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.

During the development of new vehicles, engineering efforts focus on minimizing injury risks for vulnerable road users and occupants in crash scenarios while maintaining structural integrity. To meet diverse requirements, the nonlinear behavior of passive vehicle safety systems is virtually designed and optimized using numerical Finite-Element (FE) crash simulations. However, due to the complexity of these systems, it is challenging and time-consuming for the engineers involved to understand their behavior. To reduce the time required for assessing crash simulations, we introduce a novel analysis framework that provides flexible data processing and incorporates explainable Artificial Intelligence (AI). The framework allows for examining arbitrary dependencies within parameter-, sensor-, and FE-mesh data by fitting a supervised Machine Learning (ML) model, which is then analyzed using SHapley Additive exPlanations (SHAP). To extend the SHAP methodology to the engineering domain, we introduce System and Difference SHAP values, which facilitate the aggregation of features that describe a system and enable comparisons between two simulations based on input feature contributions in the output space. This allows engineers to intuitively understand contributions to the overall system behavior and generate a data-driven understanding that can be rapidly established. Three industry use-cases from the structural and occupant vehicle safety domain are used to evaluate the framework. The observations achieved demonstrate enhanced and previously unseen insights into the behavior of the crash loaded systems by the effective use of AI within virtual engineering. Comprehensive ablation studies show reproducibility and consistency of the results obtained when using alternative ML models or sensitivity analysis methods.

To assess crashworthiness of vehicles, it is important to consider the inherent variability of physical crash tests in the virtual design, i.e., before the first physical tests, through robustness evaluations. For this, forward uncertainty propagation methods have to be established based on computational models of the vehicle for relevant load cases. In the context of occupant safety, the seating position of an anthropometric test device (ATD) and the seatbelt routing in the initial state are influential uncertain parameters for injury criteria. For this purpose, we propose a two-stage probabilistic modeling approach to incorporate these aleatoric uncertainties, based on test data, into the respective computational models for stochastic crash simulations. The ATD seating position is modeled using linear Gaussian Bayesian networks, and the seatbelt routing is represented by multitask Gaussian process models. This method enables the generation of physically consistent geometric variations that are automatically transferred into adapted finite-element models. The approach is applied in a generic frontal crash scenario with the THOR-50M ATD. The results highlight that variations in seatbelt routing affect chest compressions, while uncertainties in ATD seating position significantly influence lower body loading. The diagonal belt path is identified as the dominant factor for chest injury criteria, whereas the horizontal hip point location and correlated leg posture govern pelvis and femur loads. The findings underline the importance of probabilistic modeling geometric uncertainties in virtual robustness assessments of occupant protection systems.

Topology optimization represents a cutting-edge optimization method that facilitates the design of structures beyond the capabilities of conventional human design approaches. Topology Optimization (TO) identifies highly efficient structures capable of carrying significant loads with minimal material usage. However, most TO studies do not account for impact problems or non-linear plastic constitutive material relationships. This paper introduces the application of the Material Field Series Expansion method in conjunction with the Covariance Matrix Adaptation Evolution Strategy with the potential to optimize structures subject to crash loadings and permanent deformation. Although the results of this work are still preliminary using low impact velocities, the method shows great possibilities to address the crashworthiness problem considerably reducing the computational cost.

High temperatures and structural deformations can compromise the functionality and reliability of new components for mechatronic systems. Therefore, high-fidelity simulations (HFS) are employed during the design process, as they enable a detailed analysis of the thermal and structural behavior of the system. However, such simulations are both computationally expensive and tedious, particularly during iterative optimization procedures. Establishing a parametric reduced order model (pROM) can accelerate the design's optimization if the model can accurately predict the behavior over a wide range of material and geometric properties. However, many existing methods exhibit limitations when applied to wide design ranges. In this work, we introduce the parametric Box Reduction (pBR) method, a matrix interpolation technique that minimizes the non-physical influence of training points due to the large parameter ranges. For this purpose, we define a new interpolation function that computes a local weight for each design variable and integrates them into the global function. Furthermore, we develop an intuitive clustering technique to select the training points for the model, avoiding numerical artifacts from distant points. Additionally, these two strategies do not require normalizing the parameter space and handle every property equally. The effectiveness of the pBR method is validated through two physical applications: structural deformation of a cantilever Timoshenko beam and heat transfer of a power module of a power converter. The results demonstrate that the pBR approach can accurately capture the behavior of mechatronic components across large parameter ranges without sacrificing computational efficiency.

Uncertainty quantification (UQ) of nonlinear frequency response functions (FRFs) is computationally challenging due to high dimensionality and multi-solution responses. This study presents a systematic evaluation of model order reduction via principal component analysis (PCA) within three UQ frameworks combined with harmonic balance method (HBM) solvers: polynomial chaos expansion (PCE) via regression, Kriging, and the Chebyshev interval method (CIM). The proposed PCA-enhanced frameworks (HBM-PCE, HBM-Kriging, and HBM-CIM) are evaluated for computational efficiency, accuracy, and robustness. Two benchmark models are considered: a 2-DOF nonlinear mass-spring-damper system and a geometrically nonlinear structure, each subjected to uniform and Gaussian uncertainties. Results demonstrate that PCA significantly reduces the computational cost of surrogate model construction and uncertainty propagation across all three frameworks, while maintaining errors below 0.1% compared to 1000 Monte Carlo samples. In particular, PCA improves the stability and efficiency of HBM-PCE and HBM-Kriging, and enables HBM-CIM to produce smooth and reliable uncertainty bounds where the conventional CIM fails for strongly nonlinear FRFs. Overall, PCA provides a unified and effective strategy for enhancing both probabilistic and possibilistic UQ of nonlinear FRFs.

Uncertainty quantification (UQ) of nonlinear frequency response functions (FRFs) remains challenging due to multi-solution behavior and resonance frequency shifts. Existing surrogate modeling and model order reduction (MOR) techniques often fail to efficiently capture these nonlinear and frequency-shifted characteristics. To address this gap, we propose a novel data-driven framework that combines spline interpolation with frequency scaling to construct consistent snapshot matrices, applies proper orthogonal decomposition (POD) for dimensionality reduction, and integrates polynomial chaos Kriging (PCK) within an adaptive sampling strategy. The approach enables the efficient construction of surrogate models that accurately approximate nonlinear FRFs across uncertain parameter variations. The methodology is applied to two representative nonlinear systems, a beam with spring nonlinearity and a gear model with backlash nonlinearity, under five and six uncertain parameters with high coefficients of variation. Validation against 5000 Monte Carlo simulations demonstrates excellent accuracy in estimating P5 and P95 percentiles of the nonlinear FRFs. Moreover, generating 105 surrogate-based samples incurs negligible computational cost, highlighting the method's suitability for robust design optimization and reliability analysis in complex dynamical systems such as gear transmissions and rotor systems.

Reliable finite element simulation of orthotropic-dependent failure mechanisms is crucial for understanding the mechanical behavior and optimizing engineered composites and fiber-based materials. Such materials behave brittle under tension and strongly depend on the orthotropic material orientation. Existing non-local models can reproduce brittle fracture for isotropic materials but, in most cases, they are based on the equivalent strain concept for damage initiation, which is unsuitable for orthotropic materials. This contribution introduces a stress-based non-local damage model enhanced with an implicit gradient formulation of the failure criteria. A localizing non-local length is assumed to avoid any pathological broadening of the damage band. The methodology introduces direction-dependent damage variables driven by non-local stress-based damage criteria and can thus distinguish different failure modes. The verification and validation are shown on numerical and experimental benchmark examples. The implicit gradient-based non-local damage approach allows mesh-independent results. Furthermore, it does not require a priori known crack paths and makes it possible to simulate complex failure modes. Perspectively, its effective implementation in the commercial software Abaqus and combination with other constitutive laws, e.g. to account for plasticity or moisture, make it an attractive tool for describing the mechanical material behavior of orthotropic materials, such as wood and fiber-composites.

This paper presents a novel non-intrusive hierarchical multi-fidelity harmonic balance method (HBM) framework for efficient uncertainty quantification (UQ) of nonlinear frequency response functions (FRFs) with application to gear transmission dynamics. The framework uniquely combines proper orthogonal decomposition (POD) to reduce the dimensionality of HBM-derived Fourier coefficients, enable efficient surrogate modeling in latent space, and retain access to both time-and frequency-domain responses. A key innovation is aligning low-and high-fidelity Fourier coefficient latent spaces via Procrustes analysis to improve the construction of a multi-fidelity surrogate using hierarchical Kriging with polynomial chaos Kriging (PCK) as the trend function. Further, computational efficiency in low-fidelity evaluations is achieved by integrating POD with linear regression for rapid compliance matrix estimation in loaded tooth contact analysis (LTCA). Numerical results demonstrate significant computational savings for uncertain FRF predictions of 5000 Monte Carlo (MC) samples. The proposed novelties represent the first application of hierarchical Kriging across aligned Fourier latent spaces and accelerated LTCA within a scalable framework for moderate-dimensional, vector-valued FRF uncertainty analysis, supporting broad applicability to nonlinear dynamical systems modeled with Frequency domain solvers.

This study proposes a surrogate-based approach for estimating the Time-Varying Mesh Stiffness (TVMS) of gear pairs. TVMS is a significant source of parametric-induced vibrations in the gear transmission dynamic simulation. Traditionally, TVMS is estimated using computationally intensive finite-element-based methods like Loaded-Tooth Contact Analysis (LTCA). In a gear geometry dimensioning step, macro-geometry parameters such as module, pressure angle, and helix angle are varied within certain intervals to identify feasible gear designs. Then, LTCA calculates the TVMS for valid designs under the variation of Torque and micro-geometry parameters such as profile crowning. The TVMS response is discretized into 300 rolling positions to capture the nonlinear behavior of the gear meshing cycle. Two models are proposed: (1) ModelNN, a Fully Connected Neural Network (FCNN) that predicts TVMS of all rolling positions using the roll angle as input feature, and (2) ModelAE-NN, where an autoencoder reduces the 300-dimensional TVMS response to 5–20 low-dimensional features, which are then estimated using FCNN. Both models are validated on 200 unseen test samples, achieving R2 (coefficient of determination) values over 0.99. However, ModelAE-NN is shown to be more robust, as dimensionality reduction reduces overfitting and enhances generalization. The models accelerate TVMS estimation by 24,000 times compared to LTCA. Global sensitivity analysis shows that torque is the dominant factor influencing the nonlinear gear mesh behavior. Overall, the study demonstrates the effectiveness of surrogate modeling in predicting TVMS.

This study introduces a novel methodology for vehicle development under crashworthiness constraints. We propose coupling the solution space method (SSM) with active learning reliability (ALR) to map global requirements, i.e., safety requirements on the whole vehicle, to the design parameters associated with a component. To this purpose, we use a classifier to distinguish between the design that fulfills the requirements, the safe domain, and those that do not, the failure domain. This classifier is trained on finite element simulations, exploiting the learning strategies used by ALR to efficiently and precisely identify the border between the two domains and the information provided on these domains by the SSM. We then provide an exemplary application where the efficiency of the method is shown: the safe domain is identified with 270 samples and an average total error of 2.5%. The methodology we propose here is an efficient method to identify safe designs at a comparatively low computational budget. To the best of our knowledge, there is currently no methodology available that can identify regions in the design space that result in designs satisfying the local requirements set by the SSM due to the complexity and strong nonlinearity of crashworthiness simulations. The proposed coupling exploits the information of SSM and the capabilities of ALR to provide a fast mapping between the global requirements and the design parameters, which can, in turn, be made available to the designers to inexpensively evaluate the crashworthiness of new shapes and component features.

In turbomachinery axial compressor development, detailed multi-stage 3D blade optimizations enable better designs than their common single-stage counterparts. However, this design problem entails a prohibitive computational effort for industrial application. It is very high-dimensional and highly constrained and involves expensive aerodynamic and structural design evaluations. Bayesian optimization (BO) is well suited for the two latter characteristics but suffers from the curse of dimensionality. In contrast to previous approaches, we exploit the problem’s multi-component structure to overcome this challenge. We propose to decompose the overall optimization task into lower-dimensional component subproblems. Component interactions are fully taken into account by a sequential cooperative procedure, named cooperative components BO (CC-BO). This enormously facilitates the BO without the need to modify the overall system evaluations. Additionally, we propose a variant with random subproblem decomposition. We analyze the working mechanisms of informed and random CC-BO and compare their performance to standard BO and state-of-the-art algorithms on two problems: a 100D multi-component Branin function and a 223D 4-stage aero-structural compressor blade design. Both proposed approaches significantly enhance the originally poorly performing BO and largely outperform the compared algorithms. The informed version shows an additional advantage for the well-known multi-component structure of the analytical problem. The increased flexibility of random CC-BO makes it the best choice for an efficient global optimization of the multi-stage blade design. It can be readily applied to other high-dimensional constrained optimization tasks.

Bayesian optimization is a powerful tool for solving real-world optimization tasks under tight evaluation budgets, making it well-suited for applications involving costly simulations or experiments. However, many of these tasks are also characterized by the presence of expensive constraints whose analytical formulation is unknown and often defined in high-dimensional spaces where feasible regions are small, irregular, and difficult to identify. In such cases, a substantial portion of the optimization budget may be spent just trying to locate the first feasible solution, limiting the effectiveness of existing methods. In this work, we present a Feasibility-Driven Trust Region Bayesian Optimization (FuRBO) algorithm. FuRBO iteratively defines a trust region from which the next candidate solution is selected, using information from both the objective and constraint surrogate models. Our adaptive strategy allows the trust region to shift and resize significantly between iterations, enabling the optimizer to rapidly refocus its search and consistently accelerate the discovery of feasible and good-quality solutions. We empirically demonstrate the effectiveness of FuRBO through extensive testing on the full BBOB-constrained COCO benchmark suite and other physics-inspired benchmarks, comparing it against state-of-the-art baselines for constrained black-box optimization across varying levels of constraint severity and problem dimensionalities ranging from 2 to 60.

Uncertainty quantification (UQ) in gear dynamic simulations is often faced by the curse of dimensionality, which requires many full-order model (FOM) evaluations. This paper introduces a novel framework that sequentially integrates model order reduction (MOR) and uncertainty propagation to overcome the limitations of HBM-PCE methods. A key limitation of existing methods, such as non-intrusive HBM-PCE, is that they require separate surrogate models for each response variable (displacement, velocity, acceleration) and every frequency point, leading to high computational costs, especially for real-world, high-frequency, multi-degree-of-freedom applications. The proposed approach is demonstrated on a multi-degree-of-freedom (DOF) gear system with ten uncertain macro-geometry, micro-geometry, and torque parameters. First, component mode synthesis (CMS) reduces the system from 252 DOFs to 76. Then, the harmonic balance method (HBM) solves the reduced system for 100 training samples. Fourier coefficients are compressed via an autoencoder, and a surrogate model maps uncertain inputs to latent coordinates. A comparative study of polynomial chaos expansion (PCE), Kriging, and polynomial chaos Kriging confirms the efficiency of the surrogate models in capturing time and frequency-domain displacement, velocity, and acceleration responses. Validated against 6,000 FOM samples, the method achieves high accuracy with <0.6% error. The proposed method requires only as many surrogate models as the number of latent dimensions and allows uncertainty analysis of displacement, velocity, and acceleration to be completed in a single workflow. The approach reduces the computational cost by more than two orders of magnitude compared to traditional non-intrusive HBM-PCE methods and enables an efficient and scalable uncertainty analysis of gear transmissions.

The development of machine learning (ML) applications in deep drawing is hindered by limited data availability and the absence of open-access benchmarks for validating novel approaches, including domain generalization over distinct geometries. This paper addresses these challenges by introducing a comprehensive U-shaped dataset tailored to this manufacturing process. Our U-Channel sheet metal (UCSM) dataset combines 90 real-world meshes with an infinite number of synthetic geometry samples generated from four parametric Computer-Aided Design (CAD) models, ensuring extensive geometry variety and data quantity. Additionally, ready-to-use dataset for drawability assessment and segmentation is provided. Leveraging CAD and mesh data sources bridges the gap between sparse data availability and ML requirements. Our analysis demonstrates that the proposed parametric models are geometrically valid, and real-world and synthetic data complement each other effectively, providing robust support for ML model development. While the dataset is confined to U-shaped, thin-walled, deep drawing scenarios, it considerably aids in overcoming data scarcity. Thereby, facilitates the validation and comparison of new geometry-generalizing ML methodologies in this domain. By providing this benchmark dataset, we enhance the comparability and validation of emerging methods for ML advancements in sheet metal forming.

Benchmarking is essential for developing and evaluating black-box optimization algorithms, providing a structured means to analyze their search behavior. Its effectiveness relies on carefully selected problem sets used for evaluation. To date, most established benchmark suites for black-box optimization consist of abstract or synthetic problems that only partially capture the complexities of real-world engineering applications, thereby severely limiting the insights that can be gained for application-oriented optimization scenarios and reducing their practical impact. To close this gap, we propose a new benchmarking suite that addresses it by presenting a curated set of optimization benchmarks rooted in structural mechanics. The current implemented benchmarks are derived from vehicle crashworthiness scenarios, which inherently require the use of gradient-free algorithms due to the non-smooth, highly non-linear nature of the underlying models. Within this paper, the reader will find descriptions of the physical context of each case, the corresponding optimization problem formulations, and clear guidelines on how to employ the suite.

Modeling of subcutaneous adipose tissue (SAT) plays an important role in forensic biomechanics as blunt force trauma represents one of the most common types of injury. To better understand the involved injury mechanisms, a material model is needed that can (i) represent realistic behavior for combined loading scenarios and (ii) consider the microstructure of the SAT. Therefore, a SAT model was developed that consists of two parts for the strain-energy function - a neo-Hookean part representing the adipocytes and a part representing the surrounding reinforced basement membrane, which is modeled via three circular fiber families oriented in the three main planes, resulting in isotropic model behavior. To verify the performance of the model, the analytical and numerical model solution were compared with experimental data under biaxial tension at different stretch ratios (1:1, 1:0.5, 0.5:1) and under simple shear using an objective evaluation method. The material parameters were evaluated by fitting to the data under equibiaxial tension. For the numerical analysis, the model was implemented as a user-defined material in LS-DYNA to simulate the respective experimental setups. The analytical fitting of the model was robust. Using the resulting material parameters, both the analytical and numerical simulation results were able to represent the experimental data under biaxial tension as well as under simple shear quite well. Since the fitting was only performed with data under equibiaxial tension, these findings suggest that the model assumptions are reasonable. Therefore, the model could help to further investigate the injury mechanisms in blunt impacts.

The conventional hollow cylindrical energy absorbing structure continues to face issues due to its relatively heavier weight, resulting in a very high initial peak load during crushing, hence, lowering its overall crushing performance. To address this challenge, this paper presents a novel bio-inspired cylindrical energy absorber by introducing toothed gears to the outer part of the hollow cylindrical structure, thereby, optimising it. The novel toothed gear bio-inspired cylindrical structures (TGBCS) are additively manufactured and made from six different polymer-based materials. These TGBCS are designed to mimic the gear-like profiles and the energy absorbing capabilities in the hind legs of the issus coleoptratus insect. The TGBCS are axially compressed under quasi-static loading condition and their crashworthiness performance are investigated experimentally, numerically and analytically. Composite-like deformation mechanisms are produced by the TGBCS which lead to improved load bearing and energy absorption capacities compared to their conventional types. The results also indicate that the TGBCS made from poly-lactic acid produce the best overall crushing performance in terms of specific energy absorption (SEA), mean crushing load (MCL) and crush load efficiency (CLE). Numerical investigation further reveals that SEA and CLE of TGBCS are approximately 52.54% and 12.80% higher than those of the conventional hollow cylindrical structures, respectively. Also, by correct choice of shape-topological modification of the TGBCS, it is observed that SEA can be significantly improved. Moreover, by using the simplified super folding element theory, the composite-like deformation mechanism of the TGBCS is adopted to formulate the mean crushing load.

We present a novel approach to developing non-intrusive Reduced Order Models (ROMs) for predicting nonlinear, multivalued Frequency Response Functions (FRFs). Such multivalued behavior often arises in nonlinear dynamic systems due to phenomena like softening and hardening effects, where the FRF exhibits multiple amplitude values for the same input frequency. To handle this issue, we introduce a parametric spline interpolation technique that maps both amplitude and frequency onto an auxiliary axis. This parametrization process converts the multivalued FRF into two separate single-valued functions. Using these single-valued functions, we construct consistent snapshot matrices. The spline interpolation serves as a post-processing step for Full-Order Model (FOM) solutions obtained via the Harmonic Balance Method (HBM). An autoencoder then reduces the system's dimensionality to a latent space, while a Polynomial Chaos Kriging (PCK) surrogate models the dynamics in this space. The surrogate maps the input parameters to single-valued frequency and amplitude functions. We validate the proposed approach on a Bernoulli beam with cubic spring nonlinearity and a two-degrees-of-freedom gear model, comparing it against FOM solutions. Results demonstrate that the developed approach efficiently reduces complex nonlinear FRFs without requiring prior system knowledge. This method can accelerate the robust design optimization tasks of large-scale dynamic systems, such as gear transmissions, and facilitates uncertainty propagation through the PCK surrogate.

The design and development of crashworthy vehicles depend increasingly on credible crash simulation results. This credibility requires a careful process of verification and validation (V&V). The former relates to assessing the correctness of the calculation while the latter relates to the appropriateness of the physical and numerical test results. In this context, various sources of uncertainty exist in crash tests as well as in simulation models, which can influence the validation outcomes. Hence, it is essential to study statistical validation approaches for crash applications. In this paper, we elaborate on two important aspects of validation for crash simulation models: quantification of the credibility of simulation models, and consideration of uncertainties in the validation. Limited and noisy experimental data, typical in crash applications, impose a challenge on the investigation of validation approaches. To address these challenges, we use synthetic experimental results in our study. A crash-box model is validated using the V&V20-2009 validation standard. Two case studies are considered. The first case study aims to show the application of the V&V20 approach to a crash problem. The second case is a more complex scenario and aims to explore the impact of different experimental factors, such as experimental noise, hidden uncertainties, and the number of experiments on the validation results. In the end, we examine how underlying assumptions in the validation approach affect the conservativity of the validation results.