Original Design Space Research Articles

A New Sampling Approach for the Multi-Scale Design of Metallic Materials

Abstract We present a new sampling method for the multi-scale design of polycrystalline materials, which improves the computational time efficiency compared to the existing computational approaches. The solution strategy aims to find microstructure designs that optimize component-scale mechanical properties. The microstructure is represented with a probabilistic texture descriptor that quantifies the volume fractions of different crystallographic orientations. However, the original microstructure design space is high-dimensional and thus optimization in this domain is not favorable. Instead, we generate property closures, which are the reduced spaces of volume-averaged material properties that are computed in terms of the microstructural texture descriptors. We observe that the traditional design approaches which are based on sampling in the original microstructure space and sampling on the property closure are inefficient as they lead to highly concentrated design samples in the solution space. Therefore, we introduce a new sampling method in the property closure, which creates simplexes using the triangulation of the property hull and then generating samples for each simplex. Example problems include the optimization of Galfenol and α-titanium microstructures to improve non-linear material properties. The new sampling approach is shown to obtain better solutions while decreasing the required computational time compared to the previous microstructure design methods.

Оптимизация конструкции и экспериментальное исследование напряжённо-деформированного состояния балансира задней подвески вездеходного транспортного средства

Reducing the curb weight of wheeled vehicles has long been one of the priority areas of work of automotive engineers, since this can significantly improve the operational properties of a wheeled vehicle: improve dynamics, passability, reduce fuel consumption and emissions of harmful sub-stances. A significant proportion of the vehicle's curb weight belongs to highly loaded parts of the frame, transmission and suspension. Therefore, the creation of lightweight, highly loaded parts will make a significant contribution to reducing the curb weight of the whole vehicle. The paper describes the application of the topological optimization method based on finite ele-ment modeling in the design of highly loaded parts of the chassis of vehicle. An example of the syn-thesis of the power circuit of the rear suspension balance bar of an all-terrain vehicle with a descrip-tion of the design model, load modes and interpretation of the results is shown. The optimization problem was solved using a finite element model of varying density. Minimization of the potential energy of deformation was used as an objective function, and the target volume in fractions of the original design space was used as a limitation. A comparative analysis of the obtained design with analogous designs is presented. The formulation and results of an experimental study of the stress-strain state of the optimized balance bar are described. As a result of optimization, it was possible to achieve a reduction in the weight of the balance bar to 49% in comparison with an analogue design while maintaining the required strength. Experi-mental verification of the bearing capacity of the balance bar showed the need for more thorough verification calculations of optimized parts, including taking into account manufacturing and as-sembly errors.

Mapping the global design space of nanophotonic components using machine learning pattern recognition

Nanophotonics finds ever broadening applications requiring complex components with many parameters to be simultaneously designed. Recent methodologies employing optimization algorithms commonly focus on a single performance objective, provide isolated designs, and do not describe how the design parameters influence the device behaviour. Here we propose and demonstrate a machine-learning-based approach to map and characterize the multi-parameter design space of nanophotonic components. Pattern recognition is used to reveal the relationship between an initial sparse set of optimized designs through a significant reduction in the number of characterizing parameters. This defines a design sub-space of lower dimensionality that can be mapped faster by orders of magnitude than the original design space. The behavior for multiple performance criteria is visualized, revealing the interplay of the design parameters, highlighting performance and structural limitations, and inspiring new design ideas. This global perspective on high-dimensional design problems represents a major shift in modern nanophotonic design and provides a powerful tool to explore complexity in next-generation devices.

Energy-Efficient and Quality-Assured Approximate Computing Framework Using a Co-Training Method

Approximate computing is a promising design paradigm that introduces a new dimension—error—into the original design space. By allowing the inexact computation in error-tolerance applications, approximate computing can gain both performance and energy efficiency. A neural network (NN) is a universal approximator in theory and possesses a high level of parallelism. The emerging deep neural network accelerators deployed with NN-based approximator is thereby a promising candidate for approximate computing. Nevertheless, the approximation result must satisfy the users’ requirement, and the approximation result varies across different applications. We normally deploy an NN-based classifier to ensure the approximation quality. Only the inputs predicted to meet the quality requirement can be executed by the approximator. The potential of these two NNs, however, is fully explored; the involving of two NNs in approximate computing imposes critical optimization questions, such as two NNs’ distinct views of the input data space, how to train the two correlated NNs, and what are their topologies. In this article, we propose a novel NN-based approximate computing framework with quality insurance. We advocate a co-training approach that trains the classifier and the approximator alternately to maximize the agreement of the two NNs on the input space. In each iteration, we coordinate the training of the two NNs with a judicious selection of training data. Next, we explore different selection policies and propose to select training data from multiple iterations, which can enhance the invocation of the approximate accelerator. In addition, we optimize the classifier by integrating a dynamic threshold tuning algorithm to improve the invocation of the approximate accelerator further. The increased invocation of accelerator leads to higher energy efficiency under the same quality requirement. We propose two efficient algorithms to explore the smallest topology of the NN-based approximator and the classifier to achieve the quality requirement. The first algorithm straightforward searches the minimum topology using a greedy strategy. However, the first algorithm incurs too much training overhead. To solve this issue, the second one gradually grows the topology of NNs to match the quality requirement by transferring the learned parameters. Experimental results show significant improvement on the quality and the energy efficiency compared to the existing NN-based approximate computing frameworks.

Global aerodynamic design optimization based on data dimensionality reduction

In aerodynamic optimization, global optimization methods such as genetic algorithms are preferred in many cases because of their advantage on reaching global optimum. However, for complex problems in which large number of design variables are needed, the computational cost becomes prohibitive, and thus original global optimization strategies are required. To address this need, data dimensionality reduction method is combined with global optimization methods, thus forming a new global optimization system, aiming to improve the efficiency of conventional global optimization. The new optimization system involves applying Proper Orthogonal Decomposition (POD) in dimensionality reduction of design space while maintaining the generality of original design space. Besides, an acceleration approach for samples calculation in surrogate modeling is applied to reduce the computational time while providing sufficient accuracy. The optimizations of a transonic airfoil RAE2822 and the transonic wing ONERA M6 are performed to demonstrate the effectiveness of the proposed new optimization system. In both cases, we manage to reduce the number of design variables from 20 to 10 and from 42 to 20 respectively. The new design optimization system converges faster and it takes 1/3 of the total time of traditional optimization to converge to a better design, thus significantly reducing the overall optimization time and improving the efficiency of conventional global design optimization method.

An active learning radial basis function modeling method based on self-organization maps for simulation-based design problems

The radial basis function has been widely used in constructing metamodels as response surfaces. Yet, it often faces the challenge of accuracy if a sequential sampling strategy is used to insert samples sequentially and refine the models, especially under the constraint of computational resources. In this paper, a sensitive region pursuing based active learning radial basis function (SRP-ALRBF) metamodeling approach is proposed to sequentially exploit the already-acquired knowledge in the modeling process for obtaining a desirable estimation of the relationship between the input design variables and the output response. In this method, the leave-one-out (LOO) errors of each sample point are taken to identify the boundaries of sensitive regions. According to the obtained LOO information, the original design space is divided into some subspaces by adopting the self-organization maps (SOMs). The boundary of the most sensitive region, where the output response is multi-modal or non-smooth with abrupt changes, is determined by the topological graph generated by cluster analysis in SOMs. In the most sensitive region, infill sample point searching is performed based on an optimization formulation. Ten numerical examples are used to compare the proposed SRP-ALRBF with four existing active learning RBF metamodeling approaches. Results show the advantage of the proposed SRP-ALRBF approach in both prediction accuracy and robustness. It is also applied to three engineering cases to illustrate its ability to support complex engineering design.

Artificial neural networks to predict aerodynamic coefficients of transport airplanes

PurposeMultidisciplinary design frameworks elaborated for aeronautical applications require considerable computational power that grows enormously with the utilization of higher fidelity tools to model aeronautical disciplines like aerodynamics, loads, flight dynamics, performance, structural analysis and others. Surrogate models are a good alternative to address properly and elegantly this issue. With regard to this issue, the purpose of this paper is the design and application of an artificial neural network to predict aerodynamic coefficients of transport airplanes. The neural network must be fed with calculations from computational fluid dynamic codes. The artificial neural network system that was then developed can predict lift and drag coefficients for wing-fuselage configurations with high accuracy. The input parameters for the neural network are the wing planform, airfoil geometry and flight condition. An aerodynamic database consisting of approximately 100,000 cases calculated with a full-potential code with computation of viscous effects was used for the neural network training, which is carried out with the back-propagation algorithm, the scaled gradient algorithm and the Nguyen–Wridow weight initialization. Networks with different numbers of neurons were evaluated to minimize the regression error. The neural network featuring the lowest regression error is able to reduce the computation time of the aerodynamic coefficients 4,000 times when compared with the computing time required by the full potential code. Regarding the drag coefficient, the average error of the neural network is of five drag counts only. The computation of the gradients of the neural network outputs in a scalable manner is possible by an adaptation of back-propagation algorithm. This enabled its use in an adjoint method, elaborated by the authors and used for an airplane optimization task. The results from that optimization were compared with similar tasks performed by calling the full potential code in another optimization application. The resulting geometry obtained with the aerodynamic coefficient predicted by the neural network is practically the same of that designed directly by the call of the full potential code.Design/methodology/approachThe aerodynamic database required for the neural network training was generated with a full-potential multiblock-structured code. The training process used the back-propagation algorithm, the scaled-conjugate gradient algorithm and the Nguyen–Wridow weight initialization. Networks with different numbers of neurons were evaluated to minimize the regression error.FindingsA suitable and efficient methodology to model aerodynamic coefficients based on artificial neural networks was obtained. This work also suggests appropriate sizes of artificial neural networks for this specific application. We demonstrated that these metamodels for airplane optimization tasks can be used without loss of fidelity and with great accuracy, as their local minima might be relatively close to the minima of the original design space defined by the call of computational fluid dynamics codes.Research limitations/implicationsThe present work demonstrated the ability of a metamodel with artificial neural networks to capture the physics of transonic and subsonic flow over a wing-fuselage combination. The formulation that was used was the full potential equation. However, the present methodology can be extended to model more complex formulations such as the Euler and Navier–Stokes ones.Practical implicationsOptimum networks reduced the computation time for aerodynamic coefficient calculations by 4,000 times when compared with the full-potential code. The average absolute errors obtained were of 0.004 and 0.0005 for lift and drag coefficient prediction, respectively. Airplane configurations can be evaluated more quickly.Social implicationsIf multidisciplinary optimization tasks for airplane design become more efficient, this means that more efficient airplanes (for instance less polluting airplanes) can be designed. This leads to a more sustainable aviation.Originality/valueThis research started in 2005 with a master thesis. It was steadily improved with more efficient artificial neural networks able to handle more complex airplane geometries. There is a single work using similar techniques found in a conference paper published in 2007. However, that paper focused on the application, i.e. providing very few details of the methodology to model aerodynamic coefficients.

Optimization of an implicit constrained multi-physics system for motor wheels of electric vehicle

In this paper, implicit constrained multi-physics model of a motor wheel for an electric vehicle is built and then optimized. A novel optimization approach is proposed to solve the compliance problem between implicit constraints and stochastic global optimization. Firstly, multi-physics model of motor wheel is built from the theories of structural mechanics, electromagnetism and thermal physics. Then, implicit constraints are applied from the vehicle performances and magnetic characteristics. Implicit constrained optimization is carried out by a series of unconstrained optimization and verifications. In practice, sequentially updated subspaces are designed to completely substitute the original design space in local areas. In each subspace, a solution is obtained and is then verified by the implicit constraints. Optimal solutions which satisfy the implicit constraints are accepted as final candidates. The final global optimal solution is optimized from those candidates. Discussions are carried out to discover the differences between optimal solutions with unconstrained problem and different implicit constrained problems. Results show that the implicit constraints have significant influences on the optimal solution and the proposed approach is effective in finding the optimals.

An efficient method of solving design optimization problems with arbitrary random distributions in gradient-based design spaces

Reliability-Based Design Optimization (RBDO) algorithms, such as Reliability Index Approach (RIA) and Performance Measure Approach (PMA), have been developed to solve engineering optimization problems under design uncertainties. In some existing methods, the random design space is transformed to standard normal design space and the reliability assessment, such as reliability index from RIA or performance measure from PMA, is estimated in order to evaluate the failure probability. When the random variable is arbitrarily distributed and cannot be properly fitted to any known form of probability density function, the existing RBDO methods cannot perform reliability analysis in the original design space. This paper proposes a novel Ensemble of Gradient-based Transformed Reliability Analyses (EGTRA) to evaluate the failure probability of any arbitrarily distributed random variables in the original design space. The arbitrary distribution of the random variable is approximated by a merger of multiple Gaussian kernel functions in a single-variate coordinate that is directed toward the gradient of the constraint function. The failure probability is then estimated using the ensemble of each kernel reliability analysis. This paper further derives a linearly approximated probabilistic constraint at the design point with allowable reliability level in the original design space using the aforementioned fundamentals and techniques. Numerical examples with generated random distributions show that existing RBDO algorithms can improperly approximate the uncertainties as Gaussian distributions and provide solutions with poor assessments of reliabilities. On the other hand, the numerical results show EGTRA is capable of efficiently solving the RBDO problems with arbitrarily distributed uncertainties.

An Effective Approach to Solve Design Optimization Problems With Arbitrarily Distributed Uncertainties in the Original Design Space Using Ensemble of Gaussian Reliability Analyses

Reliability-based design optimization (RBDO) algorithms have been developed to solve design optimization problems with existence of uncertainties. Traditionally, the original random design space is transformed to the standard normal design space, where the reliability index can be measured in a standardized unit. In the standard normal design space, the modified reliability index approach (MRIA) measured the minimum distance from the design point to the failure region to represent the reliability index; on the other hand, the performance measure approach (PMA) performed inverse reliability analysis to evaluate the target function performance in a distance of reliability index away from the design point. MRIA was able to provide stable and accurate reliability analysis while PMA showed greater efficiency and was widely used in various engineering applications. However, the existing methods cannot properly perform reliability analysis in the standard normal design space if the transformation to the standard normal space does not exist or is difficult to determine. To this end, a new algorithm, ensemble of Gaussian reliability analyses (EoGRA), was developed to estimate the failure probability using Gaussian-based kernel density estimation (KDE) in the original design space. The probabilistic constraints were formulated based on each kernel reliability analysis for the optimization processes. This paper proposed an efficient way to estimate the constraint gradient and linearly approximate the probabilistic constraints with fewer function evaluations (FEs). Some numerical examples with various random distributions are studied to investigate the numerical performances of the proposed method. The results showed that EoGRA is capable of finding correct solutions in some problems that cannot be solved by traditional methods. Furthermore, experiments of image processing with arbitrarily distributed photo pixels are performed. The lighting of image pixels is maximized subject to the acceptable limit. Our implementation showed that the accuracy of the estimation of normal distribution is poor while the proposed method is capable of finding the optimal solution with acceptable accuracy.

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

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.

Dual Averaging with Adaptive Random Projection for Solving Evolving Distributed Optimization Problems

We study a sequential form of the distributed dual averaging algorithm that minimizes the sum of convex functions in a special case where the number of functions increases gradually. This is done by introducing an intermediate `pivot' stage posed as a convex feasibility problem that minimizes average constraint violation with respect to a family of convex sets. Under this approach, we introduce a version of the minimum sum optimization problem that incorporates an evolving design space. Proof of mathematical convergence of the algorithm is complemented by an application problem that involves finding the location of a noisy, mobile source using an evolving wireless sensor network. Results obtained confirm that the new designs in the evolved design space are superior to the ones found in the original design space due to the unique path followed to reach the optimum.

Application of multi-fidelity simulation modelling to integrated circuit packaging

In semiconductor manufacturing, time-to-market is critical to maintain a competitive advantage through achieving customer satisfaction. Inefficient ways of utilising production resources will lead to a long cycle time. Therefore, machine allocation becomes an essential production decision in many practical manufacturing systems, especially for integrated circuit (IC) packaging. IC packaging is the process of encasing the finished die in a package in order to prevent corrosion and physical damage. Advanced IC packaging techniques add even more complexity into the production system, so reliable average cycle time of this complex system becomes difficult to obtain. We propose a new simulation optimisation framework with multi-fidelity models to study an IC packaging case of the machine allocation problem to pursue a minimum average cycle time. This framework consists of two methodologies: ordinal transformation (OT) and optimal sampling (OS). The OT first employs the low-fidelity model to fast observe all designs, and extracts insightful information from this model by transforming the original design space into an ordinal space. It follows that OS efficiently allocates the computing budget for searching the best design via high-fidelity simulations. An empirical study based on real data was conducted to validate the practical viability of the proposed framework.

Strategies for finding the design point under bounded random variables

The First Order Reliability Method is well accepted as an efficient way to solve structural reliability problems with linear or moderately non-linear limit state functions. Bounded random variables introduce strong non-linearities in the mapping to standard Gaussian space, making search for design points more demanding. Since standard Gaussian space is unbounded, two particular problems have to be addressed: a. the limiting behavior of the probability mapping as a random variable approaches its upper or lower limits; b. how to impose the bounds if design point search leaves the problems supporting domain. Both problems have been overlooked elsewhere, and are addressed in the present article. Based on the Principle of Normal Tail Approximation, two alternatives for the mapping to standard Gaussian space are studied. Three different schemes are proposed to impose the limits of bounded random variables, in the reverse mapping to original design space. Several algorithms are investigated with respect to their ability to find the design point in highly non-linear problems involving bounded random variables. A challenging benchmark reliability problem is also presented herein, and used as a test bed to explore the proposed strategies and the performance of the optimization algorithms.

S0220301 脳動脈瘤治療用ステントにおける自己組織化マップを用いた最適形状解析

As evaluation of hemodynamics in an aneurysm, a lot of parameters have been introduced. However, the relationship between stent geometry and each hemodynamic parameter is not clear enough. We applied Self-organizing Maps (SOM) to search the detailed relationship between stent strut and hemodynamic parameters. Two thousands of stent design candidates evaluated with five hemodynamic parameters and 25 stent geometry parameters were stored during the optimization process. By using SOM software, 2,000 design candidates were visualized into 30 2D-maps while preserving their own features in the original 30-dimensional design space. Each map was divided into 20 clusters to characterize stent design and flow reduction. By comparing each cluster, critical design for reduction effect of each hemodynamic parameter is suggested. Strut placement on the inflow area is important to reduce maximum shear rate. In addition to the inflow area, strut on the outflow area is required to reduce velocity, vorticity, and average shear rate.

An efficient and practical approach to obtain a better optimum solution for structural optimization

For many structural optimization problems, it is hard or even impossible to find the global optimum solution owing to unaffordable computational cost. An alternative and practical way of thinking is thus proposed in this research to obtain an optimum design which may not be global but is better than most local optimum solutions that can be found by gradient-based search methods. The way to reach this goal is to find a smaller search space for gradient-based search methods. It is found in this research that data mining can accomplish this goal easily. The activities of classification, association and clustering in data mining are employed to reduce the original design space. For unconstrained optimization problems, the data mining activities are used to find a smaller search region which contains the global or better local solutions. For constrained optimization problems, it is used to find the feasible region or the feasible region with better objective values. Numerical examples show that the optimum solutions found in the reduced design space by sequential quadratic programming (SQP) are indeed much better than those found by SQP in the original design space. The optimum solutions found in a reduced space by SQP sometimes are even better than the solution found using a hybrid global search method with approximate structural analyses.

Selection of initial designs for multi-objective optimization using classification and regression tree

One of the major challenges for solving large-scale multi-objective optimization design problems is to find the Pareto set effectively. Data mining techniques such as classification, association, and clustering are common used in computer community to extract useful information from a large database. In this paper, a data mining technique, namely, Classification and Regression Tree method, is exploited to extract a set of reduced feasible design domains from the original design space. Within the reduced feasible domains, the first generation of designs can be selected for multi-objective optimization to identify the Pareto set. A mathematical example is used to illustrate the proposed method. Two industrial applications are used to demonstrate the proposed methodology that can achieve better performances in terms of both accuracy and efficiency.

Global optimization method using SLE and adaptive RBF based on fuzzy clustering

High fidelity analysis models, which are beneficial to improving the design quality, have been more and more widely utilized in the modern engineering design optimization problems. However, the high fidelity analysis models are so computationally expensive that the time required in design optimization is usually unacceptable. In order to improve the efficiency of optimization involving high fidelity analysis models, the optimization efficiency can be upgraded through applying surrogates to approximate the computationally expensive models, which can greately reduce the computation time. An efficient heuristic global optimization method using adaptive radial basis function (RBF) based on fuzzy clustering (ARFC) is proposed. In this method, a novel algorithm of maximin Latin hypercube design using successive local enumeration (SLE) is employed to obtain sample points with good performance in both space-filling and projective uniformity properties, which does a great deal of good to metamodels accuracy. RBF method is adopted for constructing the metamodels, and with the increasing the number of sample points the approximation accuracy of RBF is gradually enhanced. The fuzzy c-means clustering method is applied to identify the reduced attractive regions in the original design space. The numerical benchmark examples are used for validating the performance of ARFC. The results demonstrates that for most application examples the global optima are effectively obtained and comparison with adaptive response surface method (ARSM) proves that the proposed method can intuitively capture promising design regions and can efficiently identify the global or near-global design optimum. This method improves the efficiency and global convergence of the optimization problems, and gives a new optimization strategy for engineering design optimization problems involving computationally expensive models.

A Bayesian Inference based Model Interpolation and Extrapolation

Model validation is a process to assess the validity and predictive capabilities of a computer model by comparing simulation results with test data for its intended use of the model. One of the key difficulties for model validation is to evaluate the quality of a computer model at different test configurations in design space, and interpolate or extrapolate the evaluation results to untested new design configurations. In this paper, an integrated model interpolation and extrapolation framework based on Bayesian inference and Response Surface Models (RSM) is proposed to validate the designs both within and outside of the original design space. Bayesian inference is first applied to quantify the distributions' hyper-parameters of the bias between test and CAE data in the validation domain. Then, the hyper-parameters are extrapolated from the design configurations to untested new design. They are then followed by the prediction interval of responses at the new design points. A vehicle design of front impact example is used to demonstrate the proposed methodology.

Reliability-Based Design Using Two-Stage Stochastic Optimization with a Treatment of Model Prediction Errors

Design problems that involve optimization of the reliability of engineering systems are the focus of this paper. Methodologies are discussed applicable to problems that involve nonlinear systems and a large number of uncertain parameters specifying the system and excitation models. To address the complexity of these problems, stochastic simulation is considered for evaluation of the system reliability. An innovative approach, called stochastic subset optimization (SSO), is discussed for performing a sensitivity analysis with respect to the design variables of the problem as well as the uncertain model parameters. In a small number of iterations, SSO converges to a smaller subset of the original design space that has high plausibility of containing the optimal design variables and that consists of near-optimal designs. For higher accuracy, an appropriate stochastic optimization algorithm may then be used to pinpoint the optimal design variables within this subset. This produces an efficient two-stage framework for optimal reliability design. Topics related to the combination of the two different stages for overall enhanced efficiency are discussed. An example is presented that illustrates the effectiveness of the proposed two-stage methodology for a challenging dynamic reliability problem. Also, a study is performed of the influence on the optimal design decisions of the prediction error of the system model, which is introduced because no model makes perfect predictions of the system response.