Structural and Multidisciplinary Optimization

Yildirim, Anil; Jacobson, Kevin E.; Anibal, Joshua L.; Stanford, Bret K.; Gray, Justin S.; Mader, Charles A.; Martins, Joaquim R. R. A.; Kennedy, Graeme J.

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03900-0

The design of many engineering systems requires multiphysics simulations and can benefit from design optimization. Two key challenges in multidisciplinary design optimization (MDO) are coupling the models and computing analytic derivatives, which are required to solve optimization problems with many design variables. While existing multiphysics frameworks address the challenge of implementing coupled models, none of them compute analytic derivatives for large-scale simulations in a general way. The OpenMDAO framework computes coupled derivatives using analytic methods, but it lacks suitable interfaces for simulation-based coupled models. To address this gap, we introduce MPhys, a modular multiphysics simulation library built with the OpenMDAO framework. MPhys defines standard disciplinary interfaces for coupled multidisciplinary models, enabling the rapid development of coupled multiphysics models for gradient-based MDO. We demonstrate MPhys’s modularity and extensibility with two example applications: aerostructural design optimization using two different aerodynamic solvers and aeropropulsive design optimization. Since its initial development, MPhys has been successfully used with a wide range of applications with various multidisciplinary coupling strategies and fidelity levels. The MPhys library is poised to significantly accelerate the integration of existing models in multiphysics applications and the development of new multidisciplinary coupling strategies. These developments will enable a wider adoption of MDO in practical engineering design.

Bao, Yajie; Li, Honglin; Zhao, Yujie; Li, Lei; Zhang, Zhenyuan; Tang, Zhonghao

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03955-z

High-dimensional problems are common in designing complex engineering structures, and surrogate models are often used to improve optimization efficiency. However, the higher the dimension is, the lower the accuracy of the surrogate model, resulting in inaccurate optimization results. Therefore, a global‒local collaborative optimization method based on parameter sensitivity analysis is proposed in this work. This method involves decomposing high-dimensional problems into multiple low-dimensional problems. With global‒local collaborative optimization, the modeling cost is effectively reduced, and the modeling efficiency and accuracy are improved. This method is applied to the 3D optimization of a compressor blade and compared with two traditional optimization methods. Because the optimization process fully considers the global and local collaborative effects, the results show that the compressor blade efficiency is improved by 7.2%, and the pressure ratio is improved by 4.03%. In addition, the optimization efficiency is improved by 56.73% while ensuring the accuracy of the surrogate model. Therefore, the proposed method is an effective solution to the problems associated with high-dimensional optimization, and it can be effectively applied to the field of engineering.

Kundu, Rahul Dev; Zhang, Xiaojia Shelly

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03930-8

We propose a multimaterial topology optimization framework for sustainable infrastructure design with substantial mechanical and economical advantages. The framework optimally harnesses the mechanical superiority of steel and the environmentally sustainable properties of biomaterials, such as laminated bamboo and timber, to design stiff, strong, and sustainable structures. The fibrous characteristics of biomaterials are incorporated using the transversely isotropic constitutive relation and Tsai–Wu failure criterion, while steel is assumed isotropic with von Mises yield criterion. Two sustainability-oriented formulations are proposed to accommodate different design scenarios, accounting for performance, environmental impacts, and economic costs. Both formulations are capable of designing optimized steel-biomaterial hybrid structures with significant sustainability improvements. Through 2D and 3D example problems, we demonstrate that the proposed framework effectively leverages the unique advantages of steel and different biomaterials to strive an ideal balance among diverse mechanical, economic, and environmental design requirements. The results indicate that both steel and biomaterials are essential to achieve cost-effective sustainable design solutions with enhanced mechanical performance. Specifically, biomaterials are predominantly used in low or moderately stressed members, while steel is optimally utilized in high-stressed or primary load-bearing members. The proposed framework presents a rational design paradigm for high-performance and sustainable multimaterial engineering structures that can benefit construction industries from both economic and environmental perspectives.

Li, Hao; Jolivet, Pierre; Alexandersen, Joe

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03931-7

Microchannel cooling is often the preferred choice for compact heat sinks. However, widely adopted topology optimisation (TO) techniques such as density-based and level-set methods often struggle to generate very thin channel strips unless maximum length scale constraints are imposed and very fine meshes are employed. To address this limitation, multi-scale design methodologies have emerged. This paper builds upon recent advances in de-homogenisation techniques to contribute to the multi-scale design of microchannels for cooling applications. We start by selecting a single-class microstructure and employ numerical homogenisation to build an offline library. This library is then fed in online macro-scale topology optimisation, where both microstructure parameters and local orientation fields are optimised. By using a sawtooth-function-based mapping, the de-homogenised results capture fine details across different length scales through a unique homogenised design. Our findings show that the generated microchannels outperform conventional pillar arrays, offering valuable insights for heat sink designers. Additionally, imperfections observed in the de-homogenised results serve as benchmarks for future improvements, addressing concerns related to modelling accuracy, manufacturability, and overall performance enhancements.

Røstum, Heine; Gros, Sebastien; Aas-Jakobsen, Ketil

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03951-3

Designing a bridge is a complex endeavour, involving multiple variables, limitations and requirements. The design process often includes high-fidelity analyses that are computationally expensive, and the internal working tools of the analysis software are often unknown. This limits the applicability of conventional numerical optimization, especially due to time constraints. As a mean to reduce the computational burden, surrogate modelling may be applied. Surrogate models are constructed on the basis of observed results from the computationally expensive high-fidelity analyses, and serves as a fast approximation of unobserved regions in the design space. If probabilistic surrogates are applied, the probabilistic element may be exploited in the optimization phase, resulting in a scheme known as Bayesian optimization. In this article, it is described how to derive a constrained Bayesian optimization scheme in the process of bridge design, where both the goal and constraints are approximated using probabilistic surrogates. The article also presents a case study where constrained Bayesian optimization is applied to a three-span post-tensioned concrete girder and the results are compared to conventional surrogate-based optimization. The results from the case study show that the Bayesian optimization schemes converge after about six iterations, significantly faster than the conventional surrogate optimization scheme, with a consistently higher relative improvement—providing a faster and more confident process for surrogate-based optimization.

Zhao, Jingjing; Li, Yuan; Li, Yan; Liu, Jinxiang

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03959-9

In order to improve the tribological performance of camshaft bearings, a design method based on NSGA-II and TOPSIS decision methods was proposed. The structural-performance parameters sample dataset was obtained. The multi-objective optimization genetic algorithm and multi-criteria decision-making method were used to optimize the bearings structure with the goal of minimizing the total friction loss and the maximum wear height, as well as maximizing the average values of minimum oil film thickness. The optimal performance and structural parameters of camshaft bearings obtained through multi-objective optimization strategy have obvious directionality. The entropy weighted TOPSIS multi-criteria decision-making method effectively obtained the optimal solution. Compared with the original structure, the optimized structure significantly reduces the total friction loss and maximum wear height.Graphical abstract[graphic not available: see fulltext]

Li, Zhen; Lu, Zhenzhou

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03923-7

Augmented failure probability (AFP) can measure the safety degree in case of random inputs with random distribution parameters, and solving the AFP updating model on gradually newly available observations can obtain posterior AFP to calibrate safety degree. However, there lack efficient algorithms for solving posterior AFP at present. Thus, an importance sampling (IS) method is proposed for efficiently estimating the posterior AFP when the newly available observations are gradually collected. By minimizing the variance of posterior AFP estimation, the optimal IS density is derived in the proposed IS method. For the operating difficulty resulted from the implicit character of the optimal IS density, the surrogate model of performance function is established for constructing a quasi-optimal IS density to adaptively approach the optimal one. For the sampling difficulty resulted from the irregularity of the quasi-optimal IS density, a new acceptance-rejection strategy is creatively designed, and its correctness is proved analytically. Since the proposed IS method adaptively combines the economic surrogate model with the minimizing estimation variance technique, the efficiency is greatly improved for estimating the posterior AFP. The examples show that the quantitative results of the posterior AFP are consistent with the qualitative ones. And comparing with the existing methods, the proposed IS method can improve the efficiency of sequentially estimating the posterior AFP while ensuring accuracy.

Faghirnejad, Saba; Kontoni, Denise-Penelope N.; Camp, Charles V.; Ghasemi, Mohammad Reza; Mohammadi Khoramabadi, Maryam

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03948-y

Nonlinear pushover analysis involves an extremely iterative process necessary for satisfying the design requirements of performance-based codes. This analysis also demands significant computational resources and advanced scientific efforts. In this study, we introduce a computer-based method for 2D-braced steel buildings that incorporates pushover analysis, optimization techniques, and optimality criteria methods to automatically design the pushover drift performance. An ant colony metaheuristic optimization algorithm is employed to achieve optimal performance-based designs for columns, chevron braces, and beams in steel moment frames. The initial phase includes implementing optimization codes in MATLAB and OpenSees for conducting the nonlinear static analysis of the 2D-braced steel frames. Several optimal configurations are produced for each brace and frame by addressing the nonlinear optimization problem. In the second step, a nonlinear pushover analysis is conducted in accordance with the provisions of the FEMA 356 code. This analysis takes into account constraints on relative displacement and plastic hinge rotation to ensure that the structure achieves the specified performance levels. Finally, the third step involves selecting the optimal design for each frame and subsequently plotting the pushover, drift and convergence curves for each frame and performance levels. This selection process ultimately aims to satisfy the criteria of performance-based design, including life safety, collapse prevention, and immediate occupancy, while minimizing the total weight for three 2D steel chevron frames: a 5-story, a 9-story, and a 13-story configuration.

Zhao, Yue; Liu, Jie

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03958-w

Inverse problems involving high-dimensional uncertain variables may encounter the challenge of the ‘curse of dimensionality’ during the solution process. In addition, the current uncertainty inverse algorithms may encounter insufficient prior information on the parameters to be inversely solved, leading to difficulty in determining the search interval for inverse problem modeling and optimization. In light of this, this study introduces an inverse solution algorithm that considers model uncertainty based on a high-dimensional model representation (HDMR) approach. The algorithm utilizes the HDMR method to construct models of system input parameters and uncertain model parameters. Subsequently, by enhancing the traditional CV-Voronoi sequential sampling method, the algorithm ensures that the sequential sampling points in the modeling process satisfy both model prediction accuracy and the distribution forms of uncertain variables. Finally, through a stepwise modeling process based on the HDMR, efficient modeling and solution are achieved in scenarios where prior information on the parameters to be determined inversely is insufficient. The effectiveness of the proposed method is validated through several numerical and engineering examples. The method presented in this study provides an effective tool for solving inverse problems with high-dimensional model uncertainty in the field of structural design.

Zhou, Zitong; Ma, Xiangtao; Zhou, Yan; Sun, Yu; Hao, Peng; Wang, Bo

2025 Structural and Multidisciplinary Optimization

doi: 10.1007/s00158-024-03942-4

The new generation of aerospace equipment necessitates advanced design for complex load-bearing environments, which is challenging for traditional structural forms. Consequently, this paper introduces a hierarchical topology optimization method for stiffened shells. This method utilizes the proposed hierarchical model and a filtering-based reinforcement description method to facilitate adaptive reinforcement configurations, such as variable shell thickness, lattices, and stiffeners of varying spacings and sizes. Two examples are presented: for a plate scenario, under identical constraints, the proposed method can obtain a hierarchical stiffener layout and achieve a 16.8% increase in stiffness compared to the traditional optimized single-level stiffener layout. For a complex engineering curved shell, the proposed method achieves hierarchical designs that meet service conditions, which traditional single-level stiffening approaches struggle to accomplish. Additionally, several discussions illustrate the robustness of the method across various service conditions. The proposed method can provide improved performance compared to traditional methods and can adaptively optimize to obtain a single-level stiffener layout when it is more reasonable in terms of design requirements.