Local constraints are a classical challenge in topology optimization: they require the enforcement of numerous constraints, which leads to prohibitive computational costs when standard optimization strategies are employed. This work investigates an aggregation-free approach requiring minimal tuning, based on the Simultaneous Analysis and Design (SAND) reformulation of the problem. The approach is applied using the Null Space Optimizer as an optimization solver, relying on sparse quadratic programming solvers for the identification of the descent directions. When using the Null Space Optimizer, there is a quantifiable relationship between the SAND and the more classically used Nested Analysis and Design (NAND) formulation, through the considered metric which identifies descent directions from the sensitivities. In the SAND formulation, this metric introduces a dependence between optimization variables related to the shape and the state of the design and must therefore be set appropriately. The method is tested on the design of a minimal volume 2D heat sink where the maximal temperature is constrained. The method is able to successfully enforce local constraints with an accuracy that is not achieved by the aggregation method, which uses a p-norm approximation of the maximal temperature. We further investigate the influence of various factors on the optimum, such as mesh resolution, quadratic programming solver and inner product.
Tracking eigenmodes associated with multiple eigenvalues and optimizing such eigenvalues is challenging. This work presents a group-theoretic framework that integrates results from group representation theory with an effective eigenmode classification algorithm. By systematically assigning eigenmodes to irreducible representations of the underlying group, the proposed method enables robust tracking of both simple and multiple eigenmodes. A partition strategy is also introduced to distinguish group-theoretic multiple eigenvalues from convergent eigenvalues, ensuring reliable mode classification and accurate clustering. The framework considers cluster mean maximization within a bound formulation, naturally accommodating eigenmode crossings and mode order switches while requiring only a limited number of eigenpairs to be computed. Numerical studies on truss structures with all common point-group symmetries—including dihedral, tetrahedral, octahedral, and icosahedral groups—demonstrate accurate eigenmode tracking and consistent optimization performance.
Data-driven deep learning models offer rapid prediction capabilities for nonlinear mechanical responses but require extensive training data and exhibit limited physical interpretability. This study proposes a method for predicting dynamic responses based on an enhanced physics-informed gated recurrent unit (EPIGRU) neural network. By incorporating the equation of motion, Newmark kinematic assumptions, and nonlinear frictional constitutive equations into a physics-based loss function to guide the training of the GRU network, the proposed method achieves reliable prediction of the dynamic responses of friction pendulums with nonlinear friction characteristics. Additionally, a physics-consistent variable-length loss computation method is presented to effectively compute physical loss for variable-length seismic sequences. Bayesian optimization employs dual strategies of a unified baseline and independent tuning to ensure performance gains stem from physical constraints rather than hyperparameter advantages. Case studies demonstrate that EPIGRU outperforms conventional GRU and PIGRU across different data splits, reducing data-driven loss by 69.62
The skeleton structures made of thin-walled hollow members, due to their high strength-to-weight ratios, serve as critical components to enhance the overall performance of engineering structures. An explicit topology optimization approach for hollow skeleton structures is proposed. Based on the Lagrangian description, the approach parametrically represents the geometry of hollow thin-walled components and introduces bead joints to establish inter-component connections. Concurrent size, shape, and topology optimization of hollow skeleton structures is achieved by varying the geometric parameters of components and bead joints. The analytical shape sensitivities for the hollow skeleton topology optimization problem are derived based on the shape derivative method. By employing shell elements for the body-fitted mesh discretization of thin-walled hollow skeletons, the approach overcomes the imprecise control of wall thickness inherent in the traditional Eulerian description. Additionally, it circumvents the bottleneck in computational efficiency caused by solid element discretization. Two types of numerical examples, namely static optimization and nonlinear dynamic optimization under impact loads, verify the effectiveness of the proposed algorithm. These examples demonstrate that the obtained optimization results feature clear boundaries and can be directly imported into CAD/CAE systems for subsequent applications.
Reliability-based design optimization (RBDO) remains challenging in engineering applications due to its prohibitive computational cost. This paper proposes a constraints-shifting sequential method (CSS) to efficiently decouple RBDO into a series of reliability assessment and optimization processes. In CSS, the constraints are iteratively shifted to reduce the feasible region by the corresponding shifting vectors. The computation of shifting vectors avoids the inverse most probable point (MPP) solving process, as they are determined directly from samples near the real limit state functions (LSFs) and the standard deviations of the input random variables. In each cycle, Genetic Algorithm (GA) is used to search for the optimal solution in reduced feasible region, and the failure probabilities of probabilistic constraints are estimated at current optimal solution. The shifting operations proceed iteratively on unsatisfied probabilistic constraints until all constraints are satisfied and the RBDO solution is reached. To alleviate the computational burden, polynomial chaos expansion (PCE) is used to approximate the objective function and LSFs. A local variation function (LVF) is developed to refine the PCE models of LSFs, while the PCE model of objective is updated by the historical optimal solutions. The performance of the proposed CSS is demonstrated through four examples of varying complexity, including a high‑dimensional problem and a real‑world engineering application, where its accuracy and computational efficiency are compared against existing RBDO methods.
Parameter-free shape optimization, i.e., parameterizing the nodal coordinates of a finite element discretization, offers many advantages over traditional shape optimization techniques, e.g., increased design freedom and ease of implementation. A disadvantage of this method is the difficulty of imposing geometric design requirements. Here, we propose differentiable constraints used to enforce minimum feature size, preserve required CAD-like features, and reduce the dimensionality of the optimized design, e.g., to 2 1/2 dimensions. This paper also considers practical improvements to our recent parameter-free optimization algorithm. Shape optimization problems are solved to demonstrate various combinations of design constraints, gain insight into appropriate problem parameter choices, and demonstrate scalability of the methods.
Phase-field topology optimization can naturally describe diffuse interfaces, boundary migration, and topological evolution, but its application to prescribed-volume compliance minimization is still affected by pseudo-time parameter sensitivity, volume-fraction drift, and mesh-dependent filtering scales. This paper proposes SI-AC-PFTO, a semi-implicit phase-field topology optimization framework with strict volume-fraction control and scale-consistent filtering for fixed-volume linear-elastic compliance-minimization problems. The framework combines semi-implicit Allen–Cahn evolution, bisection-based phase-field volume projection, physical-length-based filtering, and a relaxed x– ϕ transfer within a unified iterative workflow, so that phase-field evolution, volume feasibility, length-scale control, and analysis-density transfer are treated in a coordinated manner. Numerical validation is mainly conducted on fixed-parameter two-dimensional benchmarks. In the 180× 120 cantilever example, SI-AC-PFTO obtains a compliance of 42.63, a gray ratio of 0.1172, strict phase-field volume feasibility with |v_ϕ -v^*|≤ 10^-12 , and an analysis-density volume residual of 5.738× 10^-13 . Compared with classical PFM-TO and a non-projected SIMP+OC baseline, the proposed framework achieves lower compliance and lower residual intermediate density under the adopted stopping protocol. Compared with SIMP+OC+H calibrated by the standard non-discreteness metric M_nd , it gives comparable structural quality at a similar final discretization level, although no general runtime advantage is claimed. Cross-mesh, initial-condition, baseline-comparison, and mechanism-ablation studies show that the observed stability and intermediate-density control result from the combined action of semi-implicit evolution, strict phase-field volume projection, relaxed density transfer, and scale-consistent filtering. Additional two-dimensional examples and preliminary three-dimensional tests provide scoped feasibility evidence within the tested settings. Overall, SI-AC-PFTO offers a reproducible and mechanism-oriented phase-field implementation pathway for fixed-volume compliance-minimization problems.
The optimization methods based on computational fluid dynamics play an important role in the shape design of autonomous underwater vehicles (AUVs). However, existing frameworks face two major challenges: the iterative process entails remeshing operations and there is a primary reliance on metaheuristic algorithms. This paper describes a Bayesian algorithm coupled simulation-driven optimization framework that can be employed to assist in the shape design of AUVs. First, a computational domain mesh morphing control method, based on a linear transformation and radial basis function interpolation, is constructed to replace remeshing operations. Considering the expensive objective and cheap constraint characteristics of AUV shape optimization problems, a feasibility-weighted expected improvement Bayesian optimization (FWEI-BO) algorithm is proposed. Testing on benchmark problems shows that FWEI-BO has a stronger search capability and lower computational cost. Second, the framework is organized into an integrated and automated workflow. Through the seamless connection of morphing, simulation, and optimization, this workflow autonomously identifies the optimal solution. Finally, the developed framework is applied to the Myring-type AUV shape optimization, namely drag minimization subject to volume constraints. The results show that the drag of the AUV is reduced by 4.5
Topology optimization of reactive acoustic devices using continuous relaxation methods, such as SIMP, frequently yields intermediate (gray) material densities. These intermediate states do not represent the physical interaction between air and a rigid solid, hindering the interpretation and the manufacturing of the optimized designs. This work presents a purely binary topology optimization framework for the design of reactive acoustic devices, in which a multi-frequency sound pressure level minimization problem, subject to volume and perimeter bounds and to local topological constraints, is solved by a Sequence of Integer Linear Programs. Robustness is obtained by an adaptive trust-region strategy fully decoupled from the volume constraint and by relaxing the local topological constraints with continuous slack variables and exact penalty. Exact gradients are provided by a complex-variable adjoint sensitivity analysis of the damped harmonic wave equation. Numerical examples in 2D and 3D produce strictly black-and-white designs with significant broadband attenuation and reveal that the volume bound acts as a budget for wall thickness and reactive inclusion sizing, remaining naturally inactive at the optimum.
The mining electric shovel plays a critical role in stripping and loading operations in open-pit mines, where excavation trajectory planning strongly influences cycle time, excavated volume, and energy consumption per unit excavated volume. However, conventional motion profile-based planning approaches often struggle to achieve a satisfactory trade-off among these competing objectives while satisfying smoothness requirements, particularly jerk continuity. To address this challenge, a multi-objective excavation trajectory planning framework based on the sinusoidal-blended S-curve was proposed. By employing sinusoidal blending to reconstruct the acceleration profile, the sinusoidal-blended S-curve achieves jerk-continuous motion and thus improves overall motion smoothness. Kinematic and dynamic models of the mining electric shovel front-end working assembly were established to evaluate excavation motion and energy usage. A multi-objective optimization problem is formulated by taking cycle time, excavated volume, and energy consumption per unit excavated volume as objectives, and a genetic algorithm was employed to jointly optimize the sinusoidal-blended S-curve trajectory parameters and coordinated hoist–crowd motor speed profiles. On a 1:7 scaled WK55 mining electric shovel prototype, the proposed method is evaluated in simulation under multiple weight ratio settings and compared with several representative trajectory planning methods, showing a favorable overall balance among the considered performance indices. Prototype-level execution results further indicate that the planned trajectories can be implemented on the 1:7 scaled WK55 platform. These results support the feasibility of the proposed SSC-based excavation trajectory planning method.
This work establishes a space–time topology optimization framework that enables morphable structure design—a fundamentally new capability where optimized topology evolves through time. By treating time as an additional spatial dimension, the method transcends traditional static topology optimization to create structures that adapt their configuration during operation. The framework is applied to transient thermal and thermal flow systems. An additional temporal regularization term is used to control topology evolution when morphability must be limited. The model is formulated with density-based, time-dependent design variables. The transient state equations are solved with a finite element formulation, and the sensitivity is calculated by automatic differentiation via the adjoint method using the unified space–time representation. Four cases are analyzed: minimization of temperature spatial gradient for all instants and at the final time step, minimization of temperature temporal variation, and thermal flow optimization targeting outlet temperature minimization. The effects of temporal regularization with varying weights are also investigated to balance performance objectives with manufacturing (or operational) constraints. Results demonstrate that morphable designs achieve different optimized solutions compared to static topology optimization approaches. This methodology enables the design of structures that adapt their topology during operation to maintain optimized performance throughout transient conditions.
With advances in topology optimization and additive manufacturing, complex multi-material structures with monolithic integration have become feasible. However, interfaces between heterogeneous materials are critical performance-sensitive regions, making it essential to incorporate interfacial effects at the structural design stage. Conventional multi-material topology optimization often overlooks the prescribed interfacial effective properties, thus this paper presents a topology optimization framework that jointly accounts for multi-materials and the effective material properties of transition interfaces between the prescribed material pairs. During the optimization process, a density-penalization scheme implicitly identifies interfacial transition regions. Building on this, a generalized multi-material interpolation model that incorporates interfacial properties enables controllable assignment of both base-material and interfacial material properties. In addition, an analytical relation between the identified interface thickness and the control parameters is established, allowing flexible control of the interface thickness. Across a suite of 2D and 3D cases with diverse boundary conditions and objectives, the proposed method produces crisp interfaces with stable convergence and yields clear topology layouts.
Topology optimization for multi-material is a powerful tool for lightweight or complex structures that require more than one material with different properties. This paper presents a new approach, called Logical Operation-based Multi-Material Level Set (LO-MMLS), that extends the multi-material level set method by using Boolean functions, specifically the AND operation, to control material generation at level set intersections with reaction–diffusion equation. Each material phase is a separate definition from the level set functions, derived by combining logic functions, which clearly and precisely define materials and avoid duplication during material regeneration in optimization. This paper also incorporates a new hyperbolic tangent (tanh) function to control intermediate density values and improve design flexibility. Numerous computational examples demonstrate the new LO-MMLS method’s ability to optimize multi-material structures with clearly distributed material regions, improving stiffness performance by up to 9.37
An alternative and cheaper way of designing structures with reduced sensitivity to manufacturing variations and uncertainties in topology optimization is investigated. Established robustness schemes involve optimizing with multiple design realizations, stochastic gradients or complex perturbations approaches. Motivated by the observation that conventional deterministic designs exhibit sharp peaks in their sensitivity fields, meaning high susceptibility to uncertainties, a simple yet computationally efficient remedy is proposed. The proposed method augments the original objective with a smooth maximum of the element-wise physical design sensitivities, directly penalizing excessive local sensitivity and promoting smoother distributions. This indirectly reduces sensitivity to manufacturing variations and uncertainties, at the cost of only one additional adjoint load for compliance minimization and three for general objectives, all using the same factorization as the primal solve. Numerical studies demonstrate that the proposed formulation significantly reduces sensitivity hot spots, improves robustness to manufacturing variations, and resolves hinge localization in compliant mechanism design, all with minimal degradation of the nominal objective. The new approach that has close ties to previously studied first-order second moment approaches is named “sensitivity hot spot penalization".
We present a topology optimization framework for anisotropic elastoplastic structures based on a new deformation plasticity formulation derived directly from Hill’s yield criterion. Conventional incremental elastoplastic approaches in topology optimization, while accurate, are computationally demanding due to their path-dependent nature and the need to store internal variables over multiple load steps. The proposed Hill-based deformation plasticity formulation enables single-step loading and direct computation of the final equilibrium state, thereby eliminating path dependence and substantially reducing computational cost and memory requirements. The formulation is embedded within a density-based topology optimization framework with stiffness maximization as the design objective. Numerical examples demonstrate the effectiveness of the proposed approach, validate the proportional loading assumption, and illustrate its applicability to realistic structural design problems. The results establish the Hill-based deformation plasticity formulation as a computationally efficient and robust alternative to conventional incremental elastoplastic methods.
Optimal elastic structures often exhibit a multiscale nature, combining global material distribution with locally optimized microstructures. In two-dimensional elasticity, theory predicts that rank-3 laminates achieve the optimal mean energy for multiple-load scenarios. However, the practical use of rank-3 laminates is limited by manufacturing complexity. This motivates the search for single-scale alternative microstructures with near-optimal performance. This work investigates a simple class of single-scale periodic microstructures consisting of a star-shaped void within a convex unit cell (triangular, square, or hexagonal). An efficient parameterization of the void geometry in terms of Fourier coefficients is introduced. A gradient-based optimization procedure is developed to tune both the cell shape and the void geometry so as to minimize the mean effective stress energy under prescribed loading scenarios, without any need for an external length-scale parameter. The resulting optimized microstructures achieve energies within 0.5–3.7
Finite element model updating is essential for reliable train–bridge coupled analysis because the bridge model should reproduce both static and dynamic structural characteristics. However, updating based on a single class of responses may lead to inconsistent estimates of structural stiffness and mass. This study proposes a surrogate-assisted static–dynamic synergistic updating framework for a long-span railway suspension bridge. Three temperature-deflection slopes and four modal frequencies are adopted as the static and dynamic updating objectives, respectively. After comparing radial basis function (RBF), Kriging, and support vector machine surrogate models, the RBF model is selected for subsequent analysis. Sobol sensitivity analysis reduces the number of candidate parameters from 11 to 4, and a reduced parameter RBF surrogate model is integrated with NSGA-III and TOPSIS to identify a preferred compromise solution from the Pareto solution set. The proposed framework is evaluated using the Wufengshan Yangtze River Bridge. After updating, the slope errors are reduced to within 2.5
Structural health monitoring (SHM) involves systematic observation of structures to detect potential changes and assess their current condition. The effectiveness of SHM systems strongly depends on the number and spatial distribution of sensors. While sensor configurations can be determined based on engineering experience for small-scale structures, this approach becomes impractical for complex and large-scale structural systems. Therefore, this study proposes an optimization framework for determining the optimal sensor placement (OSP) of uniaxial sensors in two- and three-dimensional structural systems. Unlike conventional OSP approaches, this work makes a substantive contribution to the literature by introducing an automated computational environment that unifies MATLAB and the SAP2000 Open Application Programming Interface (OAPI). Furthermore, it optimizes both sensor numbers and locations based on modal mass participation ratios, eliminating human-induced bias. The Rao-1 algorithm is employed as a representative parameter-free metaheuristic to demonstrate the general applicability and robustness of the proposed framework, rather than to claim superiority of a specific optimization algorithm. In the proposed framework, the design variables of the optimization problem correspond to the sensor locations selected from the available degrees of freedom (DOFs) of the structure. The objective functions are formulated using max-MAC, avg-MAC, and rms-MAC criteria derived from the modal assurance criterion (MAC). The effectiveness of the proposed framework is demonstrated using two benchmark space-frame structures with different complexities (4-story and 20-story models). The results show that the proposed framework can significantly reduce the number of required sensors while maintaining high modal observability. Furthermore, multiple optimal sensor configurations may exist for the same objective value, highlighting the non-uniqueness of the OSP problem and providing flexibility for practical SHM applications.
Existing probabilistic fatigue life prediction approaches using probabilistic stress-life (P-S-N) curves frequently rely on predefined fatigue life distributions, which introduces distributional bias and fails to capture the fatigue life uncertainty. Moreover, they often fail to keep physical consistency and quantify uncertainty across the entire survival-probability interval, especially under limited-sample or censored conditions. These limitations result in either overly conservative or insufficiently safe designs, posing persistent challenges in fatigue-related reliability-based design. To address these challenges, this study proposes a probabilistic fatigue life prediction method using a hierarchical Bayesian physics-informed neural network (HB-PINN). Key contributions include the following: (1) An adaptive uncertainty quantification strategy is proposed for unbiased life-distribution estimation. (2) A physics-guided standard deviation of the predicted fatigue life with credibility-weighted loss ensures accurate fatigue life scatter predictions under limited data. (3) By combining Bayesian inference with physics-constrained composite loss, the HB-PINN learns from failure and runout data to yield predictive fatigue life distributions. Validation on four fatigue datasets and wind turbine cases demonstrates that the proposed method can generate probabilistically accurate and physically consistent P-S-N curves across all reliability levels, enabling designs that satisfy various reliability requirements while avoiding overly conservative results.
In this paper, we optimize the response of elasto-plastic truss structures by tailoring their nonlinear force–displacement behavior to match a prescribed target curve. The resulting inverse design problem is addressed indirectly through a sequence of surrogate sub-problems, each based on a first-order approximation of the nonlinear response. At every iteration, the solution of the current sub-problem is used to update both the design and the response approximation, enabling an iterative progress toward the final design. The force–displacement response is evaluated using a displacement-controlled numerical scheme. We present two formulations for approximating the response: one based on equivalent static displacements and another based on a direct Taylor expansion. Both rely on first-order information computed at intermediate design points. The proposed methodologies are benchmarked against the conventional approach, which solves the reference problem by directly considering its nonlinear response. Our results indicate that the proposed approaches require fewer nonlinear response evaluations to achieve an optimal design. We also observe a stable evolution of design updates due to the use of a trust region. In some cases, our approaches converge to final designs characterized by optimized objective values comparable to those obtained by directly solving the reference problem, but with different structural layouts. This is likely motivated by the presence of multiple local minima with similar performance within the non-convex design space. The code to reproduce one of the examples is available at: https://github.com/hjalgra/ep-truss-ipopt.
