Optimization is Not Enough: Why Problem Formulation Deserves Equal Attention

Black-box optimization is increasingly used in engineering design problems where simulation-based evaluations are costly and gradients are unavailable. In this context, the optimization community has largely analyzed algorithm performance in context-free setups, while not enough attention has been devoted to how problem formulation and domain knowledge may affect the optimization outcomes. We address this gap through a case study in the topology optimization of laminated composite structures, formulated as a black-box optimization problem. Specifically, we consider the design of a cantilever beam under a volume constraint, intending to minimize compliance while optimizing both the structural topology and fiber orientations. To assess the impact of problem formulation, we explicitly separate topology and material design variables and compare two strategies: a concurrent approach that optimizes all variables simultaneously without leveraging physical insight, and a sequential approach that optimizes variables of the same nature in stages. Our results show that context-agnostic strategies consistently lead to suboptimal or non-physical designs. In contrast, the sequential strategy yields better-performing and more interpretable solutions. These findings underscore the value of incorporating, when available, domain knowledge into the optimization process and motivate the development of new black-box benchmarks that reward physically informed and context-aware optimization strategies.

Investigating the Interplay of Parameterization and Optimizer in Gradient-Free Topology Optimization: A Cantilever Beam Case Study

Gradient-free black-box optimization (BBO) is widely used in engineering design and provides a flexible framework for topology optimization (TO), enabling the discovery of high-performing structural designs without requiring gradient information from simulations. Yet, its success depends on two key choices: the geometric parameterization defining the search space and the optimizer exploring it. This study investigates this interplay through a compliance minimization problem for a cantilever beam subject to a connectivity constraint. We benchmark three geometric parameterizations, each combined with three representative BBO algorithms: differential evolution, covariance matrix adaptation evolution strategy, and heteroscedastic evolutionary Bayesian optimization, across 10D, 20D, and 50D design spaces. Results reveal that parameterization quality has a stronger influence on optimization performance than optimizer choice: a well-structured parameterization enables robust and competitive performance across algorithms, whereas weaker representations increase optimizer dependency. Overall, this study highlights the dominant role of geometric parameterization in practical BBO-based TO and shows that algorithm performance and selection cannot be fairly assessed without accounting for the induced design space.

Model-free Adaptive Output Feedback Vibration Suppression in a Cantilever Beam

This paper presents a model-free adaptive control approach to suppress vibrations in a cantilevered beam excited by an unknown disturbance. The cantilevered beam under harmonic excitation is modeled using a lumped parameter approach. Based on retrospective cost optimization, a sampled-data adaptive controller is developed to suppress vibrations caused by external disturbances. Both displacement and acceleration measurements are considered for feedback. Since acceleration measurements are more sensitive to spillover, which excites higher frequency modes, a filter is developed to extract key displacement information from the acceleration data and enhance suppression performance. The vibration suppression performance is compared using both displacement and acceleration measurements.

SMT-EX: An Explainable Surrogate Modeling Toolbox for Mixed-Variables Design Exploration

Surrogate models are of high interest for many engineering applications, serving as cheap-to-evaluate time-efficient approximations of black-box functions to help engineers and practitioners make decisions and understand complex systems. As such, the need for explainability methods is rising and many studies have been performed to facilitate knowledge discovery from surrogate models. To respond to these enquiries, this paper introduces SMT-EX, an enhancement of the open-source Python Surrogate Modeling Toolbox (SMT) that integrates explainability techniques into a state-of-the-art surrogate modelling framework. More precisely, SMT-EX includes three key explainability methods: Shapley Additive Explanations, Partial Dependence Plot, and Individual Conditional Expectations. A peculiar explainability dependency of SMT has been developed for such purpose that can be easily activated once the surrogate model is built, offering a user-friendly and efficient tool for swift insight extraction. The effectiveness of SMT-EX is showcased through two test cases. The first case is a 10-variable wing weight problem with purely continuous variables and the second one is a 3-variable mixed-categorical cantilever beam bending problem. Relying on SMT-EX analyses for these problems, we demonstrate its versatility in addressing a diverse range of problem characteristics. SMT-Explainability is freely available on Github: https://github.com/SMTorg/smt-explainability .

Heterogenous Multi-Source Data Fusion Through Input Mapping and Latent Variable Gaussian Process

Artificial intelligence and machine learning frameworks have served as computationally efficient mapping between inputs and outputs for engineering problems. These mappings have enabled optimization and analysis routines that have warranted superior designs, ingenious material systems and optimized manufacturing processes. A common occurrence in such modeling endeavors is the existence of multiple source of data, each differentiated by fidelity, operating conditions, experimental conditions, and more. Data fusion frameworks have opened the possibility of combining such differentiated sources into single unified models, enabling improved accuracy and knowledge transfer. However, these frameworks encounter limitations when the different sources are heterogeneous in nature, i.e., not sharing the same input parameter space. These heterogeneous input scenarios can occur when the domains differentiated by complexity, scale, and fidelity require different parametrizations. Towards addressing this void, a heterogeneous multi-source data fusion framework is proposed based on input mapping calibration (IMC) and latent variable Gaussian process (LVGP). In the first stage, the IMC algorithm is utilized to transform the heterogeneous input parameter spaces into a unified reference parameter space. In the second stage, a multi-source data fusion model enabled by LVGP is leveraged to build a single source-aware surrogate model on the transformed reference space. The proposed framework is demonstrated and analyzed on three engineering case studies (design of cantilever beam, design of ellipsoidal void and modeling properties of Ti6Al4V alloy). The results indicate that the proposed framework provides improved predictive accuracy over a single source model and transformed but source unaware model.

Composite Material Design for Optimized Fracture Toughness Using Machine Learning

This paper investigates the optimization of 2D and 3D composite structures using machine learning (ML) techniques, focusing on fracture toughness and crack propagation in the Double Cantilever Beam (DCB) test. By exploring the intricate relationship between microstructural arrangements and macroscopic properties of composites, the study demonstrates the potential of ML as a powerful tool to expedite the design optimization process, offering notable advantages over traditional finite element analysis. The research encompasses four distinct cases, examining crack propagation and fracture toughness in both 2D and 3D composite models. Through the application of ML algorithms, the study showcases the capability for rapid and accurate exploration of vast design spaces in composite materials. The findings highlight the efficiency of ML in predicting mechanical behaviors with limited training data, paving the way for broader applications in composite design and optimization. This work contributes to advancing the understanding of ML's role in enhancing the efficiency of composite material design processes.

A new approach for solving global optimization and engineering problems based on modified Sea Horse Optimizer

Sea Horse Optimizer (SHO) is a noteworthy metaheuristic algorithm that emulates various intelligent behaviors exhibited by sea horses, encompassing feeding patterns, male reproductive strategies, and intricate movement patterns. To mimic the nuanced locomotion of sea horses, SHO integrates the logarithmic helical equation and Levy flight, effectively incorporating both random movements with substantial step sizes and refined local exploitation. Additionally, the utilization of Brownian motion facilitates a more comprehensive exploration of the search space. This study introduces a robust and high-performance variant of the SHO algorithm named mSHO. The enhancement primarily focuses on bolstering SHO's exploitation capabilities by replacing its original method with an innovative local search strategy encompassing three distinct steps: a neighborhood-based local search, a global non-neighbor-based search, and a method involving circumnavigation of the existing search region. These techniques improve mSHO algorithm's search capabilities, allowing it to navigate the search space and converge toward optimal solutions efficiently. The comprehensive results distinctly establish the supremacy and efficiency of the mSHO method as an exemplary tool for tackling an array of optimization quandaries. The results show that the proposed mSHO algorithm has a total rank of 1 for CEC'2020 test functions. In contrast, the mSHO achieved the best value for the engineering problems, recording a value of 0.012665, 2993.634, 0.01266, 1.724967, 263.8915, 0.032255, 58507.14, 1.339956, and 0.23524 for the pressure vessel design, speed reducer design, tension/compression spring, welded beam design, three-bar truss engineering design, industrial refrigeration system, multi-Product batch plant, cantilever beam problem, multiple disc clutch brake problems, respectively.

High-dimensional mixed-categorical Gaussian processes with application to multidisciplinary design optimization for a green aircraft

Multidisciplinary design optimization (MDO) methods aim at adapting numerical optimization techniques to the design of engineering systems involving multiple disciplines. In this context, a large number of mixed continuous, integer, and categorical variables might arise during the optimization process, and practical applications involve a significant number of design variables. Recently, there has been a growing interest in mixed-categorical metamodels based on Gaussian Process (GP) for Bayesian optimization. In particular, to handle mixed-categorical variables, several existing approaches employ different strategies to build the GP. These strategies either use continuous kernels, such as the continuous relaxation or the Gower distance-based kernels, or direct estimation of the correlation matrix, such as the exponential homoscedastic hypersphere (EHH) or the Homoscedastic Hypersphere (HH) kernel. Although the EHH and HH kernels are shown to be very efficient and lead to accurate GPs, they are based on a large number of hyperparameters. In this paper, we address this issue by constructing mixed-categorical GPs with fewer hyperparameters using Partial Least Squares (PLS) regression. Our goal is to generalize Kriging with PLS, commonly used for continuous inputs, to handle mixed-categorical inputs. The proposed method is implemented in the open-source software SMT and has been efficiently applied to structural and multidisciplinary applications. Our method is used to effectively demonstrate the structural behavior of a cantilever beam and facilitates MDO of a green aircraft, resulting in a 439-kilogram reduction in the amount of fuel consumed during a single aircraft mission.

Approximated Modeling and Optimal Design for a Soft Pneumatic Actuator Considering the Force/Torque and System Controllability

Soft pneumatic actuators (SPAs) are widely employed to drive soft robots. However, their inherent flexibility offers both benefits and challenges. This property reduces their output force/torque and makes them hard to control. This paper introduces a new design method that enhances the actuator's performance and controllability. The complex structure of the soft actuator is simplified by approximating it as a cantilever beam. This allows us to derive a mechanical equation between input pressure to output torque. Additionally, a dynamical model is explored to understand the correlation between the natural frequency and dimensional parameters of the SPA. The design problem is then transformed into an optimization problem, using the mechanical equation as the objective function and the dynamical equation as a constraint. By solving this optimization problem, the optimal dimensional parameters are determined. Prior to fabrication, preliminary tests are conducted using the finite element method. Six prototypes are manufactured to validate the proposed approach. The optimal actuator successfully generates the desired force/torque, while its natural frequency remains within the constrained range. This work highlights the potential of using approximated models and optimization formulation to boost the efficiency and dynamic performance of soft pneumatic actuators.

Tumoral Angiogenic Optimizer: A new bio-inspired based metaheuristic

In this article, we propose a new metaheuristic inspired by the morphogenetic cellular movements of endothelial cells (ECs) that occur during the tumor angiogenesis process. This algorithm starts with a random initial population. In each iteration, the best candidate selected as the tumor, while the other individuals in the population are treated as ECs migrating toward the tumor's direction following a coordinated dynamics through a spatial relationship between tip and follower ECs. This algorithm has an advantage compared to other similar optimization metaheuristics: the model parameters are already configured according to the tumor angiogenesis phenomenon modeling, preventing researchers from initializing them with arbitrary values. Subsequently, the algorithm is compared against well-known benchmark functions, and the results are validated through a comparative study with Particle Swarm Optimization (PSO). The results demonstrate that the algorithm is capable of providing highly competitive outcomes. Furthermore, the proposed algorithm is applied to real-world problems (cantilever beam design, pressure vessel design, tension/compression spring and sustainable explotation renewable resource). The results showed that the proposed algorithm worked effectively in solving constrained optimization problems. The results obtained were compared with several known algorithms.