Showing papers in "Structural and Multidisciplinary Optimization in 2016"

A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model

Abstract: This paper presents a new topology optimization approach based on the so-called Moving Morphable Components (MMC) solution framework. The proposed method improves several weaknesses of the previous approach (e.g., Guo et al. in J Appl Mech 81:081009, 2014a) in the sense that it can not only allow for components with variable thicknesses but also enhance the numerical solution efficiency substantially. This is achieved by constructing the topological description functions of the components appropriately, and utilizing the ersatz material model through projecting the topological description functions of the components. Numerical examples demonstrate the effectiveness of the proposed approach. In order to help readers understand the essential features of this approach, a 188 line Matlab implementation of this approach is also provided.

Bridging topology optimization and additive manufacturing

Abstract: Topology optimization is a technique that allows for increasingly efficient designs with minimal a priori decisions. Because of the complexity and intricacy of the solutions obtained, topology optimization was often constrained to research and theoretical studies. Additive manufacturing, a rapidly evolving field, fills the gap between topology optimization and application. Additive manufacturing has minimal limitations on the shape and complexity of the design, and is currently evolving towards new materials, higher precision and larger build sizes. Two topology optimization methods are addressed: the ground structure method and density-based topology optimization. The results obtained from these topology optimization methods require some degree of post-processing before they can be manufactured. A simple procedure is described by which output suitable for additive manufacturing can be generated. In this process, some inherent issues of the optimization technique may be magnified resulting in an unfeasible or bad product. In addition, this work aims to address some of these issues and propose methodologies by which they may be alleviated. The proposed framework has applications in a number of fields, with specific examples given from the fields of health, architecture and engineering. In addition, the generated output allows for simple communication, editing, and combination of the results into more complex designs. For the specific case of three-dimensional density-based topology optimization, a tool suitable for result inspection and generation of additive manufacturing output is also provided.

Topology optimization considering overhang constraints: Eliminating sacrificial support material in additive manufacturing through design

Abstract: Additively manufactured components often require temporary support material to prevent the component from collapsing or warping during fabrication. Whether these support materials are removed chemically as in the case of many polymer additive manufacturing processes, or mechanically as in the case of (for example) Direct Metal Laser Sintering, the use of sacrificial material increases total material usage, build time, and time required in post-fabrication treatments. The goal of this work is to embed a minimum allowable self-supporting angle within the topology optimization framework such that designed components and structures may be manufactured without the use of support material. This is achieved through a series of projection operations that combine a local projection to enforce minimum length scale requirements and a support region projection to ensure a feature is adequately supported from below. The magnitude of the self-supporting angle is process dependent and is thus an input variable provided by the manufacturing or design engineer. The algorithm is demonstrated on standard minimum compliance topology optimization problems and solutions are shown to satisfy minimum length scale, overhang angle, and volume constraints, and are shown to be dependent on the allowable magnitudes of these constraints.

Simultaneous material and structural optimization by multiscale topology optimization

Abstract: This paper introduces a new approach to multiscale optimization, where design optimization is applied at two scales: the macroscale, where the structure is optimized, and the microscale, where the material is optimized. Thus, structure and material are optimized simultaneously. We approach multiscale design optimization by linearizing and formulating a new way to decompose into macro and microscale design problems in such a way that solving the decomposed problems separately lead to an overall optimum solution. In addition, the macro and microstructural designs are coupled tightly through homogenization and inverse homogenization. This approach is generic in that it allows any number of unique microstructures and can be applied to a wide range of design problems. An advantage of decomposing the problem in this physical way is that it is potentially straight forward to specify additional design requirements at a specific scale or in specific regions of the design domain. The decomposition approach also allows an easy parallelization of the computational methodology and this enables the computational time to be maintained at a practical level. We demonstrate the proposed approach using the level-set topology optimization at both scales, i.e. macrostructural topological design and microstructural topology of architected material. A series of optimization problems, minimizing compliance and compliant mechanism are solved for verification and investigation of potential benefits.

Parallel surrogate-assisted global optimization with expensive functions --- a survey

Abstract: Surrogate assisted global optimization is gaining popularity. Similarly, modern advances in computing power increasingly rely on parallelization rather than faster processors. This paper examines some of the methods used to take advantage of parallelization in surrogate based global optimization. A key issue focused on in this review is how different algorithms balance exploration and exploitation. Most of the papers surveyed are adaptive samplers that employ Gaussian Process or Kriging surrogates. These allow sophisticated approaches for balancing exploration and exploitation and even allow to develop algorithms with calculable rate of convergence as function of the number of parallel processors. In addition to optimization based on adaptive sampling, surrogate assisted parallel evolutionary algorithms are also surveyed. Beyond a review of the present state of the art, the paper also argues that methods that provide easy parallelization, like multiple parallel runs, or methods that rely on population of designs for diversity deserve more attention.

Thickness control in structural optimization via a level set method

Abstract: In the context of structural optimization via a level-set method we propose a framework to handle geometric constraints related to a notion of local thickness. The local thickness is calculated using the signed distance function to the shape. We formulate global constraints using integral functionals and compute their shape derivatives. We discuss different strategies and possible approximations to handle the geometric constraints. We implement our approach in two and three space dimensions for a model of linearized elasticity. As can be expected, the resulting optimized shapes are strongly dependent on the initial guesses and on the specific treatment of the constraints since, in particular, some topological changes may be prevented by those constraints.

Improving kriging surrogates of high-dimensional design models by Partial Least Squares dimension reduction

Abstract: Engineering computer codes are often computationally expensive. To lighten this load, we exploit new covariance kernels to replace computationally expensive codes with surrogate models. For input spaces with large dimensions, using the kriging model in the standard way is computationally expensive because a large covariance matrix must be inverted several times to estimate the parameters of the model. We address this issue herein by constructing a covariance kernel that depends on only a few parameters. The new kernel is constructed based on information obtained from the Partial Least Squares method. Promising results are obtained for numerical examples with up to 100 dimensions, and significant computational gain is obtained while maintaining sufficient accuracy.

Global sensitivity analysis-enhanced surrogate (GSAS) modeling for reliability analysis

Abstract: An essential issue in surrogate model-based reliability analysis is the selection of training points. Approaches such as efficient global reliability analysis (EGRA) and adaptive Kriging Monte Carlo simulation (AK-MCS) methods have been developed to adaptively select training points that are close to the limit state. Both the learning functions and convergence criteria of selecting training points in EGRA and AK-MCS are defined from the perspective of individual responses at Monte Carlo samples. This causes two problems: (1) some extra training points are selected after the reliability estimate already satisfies the accuracy target; and (2) the selected training points may not be the optimal ones for reliability analysis. This paper proposes a Global Sensitivity Analysis enhanced Surrogate (GSAS) modeling method for reliability analysis. Both the convergence criterion and strategy of selecting new training points are defined from the perspective of reliability estimate instead of individual responses of MCS samples. The new training points are identified according to their contribution to the uncertainty in the reliability estimate based on global sensitivity analysis. The selection of new training points stops when the accuracy of the reliability estimate reaches a specific target. Five examples are used to assess the accuracy and efficiency of the proposed method. The results show that the efficiency and accuracy of the proposed method are better than those of EGRA and AK-MCS.

On the (non-)optimality of Michell structures

Abstract: Optimal analytical Michell frame structures have been extensively used as benchmark examples in topology optimization, including truss, frame, homogenization, density and level-set based approaches. However, as we will point out, partly the interpretation of Michell's structural continua as discrete frame structures is not accurate and partly, it turns out that limiting structural topology to frame-like structures is a rather severe design restriction and results in structures that are quite far from being stiffness optimal. The paper discusses the interpretation of Michell's theory in the context of numerical topology optimization and compares various topology optimization results obtained with the frame restriction to cases with no design restrictions. For all examples considered, the true stiffness optimal structures are composed of sheets (2D) or closed-walled shell structures (3D) with variable thickness. For optimization problems with one load case, numerical results in two and three dimensions indicate that stiffness can be increased by up to 80 % when dropping the frame restriction. For simple loading situations, studies based on optimal microstructures reveal theoretical gains of +200 %. It is also demonstrated how too coarse design discretizations in 3D can result in unintended restrictions on the design freedom and achievable compliance.

Truss optimization with discrete design variables: a critical review

Abstract: This review presents developed models, theory, and numerical methods for structural optimization of trusses with discrete design variables in the period 1968 --- 2014. The comprehensive reference list collects, for the first time, the articles in the field presenting deterministic optimization methods and meta heuristics. The field has experienced a shift in focus from deterministic methods to meta heuristics, i.e. stochastic search methods. Based on the reported numerical results it is however not possible to conclude that this shift has improved the competences to solve application relevant problems. This, and other, observations lead to a set of recommended research tasks and objectives to bring the field forward. The development of a publicly available benchmark library is urgently needed to support development and assessment of existing and new heuristics and methods. Combined with this effort, it is recommended that the field begins to use modern methods such as performance profiles for fair and accurate comparison of optimization methods. Finally, theoretical results are rare in this field. This means that most recent methods and heuristics are not supported by mathematical theory. The field should therefore re-focus on theoretical issues such as problem analysis and convergence properties of new methods.

A geometry projection method for the topology optimization of plate structures

Abstract: We introduce a topology optimization method for the stiffness-based design of structures made of plates. Our method renders topologies made distinctly of plates, thereby producing designs that better conform to manufacturing processes tailored to plate structures, such as those that employ stock plates that are cut and joined by various means. To force the structural members to be plates, we employ the geometry projection method to project an analytical description of a set of fixed-thickness plates onto a continuous density field defined over a 3-dimensional, uniform finite element grid for analysis. A size variable is assigned to each plate and penalized so that the optimizer can entirely remove a plate from the design. The proposed method accommodates the case where the plates in the topology are rectangular and solid, and the case where the boundaries of the plates can change and holes can be introduced. The latter case is attained by composition with a free density field. We present examples that demonstrate the effectiveness of our method and discuss future work.

A comprehensive survey on topology optimization of phononic crystals

Abstract: The objective of this paper is to present a peer-review of the literature and trends surrounding the design of phononic crystals (PnCs) using topology optimization methods. After first providing a brief review of new developments, improvements, and applications of PnCs, this paper investigates the techniques and applications of topology optimization methods for phononic band gaps and functional structures. Both gradient-based and non-gradient-based topology optimization methods have been employed in the design of PnCs. The advantages and drawbacks of the methods used in this area are discussed in detail in this paper. Based on observations of the current state, we highlight suggestions and ideas for future research in the field of phononic crystal design optimization. The paper examines how applying topology optimization techniques into the design of PnCs yields promising performance and functions. This literature survey is designed to provide an overview of the recent advances and novel applications of popular topology optimization methods for the design of PnCs.

Topology optimization: a review for structural designs under vibration problems

Abstract: This article provides a comprehensive review of structural optimization employing topology methods for structures under vibration problems. Topology optimization allows creative and radical design modifications, compared to shape and size optimization techniques. Various works of structural topology optimization, which are subjected to vibration as the response function of the optimization process, are reviewed. Different types of calculus and numerical methods commonly used for solving structural topological optimization problems are briefly discussed. Moreover, different aspects of topology optimization related to vibration problems are explained. The articles reviewed are largely confined to linear systems that concern small vibration amplitudes. Accordingly, the works related to vibration topological optimization are classified according to the method employed (homogenization, evolutionary structural optimization, solid isotropic material with penalization, or level set). The reviewed works are tabulated according to their methodology, year, and the objective functions and applications of each work. Although the homogenization and evolutionary methods were common in the past, the solid isotropic material with penalization (SIMP) method is the most popular method applied in recent years. The advantages of the level set method show promise for future applications.

Reliability-based design optimization of wind turbine blades for fatigue life under dynamic wind load uncertainty

Abstract: This paper studies reliability-based design optimization (RBDO) of a 5-MW wind turbine blade for designing reliable as well as economical wind turbine blades. A novel dynamic wind load uncertainty model has been developed using 249 groups of wind data to consider wind load variation over a large spatiotemporal range. The probability of fatigue failure during a 20-year service life is estimated using the uncertainty model in the RBDO process and is reduced to meet a desired target reliability. Meanwhile, the cost of composite materials used in the blade is minimized by optimizing the composite laminate thicknesses of the blade. In order to obtain the RBDO optimum design efficiently, deterministic design optimization (DDO) of the 5-MW wind turbine blade is carried out first using the mean wind load obtained from the wind load uncertainty model. The RBDO is then initiated from the DDO optimum. During the RBDO iterations, fatigue hotspots for RBDO are identified among the laminate section points. For an efficient RBDO process, surrogate models of 10-min fatigue damages D10 at the hotspots are accurately created using the Kriging method. Using the wind load uncertainty model and surrogate models, probability of fatigue failure during a 20-year lifespan at the hotspots and the design sensitivities are calculated at given design points. Using the probability of fatigue failure and design sensitivity, RBDO of the 5-MW wind turbine blade has been successfully carried out, satisfying the target probability of failure of 2.275 %.

Evolutionary topological design for phononic band gap crystals

Abstract: Phononic band gap crystals are made of periodic inclusions embedded in a base material, which can forbid the propagation of elastic and acoustic waves within certain range of frequencies. In the past two decades, the systematic design of phononic band gap crystals has attracted increasing attention due to their wide practical applications such as sound insulation, waveguides, or acoustic wave filtering. This paper proposes a new topology optimization algorithm based on bi-directional evolutionary structural optimization (BESO) method and finite element analysis for the design of phononic band gap crystals. The study on the maximizing gap size between two adjacent bands has been systematically conducted for out-of-plane waves, in-plane waves and the coupled in-plane and out-of-plane waves. Numerical results demonstrate that the proposed optimization algorithm is effective and efficient for the design of phononic band gap crystals and various topological patterns of optimized phononic structures are presented. Several new patterns for phononic band gap crystals have been successfully obtained.

Structural topology optimization considering connectivity constraint

Abstract: Topology optimization has been regarded as a powerful design approach for determining optimal topology of a structure to obtain desired functional performances within a defined design domain. Considering manufacturing process constraints in topology optimization becomes increasingly important due to its potential practical applications. In this paper, we propose a novel topology optimization model with manufacturing process related connectivity constraints. A generalized method, named as virtual scalar field method (VSFM), is developed for describing and enforcing desired connectivity constraint. As an illustrative example, the connectivity constraint can be converted to an equivalent maximum temperature constraint when temperature is chosen as the scalar field. The temperature constraint is then easily integrated and implemented in routine topology optimization. The simply-connected constraint, which excludes interior closed cavities and is representative of many advanced manufacturing techniques, e.g. additive manufacturing (AM) or casting, is used as an example to demonstrate the key ideas and the efficiency of the VSF method. Some numerical examples, which consider the connectivity constraint in topology optimization, are presented to show the validity of this method.

Industrial application of topology optimization for combined conductive and convective heat transfer problems

Abstract: This paper presents an industrial application of topology optimization for combined conductive and convective heat transfer problems. The solution is based on a synergy of computer aided design and engineering software tools from Dassault Systemes. The considered physical problem of steady-state heat transfer under convection is simulated using SIMULIA-Abaqus. A corresponding topology optimization feature is provided by SIMULIA-Tosca. By following a standard workflow of design optimization, the proposed solution is able to accommodate practical design scenarios and results in efficient conceptual design proposals. Several design examples with verification results are presented to demonstrate the applicability.

Matlab codes of Subset Simulation for reliability analysis and structural optimization

Abstract: This paper presents two efficient and compact Matlab codes of Subset Simulation for reliability analysis and structural optimization. The codes for reliability analysis and structural optimization comprise of the direct Monte Carlo and Markov Chain Monte Carlo. The theoretical and numerical elements of Subset Simulation are briefly presented in this paper, as well as the detailed instructions to implement the standard codes for solving reliability analysis and structural optimization problems. The paper also discusses simple extensions of argument check, post-processing, alternative stop criterion and constraint-handling. Four examples are presented to demonstrate these codes, two for reliability analysis and two for structural optimization. This paper will be helpful for the students and newcomers both in reliability analysis and structural optimization to understand and use Subset Simulation. The complete codes are included in Appendixes 1 and 2, and they can be downloaded from https://sites.google.com/site/rasosubsim/ .

Quantile-based optimization under uncertainties using adaptive Kriging surrogate models

Abstract: Uncertainties are inherent to real-world systems. Taking them into account is crucial in industrial design problems and this might be achieved through reliability-based design optimization (RBDO) techniques. In this paper, we propose a quantile-based approach to solve RBDO problems. We first transform the safety constraints usually formulated as admissible probabilities of failure into constraints on quantiles of the performance criteria. In this formulation, the quantile level controls the degree of conservatism of the design. Starting with the premise that industrial applications often involve high-fidelity and time-consuming computational models, the proposed approach makes use of Kriging surrogate models (a.k.a. Gaussian process modeling). Thanks to the Kriging variance (a measure of the local accuracy of the surrogate), we derive a procedure with two stages of enrichment of the design of computer experiments (DoE) used to construct the surrogate model. The first stage globally reduces the Kriging epistemic uncertainty and adds points in the vicinity of the limit-state surfaces describing the system performance to be attained. The second stage locally checks, and if necessary, improves the accuracy of the quantiles estimated along the optimization iterations. Applications to three analytical examples and to the optimal design of a car body subsystem (minimal mass under mechanical safety constraints) show the accuracy and the remarkable efficiency brought by the proposed procedure.

Design optimization of a novel NPR crash box based on multi-objective genetic algorithm

Abstract: Possessing the unique properties of lower mass and higher performances, the structure with Negative Poisson's Ratio (NPR) can be widely used in aerospace and vehicle industry. By combing the NPR structure filled core and the traditional crash box, a novel NPR crash box is first proposed in this work to improve the performances of the crash box. The performances of the novel NPR crash box are fully studied by comparing to the traditional crash box and the aluminum foam filled crash box. A parameterized model of the NPR crash box, which integrates the design parameters of the basic NPR cell structure, is built to improve the analysis and optimization efficiency, the accuracy of the parameterized model is also verified by comparing to traditional FEM model. Multi-objective optimization model of the NPR crash box is established by combining the parameterized model, optimal Latin square design method and response surface model approach. Non-dominated sorting genetic algorithm-II (NSGA-II) is then applied to optimize the design parameters of the basic NPR cell structure to improve the performances of the NPR crash box. The results indicate that the novel NPR crash box can improve the performances of the crash box remarkably and the combination of parameterized model and multi-objective genetic algorithms optimize the NPR crash box efficiently. The presented new method also serves as a good example for other application and optimization of NPR structure.

Level set topology optimization of cooling and heating devices using a simplified convection model

Abstract: This paper studies topology optimization of convective heat transfer problems in two and three dimensions. The convective fluxes are approximated by Newton's Law of Cooling (NLC). The geometry is described by a Level Set Method (LSM) and the temperature field is predicted by the eXtended Finite Element Method (XFEM). A constraint on the spatial gradient of the level set field is introduced to penalize small, sub-element-size geometric features. Numerical studies show that the LSM-XFEM provides improved accuracy over previously studied density methods and LSMs using Ersatz material models. It is shown that the NLC model with an iso-thermal fluid phase may over predict the convective heat flux and thus promote the formation of very thin fluid channels, depending on the Biot number characterizing the heat transfer problem. Approximating the temperature field in the fluid phase by a diffusive model mitigates this issue but an explicit feature size control is still necessary to prevent the formation of small solid members, in particular at low Biot numbers. The proposed constraint on the gradient of the level set field is shown to suppress sub-element-size features but necessitates a continuation strategy to prevent the optimization process from stagnating as geometric features merge.

A new metaheuristic for continuous structural optimization: water evaporation optimization

Abstract: The paper proposes a novel physically inspired population-based metaheuristic algorithm for continuous structural optimization called as Water Evaporation Optimization (WEO). WEO mimics the evaporation of a tiny amount of water molecules adhered on a solid surface with different wettability which can be studied by molecular dynamics simulations. A set of six truss design problems from the small to normal scale are considered for evaluating the WEO. The most effective available state-of-the-art metaheuristic optimization methods are used as basis of comparison. The optimization results demonstrate the efficiency and robustness of the WEO and its competitive performance to other algorithms for continuous structural optimization problems.

An efficient approach to reliability-based topology optimization for continua under material uncertainty

Abstract: This contribution presents a computationally efficient method for reliability-based topology optimization for continuum domains under material properties uncertainty. Material Young's modulus is assumed to be lognormally distributed and correlated within the domain. The computational efficiency is achieved through estimating the response statistics with stochastic perturbation of second order, using these statistics to fit an appropriate distribution that follows the empirical distribution of the response, and employing an efficient gradient-based optimizer. Two widely-studied topology optimization problems are examined and the changes in the optimized topology is discussed for various levels of target reliability and correlation strength. Accuracy of the proposed algorithm is verified using Monte Carlo simulation.

Stress-based design of thermal structures via topology optimization

Abstract: The design of thermal structures in the aerospace industry, including exhaust structures on embedded engine aircraft and hypersonic thermal protection systems, poses a number of complex design challenges. These challenges are particularly well addressed by the material layout capabilities of structural topology optimization; however, no topology optimization methods are readily available with the necessary thermoelastic considerations for these problems. This is due in large part to the emphasis on cases of maximum stiffness design for structures subjected to externally applied mechanical loads in the majority of topology optimization applications. In addition, while limited work in the literature has investigated thermoelastic topology optimization, a direct treatment of thermal stresses is not well documented. Such a treatment is critical in the design of thermal structures where excessive thermal stresses are a primary failure mode. In this paper, we present a method for the topology optimization of structures with combined mechanical and thermoelastic (temperature) loads that are subject to stress constraints. We present the necessary steps needed to address both the design-dependent thermal loads and accommodate the challenges of stress-based design criteria. A relaxation technique is utilized to remove the singularity phenomenon in stresses and the large number of stress constraints is handled using a scaled aggregation technique that has been shown previously to satisfy prescribed stress limits in mechanical problems. Finally, the stress-based thermoelastic formulation is applied to two numerical example problems to demonstrate its effectiveness.

Dynamic response topology optimization in the time domain using model reduction method

Abstract: The dynamic response topology optimization problems are usually computationally expensive, so it is necessary to employ the model reduction methods to reduce computational cost. This work will investigate the effectiveness of the mode displacement method(MDM) and mode acceleration method(MAM) for time-domain response problems within the framework of density-based topology optimization. Three objective functions, the mean dynamic compliance, mean strain energy and mean squared displacement are considered. It is found that, in general cases, MDM is not suitable for time-domain response topology optimization problems due to its low accuracy of approximation, while MAM works well for problems of a wide range, and when there are many time steps, the MAM based topology optimization approach is more efficient than the direct integration based approach. So for practical applications, when the problem needs many time steps, the MAM based approach is preferred and otherwise, the direct integration based approach is suggested.

Crashworthiness design of vehicle structure with tailor rolled blank

Abstract: Lightweight and crashworthiness design have been two main challenges in the vehicle industry. These two performances often conflict with each other. To not sacrifice vehicle crashworthiness performance when performing vehicle lightweight design, a novel inner part of front longitudinal beam (FLB-inner) structure with a tailor rolled blank (TRB) concept is proposed in this work, and the corresponding design method is also proposed to minimize the weight of FLB-inner. Firstly, a full-scale vehicle finite element model is adopted and experimentally verified. Secondly, the conventional uniform thickness FLB-inner panel is replaced with a TRB structure, herein, the FLB-inner is divided into four segments with different thickness according to the crashworthiness requirements of frontal impact. Then the material constitutive model and finite element modeling for TRB is established. Thirdly, the optimal Latin hypercube sampling (OLHS) technique is used to generate sampling points and the objective and constraints function values are calculated using commercial software LS-DYNA. Based on the simulation results, the ?-SVR surrogate models are constructed. Finally, the artificial bee colony (ABC) algorithm is applied to obtain the optimal thickness distribution of FLB-inner. The results indicated that the weight of the FLB-inner is reduced by 15.21 %, while the crashworthiness is mproved in comparison with the baseline design.

Simultaneous size, shape, and topology optimization of truss structures using integrated particle swarm optimizer

Abstract: The current investigation deals with the weight minimization of truss structures accomplishing the simultaneous shape, size, and topology optimization. In this regard, this study presents an effective algorithm called integrated particle swarm optimizer (iPSO) as an optimization tool. The iPSO combines favorable features of the standard PSO with an efficient concept of `weighted particle' to improve its performance. In addition, `improved fly-back' technique is introduced to handle the problem constraints. The proposed methodology is tested on a series of benchmark problems and the obtained results are compared with those available in the technical literature. The iPSO achieves the results which are capable of competitive with those obtained by other techniques used for simultaneous optimization of truss structures and reported in the literature. Furthermore, the relative simplicity of the formulation can be considered as one of the significant features of this method.

Relaxed performance measure approach for reliability-based design optimization

Abstract: The efficiency and robustness of reliability analysis methods are important factors to evaluate the probabilistic constraints in reliability-based design optimization (RBDO). In this paper, a relaxed mean value (RMV) approach is proposed in order to evaluate probabilistic constraints including convex and concave functions in RBDO using the performance measure approach (PMA). A relaxed factor is adaptively determined in the range from 0 to 2 using an inequality criterion to improve the efficiency and robustness of the inverse first-order reliability methods. The performance of the proposed RMV is compared with six existing reliability methods, including the advanced mean value (AMV), conjugate mean value (CMV), hybrid mean value (HMV), chaos control (CC), modified chaos control (MCC), and conjugate gradient analysis (CGA) methods, through four nonlinear concave and convex performance functions and three RBDO problems. The results demonstrate that the proposed RMV is more robust than the AMV, CMV, and HMV for highly concave problems, and slightly more efficient than the CC, MCC, and CGA methods. Furthermore, the proposed relaxed mean value guarantees robust and efficient convergence for RBDO problems with highly nonlinear performance functions.

Topology optimization for crashworthiness of thin-walled structures under axial impact using hybrid cellular automata

Abstract: Although topology optimization is well established in most engineering fields, it is still in its infancy concerning highly non-linear structural applications like vehicular crashworthiness. One of the approaches recently proposed and based on Hybrid Cellular Automata is modified here such that it can be applied for the first time to thin-walled structures. Classical methods based on voxel techniques, i.e., on solid three-dimensional volume elements, cannot derive structures made from thin metal sheets where the main energy absorption mode is related to plastic buckling, folding and failure. Because the main components of car structures are made from such thin-walled beams and panels, a special approach using SFE CONCEPT was developed, which is presented in this paper.

Multi-start Space Reduction (MSSR) surrogate-based global optimization method

Abstract: In this work, a new Multi-start Space Reduction (MSSR) and surrogate-based search algorithm is introduced for solving global optimization problems with computationally expensive black-box objective and/or expensive black-box constraint functions. In this new algorithm, the design space is classified into: the original design space or global space (GS), the reduced medium space (MS) that contains the promising region, and the local space (LS) that is a local area surrounding the present best solution in the search. During the search, a kriging-based multi-start optimization process is used for local optimization, sample selection and exploration. In this process, Latin hypercube sampling is used to acquire the starting points and sequential quadratic programming (SQP) is used for the local optimization. Based upon a newly introduced selection strategy, better sample points are obtained to supplement the kriging model, and the estimated mean square error of kriging is used to guide the search of the unknown areas. The multi-start search process is carried out alternately in GS, MS and LS until the global optimum is identified. The newly introduced MSSR algorithm was tested using various optimization benchmark problems, including fifteen bound constrained examples, two nonlinear constrained optimization problems, and four nonlinear constrained engineering applications. The test results revealed noticeable advantages of the new algorithm in dealing with computationally expensive black-box problems. In comparison with two nature-inspired algorithms, three space exploration methods, and two recently introduced surrogate-based global optimization algorithms, MSSR showed improved search efficiency and robustness.