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This study investigates response surface methods for computer experiments and discusses some of the approaches available. Orthogonal arrays constructed for computer experiments are studied and an example application to a technology selection and optimization study for a reusable launch vehicle is presented.
Neutron diffraction is uniquely sensitive to hydrogen positions and protonation state. In that context structural information from neutron data is complementary to that provided through X-ray diffraction. However, there are practical obstacles to overcome in fully exploiting the potential of neutron diffraction, Le. low flux and weak scattering. Several approaches are available to overcome these obstacles and we have investigated the simplest: increasing the diffracting volume of the crystals. Volume is a quantifiable metric that is well suited for experiment design and optimization techniques. By using response surface methods we have optimized xylose isomerase crystal volume, enabling neutron diffraction while we determined the crystallization parameters with the minimum of experiments. Our results suggest a systematic means of enabling neutron diffraction studies for a larger number of samples that require information on hydrogen position and/or protonation state.
As NASA moves towards developing technologies needed to implement its new Exploration program, studies conducted for Apollo in the 1960's to understand the rollover stability of capsules landing are being revisited. Although rigid body kinematics analyses of the roll-over behavior of capsules on impact provided critical insight to the Apollo problem, extensive ground test programs were also used. For the new Orion spacecraft being developed to implement today's Exploration program, new air-bag designs have improved sufficiently for NASA to consider their use to mitigate landing loads to ensure crew safety and to enable re-usability of the capsule. Simple kinematics models provide only limited understanding of the behavior of these air bag systems, and more sophisticated tools must be used. In particular, NASA and its contractors are using the LS-Dyna nonlinear simulation code for impact response predictions of the full Orion vehicle with air bags by leveraging the extensive air bag prediction work previously done by the automotive industry. However, even in today's computational environment, these analyses are still high-dimensional, time consuming, and computationally intensive. To alleviate the computational burden, this paper presents an approach that uses deterministic sampling techniques and an adaptive response surface method to not only use existing LS-Dyna solutions but also to interpolate from LS-Dyna solutions to predict the stability boundaries for a capsule on airbags. Results for the stability boundary in terms of impact velocities, capsule attitude, impact plane orientation, and impact surface friction are discussed.
Selected experimental results from a wind tunnel study of a subscale VTOL concept with distributed propulsion and tilt lifting surfaces are presented. The vehicle complexity and automated test facility were ideal for use with a randomized designed experiment. Design of Experiments and Response Surface Methods were invoked to produce run efficient, statistically rigorous regression models with minimized prediction error. Static tests were conducted at the NASA Langley 12-Foot Low-Speed Tunnel to model all six aerodynamic coefficients over a large flight envelope. This work supports investigations at NASA Langley in developing advanced configurations, simulations, and advanced control systems.
Due to the growing problem of noise in today's air transportation system, there have arisen needs to incorporate noise considerations in the conceptual design of revolutionary aircraft. Through the use of response surfaces, complex noise models may be converted into polynomial equations for rapid and simplified evaluation. This conversion allows many of the commonly used response surface-based trade space exploration methods to be applied to noise analysis. This methodology is demonstrated using a noise model of a notional 300 passenger Blended-Wing-Body (BWB) transport. Response surfaces are created relating source noise levels of the BWB vehicle to its corresponding FAR-36 certification noise levels and the resulting trade space is explored. Methods demonstrated include: single point analysis, parametric study, an optimization technique for inverse analysis, sensitivity studies, and probabilistic analysis. Extended applications of response surface-based methods in noise analysis are also discussed.
Turbine performance directly affects engine specific impulse, thrust-to-weight ratio, and cost in a rocket propulsion system. A global optimization framework combining the radial basis neural network (RBNN) and the polynomial-based response surface method (RSM) is constructed for shape optimization of a supersonic turbine. Based on the optimized preliminary design, shape optimization is performed for the first vane and blade of a 2-stage supersonic turbine, involving O(10) design variables. The design of experiment approach is adopted to reduce the data size needed by the optimization task. It is demonstrated that a major merit of the global optimization approach is that it enables one to adaptively revise the design space to perform multiple optimization cycles. This benefit is realized when an optimal design approaches the boundary of a pre-defined design space. Furthermore, by inspecting the influence of each design variable, one can also gain insight into the existence of multiple design choices and select the optimum design based on other factors such as stress and materials considerations.
Aerospace conceptual vehicle design is a complex process which involves multidisciplinary studies of configuration and technology options considering many parameters at many values. NASA Langley's Vehicle Analysis Branch (VAB) has detailed computerized analysis capabilities in most of the key disciplines required by advanced vehicle design. Given a configuration, the capability exists to quickly determine its performance and lifecycle cost. The next step in vehicle design is to determine the best settings of design parameters that optimize the performance characteristics. Typical approach to design optimization is experience based, trial and error variation of many parameters one at a time where possible combinations usually number in the thousands. However, this approach can either lead to a very long and expensive design process or to a premature termination of the design process due to budget and/or schedule pressures. Furthermore, one variable at a time approach can not account for the interactions that occur among parts of systems and among disciplines. As a result, vehicle design may be far from optimal. Advanced multidisciplinary design optimization (MDO) methods are needed to direct the search in an efficient and intelligent manner in order to drastically reduce the number of candidate designs to be evaluated. The payoffs in terms of enhanced performance and reduced cost are significant. A literature review yields two such advanced MDO methods used in aerospace design optimization; Taguchi methods and response surface methods. Taguchi methods provide a systematic and efficient method for design optimization for performance and cost. However, response surface method (RSM) leads to a better, more accurate exploration of the parameter space and to estimated optimum conditions with a small expenditure on experimental data. These two methods are described.
A method and system for design optimization that incorporates the advantages of both traditional response surface methodology (RSM) and neural networks is disclosed. The present invention employs a unique strategy called parameter-based partitioning of the given design space. In the design procedure, a sequence of composite response surfaces based on both neural networks and polynomial fits is used to traverse the design space to identify an optimal solution. The composite response surface has both the power of neural networks and the economy of low-order polynomials (in terms of the number of simulations needed and the network training requirements). The present invention handles design problems with many more parameters than would be possible using neural networks alone and permits a designer to rapidly perform a variety of trade-off studies before arriving at the final design.
A method and system for data modeling that incorporates the advantages of both traditional response surface methodology (RSM) and neural networks is disclosed. The invention partitions the parameters into a first set of s simple parameters, where observable data are expressible as low order polynomials, and c complex parameters that reflect more complicated variation of the observed data. Variation of the data with the simple parameters is modeled using polynomials; and variation of the data with the complex parameters at each vertex is analyzed using a neural network. Variations with the simple parameters and with the complex parameters are expressed using a first sequence of shape functions and a second sequence of neural network functions. The first and second sequences are multiplicatively combined to form a composite response surface, dependent upon the parameter values, that can be used to identify an accurate mode
Space transportation system conceptual design is a multidisciplinary process containing considerable element of risk. Risk here is defined as the variability in the estimated (output) performance characteristic of interest resulting from the uncertainties in the values of several disciplinary design and/or operational parameters. Uncertainties from one discipline (and/or subsystem) may propagate to another, through linking parameters and the final system output may have a significant accumulation of risk. This variability can result in significant deviations from the expected performance. Therefore, an estimate of variability (which is called design risk in this study) together with the expected performance characteristic value (e.g. mean empty weight) is necessary for multidisciplinary optimization for a robust design. Robust design in this study is defined as a solution that minimizes variability subject to a constraint on mean performance characteristics. Even though multidisciplinary design optimization has gained wide attention and applications, the treatment of uncertainties to quantify and analyze design risk has received little attention. This research effort explores the dual response surface approach to quantify variability (risk) in critical performance characteristics (such as weight) during conceptual design.
The usefulness of piecewise polynomials with C1 and C2 derivative continuity for response surface construction method is examined. A Moving Least Squares (MLS) method is developed and compared with four other interpolation methods, including kriging. First the selected methods are applied and compared with one another in a two-design variables problem with a known theoretical response function. Next the methods are tested in a four-design variables problem from a reliability-based design application. In general the piecewise polynomial with higher order derivative continuity methods produce less error in the response prediction. The MLS method was found to be superior for response surface construction among the methods evaluated.
Accounting for the statistical geometric and material variability of structures in analysis has been a topic of considerable research for the last 30 years. The determination of quantifiable measures of statistical probability of a desired response variable, such as natural frequency, maximum displacement, or stress, to replace experience-based "safety factors" has been a primary goal of these studies. There are, however, several problems associated with their satisfactory application to realistic structures, such as bladed disks in turbomachinery. These include the accurate definition of the input random variables (rv's), the large size of the finite element models frequently used to simulate these structures, which makes even a single deterministic analysis expensive, and accurate generation of the cumulative distribution function (CDF) necessary to obtain the probability of the desired response variables. The research presented here applies a methodology called probabilistic dynamic synthesis (PDS) to solve these problems. The PDS method uses dynamic characteristics of substructures measured from modal test as the input rv's, rather than "primitive" rv's such as material or geometric uncertainties. These dynamic characteristics, which are the free-free eigenvalues, eigenvectors, and residual flexibility (RF), are readily measured and for many substructures, a reasonable sample set of these measurements can be obtained. The statistics for these rv's accurately account for the entire random character of the substructure. Using the RF method of component mode synthesis, these dynamic characteristics are used to generate reduced-size sample models of the substructures, which are then coupled to form system models. These sample models are used to obtain the CDF of the response variable by either applying Monte Carlo simulation or by generating data points for use in the response surface reliability method, which can perform the probabilistic analysis with an order of magnitude less computational effort. Both free- and forced-response analyses have been performed, and the results indicate that, while there is considerable room for improvement, the method produces usable and more representative solutions for the design of realistic structures with a substantial savings in computer time.
The Probabilistic Dynamic Synthesis method is a new technique for obtaining the statistics of a desired response engineering quantity for a structure with non-deterministic parameters. The method uses measured data from modal testing of the structure as the input random variables, rather than more "primitive" quantities like geometry or material variation. This modal information is much more comprehensive and easily measured than the "primitive" information. The probabilistic analysis is carried out using either response surface reliability methods or Monte Carlo simulation. A previous work verified the feasibility of the PDS method on a simple seven degree-of-freedom spring-mass system. In this paper, extensive issues involved with applying the method to a realistic three-substructure system are examined, and free and forced response analyses are performed. The results from using the method are promising, especially when the lack of alternatives for obtaining quantitative output for probabilistic structures is considered.
The Probabilistic Dynamic Synthesis method is a technique for obtaining the statistics of a desired response engineering quantity for a structure with non-deterministic parameters. The method uses measured data from modal testing of the structure as the input random variables, rather than more "primitive" quantities like geometry or material variation. This modal information is much more comprehensive and easily measured than the "primitive" information. The probabilistic analysis is carried out using either response surface reliability methods or Monte Carlo simulation. In previous work, the feasibility of the PDS method applied to a simple seven degree-of-freedom spring-mass system was verified. In this paper, extensive issues involved with applying the method to a realistic three-substructure system are examined, and free and forced response analyses are performed. The results from using the method are promising, especially when the lack of alternatives for obtaining quantitative output for probabilistic structures is considered.
The preliminary conceptual design of the Korean fusion demonstration reactor (K-DEMO) with a major radius of 6.8 m and the fusion power of 2200 MW has been studied since 2012. The overall configuration of the K-DEMO divetor system based on the ITER-like water-cooled tungsten technology is a double-null type symmetric divertor subdivided into 32 toroidal modules for the vertical maintenance. A detached divertor scenario with impurity seeding was considered as the primary approach for the power exhaust to reduce the peak heat flux lower than the engineering limit of 10 MW/m 2 . The power exhaust performance at the scrap off layer was estimated by using UEDGE-2D code, a two-dimensional fluid transport code for collisional edge plasma and neutral species like N, Ne, and Ar. Particle and heat flux on inboard and outboard divertor targets were calculated for the detached cased depending on parameters such as the impurity seeding rate, pumping rate, and the pedestal density. On the other hand, a magnetic solution like X-divertor, snowflake divertor, and super X-divertor to expand the plasma wet area was considered for K-DEMO since the detached divertor increasing a radiation fraction by impurity seeding might be able to be unstable. However, the extremely high current of poloidal coils was required more than the engineering limit, 20 MA, to form magnetic field lines for the alternative divertors. Based on the physical calculation of the edge plasma, engineering analyses were carried out to find out the thermal and structural reliability. The thermo-hydraulic analysis confirmed thermal stability, whether all comprising materials are operating within their allowable temperature windows when the case of the peak heat flux is set to 10 MW/m 2 on the outboard divertor target. The response surface optimization method derived two optimal design candidates employing two kinds of heat sink materials, respectively: the reduced activation ferritic martensitic (RAFM) steel and CuCrZr alloy. Additionally, the drawbacks and merits of the two materials were definite. The optimal design with applying RAFM steel was vulnerable to withstand thermal and mechanical loads since low thermal conductivity caused too thin thickness of the heat sink. On the other hand, the CuCrZr alloy has critical drawbacks in terms of activation and radioactive waste despite its high thermal conductivity. Meanwhile, preliminary electromagnetic (EM) analysis was carried out to estimate the EM loads caused by the abnormal behaviors of plasma since EM loads are one of the most critical external loads for designing a DEMO divertor.
The concept of competitiveness is considered as a function of both high quality and low cost, and the techniques which drive a design simultaneously in both directions. In order to examine these techniques, the term 'genopersistation', which is derived from the words 'genesis' and 'persistence' and defined to mean the bringing forth, sustaining, and eventual annihilation of something. For a product, the term includes the conceptual design, design development, test and evaluation, production, deployment, operation, support, evolution, and retirement of product. The techniques are summarized which can be used to simultaneously genopersistate both the high quality and low cost into a product. Attention is given to Taguchi (1986) methods, the response surface methodology (Montgomery, 1984), Quality Function Deployment (Akao, 1990), Hoshin Kanari (Akao, 1991), activity based costing (O'Guin, 1991), concurrent engineering (Carter and Baker, 1991), and multidisciplinary optimization (Evans, 1984).
A low-cost, portable, and simplified system has been developed that is suitable for in-situ calibration and/or evaluation of multi-axis inertial measurement instruments. This system overcomes facility restrictions and maintains or improves the calibration quality for users of accelerometer-based instruments with applications in avionics, experimental wind tunnel research, and force balance calibration applications. The apparatus quickly and easily positions a multi-axis accelerometer system into a precisely known orientation suitable for in-situ quality checks and calibration. In addition, the system incorporates powerful and sophisticated statistical methods, known as response surface methodology and statistical quality control. These methods improve calibration quality, reduce calibration time, and allow for increased calibration frequency, which enables the monitoring of instrument stability over time.
This paper documents an approach to conceptual and preliminary aircraft design in which system synthesis is achieved using statistical methods, specifically design of experiments (DOE) and response surface methodology (RSM). These methods are employed in order to more efficiently search the design space for optimum configurations. In particular, a methodology incorporating three uses of these techniques is presented. First, response surface equations are formed which represent aerodynamic analyses, in the form of regression polynomials, which are more sophisticated than generally available in early design stages. Next, a regression equation for an overall evaluation criterion is constructed for the purpose of constrained optimization at the system level. This optimization, though achieved in a innovative way, is still traditional in that it is a point design solution. The methodology put forward here remedies this by introducing uncertainty into the problem, resulting a solutions which are probabilistic in nature. DOE/RSM is used for the third time in this setting. The process is demonstrated through a detailed aero-propulsion optimization of a high speed civil transport. Fundamental goals of the methodology, then, are to introduce higher fidelity disciplinary analyses to the conceptual aircraft synthesis and provide a roadmap for transitioning from point solutions to probabalistic designs (and eventually robust ones).