Hierarchical tree search tames sprawling truss design spaces

The work, led by Arvan Sedighzadeh, Matteo Torzoni, and Alberto Corigliano, formulates truss design synthesis as a discrete Markov decision process in which grammar-constrained actions generate feasible intermediate layouts and Monte Carlo Tree Search learns an optimal design policy.

The core problem the researchers attack is computational scalability. Monte Carlo Tree Search has already been shown to outperform metaheuristic methods such as genetic algorithms and alternative reinforcement learning approaches including Q-learning and deep Q-learning on truss benchmarks. Yet its computational scalability is limited by the rapid growth of admissible configurations in dense grid design domains. Every additional node in the design domain expands the set of valid truss topologies, causing an exponential growth in the branching of the search tree and, accordingly, substantial computational costs. The team’s answer is a Hierarchical Monte Carlo Tree Search framework, or H-MCTS, in which staged grid refinements focus computational resources on promising regions of the domain, thereby alleviating the curse of dimensionality.

The optimization problem and the initial seed configuration are preserved across all stages, but they are embedded within progressively denser grid-world environments. In this study the authors set the number of refinement levels to four, empirically identified as a suitable compromise between solution quality and computational cost. At each stage, a coarse Monte Carlo Tree Search identifies active, structurally relevant nodes. For each active node, excluding those serving as structural supports, a circular refinement region is defined on the next finer grid, centered at that node with a radius determined from the node spacing. Only inactive nodes located within or on the boundary of these regions are retained; all other inactive nodes are removed. The process repeats across grids, progressively narrowing the design space, and the final candidate solution is selected from all stages based on the lowest achieved design objective.

The objective itself is deliberately practical. The study seeks the truss geometry that minimizes the maximum absolute displacement under static loading, a formulation analogous to compliance minimization in topology optimization, subject to linear elastic equilibrium, prescribed boundary displacements, and a cap on total structural volume. Grammar rules keep every intermediate layout mechanically admissible by enforcing the formation of hinged triangular subassemblies: an allowed action selects an inactive node, selects an element already part of the current layout, and applies one of two legal operators, one that adds the node without removing anything and one that removes the selected element before linking the new node. New connections always avoid intersections with existing elements, so no nonsensical geometry ever enters the search tree.

On a previously studied case with a state space of 1,433,800 valid terminal states, the authors compared two selection strategies within the tree search. The MixMax strategy, which balances average accumulated reward against the best reward seen through each branch, converged to the global optimum within 400 episodes, whereas the alpha-only strategy did not reach it even after 10,000 episodes. In quantitative terms, the MixMax policy achieved an objective ratio of 100 percent, meaning it recovered the global optimum, while requiring an average of 780 finite element evaluations and about 23 seconds. The alpha-only formulation needed 19,570 finite element evaluations and about 320 seconds to attain a lower performance. The authors attribute the earlier difficulty a prior study had in recovering the optimal layout to its exclusive use of the alpha-only selection rule.

The hierarchical extension then delivers on its promise of scalability. Applying single-stage Monte Carlo Tree Search to one case study on a finer 17 by 21 grid reduced the maximum displacement from 0.2621 on the coarse 3 by 9 grid to 0.2426, but the resulting layout lost the symmetry that the symmetric boundary conditions demanded, a symptom of suboptimal exploration in high-resolution design spaces. Attempts to run standard MCTS on finer grids also exhibited premature termination or failure to complete the optimization process. H-MCTS, by contrast, progressed stably through increasingly finer grids up to a resolution of 26 by 21, achieving a 19.1 percent reduction in the maximum absolute displacement compared to single-stage MCTS while cutting computational time by nearly two hours despite the finer grid.

Across further benchmark case studies adapted from prior research using Monte Carlo Tree Search, Q-learning, and deep Q-learning, the hierarchical framework matched the performance of those methods in the first stage and surpassed them in subsequent ones. Case Study C showed the largest improvement, with the best policy at the fourth stage yielding a 27.6 percent reduction in the design objective. For one case the method could not reproduce the first-stage optimum found by exhaustive search, attaining an objective ratio of 92.72 percent, but this still exceeded the 90.44 percent achieved in earlier work, an advantage credited to the MixMax tree policy, and subsequent refinements ultimately produced a 12.6 percent reduction in maximum displacement by the final stage. Computational effort generally rose with each stage as refinements added inactive nodes and enlarged the branching factor, though one low-complexity case bucked the trend, with finite element evaluations falling from 1609 to 415 to 295 across its three stages.

The most visually compelling demonstration involves a bridge-like truss built progressively, mimicking additive manufacturing, with members made of Eurocode IPE 80 steel profiles. Because the seed layout is not pre-connected to the target support node, the structure must grow toward it through sequential assembly while the algorithm balances stiffness gained from new members against added weight. The progressive construction reached the global optimum obtained via exhaustive search. The team then tackled moving loads and localized damage through an offline-online strategy: offline, H-MCTS is executed for several discrete load positions and potential damage scenarios, producing a database of optimized configurations that satisfy yield and buckling constraints; online, new load positions are handled by interpolating between stored topologies while preventing member intersections and ensuring mechanical admissibility. For a 5000-kilogram point mass mimicking a moving vehicle, interpolated designs for unseen intermediate load positions were generated instantaneously, achieving an average accuracy of 88.83 percent relative to full H-MCTS solutions, with the best case reaching 90.87 percent.

The authors model thermal degradation by reducing the Young’s modulus and yield strength of damaged members to 84 gigapascals and 141 megapascals respectively, while acknowledging that this simplified representation omits residual stresses, creep, and geometric distortions. When a damaged member exceeded its critical buckling stress, the algorithm classified it as critical, defined a circular region around it, and reinforced both the damaged member and intact slender elements that would otherwise buckle; one slender member’s capacity rose from 36.56 to 146.24 megapascals after being subdivided into shorter segments. For an exemplary intermediate load case, the adapted design reduced the axial stress in the damaged element by 68.80 percent and lowered the maximum absolute displacement from 2.19 to 1.82 millimeters. Across all critical cases the stress in the damaged element was substantially reduced after optimization, and the paper reports stress reductions in compromised members of up to 72.8 percent.

Future developments may extend the framework to variable member profiles, potentially exploiting reusable component inventories to support circular and resource-efficient design strategies, and to three-dimensional domains where deep learning-assisted policy networks may guide the search in significantly larger action spaces. Additional directions include continuous-domain representations via kernel regression, dynamic loading scenarios, and robustness to uncertainties in material properties and external loads. For now, the study demonstrates on a personal computer with an Intel Core i7 processor and 16 gigabytes of RAM that staged refinement, grammar-constrained actions, and a smarter exploitation rule can together make tree search a credible engine for designing trusses that adapt on the fly to shifting loads and degrading members, all while keeping every intermediate configuration buildable.

Subject of Research: Hierarchical Monte Carlo Tree Search for scalable truss topology optimization with adaptive morphing under moving loads and damage

Article Title: Truss structure optimization via hierarchical tree search

Article References: Sedighzadeh, A., Torzoni, M., & Corigliano, A. (2026). Truss structure optimization via hierarchical tree search. Advanced Modeling and Simulation in Engineering Sciences, 13(1), Article 4. https://doi.org/10.1186/s40323-026-00320-1

Image Credits: AI Generated

DOI: 10.1186/s40323-026-00320-1

Keywords: truss optimization, Monte Carlo Tree Search, hierarchical grid refinement, structural design, reinforcement learning, topology optimization, moving loads, structural damage, finite element analysis, generative grammar, offline-online strategy, computational design synthesis