for automating not only the design process but also decision-making during the key stages of the topology optimization process. TopOptAgents consists of six LLM-based agents collaborating through iterative self-refinement cycles spanning problem formulation, validation, code generation and execution, and quality assessment of the optimized structure. This process enables error correction and progressive improvement of both the optimization setup and resulting design. The framework is demonstrated on optimization problems selected to cover a range of settings that differ in their literature coverage and numerical characteristics The benefits of iterative self-refinement are found to be particularly pronounced for problem classes where the pretrained language model has limited prior exposure, such as formulations whose literature and open-source implementations are comparatively sparse. In such cases, the proposed framework reliably produces converged designs where a single state-of-the-art LLM struggles, suggesting that self-refinement broadens the range of topology optimization problems that LLM-based automation can reliably address.

Hyunjee Park, Hayoung Chung

Department of Mechanical Engineering, Ulsan National Institute of Science and Technology, 50 UNIST-gil, Ulju-gun, Ulsan, 44919, Republic of Korea

Topology Optimization Multi-Agent LLM Systems Multi-Agent System Optimization LLM Self-Refinement

Ask PDF

Sep 29, 2026

OptiCom : A Unified Framework for State-Conditioned Composition in LLM-Driven Optimization

Large language models (LLMs) are increasingly deployed to solve complex scientific and practical problems via iterative optimization. However, dynamically coordinating diverse search mechanisms as candidate quality, failure modes, and resource budgets evolve remains a critical open challenge. Targeted empirical diagnostics reveal that mechanism effectiveness is highly state-dependent. Motivated by this, we analyze how individual decisions drive final outcomes, decomposing the expected terminal improvement under a shared budget into cumulative decision opportunities minus cumulative selection losses. Guided by this opportunity-loss theoretical foundation, we propose OptiCom, a unified framework that represents LLM-driven optimizers within a shared configuration space: C=(A,Q,O,E,M,S), corresponding to artifact, query, operator, evaluation, memory, and strategy. Operating within this space, a fast LLM-based Optimization Controller dynamically composes immediate mechanisms through structured Action Packages, while a slower Strategy Adapter refines long-term selection preferences, operator weights, and templates based on accumulated trajectory feedback. Comprehensive evaluations across 32 benchmark groups demonstrate the superiority of framework: OptiCom achieves an average Max-score rank of 1.72 among 14 evaluated configurations, securing the top score in 23 groups. Ultimately, these results highlight the broad applicability and high extensibility of OptiCom as a general-purpose paradigm for robust LLM test-time scaling.

Chenxing Wei, Sichen Liu, Lizhao Liu +6

School of Computing and Data Science, The University of Hong Kong · Shenzhen Loop Area Institute · ByteDance +3

Large Language Model-Guided Optimization Test-Time Scaling

Ask PDF

Abstract

Topology Optimization

Explore similar work

AutoSiMP: Autonomous Topology Optimization from Natural Language via LLM-Driven Problem Configuration and Adaptive Solver Control

Self-Refining Topology Optimization via an LLM-Based Multi-Agent Framework

OptiCom : A Unified Framework for State-Conditioned Composition in LLM-Driven Optimization