Large Number Of Design Variables Research Articles

Intuitive and feasible geometric representation of airfoil using variational autoencoder

Abstract Airfoil shape optimization is crucial for improving aerodynamic performance in advanced aircraft designs. Given the extensive functional evaluations required for optimization, surrogate modeling is widely used to alleviate computational burden. However, greater flexibility in airfoil parameterization often requires a larger number of design variables, leading to the challenge known as the curse of dimensionality in surrogate modeling. In recent years, generative models such as generative adversarial networks and variational autoencoders have shown potential to represent large design spaces with compact design variables. However, these models still exhibit limited feasibility and intuitiveness due to their high model capacity, which in turn degrades the efficiency of design optimization. To address this issue, we have developed a novel airfoil parameterization method using a variational autoencoder. The proposed method improves feasibility by using architecture modeling to separate the generation of thickness and camber distributions, resulting in smooth and nonintersecting airfoils. It also improves intuitiveness by using a physics loss function that aligns latent dimensions with geometric features of the airfoils. Notably, extensive comparative analyses validate the effectiveness of our method in terms of flexibility, parsimony, feasibility, and intuitiveness, leading to increased efficiency in aerodynamic design optimization.

An enhanced guided stochastic search with repair deceleration mechanism for very high-dimensional optimization problems of steel double-layer grids

Finding reasonably good solutions using a fewer number of objective function evaluations has long been recognized as a good attribute of an optimization algorithm. This becomes more important, especially when dealing with very high-dimensional optimization problems, since contemporary algorithms often need a high number of iterations to converge. Furthermore, the excessive computational effort required to handle the large number of design variables involved in the optimization of large-scale steel double-layer grids with complex configurations is perceived as the main challenge for contemporary structural optimization techniques. This paper aims to enhance the convergence properties of the standard guided stochastic search (GSS) algorithm to handle computationally expensive and very high-dimensional optimization problems of steel double-layer grids. To this end, a repair deceleration mechanism (RDM) is proposed, and its efficiency is evaluated through challenging test examples of steel double-layer grids. First, parameter tuning based on rigorous analyses of two preliminary test instances is performed. Next, the usefulness of the proposed RDM is further investigated through two very high-dimensional instances of steel double-layer grids, namely a 21,212-member free-form double-layer grid, and a 25,514-member double-layer multi-dome, with 21,212 and 25,514 design variables, respectively. The obtained numerical results indicate that the proposed RDM can significantly enhance the convergence rate of the GSS algorithm, rendering it an efficient tool to handle very high-dimensional sizing optimization problems.

Optimization of acoustic porous material absorbers modeled as rigid multiple microducts networks: Metamaterial design using additive manufacturing

This research presents a strategy to enhancing sound absorption in porous acoustic metamaterials through the optimization of micro-duct networks. The study employs a combination of analytical and numerical methods, systematically adjusting micro-duct diameters within a 2D grid to optimize absorption coefficients at specific frequency bands. The Finite Element Transfer Method (FETM) is utilized for modeling, supported by a multi-objective function influenced by the surface impedance of the network. This approach ensures practical applicability and manufacturability of the designed metamaterial. A significant aspect of the study is the development of an analytical mobility matrix for each microduct, integrated through the Transfer Matrix Method using the visco-thermal dissipation theory. This integration results in a physically coherent model, for which a semi-analytical sensitivity analysis related to the microduct diameters can be directly performed. The optimization employs the Method of Moving Asymptotes (MMA), effectively managing the complexity associated with a large number of design variables. Subsequently, the optimized structure is adapted into a 3D grid, facilitating prototype creation using Additive Manufacturing Technology (AMT) with a specific polymer material. The research methodology is validated through four distinct test cases, each demonstrating the efficacy and adaptability of the optimization tools and techniques. Experimental validation, conducted using an impedance tube, indicates a significant shift in the first absorption maximum from 2500 Hz to 1000 Hz, achieved without altering the material thickness. Additional validation through 3D Finite Element Method (FEM) modeling, which includes visco-thermal effects, further confirms the acoustic efficiency of the final designs. While the potential for achieving lower sub-wavelength conditions is recognized, the study opts for simpler structures, considering the current limitations of 3D printing technology. This study contributes to both theoretical understanding and practical application in acoustic material design, emphasizing the potential for customizing materials to specific frequency ranges. The integration of FETM modeling with MMA provides a systematic and effective approach for optimizing acoustic materials, particularly in enhancing absorption at lower frequencies.

Optimal design of selected features of exhaust system shields using different optimization methods and artificial neural networks

The article presents problems related to methods of optimal design of heat shields used in exhaust systems of internal combustion engines. The optimization method proposed in this paper goes well beyond the scope of the standard design process. The paper uses a variety of local and global optimization algorithms, those both built into numerical simulation systems and in-house and external algorithms. An optimization criterion was defined and numerically implemented, together with constraints derived from the real requirements for this type of shielding. A simplified numerical model of finite element method providing the required accuracy adapted to the optimization task was developed. In addition, the work also presents a method for creating finite element surrogate models using artificial neural networks. The process of selecting the network topology and its learning allowed the development of a metamodel characterized by very good quality, for which, despite the relatively large number of design variables, the response errors are completely acceptable from a practical point of view. Numerical results were compared and developed for the used methods and algorithms.

Efficient strategy for topology optimization of stochastic viscoelastic damping structures

In this study, the dynamic topology optimization (TO) of stochastic viscoelastic damping structures (VDSs) is performed for the first time. However, the expensive computation cost seriously hinders the TO process. To address this problem, an efficient strategy is proposed. On the one hand, in the stochastic structural response analysis, a fully adaptive method based on direct probability integral method is presented to determine the number and locations of samples. Meanwhile, to improve the computational efficiency of structural response for each sample, a piecewise model-order reduction method based on Krylov subspace is adopted to generate the orthonormal basis and project the original large-scale system onto a low-order system. On the other hand, to overcome the optimization challenge arising from large number of design variables in the density-based topology method, a material-field series-expansion method is employed to describe the topology layout and significantly reduce the number of design variables. Moreover, the sensitivity of the optimization model is derived by the adjoint method and the method of moving asymptotes (MMA) is used to efficiently update the design variables. Some numerical examples comprehensively demonstrate the effectiveness and efficiency of the proposed strategy. The results indicate that the uncertainty of structural parameters greatly affect both the topology layout of VDS and vibration damping capacity.

An Incremental Interpolation Scheme With Discrete Cosine Series Expansion for Multimaterial Topology Optimization

Abstract Topology optimization is a powerful tool for structural design, while its computational cost is quite high due to the large number of design variables, especially for multilateral systems. Herein, an incremental interpolation approach with discrete cosine series expansion (DCSE) is established for multilateral topology optimization. A step function with shape coefficients (i.e., ensuring that no extra variables are required as the number of materials increases) and the use of the DCSE together reduces the number of variables (e.g., from 8400 to 120 for the optimization of the clamped–clamped beam with four materials). Remarkably, the proposed approach can effectively bypass the checkerboard problem without using any filter. The enhanced computational efficiency (e.g., a ∼89.2% reduction in computation time from 439.1 s to 47.4 s) of the proposed approach is validated via both 2D and 3D numerical cases.

A domain knowledge-informed design space exploration methodology for mechanical layout design

Layout designs have a large number of design variables and various physical constraints. Conventional design exploration approaches are time-consuming and may require human intervention. A unique feature about physical layout designs is the availability of domain knowledge, which can be utilised to speed up the design process. In this paper, we propose a generative deep learning-based design space exploration (DSE) methodology that is capable of learning design constraints in the layout design without explicit supervision. Moreover, it can incorporate domain knowledge in the generated layouts, thereby speeding up the design process. This is realised by constructing a layout generation variational autoencoder (LGVAE) model, which uses a latent space as an interface to generate the layouts. By training the LGVAE model, significantly lower-dimensional representations can be learned compared to the original dimensionality of the design space. Therefore, the number of design variables is greatly reduced. We showcase the performance of the proposed DSE approach by solving the heat source layout design problem encountered in thermal management of chips. Experiments demonstrate that the LGVAE model is capable of generating compressed latent representations that capture the characteristics of the input samples, which makes the DSE cost-effective.

An efficient aerodynamic optimization method based on approximate gradient analysis

With the rapid development of the computer capability and the continuous improvement of CFD (Computational Fluid Dynamics) technology, optimization design has gradually become a powerful and reliable means to solve the aerodynamic design problems. Among all kinds of optimization methods, gradient-based optimization algorithm can find the local optimal solution quickly, which is widely used in engineering problems. However, the computation of gradient is very expensive when an engineering problem involves large number of design variables. In this paper, an optimization method based on approximate gradient analysis is introduced, where the one-dimensional linear searches during the optimization process are evaluated with high-fidelity flow field analyses, while the gradients are evaluated with low-fidelity flow field analyses, leading to a significant saving in the computational cost of high-fidelity analysis. This method is demonstrated using a numerical example and RAE2822 airfoil optimization, and it is finally applied to the aerodynamic optimization of a wing-body configuration, which obtains good optimization results.

Geometrically exact beam theory for gradient-based optimization

Decades of research have progressed geometrically exact beam theory to the point where it is now an invaluable resource for analyzing and modeling highly flexible slender structures. Large-scale optimization using geometrically exact beam theory remains nontrivial, however, due to the inability of gradient-free optimizers to handle large numbers of design variables in a computationally efficient manner and the difficulties associated with obtaining smooth, accurate, and efficiently calculated design sensitivities for gradient-based optimization. To overcome these challenges, this paper presents a finite-element implementation of geometrically exact beam theory which has been developed specifically for gradient-based optimization. A key feature of this implementation of geometrically exact beam theory is its compatibility with forward and reverse-mode automatic differentiation. Another key feature is its support for both continuous and discrete adjoint sensitivity analysis. Other features are also presented which build upon previous implementations of geometrically exact beam theory, including a singularity-free rotation parameterization based on Wiener-Milenković parameters, an implementation of stiffness-proportional structural damping using a discretized form of the compatibility equations, and a reformulation of the equations of motion for geometrically exact beam theory as a semi-explicit system. Several examples are presented which verify the utility and validity of each of these features.

Design of metamaterial-based heat manipulators using isogeometric level-set topology optimization

We exploit level-set topology optimization to find the optimal material distribution for metamaterial-based heat manipulators. The level-set function, geometry, and solution field are parameterized using the Non-Uniform Rational B-Spline (NURBS) basis functions to take advantage of easy control of smoothness and continuity. In addition, NURBS approximations can produce conic geometries exactly and provide higher efficiency for higher-order elements. The values of the level-set function at the control points (called expansion coefficients) are utilized as design variables. For optimization, we use an advanced mathematical programming technique, Sequential Quadratic Programming. Taking into account a large number of design variables and the small number of constraints associated with our optimization problem, the adjoint method is utilized to calculate the required sensitivities with respect to the design variables. The efficiency and robustness of the proposed method are demonstrated by solving three numerical examples. We have also shown that the current method can handle different geometries and types of objective functions. In addition, regularization techniques such as Tikhonov regularization and volume regularization have been explored to reduce unnecessary complexity and increase the manufacturability of optimized topologies.

Efficient Design Optimization of Cable-Stayed Bridges: A Two-Layer Framework with Surrogate-Model-Assisted Prediction of Optimum Cable Forces

Cable-stayed bridges have commonly been built for crossing large-span obstacles, such as rivers, valleys, and existing structures. Obtaining an optimum design for a cable-stayed bridge is challenging, due to the large number of design variables and design constraints that are typically nonlinear and usually conflict with each other. Therefore, it is a reasonable alternative to turn the large and complex optimization problem into two sub-problems, i.e., optimizing the internal force distribution by adjusting the cable prestressing forces, and optimizing the other sizing or geometrical parameters. However, conventional methods are lacking in efficiency when dealing with the problem of optimization of cable forces in the first sub-problem, under the circumstance that iteration between the two sub-problems is required. To address this, this paper presents a surrogate-model-assisted method to construct a cable forces predictor ahead of the structural optimization process, so that cable forces can be effectively predicted rather than optimized in each iterative round. Additionally, B-spline interpolation curve is adopted for variable condensation when sampling for the surrogate model. Finally, the structure optimization in the second sub-problem is performed by leveraging an optimization program based on particle swarm optimization method. The performance of the proposed framework is tested with a practical engineering application. Results show that the proposed method showcases good efficiency and accuracy. The theoretical raw material consumption of the towers and the cables is 32% lower than the original design.

Optimal Design of Shear Walls for Minimizing the Structural Torsion Using Cascade Optimization Algorithm

Due to a large number of design variables involved in the optimization of RC structures, a multi stage cascade optimization is used. This algorithm speeds up an accurate optimal design for the large-scale structures reducing the number of variables by dividing single optimization to a number of stages such that the optimization of each stage starts with the optimum results of the previous one. Here, the first stage of cascade optimization method uses the assembling of the stiffness matrix of the entire RC structure for minimizing the torsion of the stories as an objective function. By assembling the stiffness matrix of the RC frame without shear walls and the shear wall alone, the length and thickness of the shear wall on each story are taken as design variables of the first stage. The optimized reinforcement arrangement of walls according to the required wall rebar area is the goal of the next stage. Using this method, designing the RC structure and minimizing the structural torsion of each floor simultaneously, can result in different length and thickness for the shear walls in different stories. Reducing the structural torsion leads to economical structure. Here, the MATLAB and ETABS interfacing are utilized.

Development of Building Design Optimization Methodology: Residential Building Applications

Building design optimization is a highly complex problem, requiring long computational running processes because of the many options that exist when a building is being designed. This paper introduces an integrated approach through which to perform this optimization within an acceptable time frame. The approach includes the methods of variable selection, model simplification, and a sequential optimization process. Using singular value decomposition, a large number of design variables is reduced to a smaller subset that can be solved more quickly through the optimization algorithm. To expedite the variable selection process, a modeling approach that quickly simulates annual energy consumption was developed to replace full annual energy simulations. The developed methodology was applied to two residential buildings in the US, and the results are discussed herein. To assess the accuracy of the integrated optimization methodology, the optimized life cycle costs are compaa variables demonstrating the strongest contributions in the optimization study were identified. The proposed methodology significantly shortened the time requirements for the optimization processes of the two case studies by 74% and 84%; the optimized life cycle costs were within 0.05% and 0.06%, respectively, of the optimum point.

A BEM broadband topology optimization strategy based on Taylor expansion and SOAR method—Application to 2D acoustic scattering problems

AbstractIn this article, an innovative method is proposed for broadband topology optimization of sound‐absorbing materials adhering to the surface of a sound barrier structure. Helmholtz equation for the acoustic problems is solved using the boundary element method, while sensitivities of the objective function with respect to a large number of design variables are calculated via a sensitivity analysis method originated from the adjoint variable method (AVM), and the optimal solution is obtained by the method of moving asymptotes (MMA). Since the traditional single‐frequency topology optimization is frequency dependent, that is, the optimal solution at a certain frequency may not be optimal at other frequencies, broadband topology optimization of sound‐absorbing materials is conducted in this work. To address the problem of tremendous computational cost due to repetitive calculations at each discrete frequency point during broadband optimization, Taylor expansion of the Hankel function is performed to decouple the frequency dependent and independent terms in the BEM equations. For large‐scale problems, a reduced‐order model that retains the essential structures and key properties of the original model is built using the second‐order Arnoldi (SOAR) method. The accuracy and feasibility of the proposed algorithms are finally validated via some two‐dimensional numerical examples.

Complex Standard Eigenvalue Problem Derivative Computation for Laminar–Turbulent Transition Prediction

As a high-fidelity approach to transition prediction, the coupled Reynolds-averaged Navier–Stokes (RANS) and linear stability theory (LST)-based method is widely used in engineering applications and is the preferred method for laminar flow optimization. However, the further development of gradient-based laminar flow wing optimization schemes is hindered by a lack of efficient and accurate derivative computation methods for LST-based eigenvalue problems with a large number of design variables. To address this deficiency and to compute the derivatives in the LST-based solution solver, we apply the adjoint method and analytical reverse algorithm differentiation (RAD), which scale well with the number of inputs. The core of this paper is the computation of the standard eigenvalue and eigenvector derivatives for the LST problem, which involves a complex matrix. We develop an adjoint method to compute these derivatives, and we couple this method with RAD to reduce computational costs. In addition, we incorporate the LST-based partial derivatives into the laminar–turbulent transition prediction framework for the computation of total derivatives. We verify our proposed method with reference to finite difference (FD) results for an infinite swept wing. Both the intermediate derivatives from the transition module and total derivatives agree with the FD reference results to at least three digits, demonstrating the accuracy of our proposed approach. The fully adjoint and the coupled adjoint–RAD methods both have considerable advantages in terms of computational efficiency compared with iterative RAD and FD methods. The LST-based transition method and the proposed method for efficient and accurate derivative computations have prospects for wide application to laminar flow optimization in aerodynamic design.

An Evolutionary Approach of Grasp Synthesis for Sheet Metal Parts With Multitype Grippers

AbstractRobot-mounted grippers are used to position, immobilize, and manipulate parts and assemblies during manufacturing. In the design of these systems, the gripper assembly, comprising the grippers that grasp the part and the frame that holds them together, is customized to each part. Due to the large number of design variables and unique design needs for each gripper, automation of gripper assemblies has been limited, especially where multiple gripper types are used to grasp a part. To this end, this article presents an evolutionary approach that synthesizes and optimizes grasps and gripper assembly layouts using two different gripper types—suction cups and magnets—from the geometric models of sheet metal parts. The method first generates an option space of gripper placement on the suitable faces of the part model. Then, a genetic algorithm generates grasps on this option space by varying both the count and locations of each gripper type. Through generations, these grasps are optimized against five criteria and one constraint: factor of safety, cost, residual moment, deflection, frame weight, and gripper clearance. These criteria are then combined into a single criterion that represents a pareto condition for assessing the grasps. The algorithm is implemented in software code for validation, and the article presents detailed validation of the algorithm using four sheet metal parts. The results show that the algorithm improves the grasp from all six aspects, when started from either program-assigned or user-defined initial grasps. The high agreement between the final grasp designs resulting from multiple runs of the algorithm on a part illustrates the stability and repeatability of the algorithm.

Data-Driven Surrogate-Assisted Optimization of Metamaterial-Based Filtenna Using Deep Learning

In this work, a computationally efficient method based on data-driven surrogate models is proposed for the design optimization procedure of a Frequency Selective Surface (FSS)-based filtering antenna (Filtenna). A Filtenna acts as a module that simultaneously pre-filters unwanted signals, and enhances the desired signals at the operating frequency. However, due to a typically large number of design variables of FSS unit elements, and their complex interrelations affecting the scattering response, FSS optimization is a challenging task. Herein, a deep-learning-based algorithm, Modified-Multi-Layer-Perceptron (M2LP), is developed to render an accurate behavioral model of the unit cell. Subsequently, the M2LP model is applied to optimize FSS elements being parts of the Filtenna under design. The exemplary device operates at 5 GHz to 7 GHz band. The numerical results demonstrate that the presented approach allows for an almost 90% reduction of the computational cost of the optimization process as compared to direct EM-driven design. At the same time, physical measurements of the fabricated Filtenna prototype corroborate the relevance of the proposed methodology. One of the important advantages of our technique is that the unit cell model can be re-used to design FSS and Filtenna operating various operating bands without incurring any extra computational expenses.

Multi-level intelligent design of variable angle tow laminates via image-driven method

Compared with the traditional design of constant stiffness composite laminates, the fiber variable angle-tow (VAT) laminates as variable stiffness composites have more flexible design space. In order to obtain continuous fiber paths and high-precision analysis model, the design of VAT requires more detail features, yet the large number of design variables presents a significant obstacle to conventional optimization methods. In this study, an image-driven intelligent optimization method is proposed. The VAT fiber angle vector field is accurately described by the B-spline surface with different control points, and is mapped into a gray image with fixed size. Through the isogeometric analysis (IGA) method, the buckling load is gained as the label of the images, and the dataset is obtained to train convolutional neural networks (CNNs) for deep learning models. Then, a multi-level refinement of the local key design space effectively optimizes the VAT laminates while taking into account manufacturing constraints. Due to the images providing as a single unit of input data, the CNNs model can be applied in variable design space without resampling. Three numerical examples demonstrate the effectiveness, and obtain higher buckling load of VAT laminates in different level design spaces.

Optimal Design of Reinforced Concrete Frame Structures Using Cascade Optimization Method

Optimal design of structures is one of the goals of a structural engineer. Since a large number of design variables are involved, the high computational time for optimization process is one of the main problems of using single heuristic or meta-heuristic algorithms utilized for this purpose, especially in large-scale structures. Cascade optimization replaces the single optimization problem with a multiple stage process. Each step starts with the previous optimum design, with the number of variables being reduced thus increasing the speed of the convergence in each stage of optimization. In this article a Cascade optimization method is developed for optimum design of RC frame structures. By using this algorithm in place of a single optimization method and ensuring its efficiency, the optimum section dimensions are determined by a new method. Then, genetic algorithm is utilized for optimizing of rebar arrangement. According to the results, this method performs optimum design and reduces the elapsed time considerably. Optimization of large-scale concrete structure is carried out by the developed method. It should be noted that the present method works with MATLAB and ETABS interfacing.

Simultaneous wing shape and actuator parameter optimization using the adjoint method

Increasing attention to urban air mobility with distributed electric propulsion causes an increased interest in propeller aircraft design and optimization. Various studies have focused on designing and optimizing the aerodynamic performance of wings and propellers. However, existing studies optimized the wing and propeller separately due to the numerical challenges in simulations and optimization. It is not clear to what extent high-fidelity coupled wing-propeller aerodynamic optimization can benefit the propeller aircraft design. As a first step to answer the above question, this study develops the capability to simultaneously optimize the wing shape and actuator parameters. We use a high-fidelity computational fluid dynamics solver to simulate the wing aerodynamics, and the propeller is modeled as an actuator disk. We develop a smoothed actuator disk formulation that allows us to change the actuator's location and radius during the optimization, along with the wing shape. We use the discrete adjoint approach to compute the derivatives and couple them with a gradient-based optimization framework. The benefit of the above adjoint-based optimization framework is that it can use a large number of design variables to allow large design freedom for both wing and actuator. We perform 15 optimizations to evaluate how various actuator parameters impact the wing aerodynamic performance; both single and double actuator configurations are considered. We observe that the spanwise location and outer radius are the two most important actuator parameters, and using more actuator parameters as the design variables result in better performance. For example, optimizing the wing shape while allowing the actuator location and outer radius to move exhibits 11.4% more drag reduction compared with optimizing the wing shape while fixing the actuator parameters. This study serves as the starting point for more detailed high-fidelity coupled wing-propeller aerodynamic optimizations.