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Showing papers in "Structural and Multidisciplinary Optimization in 2021"


Journal ArticleDOI
TL;DR: In this paper, the authors present a review of existing approaches for topology optimization of multi-scale structures, explaining the principles of each category, analyzing their strengths and applicability, and discussing open research questions.
Abstract: Multi-scale structures, as found in nature (e.g., bone and bamboo), hold the promise of achieving superior performance while being intrinsically lightweight, robust, and multi-functional. Recent years have seen a rapid development in topology optimization approaches for designing multi-scale structures, but the field actually dates back to the seminal paper by Bendsoe and Kikuchi from 1988 (Computer Methods in Applied Mechanics and Engineering 71(2): pp. 197–224). In this review, we intend to categorize existing approaches, explain the principles of each category, analyze their strengths and applicabilities, and discuss open research questions. The review and associated analyses will hopefully form a basis for future research and development in this exciting field.

134 citations


Journal ArticleDOI
TL;DR: This paper demonstrates that one can directly execute topology optimization (TO) using neural networks (NN) and uses the NN’s activation functions to represent the popular Solid Isotropic Material with Penalization (SIMP) density field.
Abstract: Neural networks, and more broadly, machine learning techniques, have been recently exploited to accelerate topology optimization through data-driven training and image processing. In this paper, we demonstrate that one can directly execute topology optimization (TO) using neural networks (NN). The primary concept is to use the NN’s activation functions to represent the popular Solid Isotropic Material with Penalization (SIMP) density field. In other words, the density function is parameterized by the weights and bias associated with the NN, and spanned by NN’s activation functions; the density representation is thus independent of the finite element mesh. Then, by relying on the NN’s built-in backpropogation, and a conventional finite element solver, the density field is optimized. Methods to impose design and manufacturing constraints within the proposed framework are described and illustrated. A byproduct of representing the density field via activation functions is that it leads to a crisp and differentiable boundary. The proposed framework is simple to implement and is illustrated through 2D and 3D examples. Some of the unresolved challenges with the proposed framework are also summarized.

90 citations


Journal ArticleDOI
TL;DR: Results show that the proposed active-learning method can provide an accurate failure probability estimation with a less computational cost and is compared with several state-of-the-art methods.
Abstract: Active-learning surrogate model–based reliability analysis is widely employed in engineering structural reliability analysis to alleviate the computational burden of the Monte Carlo method. To date, most of these methods are built based on the single-fidelity surrogate model, such as the Kriging model. However, the computational burden of constructing a fine Kriging model may be still expensive if the high-fidelity (HF) simulation is extremely time-consuming. To solve this problem, an active-learning method based on the multi-fidelity (MF) Kriging model for structural reliability analysis (abbreviated as AMK-MCS+AEFF), which is an online data-driven method fusing information from different fidelities, is proposed in this paper. First, an augmented expected feasibility function (AEFF) is defined by considering the cross-correlation, the sampling density, and the cost query between HF and low-fidelity (LF) models. During the active-learning process of AMK-MCS+AEFF, both the location and fidelity level of the updated sample can be determined objectively and adaptively by maximizing the AEFF. Second, a new stopping criterion that associates with the estimated relative error is proposed to ensure that the iterative process terminates in a proper iteration. The proposed method is compared with several state-of-the-art methods through three numerical examples and an engineering case. Results show that the proposed method can provide an accurate failure probability estimation with a less computational cost.

39 citations


Journal ArticleDOI
TL;DR: A systematic and comprehensive review of educational articles and codes in SMO, including topology, sizing, and shape optimization and building blocks, is presented in this article, where the authors provide guidance for beginners to approach various optimization methods.
Abstract: Ever since the publication of the 99-line topology optimization MATLAB code (top99) by Sigmund in 2001, educational articles have emerged as a popular category of contributions within the structural and multidisciplinary optimization (SMO) community. The number of educational papers in the field of SMO has been growing rapidly in recent years. Some educational contributions have made a tremendous impact on both research and education. For example, top99 (Sigmund in Struct Multidisc Optim 21(2):120–127, 2001) has been downloaded over 13,000 times and cited over 2000 times in Google Scholar. In this paper, we attempt to provide a systematic and comprehensive review of educational articles and codes in SMO, including topology, sizing, and shape optimization and building blocks. We first assess the papers according to the adopted methods, which include density-based, level-set, ground structure, and more. We then provide comparisons and evaluations on the codes from several key aspects, including techniques, efficiency, usability, readability, environment, and compatibility. In addition, we conduct numerical experiments on the reviewed codes using the benchmark cantilever beam example to provide feedback on the overall user experience. With a systematic review and comparison, this paper aims to offer insights on the educational values and practicality for employing these codes. We try to provide not only guidance for beginners to approach various optimization methods, but also a dictionary to direct readers to effectively target the relevant codes and building blocks based on their demands. Finally, based on the findings in this review paper, we provide some perspectives and recommendations for future educational contributions.

38 citations


Journal ArticleDOI
TL;DR: A deep learning-based CAD/CAE framework in the conceptual design phase that automatically generates 3D CAD designs and evaluates their engineering performance is proposed and indicates that AI can be practically incorporated into an end-use product design project.
Abstract: Engineering design research integrating artificial intelligence (AI) into computer-aided design (CAD) and computer-aided engineering (CAE) is actively being conducted. This study proposes a deep learning-based CAD/CAE framework in the conceptual design phase that automatically generates 3D CAD designs and evaluates their engineering performance. The proposed framework comprises seven stages: (1) 2D generative design, (2) dimensionality reduction, (3) design of experiment in latent space, (4) CAD automation, (5) CAE automation, (6) transfer learning, and (7) visualization and analysis. The proposed framework is demonstrated through a road wheel design case study and indicates that AI can be practically incorporated into an end-use product design project. Engineers and industrial designers can jointly review a large number of generated 3D CAD models by using this framework along with the engineering performance results estimated by AI and find conceptual design candidates for the subsequent detailed design stage.

38 citations


Journal ArticleDOI
TL;DR: In this paper, a two-stage network model is proposed using convolutional encoder-decoder networks that incorporate a new way of loss functions to reduce the number of structural disconnection cases as well as to reduce pixel-wise error to enhance the predictive performance of DNNs for topology optimization.
Abstract: A vital necessity when employing state-of-the-art deep neural networks (DNNs) for topology optimization is to predict near-optimal structures while satisfying pre-defined optimization constraints and objective function. Existing studies, on the other hand, suffer from the structural disconnections which result in unexpected errors in the objective and constraints. In this study, a two-stage network model is proposed using convolutional encoder-decoder networks that incorporate a new way of loss functions to reduce the number of structural disconnection cases as well as to reduce pixel-wise error to enhance the predictive performance of DNNs for topology optimization without any iteration. In the first stage, a single DNN model architecture is proposed and used in two parallel networks using two different loss functions for each called the mean square error (MSE) and mean absolute error (MAE). Once the priori information is generated from the first stage, it is instantly fed into the second stage, which acts as a rectifier network over the priori predictions. Finally, the second stage is trained using the binary cross-entropy (BCE) loss to provide the final predictions. The proposed two-stage network with the proposed loss functions is implemented for both two-dimensional (2D) and three-dimensional (3D) topology optimization datasets to observe its generalization ability. The validation results showed that the proposed two-stage framework could improve network prediction ability compared to a single network while significantly reducing compliance and volume fraction errors.

37 citations


Journal ArticleDOI
TL;DR: An efficient multi-objective optimization method based on the adaptive approximation model using the Latin hypercube design for obtaining the initial sample points and the reverse shape parameter analysis method is proposed to obtain improved approximation models.
Abstract: Considering the high computational cost caused by solving multi-objective optimization (MOO) problems, an efficient multi-objective optimization method based on the adaptive approximation model is developed. Firstly, the Latin hypercube design (LHD) is employed for obtaining the initial sample points. Secondly, initial approximation models of objective functions and constraints are established by using the radial basis function (RBF). For ensuring the accuracy of the approximation models, the reverse shape parameter analysis method (RSPAM) is proposed to obtain improved approximation models. Thirdly, the micro multi-objective genetic algorithm (μMOGA) is adopted to solve the Pareto optimal set and the local-densifying approximation method is also applied to strengthen the ability of solving accurate Pareto optimal sets. Finally, the effectiveness and practicability of the proposed method is demonstrated by two numerical examples and two engineering examples.

36 citations


Journal ArticleDOI
TL;DR: Dual-model artificial neural networks that are trained with both forward and sensitivity data are constructed and integrated into the Solid Isotropic Material with Penalization (SIMP) method to replace the FEA.
Abstract: Topology optimization (TO) is a common technique used in free-form designs. However, conventional TO-based design approaches suffer from high computational cost due to the need for repetitive forward calculations and/or sensitivity analysis, which are typically done using high-dimensional simulations such as finite element analysis (FEA). In this work, artificial neural networks are used as efficient surrogate models for forward and sensitivity calculations in order to greatly accelerate the design process of topology optimization. To improve the accuracy of sensitivity analyses, dual-model artificial neural networks that are trained with both forward and sensitivity data are constructed and are integrated into the Solid Isotropic Material with Penalization (SIMP) method to replace the FEA. The performance of the accelerated SIMP method is demonstrated on two benchmark design problems namely minimum compliance design and metamaterial design. The efficiency gained in the problem with size of 64 × 64 is 137 times in forward calculation and 74 times in sensitivity analysis. In addition, effective data generation methods suitable for TO designs are investigated and developed, which lead to a great saving in training time. In both benchmark design problems, a design accuracy of 95% can be achieved with only around 2000 training data.

34 citations


Journal ArticleDOI
TL;DR: This paper presents a MATLAB code with the implementation of the Topology Optimization of Binary Structures (TOBS) method, to show that topology optimization with integer linear programming can be efficiently carried out, contrary to the previous reports in the literature.
Abstract: This paper presents a MATLAB code with the implementation of the Topology Optimization of Binary Structures (TOBS) method first published by Sivapuram and Picelli (Finite Elem Anal Des 139: pp. 49–61, 2018). The TOBS is a gradient-based topology optimization method that employs binary design variables and formal mathematical programming. Besides its educational purposes, the 101-line code is provided to show that topology optimization with integer linear programming can be efficiently carried out, contrary to the previous reports in the literature. Compliance minimization subject to a volume constraint is first solved to highlight the main features of the TOBS method. The optimization parameters are discussed. Then, volume minimization subject to a compliance constraint is solved to illustrate that the method can efficiently deal with different types of constraints. Finally, simultaneous volume and displacement constraints are investigated in order to expose the capabilities of the optimizer and to serve as a tutorial of multiple constraints. The 101-line MATLAB code and some simple enhancements are elucidated, keeping only the integer programming solver unmodified so that it can be tested and extended to other numerical examples of interest.

33 citations


Journal ArticleDOI
TL;DR: A scheme based on the augmented Lagrangian method is adopted, which treats the problem consistently with the local definition of stress without employing traditional constraint aggregation techniques and can handle nonlinear material models with a given strain energy density function.
Abstract: We present PolyStress, a Matlab implementation for topology optimization with local stress constraints considering linear and material nonlinear problems. The implementation of PolyStress is built upon PolyTop, an educational code for compliance minimization on unstructured polygonal finite elements. To solve the nonlinear elasticity problem, we implement a Newton-Raphson scheme, which can handle nonlinear material models with a given strain energy density function. To solve the stress-constrained problem, we adopt a scheme based on the augmented Lagrangian method, which treats the problem consistently with the local definition of stress without employing traditional constraint aggregation techniques. The paper discusses several theoretical aspects of the stress-constrained problem, including details of the augmented Lagrangian-based approach implemented herein. In addition, the paper presents details of the Matlab implementation of PolyStress, which is provided as electronic supplementary material. We present several numerical examples to demonstrate the capabilities of PolyStress to solve stress-constrained topology optimization problems and to illustrate its modularity to accommodate any nonlinear material model. Six appendices supplement the paper. In particular, the first appendix presents a library of benchmark examples, which are described in detail and can be explored beyond the scope of the present work.

33 citations


Journal ArticleDOI
TL;DR: A novel formulation of the frequency band constraint based on a modified Heaviside function is proposed, which is continuous and differentiable and derived and used in a gradient-based optimization method, which validates the effectiveness of the developed method.
Abstract: Engineering structures usually operate in some specific frequency bands. An effective way to avoid resonance is to shift the structure’s natural frequencies out of these frequency bands. However, in the optimization procedure, which frequency orders will fall into these bands are not known a priori. This makes it difficult to use the existing frequency constraint formulations, which require prescribed orders. For solving this issue, a novel formulation of the frequency band constraint based on a modified Heaviside function is proposed in this paper. The new formulation is continuous and differentiable; thus, the sensitivity of the constraint function can be derived and used in a gradient-based optimization method. Topology optimization for maximizing the structural fundamental frequency while circumventing the natural frequencies located in the working frequency bands is studied. For eliminating the frequently happened numerical problems in the natural frequency topology optimization process, including mode switching, checkerboard phenomena, and gray elements, the “bound formulation” and “robust formulation” are applied. Three numerical examples, including 2D and 3D problems, are solved by the proposed method. Frequency band gaps of the optimized results are obtained by considering the frequency band constraints, which validates the effectiveness of the developed method.

Journal ArticleDOI
TL;DR: This paper takes into consideration the excellent energy absorption ability of hierarchical honeycombs and auxetic structures and proposes a novel auxetic hierarchical crash box assembled by the auxetics hierarchical filling cores and the outer square thin-walled tube.
Abstract: This paper takes into consideration the excellent energy absorption ability of hierarchical honeycombs and auxetic structures and proposes a novel auxetic hierarchical crash box assembled by the auxetic hierarchical filling cores and the outer square thin-walled tube. The crushing performance of the auxetic hierarchical crash box is systematically investigated. The comparisons of energy absorption ability are made among the auxetic hierarchical crash box, aluminum foam-filled crash box, and the traditional crash box. In addition, a multi-objective optimization design is conducted based on the surrogate model with higher accuracy. The non-dominated sorting genetic algorithm (NSGA-II) and archive-based micro genetic algorithm (AMGA) are, respectively, employed to obtain the pareto sets. The results show that the optimum solution with AMGA has a smaller relative error, and the multi-objective optimization successfully improves the crushing performance of the auxetic hierarchical crash box. The electric vehicle crashworthiness is remarkably improved by the application of the auxetic hierarchical crash box. The conclusions of this paper can provide a new solution for the design of the crash box.

Journal ArticleDOI
TL;DR: A review of the uncertainty treatment practices in design optimization of structural and multidisciplinary systems under uncertainties can be found in this paper, where uncertainties of a structural or multi-disciplinary system are taken into account by using safety factors specified in the regulations or design codes.
Abstract: Design optimization of structural and multidisciplinary systems under uncertainty has been an active area of research due to its evident advantages over deterministic design optimization. In deterministic design optimization, the uncertainties of a structural or multidisciplinary system are taken into account by using safety factors specified in the regulations or design codes. This uncertainty treatment is a subjective and indirect way of dealing with uncertainty. On the other hand, design under uncertainty approaches provide an objective and direct way of dealing with uncertainty. This paper provides a review of the uncertainty treatment practices in design optimization of structural and multidisciplinary systems under uncertainties. To this end, the activities in uncertainty modeling are first reviewed, where theories and methods on uncertainty categorization (or classification), uncertainty handling (or management), and uncertainty characterization are discussed. Second, the tools and techniques developed and used for uncertainty modeling and propagation are discussed under the broad two classes of probabilistic and non-probabilistic approaches. Third, various design optimization methods under uncertainty which incorporate all the techniques covered in uncertainty modeling and analysis are reviewed. In addition to these in-depth reviews on uncertainty modeling, uncertainty analysis, and design optimization under uncertainty, some real-life engineering applications and benchmark test examples are provided in this paper so that readers can develop an appreciation on where and how the discussed techniques can be applied and how to compare them. Finally, concluding remarks are provided, and areas for future research are suggested.

Journal ArticleDOI
TL;DR: A modified advancedmean value (MAMV) method is proposed to improve efficiency and robustness of the advanced mean value method, which encounters inefficiency and numerical instability for the concave or highly nonlinear performance measure functions.
Abstract: Due to the limited joint position space and the consideration of reducing the moment of inertia and vibration for industrial robot, the design optimization for rotate vector (RV) reducer is becoming a new and urgent problem in industry. Currently, the existing researches focus on deterministic design optimization, which may cause unreliable designs without considering uncertainties. Therefore, the study focuses on the implementation of reliability-based design optimization (RBDO) to the RV reducer. The aim is to make the RV reducer smaller in size while ensuring a higher reliability. Firstly, a modified advanced mean value (MAMV) method is proposed to improve efficiency and robustness of the advanced mean value method, which encounters inefficiency and numerical instability for the concave or highly nonlinear performance measure functions. Secondly, a mathematical model of RBDO for the RV reducer is established. Thirdly, the proposed MAMV method is integrated into double-loop method to optimize the established mathematical model of RBDO with different target reliability. The results show that the proposed MAMV method is efficient compared with other methods. In addition, the volume of the RV reducer is correspondingly reduced by 9.44%, 7.89%, and 5.66% compared with that before optimization when the target reliabilities are 99.38%, 99.87%, and 99.98%.

Journal ArticleDOI
TL;DR: In this article, the authors proposed windowed shaped cuttings as a mechanism to reduce the high peak collapsing force (PCL) of the multi-cell hexagonal tubes and systemically investigated the axial crushing of different windowed multicell tubes and also sought for their optimal crashworthiness design.
Abstract: Recently, multi-cell structures have received increased attention for crashworthiness applications due to their superior energy absorption capability. However, such structures were featured with high peak collapsing force (PCL) forming a serious safety concern, and this limited their application for vehicle structures. Accordingly, this paper proposes windowed shaped cuttings as a mechanism to reduce the high PCL of the multi-cell hexagonal tubes and systemically investigates the axial crushing of different windowed multi-cell tubes and also seeks for their optimal crashworthiness design. Three different multi-cell configurations were constructed using wall-to-wall (WTW) and corner-to-corner (CTC) connection webs. Validated finite element models were generated using explicit finite element code, LS-DYNA, and were used to run crush simulations on the studied structures. The crashworthiness responses of the multi-cell standard tubes (STs), i.e., without windows, and multi-cell windowed tubes (WTs) were determined and compared. The WTW connection type was found to be more effective for STs and less favorable for WTs. Design of experiments (DoE), response surface methodology (RSM), and multiple objective particle swarm optimization (MOPSO) tools were employed to find the optimal designs of the different STs and WTs. Furthermore, parametric analysis was conducted to uncover the effects of key geometrical parameters on the main crashworthiness responses of all studied structures. The windowed cuttings were found to be able to slightly reduce the PCL of the multi-cell tubes, but this reduction was associated with a major negative implication on their energy absorption capability. This work provides useful insights on designing effective multi-cell structures suitable for vehicle crashworthiness applications.

Journal ArticleDOI
TL;DR: The results show that the optimized design obtained by the method has slightly higher compliance but significantly lower stress level than the solution without considering the FPS constraint.
Abstract: Previous studies on topology optimization subject to stress constraints usually considered von Mises or Drucker–Prager criterion. In some engineering applications, e.g., the design of concrete structures, the maximum first principal stress (FPS) must be controlled in order to prevent concrete from cracking under tensile stress. This paper presents an effective approach to dealing with this issue. The approach is integrated with the bi-directional evolutionary structural optimization (BESO) technique. The p-norm function is adopted to relax the local stress constraint into a global one. Numerical examples of compliance minimization problems are used to demonstrate the effectiveness of the proposed algorithm. The results show that the optimized design obtained by the method has slightly higher compliance but significantly lower stress level than the solution without considering the FPS constraint. The present methodology will be useful for designing concrete structures.

Journal ArticleDOI
TL;DR: This study develops two machine learning (ML)-based approaches for the design of one-dimensional periodic and non-periodic metamaterial systems and demonstrates the robustness of the proposed two approaches through numerical simulations and design examples.
Abstract: Metamaterial systems have opened new, unexpected, and exciting paths for the design of acoustic devices that only few years ago were considered completely out of reach. However, the development of an efficient design methodology still remains challenging due to highly intensive search in the design space required by the conventional optimization-based approaches. To address this issue, this study develops two machine learning (ML)-based approaches for the design of one-dimensional periodic and non-periodic metamaterial systems. For periodic metamaterials, a reinforcement learning (RL)-based approach is proposed to design a metamaterial that can achieve user-defined frequency band gaps. This RL-based approach surpasses conventional optimization-based methods in the reduction of computation cost when a near-optimal solution is acceptable. Leveraging the capability of exploration in RL, the proposed approach does not require any training datasets generation and therefore can be deployed for online metamaterial design. For non-periodic metamaterials, a neural network (NN)-based approach capable of learning the behavior of individual material units is presented. By assembling the NN representation of individual material units, a surrogate model of the whole metamaterial is employed to determine the properties of the resulting assembly. Interestingly, the proposed approach is capable of modeling different metamaterial assemblies satisfying user-defined properties while requiring only a one-time network training procedure. Also, the NN-based approach does not need a pre-defined number of material unit cells, and it works when the physical model of the unit cell is not well understood, or the situation where only the sensor measurements of the unit cell are available. The robustness of the proposed two approaches is validated through numerical simulations and design examples.

Journal ArticleDOI
TL;DR: In this paper, a topology optimization framework based on the genetic algorithm and finite element method was developed to design periodic barriers embedded in semi-infinite space for reducing surface waves on demand.
Abstract: Dispersion engineering is always the important topic in the field of artificial periodic structures. In particular, topology optimization of composite structures with expected bandgaps plays a key role. However, most reported studies focused on topology optimization for bulk waves, and the optimization for surface wave bandgaps (SWBGs) is still missing. In this paper, we develop a topology optimization framework based on the genetic algorithm and finite element method to design periodic barriers embedded in semi-infinite space for reducing surface waves on demand. The objective functions for SWBGs are proposed based on the energy distribution properties of surface waves. The numerical results show that the optimization framework has stable convergence for this problem and is effective to optimize SWBGs. Considering large SWBGs and low filling fraction of solids, we investigate single- and multi-objective optimizations, respectively, and obtain novel wave barriers with good performance. The beneficial configuration features and mechanism of broadband SWBGs from the optimized results are explored. The results indicate that for the first-order SWBG, most optimized structures consist of a main scatterer at the center and some subsidiary scatterers near the surface. The rigid body resonance of the main scatterer determines the lower edge of SWBG, and the subsidiary scatterers can regulate the upper edge. Higher-order SWBGs are generated from the interaction of multiple scatterers, whose relative distance has a great influence on the position of SWBGs. The optimized structures can make the surface waves propagate far away from the surface within the frequencies of bandgaps, leading to a strong attenuation of surface vibration. In practice, our topological optimization framework is promising in designing high-performance surface wave devices and novel isolating structures for earthquake or environmental vibration in civil engineering.

Journal ArticleDOI
TL;DR: It is proved that the proposed SIMP model is effective for topology optimization of continuum structures including self-weight, and it is found that the optimal structural topology is affected by the ratio of the external force to self- weight.
Abstract: This work proposes an improved topology optimization model for optimizing continuum structures with self-weight loading conditions. A modified Solid Isotropic Material with Penalization (SIMP) model is proposed to avoid the parasitic effect. At the same time, the penalty factor of the SIMP model is increased to maintain the activeness of the prescribed volume constraint and to drive the design domain to binary distribution. The optimization objectives include minimizing the total strain energy of the design domain and minimizing the total displacement of the fixed domain. The shape optimization procedure is used to furtherly enhance structural performance. The whole optimization procedure is implemented with a two-dimensional model under the loading conditions of self-weight and external force. The classic MBB beam and self-weight arch are utilized to verify the proposed method, and conceptual layout designs of the steel structure bridge are conducted. It is proved that the proposed model is effective for topology optimization of continuum structures including self-weight. And it is found that the optimal structural topology is affected by the ratio of the external force to self-weight.

Journal ArticleDOI
TL;DR: Two separate topology optimization MATLAB codes are proposed for a piezoelectric plate in actuation and energy harvesting based on 2D finite element modeling and the optimality criteria and method of moving asymptotes (MMA) are utilized as optimization algorithms.
Abstract: In this paper, two separate topology optimization MATLAB codes are proposed for a piezoelectric plate in actuation and energy harvesting. The codes are written for one-layer piezoelectric plate based on 2D finite element modeling. As such, all forces and displacements are confined in the plane of the piezoelectric plate. For the material interpolation scheme, the extension of solid isotropic material with penalization approach known as PEMAP-P (piezoelectric material with penalization and polarization) which considers the density and polarization direction as optimization variables is employed. The optimality criteria and method of moving asymptotes (MMA) are utilized as optimization algorithms to update the optimization variables in each iteration. To reduce the numerical instabilities during optimization iterations, finite element equations are normalized. The efficiencies of the codes are illustrated numerically by illustrating some basic examples of actuation and energy harvesting. It is straightforward to extend the codes for various problem formulations in actuation, energy harvesting and sensing. The finite element modeling, problem formulation and MATLAB codes are explained in detail to make them appropriate for newcomers and researchers in the field of topology optimization of piezoelectric material.

Journal ArticleDOI
TL;DR: A synthesis approach in a density-based topology optimization setting to design large deformation compliant mechanisms for inducing desired strains in biological tissues using an extended robust formulation based on geometrical nonlinearity and total Lagrangian finite element formulation.
Abstract: This paper presents a synthesis approach in a density-based topology optimization setting to design large deformation compliant mechanisms for inducing desired strains in biological tissues. The modelling is based on geometrical nonlinearity together with a suitably chosen hypereleastic material model, wherein the mechanical equilibrium equations are solved using the total Lagrangian finite element formulation. An objective based on least-square error with respect to target strains is formulated and minimized with the given set of constraints and the appropriate surroundings of the tissues. To circumvent numerical instabilities arising due to large deformation in low stiffness design regions during topology optimization, a strain-energy-based interpolation scheme is employed. The approach uses an extended robust formulation, i.e., the eroded, intermediate, and dilated projections for the design description as well as variation in tissue stiffness. Efficacy of the synthesis approach is demonstrated by designing various compliant mechanisms for providing different target strains in biological tissue constructs. Optimized compliant mechanisms are 3D printed and their performances are recorded in a simplified experiment and compared with simulation results obtained by a commercial software.

Journal ArticleDOI
TL;DR: This work develops a Bayesian optimization framework explicitly exploiting the features inherent to structural design problems, that is, expensive constraints and cheap objectives, which suggests it is well-suited for multi-objective constrained optimization problems in structural design.
Abstract: The planning and design of buildings and civil engineering concrete structures constitutes a complex problem subject to constraints, for instance, limit state constraints from design codes, evaluated by expensive computations such as finite element (FE) simulations. Traditionally, the focus has been on minimizing costs exclusively, while the current trend calls for good trade-offs of multiple criteria such as sustainability, buildability, and performance, which can typically be computed cheaply from the design parameters. Multi-objective methods can provide more relevant design strategies to find such trade-offs. However, the potential of multi-objective optimization methods remains unexploited in structural concrete design practice, as the expensiveness of structural design problems severely limits the scope of applicable algorithms. Bayesian optimization has emerged as an efficient approach to optimizing expensive functions, but it has not been, to the best of our knowledge, applied to constrained multi-objective optimization of structural concrete design problems. In this work, we develop a Bayesian optimization framework explicitly exploiting the features inherent to structural design problems, that is, expensive constraints and cheap objectives. The framework is evaluated on a generic case of structural design of a reinforced concrete (RC) beam, taking into account sustainability, buildability, and performance objectives, and is benchmarked against the well-known Non-dominated Sorting Genetic Algorithm II (NSGA-II) and a random search procedure. The results show that the Bayesian algorithm performs considerably better in terms of rate-of-improvement, final solution quality, and variance across repeated runs, which suggests it is well-suited for multi-objective constrained optimization problems in structural design.

Journal ArticleDOI
TL;DR: The present study shows that the optimal network is not always optimal in terms of reproducing the aerodynamic performance, and can be applied to a real turbine design problem and reduce the design time.
Abstract: An objective of mechanical design is to obtain a shape that satisfies specific requirements. In the present work, we achieve this goal using a conditional variational autoencoder (CVAE). The method enables us to analyze the relationship between aerodynamic performance and the shape of aerodynamic parts, and to explore new designs for the parts. In the CVAE model, a shape is fed as an input and the corresponding aerodynamic performance index is fed as a continuous label. Then, shapes are generated by specifying the continuous label and latent vector. When CVAE is applied to mechanical design, it is desired to draw shapes that reproduce the specified aerodynamic performance. In ordinal CVAE, the model is trained to minimize reconstruction loss and latent loss, and it is usually optimized considering the sum of these losses. However, the present study shows that the optimal network is not always optimal in terms of reproducing the aerodynamic performance. The proposed method is verified using two numerical examples: a two-dimensional (2D) airfoil and a turbine blade. In the airfoil example, we demonstrate the effects of latent dimension, and in the turbine design example, we demonstrate that the proposed method can be applied to a real turbine design problem and reduce the design time.

Journal ArticleDOI
TL;DR: This work summarizes various advanced, yet often straightforward, statistical tools that help surrogate modeling practitioners achieve maximum accuracy within limited resources.
Abstract: Tasks such as analysis, design optimization, and uncertainty quantification can be computationally expensive. Surrogate modeling is often the tool of choice for reducing the burden associated with such data-intensive tasks. However, even after years of intensive research, surrogate modeling still involves a struggle to achieve maximum accuracy within limited resources. This work summarizes various advanced, yet often straightforward, statistical tools that help. We focus on four techniques with increasing popularity in the surrogate modeling community: (i) variable screening and dimensionality reduction in both the input and the output spaces, (ii) data sampling techniques or design of experiments, (iii) simultaneous use of multiple surrogates, and (iv) sequential sampling. We close the paper with some suggestions for future research.

Journal ArticleDOI
TL;DR: This paper reviews and assesses the topic of topology optimization within concrete construction, carrying out an extensive quantitative as well as qualitative review on practical and numerical applications and serves as a guidance for subsequent research.
Abstract: Structural optimization within concrete construction has been increasingly taken up in research within the last two decades. Possible drivers are the need for material-reduced and thus resource-efficient structures as well as recent advancements in automated concrete construction. However, structural concrete is characterized by nonlinear material behavior. Consequently, the merge of structural concrete design and topology optimization is not trivial. This paper reviews and assesses the topic of topology optimization within concrete construction, carrying out an extensive quantitative as well as qualitative review on practical and numerical applications. The following research areas are identified: Multimaterial modeling, stress constraints, concrete damage modeling, strut and tie modeling, combined truss-continuum topology optimization, the consideration of multiple load cases, a focus on construction techniques and alternative approaches. Although the number of research papers dealing with the topic of topology optimization in concrete construction is numerous, there are only few that actually realized topology optimized concrete structures. In addition, only a little number of experiments was performed for an objective evaluation of the found geometries so far. Concluding this review, a list of future challenges, like the incorporation of sustainability measurements within the optimization process, is given and thus serves as a guidance for subsequent research.

Journal ArticleDOI
TL;DR: The results show that the optimization leads to an increase of the gliding range of Petrel-L as much as 83.3% when the hotel load is 0.5 W, which is verified by the sea trial.
Abstract: Underwater glider (UG) is widely applied for long-term ocean observation, the gliding range of which is mainly influenced by its design. In this paper, the design parameters that have obvious influence on the gliding range, including the buoyancy factor, compressibility of the pressure hull, hydrodynamic coefficients, and motion parameters, are selected based on the gliding range model of UG. Due to their complicate coupling relationship in the design of the UG, the multidisciplinary optimization (MDO) design framework integrating the collaborative optimization (CO) method and approximate model technology is adopted to optimize the key parameters by taking the gliding range as the optimization target. The results show that the optimization leads to an increase of the gliding range of Petrel-L as much as 83.3% when the hotel load is 0.5 W, which is verified by the sea trial. The optimization is applicable to other types of underwater gliders.

Journal ArticleDOI
TL;DR: An effective B-spline parameterization method for stiffener layout optimization of shell structures that produces checkerboard-free design results with a clear layout and naturally avoids overhang stiffeners is developed.
Abstract: Thin-walled shell structures are widely used in aeronautical and aerospace engineering. This paper develops an effective B-spline parameterization method for stiffener layout optimization of shell structures. Height variables are defined by B-spline control parameters to characterize the stiffener layout reinforcing the shell structure. A continuous height field is subsequently generated via B-spline and basis functions. In view of possible curvatures of shell structures, the height field is projected from parametric space onto the shell structure by means of the parametric mapping. In this work, the finite element method is adopted with the solid-shell coupling method used for structural analysis. Pseudo-densities associated with solid elements are determined based on the B-spline parameterization and Heaviside function. Several numerical examples are dealt with to demonstrate the proposed method. Compared with the standard density-based method, the proposed method produces checkerboard-free design results with a clear layout and naturally avoids overhang stiffeners.

Journal ArticleDOI
TL;DR: A 250-line Matlab code is presented to handle stiffness, volume and buckling load factors either as the objective function or as constraints and uses the Kreisselmeier-Steinhauser aggregation function to reduce multiple objectives to a single, differentiable one.
Abstract: We present a 250-line Matlab code for topology optimization for linearized buckling criteria. The code is conceived to handle stiffness, volume and buckling load factors (BLFs) either as the objective function or as constraints. We use the Kreisselmeier-Steinhauser aggregation function in order to reduce multiple objectives (viz. constraints) to a single, differentiable one. Then, the problem is sequentially approximated by using MMA-like expansions and an OC-like scheme is tailored to update the variables. The inspection of the stress stiffness matrix leads to a vectorized implementation for its efficient construction and for the sensitivity analysis of the BLFs. This, coupled with the efficiency improvements already presented by Ferrari and Sigmund (Struct Multidiscip Optim 62:2211–2228, 2020a), cuts all the computational bottlenecks associated with setting up the buckling analysis and allows buckling topology optimization problems of an interesting size to be solved on a laptop. The efficiency and flexibility of the code are demonstrated over a few structural design examples and some ideas are given for possible extensions.

Journal ArticleDOI
TL;DR: A MATLAB implementation of the geometrically nonlinear topology optimization code for compliance minimization of statically loaded structures and the complete 137-line code is included as an Appendix.
Abstract: Topology optimization, as a powerful conceptual design method, has been widely adopted in both academic research and industrial applications. To further promote the development of topology optimization, many computer programs have been published for educational purposes over the past decades. However, most of the computer programs are constructed based on a linear assumption. On the basis of bi-directional evolutionary structural optimization (BESO) method, the paper presents a MATLAB implementation of the geometrically nonlinear topology optimization code for compliance minimization of statically loaded structures. Excluding 19 lines which are used for explanation, only 118 lines are needed for the initialization of the design parameters, nonlinear finite element analysis, sensitivity calculation, sensitivity filtration, and topological design variables update. Different design problems can be solved by modifying several lines in the proposed program. The complete 137-line code is included as an Appendix and is intended for educational purposes only.

Journal ArticleDOI
TL;DR: It is demonstrated how implicit geometry modeling using nTop platform can help to interpret these multi-scale designs as single-scale manufacturable designs (de-homogenization) and at least 3 orders of magnitude more efficient compared to standard density-based topology optimization, allowing for high-resolution 3D structures to be obtained on a standard workstation PC.
Abstract: In this article, we demonstrate the state-of-the-art of multi-scale topology optimization for 3D structural design. Many structures designed for additive manufacturing consist of a solid shell surrounding repeated microstructures, so-called infill material. We demonstrate the performance of different types of infill microstructures, such as isotropic truss or plate lattice structures and show that the best results can be obtained using spatially varying and oriented orthotropic microstructures. Furthermore, we demonstrate how implicit geometry modeling using nTop platform can help to interpret these multi-scale designs as single-scale manufacturable designs (de-homogenization). More importantly, we demonstrate the small difference in performance between these multi-scale and single-scale designs through extensive numerical testing. The presented method is at least 3 orders of magnitude more efficient compared to standard density-based topology optimization, allowing for high-resolution 3D structures to be obtained on a standard workstation PC.