Consequence-guided clarification: releasing minimum-compliance topology-optimized designs from incomplete specifications

1 Introduction

• Solver convergence cannot establish that an underspecified problem was interpreted as intended, because different loads, supports, or domains can change the material distribution.A solver can satisfy equilibrium and the volume constraint for any of those choices.Source: §1 · p. 1
• Asking every slot or committing to a default ignores that answers vary in how much they change the design.CGC instead bases its decision on the loss of designs it could commit to, not only on each reading’s optimal compliance.Source: §1 · p. 2
• CGC releases a design only when its loss is within 5% at every open reading; otherwise it asks the question whose least-favorable answer leaves the smallest worst-case loss.The method uses no probabilities and evaluates the worst remaining reading.Source: §1 · p. 2
• CGC-S delays some reference optimizations, and the study also evaluates robust designs, numerical checks, and a release-check contribution.The evaluated scope is single-material, isotropic, linear-elastic minimum-compliance reconstructions; stress limits, nonlinear response, manufacturing requirements, and human interaction costs are outside it.Source: §1 · pp. 2–3
• The benchmark comprises 36 tasks reconstructed from underspecified arXiv examples, with each listed reading replayed as intended and answered exactly.The replay simulates an exact answerer, not a human clarification workflow.Source: §1 · p. 2
• 105 questions for CGC versus 188 for ask-all; checking only the default design took 117, and default designs failed another reading in 26 of 36 tasks.The reported counts exclude mandatory questions about load direction, sign, and material budget.Source: §1 · p. 3

2 Clarification as a design decision

• The decision has three steps: enumerate possible complete specifications, compare a candidate with a reference at each, then ask or accept according to the largest relative excess compliance.Sections 3.3 and 3.4 turn the check into question policies.Source: §2 · p. 3

2.1 Slots, readings, and open readings

• A reading assigns admissible values to every ambiguity slot, and answers leave open only readings consistent with those values.The default reading is a baseline for silent-default workflows, not an estimate of the intended specification; the replay considers each admissible reading as intended and returns exact answers.Source: §2.1 · pp. 3–4

2.2 Compliance, feasibility, and computed regret

• Feasibility requires the volume bound, required passive solids, and the cross-radius convention; the best-known reference is computed from feasible designs and can overstate the true optimum.Consequently, a computed loss can understate loss against the optimum, making a passed check optimistic.Source: §2.2 · p. 4

2.3 Why the regret of a design matters

• For every spring-example reading, optimal compliance is 1, yet the design committed at θ = 0.5 has 228.96% excess compliance at θ = 0.2.Comparing optimal objective values alone can therefore miss a large commitment error.Source: §2.3 · p. 4

2.4 Mandatory questions and the meaning of a passed check

• Load direction, sign, and volume fraction are mandatory questions by choice, even when compliance consequences alone would not require asking about them.A reversed single load leaves compliance and the optimal density field unchanged, but its direction matters for uses such as displacements or stresses.Source: §2.4 · pp. 4–5
• A passed check covers only the loss tolerance at every open enumerated reading under the references and feasibility convention used; it does not establish absolute performance, cover omitted readings, or verify manufacturability.An absolute requirement, such as a displacement limit, must be checked separately at every open reading.Source: §2.4 · p. 5

3.1 Design libraries and cross-evaluation

• CGC-S records computed reading bundles and cross-evaluates their designs; its partial-library bound can reject a candidate but cannot establish that no other design covers the readings.The local check can differ from the offline full-table check because an eliminated, uncomputed reading may yield a better reference.Source: §3.1 · p. 5, §3.2 · p. 5, §3.4 · p. 7

3.2 A pointwise bound from a partial library

• A partial library’s worst-case statistic is restricted to its computed designs, so it is not generally a lower bound on minimax loss over a larger design set.In the stated example, two library designs have loss 9, while an additional feasible design covers both readings.Source: §3.2 · p. 5

3.3 Greedy minimax question choice

• For each varying slot, the greedy rule scores the worst answer by the best remaining candidate’s maximum loss, then asks the slot with the smallest score.It is a one-step rule without a claimed optimality or approximation guarantee.Source: §3.3 · p. 6

3.4 CGC-S with an information ledger

• CGC-S seeds a smaller library, postpones missing reference bundles until screening cannot continue or a candidate needs verification, then releases a passing design or asks again.If the check fails and no question remains within the budget, it withholds release; the local check does not guarantee the offline full-table result.Source: §3.4 · p. 7

3.5 Cost and robust alternatives

• CGC-S can save optimization work when answers prune readings, but if all readings remain open it requests all their bundles before release.Asking every slot can identify a reading and require only one bundle; the method trades computation against interaction.Source: §3.5 · pp. 7–8

4.1 Finite elements, filtering, and optimization

• The solver uses SIMP with quadrilateral plane-stress elements in 2D and hexahedral elements in 3D, plus density filtering and projection.Passive solid and void sets are enforced, and volume is constrained over physical density in the active domain.Source: §4.1 · p. 8
• Independent loads are solved separately, while raised-cosine traction patches are normalized to their prescribed resultants; the stored optimizations use Emin/E0 = 10^-6.A separate void-stiffness experiment re-evaluates frozen designs at 10^-9.Source: §4.1 · p. 8

4.2 Verification and reference quality

• Acceptance checks test convergence, equilibrium, and volume but not stationarity or global optimality; projected-gradient residual exceeded 0.1 in 64 of 134 selected references.The difference between two starts measures observed numerical variation, not an error bound.Source: §4.2 · p. 9
• A Frank–Wolfe tangent-plane certificate supplies a lower bound for the SIMP optimum, keeping numerical acceptance, reference quality, and discretization as separate checks.The bound follows from linear stiffness being at least as large as SIMP cubic stiffness for physical densities between zero and one.Source: §4.2 · p. 9

5.1 Controlled ambiguity tracks

• Controlled tracks vary load position, patch width, or load split across 2D and 3D designs, with additional matched controls and an 18-reading three-slot cantilever.The 2D tracks use nine values and two optimization starts per value; the 3D refinement uses one start and tests persistence rather than reference convergence.Source: §5.1 · p. 9

5.2 Frozen literature frame

• The frozen arXiv frame began with 247 papers, of which 185 examples met eligibility criteria and 132 fit the reconstruction scope; Figure 2 separates open-slot categories by dimension.The frame is explicitly not representative of all engineering specifications or published topology-optimization papers, and screening spot checks found record errors.Source: §5.2 · p. 10, Figure 2 · p. 10

5.3 Annotation, adjudication, and reconstruction

• After adjudication, the primary benchmark contained 36 tasks—29 two-dimensional and seven three-dimensional—with 134 readings, based on author judgments about finite reconstruction sets.The solver and reconstruction conventions were standardized rather than reproducing every source method; mandatory alternatives were not exhaustively replayed.Source: §5.3 · p. 11

5.4 Policies, denominators, and evaluation

• Each task was replayed once per admissible reading with exact answers, and question totals exclude mandatory questions unless stated; the policies and ledger were revised using these tasks, with no held-out evaluation.Offline evaluation checks each release against all open readings using exhaustive best-known references unavailable to the policy.Source: §5.4 · p. 11

6.1 The consequence of an ambiguity depends on geometry and load introduction

• Patch-width loss stayed at most about 0.54% in three geometries, while load-center consequences varied sharply with geometry and load-introduction convention.The MBB full cross-table maximum for width-specific designs was 12.4%, versus 2.1% for its single default-width design; the largest center-position loss remains regularization-dependent.Source: §6.1 · pp. 11–12, Figure 3 · p. 12, Figure 4 · p. 12

6.2 Silent defaults often exceed the tolerance within the reading pool

• The silent default failed at least one open reading in 26 tasks and 101 of 134 cases, and failed the selected replay reading in 65 cases.Failure categories include feasibility violations and loads placed on numerically void material; these are diagnostic categories, not established causes.Source: §6.2 · p. 13, Figure 5 · p. 13

6.4 Most releases hold against better references; decisions near the tolerance depend on the mesh

• At τ = 0.05, CGC and CGC-S released all 134 cases without pool-unsafe releases, asking 105 questions versus 188 for ask-all.The comparison excludes mandatory questions; the zero pool-unsafe count is by construction for full CGC and an observed outcome for CGC-S.Source: §6.3 · p. 13, Figure 8 · p. 15, Table 5 · p. 15

6.3 The release check accounts for most of the saving

• Applying the release check only to the default design asked 117 questions without a pool-unsafe release, accounting for 71 of 83 avoided questions.Other computed candidates supplied the remaining 12 avoided questions; default-only checking still required reference values.Source: §6.3 · p. 13

6.4 Most releases hold against better references; decisions near the tolerance depend on the mesh

• CGC-S’s zero pool-unsafe releases at τ = 0.05 were an observed audit outcome, while local and full-table references agreed at 313 of 314 open readings.The benchmark therefore offers only a weak test of the local-reference approximation.Source: §6.4 · pp. 14–15, Figure 8 · p. 15

6.4.1 Additional starts and mesh refinement

• Additional starts changed decisions only in L25, while refinement changed decisions in L18, L21, and L25; better references pushed three L18 releases above τ to 0.0504.Replaying CGC and CGC-S with improved references added one question to each L18 case, for 108 questions total, while releasing in all 134 cases without a pool-unsafe release.Source: §6.4.1 · p. 15, Table 6 · p. 16

6.4.2 Conservative certification and coverage checks

• A certified replay released 37 of 134 cases at τ = 0.05, while CGC-S and ask-all each had one pool-unsafe release at tolerances 0.01 and 0.02.The latter occurred in L11 because a design from an eliminated reading lowered the best-known reference compliance; midpoint and held-out tests did not cover every intermediate value.Source: §6.4.2 · p. 16, Table 7 · p. 16

6.5 Screening saves references when questions prune readings

• 818 to 514 two-start reference bundles: CGC-S reduced requests by 37.2% versus CGC on the primary replay.The reduction was 18.7% with tasks weighted equally and 9.9% without L22; savings came when answers removed readings, while harmless alternatives remained costly.Source: §6.5 · p. 16
• 155 s versus 36 s: median primary-case batch time was higher for CGC-S than ask-all, with shared-processor contention in the measurements.Each avoided question cost about 4.6 extra two-start reference bundles; cross-evaluation and human-answering time were excluded, so the measurements do not establish workflow break-even.Source: §6.5 · p. 16

6.6 Richer reading sets make question order matter

• On the 41-task, 240-reading union set, CGC asked 336 questions and CGC-S 380, versus 531 for ask-all; neither policy made a pool-unsafe release at τ = 0.05.CGC released in 232 cases and CGC-S in 224; both withheld in some tasks with four open slots under budget three.Source: §6.6 · p. 17
• On the richer union set, full CGC asked fewer questions and released more often than either fixed-order policy, but greedy selection lacks a general question-optimality guarantee.A fixed slot order chosen after seeing all replay outcomes could use fewer questions than greedy selection.Source: §6.6 · p. 17
• On the union set, CGC-S requested 1,291 bundles versus 2,686 for CGC, a 51.9% reduction, but asked more questions and withheld more releases.At τ = 0.02, CGC-S made three pool-unsafe releases; the comparison was not at equal decisions.Source: §6.6 · p. 17

7.1 What consequence-guided clarification changes

• The acceptance check supplied 71 of 83 avoided questions when applied to the default design; other computed candidates supplied the remaining 12.It asks whether a design covers every open reading rather than requiring every descriptive detail to be settled; question order added no saving on mostly one-slot tasks.Source: §7.1 · p. 17
• Each avoided question cost CGC-S about 4.6 additional two-start reference bundles versus ask-all, excluding cross-evaluation and human-answering time.The comparison does not establish overall workflow break-even; where answers are cheap and immediate, ask-all is preferable.Source: §7.1 · pp. 17–18

7.2 Coverage and numerical limits

• A passed check covers only enumerated readings; a design can pass throughout its pool yet fail severely just outside it.The finite reading set still needs expert review, and neither minimax computation nor a language-model front end certifies an unrepresented intent.Source: §7.2 · p. 18
• Computed references have no global-optimality certificate; better references moved three L18 releases above the primary tolerance, and mesh refinement changed decisions in three selected tasks.The two-start spread is a heuristic, not an error bound; CGC-S's zero exceedance count at τ = 0.05 did not persist at stricter tolerances.Source: §7.2 · p. 18

7.3 Benchmark scope and provenance

• The benchmark is an arXiv reconstruction set used to develop the policies, with no held-out or prospective test.Reconstruction choices altered individual outcomes, though frequent default failures remained; a set with more graded consequences would test the check more severely.Source: §7.3 · p. 18

7.4 Implications for computational design workflows

• A practical interface could show open readings, candidate designs, proposed questions, and the basis of the acceptance test; scaling remains a constraint.With n readings, full CGC requires n reference bundles and n^2 design–reading evaluations; the benchmark had at most 18 readings per task, and no cheaper screen for larger sets was tested.Source: §7.4 · p. 18
• Within these limits, the check makes design decisions inspectable by showing which enumerated specifications a computed design covers and which unanswered details prevent acceptance.With an exact answerer it can guide whether to ask, compute, or commit; without one, it can still test coverage of listed alternatives.Source: §7.4 · pp. 18–19

8 Conclusion

• 26 of 36 encoded literature tasks had a silent-default design fail the 5% loss tolerance at another reading by excess loss or infeasibility.CGC and CGC-S each asked 105 non-mandatory questions, compared with 188 for ask-all; checking only the default design took 117.Source: §8 · p. 19
• Better references moved three accepted L18 cases above tolerance, and mesh refinement changed decisions in three selected tasks.The lower-bound replay certified far fewer releases; most decisions compare against computed designs without a certificate against the unknown optimum.Source: §8 · p. 19
• Screening reduced requests from 818 to 514, but each avoided question cost about 4.6 additional two-start reference bundles versus ask-all.Cross-evaluation and human-answering time were excluded; the tasks were also used to develop the policies.Source: §8 · p. 19

Data and code availability

• The planned release includes study records and scripts to regenerate reported results from stored optimization records; third-party paper texts and language-model copies are excluded.The data and code are to be released at the stated repository when the preprint is posted, with an archival identifier assigned at release.Source: Data and code availability · p. 19

• Both authors contributed to conceptualization, methodology, validation, data curation, and writing – review & editing.Source: CRediT authorship contribution statement · p. 19
• Shaoliang Yang also contributed software, formal analysis, investigation, visualization, and writing – original draft.Source: CRediT authorship contribution statement · p. 19
• Jun Wang also contributed supervision and funding acquisition.Source: CRediT authorship contribution statement · p. 19

• Language models were used for literature screening, first task encodings, blind re-annotation, and a supplementary front-end experiment.Source: Declaration of generative AI and… · p. 20
• The authors conceived and directed the study, made its protocol and design decisions, collected data, analyzed results, wrote the manuscript, and take responsibility for its conclusions.Source: Declaration of generative AI and… · p. 20
• Claude Code helped implement solver, experiment, and plotting scripts, while Claude Code and OpenAI Codex edited and checked manuscript text under author review.Source: Declaration of generative AI and… · p. 20
• No AI system is an author.Source: Declaration of generative AI and… · p. 20