structural design and optimization

This work addresses the lack of a unified, open-source, and modular platform for collaboratively exploring shape and topology optimization methods in both teaching and research. The authors present an object-oriented, MATLAB-based open-source framework that employs abstract base classes to define core interfaces, enabling seamless integration of parametric and level-set-based shape optimization alongside density-based, level-set, and topological sensitivity approaches to topology optimization. By directly mapping mathematical formulations to executable code, the framework allows users to extend objective functionals or constraints simply by deriving new classes without modifying the core implementation. Highly modular and reproducible, the framework bridges the gap between shape and topology optimization, offering a continuous research pathway. Its effectiveness and flexibility are demonstrated through diverse numerical examples in both educational and research contexts.

This work addresses self-weight-dominated gridshell structures by simultaneously optimizing topological connectivity and surface elevation to ensure all members carry purely compressive or tensile forces, thereby achieving both mechanical efficiency and geometric buildability. Methodologically, self-weight is innovatively modeled as a design-variable-dependent load, enabling—for the first time—the joint convex optimization of topology and geometry, thus overcoming the limitations of conventional sequential design paradigms. The approach integrates parametric surface discretization with the force density method within a second-order cone programming (SOCP) framework, guaranteeing global optimality and computational efficiency. Compared to standard 3D layout optimization, the proposed method achieves speedups of several orders of magnitude while delivering higher accuracy. Furthermore, it uncovers the nonlinear morphological evolution of optimal forms under increasing self-weight, directly generating lightweight, buildable shell geometries.

This work addresses the limitations of black-box optimization in structural design, which often yields suboptimal or physically implausible solutions due to its neglect of problem modeling and domain knowledge. Focusing on the topology optimization of laminated composite structures, the study proposes an explicit decoupling of topological and fiber orientation design variables, combined with a physics-informed sequential optimization strategy. This approach departs from conventional context-agnostic black-box paradigms by leveraging domain-specific insights. Compared to concurrent optimization of all variables, the proposed sequential method significantly improves compliance minimization under volume constraints, yielding superior and physically interpretable designs. The results underscore the critical role of integrating domain knowledge into the optimization process to enhance both performance and solution plausibility.

Traditional mechanical structural optimization relies heavily on expert knowledge and computationally expensive finite element analysis (FEM), while existing machine learning approaches suffer from poor generalizability and dependence on large-scale labeled datasets. Method: We propose the first zero-shot LLM-FEM collaborative framework for autonomous 2D truss generation, multi-objective evaluation, and iterative optimization—requiring no fine-tuning. A general-purpose large language model (GPT-4.1 and its lightweight variant) serves as the natural language reasoning engine, tightly coupled with a physics-based FEM module for structural validation. Convergence is guided via temperature-controlled sampling and multi-objective feedback-driven prompt engineering. Contribution/Results: Compared to NSGA-II, our method drastically reduces FEM evaluations and accelerates convergence. GPT-4.1-mini (temperature=0.5) achieves the highest constraint satisfaction rate and optimization step efficiency. This work establishes, for the first time, the feasibility of LLMs as cross-task general-purpose structural optimizers.

To address the pedagogical need for accessible topology optimization education, this paper introduces PyHexTop—a lightweight, open-source Python implementation for compliance minimization under a volume constraint. Method: PyHexTop discretizes the design domain using regular hexagonal elements, the first such lightweight implementation to do so; this inherently suppresses checkerboard patterns without requiring additional filtering. Built on NumPy and SciPy, it extends the HoneyTop90 framework and integrates the Solid Isotropic Material with Penalization (SIMP) interpolation scheme, analytical sensitivity analysis, and the Optimality Criteria (OC) optimizer. Contributions/Results: (1) Its minimal, highly readable codebase significantly lowers the learning barrier; (2) it robustly generates high-quality, checkerboard-free topologies on benchmark problems (e.g., the MBB beam); (3) the fully documented, open-source code has been validated in undergraduate and graduate courses across multiple universities, demonstrating both pedagogical effectiveness and practical engineering utility.

Addressing the challenge of simultaneously achieving structural performance and manufacturability in two-dimensional topology optimization, this paper proposes a multi-thickness density-based optimization method. The approach integrates hierarchical penalization, smoothed Heaviside projection, parameter continuation, and adaptive mesh refinement to effectively suppress spurious thin features. With only three discrete thickness levels, it closely approximates the performance of continuous thickness-varying designs. The method inherently accommodates both additive manufacturing and conventional machining constraints, significantly enhancing convergence stability and geometric resolution. Benchmark tests on cantilever and MBB beams demonstrate compliance errors below 2% and stiffness values nearly matching those of variable-thickness optimization—substantially outperforming the standard SIMP method in both accuracy and manufacturability.

This work addresses key challenges in multi-material topology optimization—namely, limitations on the number of candidate materials, redundant design spaces, and difficulties in ensuring physically valid material interpolation—by introducing a generalized shape function (gSF) method. The approach employs an n-dimensional linear shape function to map the multi-material simplex domain onto a compact design space, using natural coordinates as design variables to determine material densities. By integrating density filtering with a barycentric-projection strategy, the method rigorously enforces barycentric coordinate properties. For the first time, it establishes a generalized n-linear shape function applicable to arbitrary dimensions, thereby overcoming conventional constraints on material count and enabling a highly scalable optimization framework. The method successfully optimizes 2D and 3D structures—including compliant mechanisms—with up to 24 and 15 materials, respectively, achieving smooth convergence of objective functions and demonstrating its efficiency, generality, and engineering applicability.

Existing topology optimization methods rely on post-processing—such as mesh smoothing and feature extraction—to obtain manufacturable, smooth boundaries, thereby increasing computational cost and workflow complexity. This paper proposes a Gaussian Function Integration (GFI) topology optimization framework: structural geometry is explicitly represented via superposition of anisotropic Gaussian functions, coupled with a Heaviside-type level-set projection for mesh-independent parametric modeling. To the best of our knowledge, this is the first work to introduce explicit Gaussian function integration into topology optimization; the representation inherently ensures curvature continuity and manufacturing compatibility, while enabling flexible control over smoothness, discreteness, and geometric complexity through tunable parameters. Numerical experiments—including 2D/3D stiffness maximization and compliant mechanism design—demonstrate performance competitive with state-of-the-art Moving Morphable Components (MMC) methods, while yielding designs with sharper boundaries and superior geometric consistency—entirely eliminating the need for post-processing.

This study investigates how the choice of norm in compliance-based objective functions influences structural topology in topology optimization. While the form of the objective function is known to significantly affect optimization outcomes, the mechanistic differences among compliance formulations derived from various norms—specifically ℓ², √ℓ², and spectral ℓ¹ norms induced by the stiffness matrix—remain unclear. Through systematic numerical experiments within a finite element framework, the work reveals that the classical quadratic (ℓ²) compliance promotes uniform load paths, whereas the spectral ℓ¹ norm yields sparse and highly localized structural members. These findings demonstrate that, under identical physical constraints, distinct norm-based formulations can produce markedly different optimization landscapes and topological configurations, offering a principled pathway toward tailoring structural performance to specific design requirements.

This study addresses the high computational cost and limited human–computer interaction efficiency of traditional topology optimization, as well as the cumbersome region-selection procedures in existing human-in-the-loop approaches. To overcome these limitations, the authors propose a novel human-in-the-loop optimization framework integrated with artificial intelligence, which employs a U-Net image segmentation model to predict regions where users prefer structural modifications, thereby enabling intelligent guidance of design iterations. Trained on synthetic data, the model demonstrates cross-task generalization and exhibits emergent behavior surpassing that achievable with single-region training data. Experimental results show that, with only a 15-second increase in user interaction time, the method significantly enhances structural manufacturability or improves linear buckling load capacity by up to 39%.