Large Number Of Design Variables Research Articles

2312 単純化に基づく機械システムの階層的最適設計(OS7-1 システム最適化)

The design of mechanical products is a complicated process, due to the large number of design variables and performance characteristics, so it is difficult for design engineers to obtain global optimum solutions. This paper proposes an optimization method for such product designs in which hierarchical product optimization methodologies are constructed based on simplification of the optimization problem. We simplify design variables and/or performance characteristics, and arrive at simplified optimization problems. The subsequent optimizations are started at the lowest, simplest levels of the hierarchical optimization structure, and proceed to higher levels. Simplification facilitates obtaining global optimum solutions and clarifies the focal issues of the optimization problem. Some applied examples demonstrate the effectiveness of the proposed method.

On the effectiveness of assessing model accuracy at design points for radial basis functions

AbstractA study is performed on the effectiveness of assessing accuracies of radial basis functions (RBFs) at design (or sampling) points. The RBF has been shown to be appropriate for creating approximation functions for linear, quadratic, and higher‐order nonlinear responses even with a relatively small size of design samples. One favourable feature of RBFs is that all RBF models pass exactly through design points; therefore, an RBF model has no error at design points. However, this feature prevents the use of statistical parameters, which are commonly used in polynomial‐regression approximations, to obtain model accuracy based on evaluations at design points. Since obtaining additional samples for many complex engineering applications is expensive, it is desirable to have some knowledge of the model accuracy based on evaluations at design points so that the best RBF model(s) can be selected. In this study, the effectiveness of using statistical parameter R2 for prediction (R) is evaluated for two highly accurate RBFs using mathematical functions and real engineering applications. The results show that R is inappropriate for RBF models, and should be used cautiously for highly nonlinear responses with a small number of sampling points and/or a large number of design variables. Copyright © 2006 John Wiley & Sons, Ltd.

Aerofoil profile and sweep optimisation for a blended wing-body aircraft using a discrete adjoint method

AbstractAerodynamic optimisations of a blended wing-body (BWB) aircraft are presented. A discrete adjoint solver is used to calculate efficiently the gradients, which makes it possible to optimise for a large number of design variables. The optimisations employ either a variable-fidelity method that combines low- and high-fidelity models or a direct sequential quadratic programming (SQP) method. Four Euler optimisations of a BWB aircraft are then presented. The optimisation is allowed to change a series of master sections defining the aircraft geometry as well as the sweep angle on the outer wing for two of the optimisations. Substantial improvements are obtained, not only in the Euler mode but also when the optimised geometries are evaluated using Reynolds-averaged Navier-Stokes solutions. Some interesting features of the optimised wing profiles are discussed.

Update strategies for kriging models used in variable fidelity optimization

Many optimization methods for simulation-based design rely on the sequential use of metamodels to reduce the associated computational burden. In particular, kriging models are frequently used in variable fidelity optimization. Nevertheless, such methods may become computationally inefficient when solving problems with large numbers of design variables and/or sampled data points due to the expensive process of optimizing the kriging model parameters in each iteration. One solution to this problem would be to replace the kriging models with traditional Taylor series response surface models. Kriging models, however, were shown to provide good approximations of computer simulations that incorporate larger amounts of data, resulting in better global accuracy. In this paper, a metamodel update management scheme (MUMS) is proposed to reduce the cost of using kriging models sequentially by updating the kriging model parameters only when they produce a poor approximation. The scheme uses the trust region ratio (TR-MUMS), which is a ratio that compares the approximation to the true model. Two demonstration problems are used to evaluate the proposed method: an internal combustion engine sizing problem and a control-augmented structural design problem. The results indicate that the TR-MUMS approach is very effective; on the demonstration problems, it reduced the number of likelihood evaluations by three orders of magnitude compared to using a global optimizer to find the kriging parameters in every iteration. It was also found that in trust region-based method, the kriging model parameters need not be updated using a global optimizer—local methods perform just as well in terms of providing a good approximation without affecting the overall convergence rate, which, in turn, results in a faster execution time.

Evolutionary multiobjective industrial design: the case of a racing car tire-suspension system

When dealing with multiobjective optimization (MO) of the tire-suspension system of a racing car, a large number of design variables and a large number of objectives have to be taken into account. Two different models have been used, both validated on data coming from an instrumented car, a differential equation-based physical model, and a neural network purely numerical model. Up to 23 objective functions have been defined, at least 14 of which are in strict conflict of each other. The equivalent scalar function based and the objective-as-constraint formulations are intentionally avoided due to their well-known limitations. A fuzzy definition of optima, being a generalization of Pareto optimality, is applied to the problem. The result of such an approach is that subsets of Pareto optimal solutions (on such a problem, a big portion of the entire search space) can be properly selected as a consequence of input from the designer. The obtained optimal solutions are compared with the reference vehicle and with the optima previously obtained with design of experiment techniques and different MO optimization strategies. The proposed strategy improves both the reference (actual) car and previously obtained optima (scalar preference function) in the majority of objectives with technically significant improvements. Moreover, the strategy offers an univoque criterion for the choice among tradeoff solutions in the 14-dimensional objective space. The problem is used as a test of a proposed optimal design strategy for industrial problems, integrating differential equation and neural networks modeling, design of experiments, MO, and fuzzy optimal-based decision making. Such a linked approach gives also a unified view of where to concentrate the computational effort.

Nearly-orthogonal sampling and neural network metamodel driven conceptual design of multistage space launch vehicle

This paper presents a new design methodology for efficient conceptual design of complex systems involving multidisciplinary and computationally intensive analysis with large number of design variables. A nearly-orthogonal sampling of design space is proposed with good space filling properties to extract maximum useful information about system behavior using much lower number of trial designs. This sampled data is then used as training data for artificial neural network, which will act as a metamodel to represent the time consuming disciplinary analyses. A stage-wise interconnection of separate neural networks is also proposed for trajectory metamodel to offset dimensionality curse of neural networks. Genetic Algorithm performs global optimization by utilizing this metamodel and subsequently sequential quadratic programming performs the local optimization utilizing exact analyses. The performance of proposed methodology is investigated in this paper for the conceptual design optimization of multistage solid fueled space launch vehicle. The results show excellent approximation of highly non-linear functions using proposed sampling and drastic reduction in overall design optimization time, due to greatly reduced number of exact disciplinary analyses.

Optimal placement and tuning of multiple tuned mass dampers for suppressing multi-mode structural response

The optimal design of multiple tuned mass dampers (multiple TMD\'s) to suppress multi-mode structural response of beams and floor structures was investigated. A new method using a numerical optimizer, which can effectively handle a large number of design variables, was employed to search for both optimal placement and tuning of TMD\'s for these structures under wide-band loading. The first design problem considered was vibration control of a simple beam using 10 TMD\'s. The results confirmed that for structures with widely-spaced natural frequencies, multiple TMD\'s can be adequately designed by treating each structural vibration mode as an equivalent SDOF system. Next, the control of a beam structure with two closely-spaced natural frequencies was investigated. The results showed that the most effective multiple TMD\'s have their natural frequencies distributed over a range covering the two controlled structural frequencies and have low damping ratios. Moreover, a single TMD can also be made effective in controlling two modes with closely spaced frequencies by a newly identified control mechanism, but the effectiveness can be greatly impaired when the loading position changes. Finally, a realistic problem of a large floor structure with 5 closely spaced frequencies was presented. The acceleration responses at 5 positions on the floor excited by 3 wide-band forces were simultaneously suppressed using 10 TMD\'s. The obtained multiple TMD\'s were shown to be very effective and robust.

A general and effective approach for the optimal design of fiber reinforced composite structures

This paper presents a general and effective procedure based on a mathematical programming approach for composite structures optimal design, under weight, stiffness and strength criteria. Effective means that the developed approach is able to find a local optimum in few iterations, even with a large number of design variables. In addition, as the formulation may include strength constraints, it is shown that not only plies thickness but also fiber orientations should be considered as design variables. Besides this, it is explained how these angular variables can improve the structural performance. The generality of the approach is related to the fact that the developed optimization technique is not only dedicated to the design of particular laminates configurations, but is also reliable for general problems including isotropic and/or anisotropic materials, as well as other structural responses. The design problem formulation relies on a direct parameterization of the laminates in terms of the physical design variables, that is orientations and thicknesses, and not on alternative ones like the lamination parameters. Numerical applications, including a non-homogeneous composite membrane design problem and an industrial test case, are presented.

A Coupled-Adjoint Sensitivity Analysis Method for High-Fidelity Aero-Structural Design

This paper presents an adjoint method for sensitivity analysis that is used in an aero-structural aircraft design framework. The aero-structural analysis uses high-fidelity models of both the aerodynamics and the structures. Aero-structural sensitivities are computed using a coupled-adjoint approach that is based on previously developed single discipline sensitivity analysis. Alternative strategies for coupled sensitivity analysis are also discussed. The aircraft geometry and a structure of fixed topology are parameterized using a large number of design variables. The aero-structural sensitivities of aerodynamic and structural functions with respect to these design variables are computed and compared with results given by the complex-step derivative approximation. The coupled-adjoint procedure is shown to yield very accurate sensitivities and to be computationally efficient, making high-fidelity aero-structural design feasible for problems with thousands of design variables.

Comprehensive thermal optimization of liquid composite molding to reduce cycle time and processing stresses

AbstractLiquid composite molding (LCM) is a well‐established and flexible composite manufacturing technology capable of producing large parts at a relatively low cost. In this family of related injection processes, a large number of design variables have strong impact on manufacturing performance. The determination of adequate process parameters is key to yield successful molding conditions and reduce cycle time. In addition, properties and durability of composite parts are strongly affected by internal stresses. Excessive stress levels may lead to important defects in the part at the curing stage and after processing, when the part is cooled to room temperature. In this investigation, a comprehensive curing optimization algorithm is proposed to reduce internal stresses during composite processing. This study focuses on the minimization of the macroscopic residual stresses that appear during cure and cooling in thermoset composite laminates as a result of temperature and degree of cure gradients. The proposed fitness function to be minimized is based on the physics of the matrix material transformation and on the mechanical behavior of the composite material. An evolutionary strategy based on genetic algorithms (GA) is implemented for the minimization of the fitness function. Optimization is carried out for thin and thick glass/polyester laminated composites. Different optimization schemes with thermo‐elastic and viscoelastic models of the composite mechanical properties are studied. The advantages and drawbacks of each model are stated and discussed. POLYM. COMPOS., 26:209–230, 2005. © 2005 Society of Plastics Engineers

Variable chromosome length genetic algorithm for progressive refinement in topology optimization

This article introduces variable chromosome lengths (VCL) in the context of a genetic algorithm (GA). This concept is applied to structural topology optimization but is also suitable to a broader class of design problems. In traditional genetic algorithms, the chromosome length is determined a priori when the phenotype is encoded into the corresponding genotype. Subsequently, the chromosome length does not change. This approach does not effectively solve problems with large numbers of design variables in complex design spaces such as those encountered in structural topology optimization. We propose an alternative approach based on a progressive refinement strategy, where a GA starts with a short chromosome and first finds an ‘optimum’ solution in the simple design space. The ‘optimum’ solutions are then transferred to the following stages with longer chromosomes, while maintaining diversity in the population. Progressively refined solutions are obtained in subsequent stages. A strain energy filter is used in order to filter out inefficiently used design cells such as protrusions or isolated islands. The variable chromosome length genetic algorithm (VCL-GA) is applied to two structural topology optimization problems: a short cantilever and a bridge problem. The performance of the method is compared to a brute-force approach GA, which operates ab initio at the highest level of resolution.

Optimal design of vehicle target pulse and restraint system

This paper presents an occupant-first approach, which allows for the simultaneous consideration of vehicle structure and the design of the occupant restraint system in the early stages. There are many difficulties in solving this problem, such as the large number of design variables, the large number of constraints and the highly non-linear nature of the occupant's responses. An efficient genetic algorithm (GA) is applied to deal with the challenges involved in the optimal design of the vehicle target pulse and the occupant restraint system. The methodology is demonstrated through a vehicle design problem. The results show that the occupant-first approach is effective and the GA is a valuable optimisation tool which can be used to achieve design targets.

Design of multiple tuned mass dampers by using a numerical optimizer

A new method to design multiple tuned mass dampers (multiple TMDs) for minimizing excessive vibration of structures has been developed using a numerical optimizer. It is a very powerful method by which a large number of design variables can be effectively handled without imposing any restriction before the analysis. Its framework is highly flexible and can be easily extended to general structures with different combinations of loading conditions and target controlled quantities. The method has been used to design multiple TMDs for SDOF structures subjected to wide-band excitation. Some novel results have been obtained. To reduce displacement response of the structure, the optimally designed multiple TMDs have distributed natural frequencies and distinct damping ratios at low damping level. The obtained optimal configuration of TMDs was different from the earlier analytical solutions and was proved to be the most effective. A robustness design of multiple TMDs has also been presented. Robustness is defined as the ability of TMDs to function properly despite the presence of uncertainties in the parameters of the system. Numerical examples of minimizing acceleration structural response have been given where the system parameters are uncertain and are modeled as independent normal variates. It was found that, in case of uncertainties in the structural properties, increasing the TMD damping ratios along with expanding the TMD frequency range make the system more robust. Meanwhile, if TMD parameters themselves are uncertain, it is necessary to design TMDs for higher damping ratios and a narrower frequency range. Copyright © 2004 John Wiley & Sons, Ltd.

A method for using legacy data for metamodel-based design of large-scale systems

Despite a steady increase in computing power, the complexity of engineering analyses seems to advance at the same rate. Traditional parametric design analysis is inadequate for the analysis of large-scale engineering systems because of its computational inefficiency; therefore, a departure from the traditional parametric design approach is required. In addition, the existence of legacy data for complex, large-scale systems is commonplace. Approximation techniques may be applied to build computationally inexpensive surrogate models for large-scale systems to replace expensive-to-run computer analysis codes or to develop a model for a set of nonuniform legacy data. Response-surface models are frequently utilized to construct surrogate approximations; however, they may be inefficient for systems having with a large number of design variables. Kriging, an alternative method for creating surrogate models, is applied in this work to construct approximations of legacy data for a large-scale system. Comparisons between response surfaces and kriging are made using the legacy data from the High Speed Civil Transport (HSCT) approximation challenge. Since the analysis points already exist, a modified design-of-experiments technique is needed to select the appropriate sample points. In this paper, a method to handle this problem is presented, and the results are compared against previous work.

다단계 반응표면법을 이용한 치과용 임플란트의 3차원 형상최적설계

In this paper, an optimum design problem for endosseous implant in dentistry is studied to find best implant design. An optimum design problem is formulated to reduce stresses arising at the cortical as well as cancellous bones, in which sufficient design parameters are chosen for design definition that encompasses major implants in popular use. Optimization at once (OAO) with the large number of design variables, however, causes too costly solution or even failure to converge. A concept of multilevel optimization (MLO) is employed to this end, which is to group the design variables of similar nature, solve the sub-problem of smaller size for each group in sequence, and this is iterated until convergence. Each sub-problem is solved based on the response surface method (RSM) due to its efficiency for small sized problem.<br/>

High-Fidelity Aerostructural Design Optimization of a Supersonic Business Jet

This paper focuses on the demonstration of an integrated aerostructural method for the design of aerospace vehicles. Both aerodynamics and structures are represented using high-fidelity models such as the Euler equations for the aerodynamics and a detailed finite element model for the primary structure. The aerodynamic outer-mold line and a structure of fixed topology are parameterized using a large number of design variables. The aerostructural sensitivities of aerodynamic and structural cost functions with respect to both outer-mold line shape and structural variables are computed using an accurate and efficient coupled-adjoint procedure. Kreisselmeier‐ Steinhauser functions are used to reduce the number of structural constraints in the problem. Results of the aerodynamic shape and structural optimization of a natural laminar-flow supersonic business jet are presented together with an assessment of the accuracy of the sensitivity information obtained using the coupled-adjoint procedure.

Topology Designs of Slider Air Bearings

A new approach for topology designs of slider air bearings in magnetic recording disk drives is suggested by using large-scale discrete variable optimization techniques. Conventional optimization techniques are restricted to the original topology of the slider by modifying the initial designs. To overcome the restriction, a new topology design approach is presented with enhanced mathematical techniques. Topology optimization of slider air bearings typically has a large number of design variables because the finite mesh must be fine enough to represent the shape of the air bearing surface (ABS). To handle a large number of design variables, an efficient strategy for the optimization including the sensitivity analysis must be established. As a gradient-based local optimization algorithm, the sequential unconstrained minimization technique (SUMT) using an exterior penalty function is used, which requires little computational effort and computer memory. For the gradient calculation, the analytical design sensitivity analysis method introducing an adjoint variable is employed. A topology design problem is formulated as a function of the residuals which is calculated by solving the generalized Reynolds equation. A very large number of discrete design variables (=9409) are dealt with, which denote the rail heights at grid cells. To validate the suggested design methodology, a developed program is applied to two slider models with one and three trailing rails. The simulation results demonstrated the effectiveness of the proposed design methodology by showing that the optimized topologies have reasonable shapes without any initial designs.

Size/geometry optimization of trusses by the force method and genetic algorithm

AbstractIn this article size/geometry optimization of trusses is performed using the force method and genetic algorithm. A large number of design variables consisting of cross‐sectional areas and nodal coordinates are involved in such an optimization, and due to a large number of constraints, the dimensions of the design space are often numerous and in the case of discrete values for cross sections usually discontinuous. In order to avoid local optima, modified genetic algorithms are developed. Furthermore, the force method is employed to improve the speed of the optimization. In the first phase of the described method, the initial geometry of the truss is fixed and near optimum ranges for the cross‐section areas are obtained using the relationships from the force method. In the second phase, the geometry of the structure is altered with the aim of designing lower‐weight structures. Within the Genetic Algorithm a new dynamic penalty function is defined and a modified process of reproduction is presented. A contraction process is also employed for the design space using shorter substrings for the design variables.

Power converter design optimization

This article present GA-based design approach to optimization of power electronics circuits is shown to be a very effective and powerful tool for obtaining improved solutions compared to traditional design procedures. In the GA procedure, each design is represented using a for the converter design problem, each gene string is used to represent the set of electrical components that define one possible converter design. The design of power electronics systems involves a large number of design variables and the application of knowledge from several different engineering fields.

Implementation of a multi-level optimisation methodology within the e-design of a blended wing body

The modern requirement for low weight/high strength aircraft structures is placing an every increasing emphasise on the direct use of optimisation methods in the overall design process. In certain cases, such as that of the Blended Wing- Body (BWB) aircraft, the design cannot be successfully completed without the application of optimisation methods. This type of complex design has inevitably led to the growth in the size of the optimisation problem with large numbers of design variables and constraints being required in a design with strong interactions between the various structural components. Treating the problem in the conventional manner with all components considered simultaneously leads to excessive computer run times. In order to control this growth in problem size a multi-level approach has to be used in order to break down the problem into several sub-problems taking into account the structural coupling effects. Such an approach enables the use of parallel computer processing leading to a significant reduction of the computer run time and decreases the design cycle time.