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All sourcesNASA NTRSDOE OSTI
NASA NTRS2004

Progressive Response Surfaces

Response surface functions are often used as simple and inexpensive replacements for computationally expensive computer models that simulate the behavior of a complex system over some parameter space. Progressive response surfaces are ones that are built up progressively as global information is added from new sample points in the parameter space. As the response surfaces are globally upgraded based on new information, heuristic indications of the convergence of the response surface approximation to the exact (fitted) function can be inferred. Sampling points can be incrementally added in a structured fashion, or in an unstructured fashion. Whatever the approach, at least in early stages of sampling it is usually desirable to sample the entire parameter space uniformly. At later stages of sampling, depending on the nature of the quantity being resolved, it may be desirable to continue sampling uniformly over the entire parameter space (Progressive response surfaces), or to switch to a focusing/economizing strategy of preferentially sampling certain regions of the parameter space based on information gained in early stages of sampling (Adaptive response surfaces). Here we consider Progressive response surfaces where a balanced indication of global response over the parameter space is desired.We use a variant of Moving Least Squares to fit and interpolate structured and unstructured point sets over the parameter space. On a 2-D test problem we compare response surface accuracy for three incremental sampling methods: Progressive Lattice Sampling; Simple-Random Monte Carlo; and Halton Quasi-Monte-Carlo sequences. We are ultimately after a system for constructing efficiently upgradable response surface approximations with reliable error estimates.

Romero, V. J.↗
NASA NTRS1992

Tuning a fuzzy controller using quadratic response surfaces

Response surface methodology, an alternative method to traditional tuning of a fuzzy controller, is described. An example based on a simulated inverted pendulum 'plant' shows that with (only) 15 trial runs, the controller can be calibrated using a quadratic form to approximate the response surface.

Schott, Brian↗
NASA NTRS2010

Estimation of Supersonic Stage Separation Aerodynamics of Winged-Body Launch Vehicles Using Response Surface Methods

Response surface methodology was used to estimate the longitudinal stage separation aerodynamic characteristics of a generic, bimese, winged multi-stage launch vehicle configuration at supersonic speeds in the NASA LaRC Unitary Plan Wind Tunnel. The Mach 3 staging was dominated by shock wave interactions between the orbiter and booster vehicles throughout the relative spatial locations of interest. The inference space was partitioned into several contiguous regions within which the separation aerodynamics were presumed to be well-behaved and estimable using central composite designs capable of fitting full second-order response functions. The underlying aerodynamic response surfaces of the booster vehicle in belly-to-belly proximity to the orbiter vehicle were estimated using piecewise-continuous lower-order polynomial functions. The quality of fit and prediction capabilities of the empirical models were assessed in detail, and the issue of subspace boundary discontinuities was addressed. Augmenting the central composite designs to full third-order using computer-generated D-optimality criteria was evaluated. The usefulness of central composite designs, the subspace sizing, and the practicality of fitting lower-order response functions over a partitioned inference space dominated by highly nonlinear and possibly discontinuous shock-induced aerodynamics are discussed.

Erickson, Gary E.↗
NASA NTRS1995

Application of response surface techniques to helicopter rotor blade optimization procedure

In multidisciplinary optimization problems, response surface techniques can be used to replace the complex analyses that define the objective function and/or constraints with simple functions, typically polynomials. In this work a response surface is applied to the design optimization of a helicopter rotor blade. In previous work, this problem has been formulated with a multilevel approach. Here, the response surface takes advantage of this decomposition and is used to replace the lower level, a structural optimization of the blade. Problems that were encountered and important considerations in applying the response surface are discussed. Preliminary results are also presented that illustrate the benefits of using the response surface.

Henderson, Joseph Lynn↗
NASA NTRS1998

Comparison of Response Surface and Kriging Models in the Multidisciplinary Design of an Aerospike Nozzle

The use of response surface models and kriging models are compared for approximating non-random, deterministic computer analyses. After discussing the traditional response surface approach for constructing polynomial models for approximation, kriging is presented as an alternative statistical-based approximation method for the design and analysis of computer experiments. Both approximation methods are applied to the multidisciplinary design and analysis of an aerospike nozzle which consists of a computational fluid dynamics model and a finite element analysis model. Error analysis of the response surface and kriging models is performed along with a graphical comparison of the approximations. Four optimization problems are formulated and solved using both approximation models. While neither approximation technique consistently outperforms the other in this example, the kriging models using only a constant for the underlying global model and a Gaussian correlation function perform as well as the second order polynomial response surface models.

Simpson, Timothy W.↗
NASA NTRS1998

Comparison of Response Surface and Kriging Models for Multidisciplinary Design Optimization

In this paper, we compare and contrast the use of second-order response surface models and kriging models for approximating non-random, deterministic computer analyses. After reviewing the response surface method for constructing polynomial approximations, kriging is presented as an alternative approximation method for the design and analysis of computer experiments. Both methods are applied to the multidisciplinary design of an aerospike nozzle which consists of a computational fluid dynamics model and a finite-element model. Error analysis of the response surface and kriging models is performed along with a graphical comparison of the approximations, and four optimization problems m formulated and solved using both sets of approximation models. The second-order response surface models and kriging models-using a constant underlying global model and a Gaussian correlation function-yield comparable results.

Simpson, Timothy W.↗
NASA NTRS1993

Effect of design selection on response surface performance

The mathematical formulation of the engineering optimization problem is given. Evaluation of the objective function and constraint equations can be very expensive in a computational sense. Thus, it is desirable to use as few evaluations as possible in obtaining its solution. In solving the equation, one approach is to develop approximations to the objective function and/or restraint equations and then to solve the equation using the approximations in place of the original functions. These approximations are referred to as response surfaces. The desirability of using response surfaces depends upon the number of functional evaluations required to build the response surfaces compared to the number required in the direct solution of the equation without approximations. The present study is concerned with evaluating the performance of response surfaces so that a decision can be made as to their effectiveness in optimization applications. In particular, this study focuses on how the quality of approximations is effected by design selection. Polynomial approximations and neural net approximations are considered.

Carpenter, William C.↗
NASA NTRS2010

Advanced Response Surface Modeling of Ares I Roll Control Jet Aerodynamic Interactions

The Ares I rocket uses roll control jets. These jets have aerodynamic implications as they impinge on the surface and protuberances of the vehicle. The jet interaction on the body can cause an amplification or a reduction of the rolling moment produced by the jet itself, either increasing the jet effectiveness or creating an adverse effect. A design of experiments test was planned and carried out using computation fluid dynamics, and a subsequent response surface analysis ensued on the available data to characterize the jet interaction across the ascent portion of the Ares I flight envelope. Four response surface schemes were compared including a single response surface covering the entire design space, separate sector responses that did not overlap, continuously overlapping surfaces, and recursive weighted response surfaces. These surfaces were evaluated on traditional statistical metrics as well as visual inspection. Validation of the recursive weighted response surface was performed using additionally available data at off-design point locations.

Favaregh, Noah M.↗
NASA NTRS2001

Response Surface Modeling Using Multivariate Orthogonal Functions

A nonlinear modeling technique was used to characterize response surfaces for non-dimensional longitudinal aerodynamic force and moment coefficients, based on wind tunnel data from a commercial jet transport model. Data were collected using two experimental procedures - one based on modem design of experiments (MDOE), and one using a classical one factor at a time (OFAT) approach. The nonlinear modeling technique used multivariate orthogonal functions generated from the independent variable data as modeling functions in a least squares context to characterize the response surfaces. Model terms were selected automatically using a prediction error metric. Prediction error bounds computed from the modeling data alone were found to be- a good measure of actual prediction error for prediction points within the inference space. Root-mean-square model fit error and prediction error were less than 4 percent of the mean response value in all cases. Efficacy and prediction performance of the response surface models identified from both MDOE and OFAT experiments were investigated.

Morelli, Eugene A.↗
NASA NTRS2011

Autonomous Aerobraking: Thermal Analysis and Response Surface Development

A high-fidelity thermal model of the Mars Reconnaissance Orbiter was developed for use in an autonomous aerobraking simulation study. Response surface equations were derived from the high-fidelity thermal model and integrated into the autonomous aerobraking simulation software. The high-fidelity thermal model was developed using the Thermal Desktop software and used in all phases of the analysis. The use of Thermal Desktop exclusively, represented a change from previously developed aerobraking thermal analysis methodologies. Comparisons were made between the Thermal Desktop solutions and those developed for the previous aerobraking thermal analyses performed on the Mars Reconnaissance Orbiter during aerobraking operations. A variable sensitivity screening study was performed to reduce the number of variables carried in the response surface equations. Thermal analysis and response surface equation development were performed for autonomous aerobraking missions at Mars and Venus.

Dec, John A.↗
NASA NTRS2000

Neural Network and Response Surface Methodology for Rocket Engine Component Optimization

The goal of this work is to compare the performance of response surface methodology (RSM) and two types of neural networks (NN) to aid preliminary design of two rocket engine components. A data set of 45 training points and 20 test points obtained from a semi-empirical model based on three design variables is used for a shear coaxial injector element. Data for supersonic turbine design is based on six design variables, 76 training, data and 18 test data obtained from simplified aerodynamic analysis. Several RS and NN are first constructed using the training data. The test data are then employed to select the best RS or NN. Quadratic and cubic response surfaces. radial basis neural network (RBNN) and back-propagation neural network (BPNN) are compared. Two-layered RBNN are generated using two different training algorithms, namely solverbe and solverb. A two layered BPNN is generated with Tan-Sigmoid transfer function. Various issues related to the training of the neural networks are addressed including number of neurons, error goals, spread constants and the accuracy of different models in representing the design space. A search for the optimum design is carried out using a standard gradient-based optimization algorithm over the response surfaces represented by the polynomials and trained neural networks. Usually a cubic polynominal performs better than the quadratic polynomial but exceptions have been noticed. Among the NN choices, the RBNN designed using solverb yields more consistent performance for both engine components considered. The training of RBNN is easier as it requires linear regression. This coupled with the consistency in performance promise the possibility of it being used as an optimization strategy for engineering design problems.

Vaidyanathan, Rajkumar↗
NASA NTRS2003

Method for Constructing Composite Response Surfaces by Combining Neural Networks with other Interpolation or Estimation Techniques

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.

Rai, Man Mohan↗
NASA NTRS1998

Response Surface Model Building and Multidisciplinary Optimization Using D-Optimal Designs

This paper discusses response surface methods for approximation model building and multidisciplinary design optimization. The response surface methods discussed are central composite designs, Bayesian methods and D-optimal designs. An over-determined D-optimal design is applied to a configuration design and optimization study of a wing-body, launch vehicle. Results suggest that over determined D-optimal designs may provide an efficient approach for approximation model building and for multidisciplinary design optimization.

Unal, Resit↗
NASA NTRS2008

Estimation of Supersonic Stage Separation Aerodynamics of Winged-Body Launch Vehicles Using Response Surface Methods

A collection of statistical and mathematical techniques referred to as response surface methodology was used to estimate the longitudinal stage separation aerodynamic characteristics of a generic, bimese, winged multi-stage launch vehicle configuration using data obtained on small-scale models at supersonic speeds in the NASA Langley Research Center Unitary Plan Wind Tunnel. The simulated Mach 3 staging was dominated by multiple shock wave interactions between the orbiter and booster vehicles throughout the relative spatial locations of interest. This motivated a partitioning of the overall inference space into several contiguous regions within which the separation aerodynamics were presumed to be well-behaved and estimable using cuboidal and spherical central composite designs capable of fitting full second-order response functions. The primary goal was to approximate the underlying overall aerodynamic response surfaces of the booster vehicle in belly-to-belly proximity to the orbiter vehicle using relatively simple, lower-order polynomial functions that were piecewise-continuous across the full independent variable ranges of interest. The quality of fit and prediction capabilities of the empirical models were assessed in detail, and the issue of subspace boundary discontinuities was addressed. The potential benefits of augmenting the central composite designs to full third order using computer-generated D-optimality criteria were also evaluated. The usefulness of central composite designs, the subspace sizing, and the practicality of fitting low-order response functions over a partitioned inference space dominated by highly nonlinear and possibly discontinuous shock-induced aerodynamics are discussed.

Erickson, Gary E.↗
NASA NTRS2005

Comparison of Response Surface Construction Methods for Derivative Estimation Using Moving Least Squares, Kriging and Radial Basis Functions

Response construction methods using Moving Least Squares (MLS), Kriging and Radial Basis Functions (RBF) are compared with the Global Least Squares (GLS) method in three numerical examples for derivative generation capability. Also, a new Interpolating Moving Least Squares (IMLS) method adopted from the meshless method is presented. It is found that the response surface construction methods using the Kriging and RBF interpolation yields more accurate results compared with MLS and GLS methods. Several computational aspects of the response surface construction methods also discussed.

Krishnamurthy, Thiagarajan↗
NASA NTRS1991

Comparison of polynomial approximations and artificial neural nets for response surfaces in engineering optimization

Engineering optimization problems involve minimizing some function subject to constraints. In areas such as aircraft optimization, the constraint equations may be from numerous disciplines such as transfer of information between these disciplines and the optimization algorithm. They are also suited to problems which may require numerous re-optimizations such as in multi-objective function optimization or to problems where the design space contains numerous local minima, thus requiring repeated optimizations from different initial designs. Their use has been limited, however, by the fact that development of response surfaces randomly selected or preselected points in the design space. Thus, they have been thought to be inefficient compared to algorithms to the optimum solution. A development has taken place in the last several years which may effect the desirability of using response surfaces. It may be possible that artificial neural nets are more efficient in developing response surfaces than polynomial approximations which have been used in the past. This development is the concern of the work.

Carpenter, William C.↗
NASA NTRS1999

Development of Response Surface Models for Rapid Analysis & Multidisciplinary Optimization of Launch Vehicle Design Concepts

Multdisciplinary design optimization (MDO) is an important step in the design and evaluation of launch vehicles, since it has a significant impact on performance and lifecycle cost. The objective in MDO is to search the design space to determine the values of design parameters that optimize the performance characteristics subject to system constraints. Vehicle Analysis Branch (VAB) at NASA Langley Research Center has computerized analysis tools in many of the disciplines required for the design and analysis of launch vehicles. Vehicle performance characteristics can be determined by the use of these computerized analysis tools. The next step is to optimize the system performance characteristics subject to multidisciplinary constraints. However, most of the complex sizing and performance evaluation codes used for launch vehicle design are stand-alone tools, operated by disciplinary experts. They are, in general, difficult to integrate and use directly for MDO. An alternative has been to utilize response surface methodology (RSM) to obtain polynomial models that approximate the functional relationships between performance characteristics and design variables. These approximation models, called response surface models, are then used to integrate the disciplines using mathematical programming methods for efficient system level design analysis, MDO and fast sensitivity simulations. A second-order response surface model of the form given has been commonly used in RSM since in many cases it can provide an adequate approximation especially if the region of interest is sufficiently limited.

Unal, Resit↗
NASA NTRS2010

The Role of Hierarchy in Response Surface Modeling of Wind Tunnel Data

This paper is intended as a tutorial introduction to certain aspects of response surface modeling, for the experimentalist who has started to explore these methods as a means of improving productivity and quality in wind tunnel testing and other aerospace applications. A brief review of the productivity advantages of response surface modeling in aerospace research is followed by a description of the advantages of a common coding scheme that scales and centers independent variables. The benefits of model term reduction are reviewed. A constraint on model term reduction with coded factors is described in some detail, which requires such models to be well-formulated, or hierarchical. Examples illustrate the consequences of ignoring this constraint. The implication for automated regression model reduction procedures is discussed, and some opinions formed from the author s experience are offered on coding, model reduction, and hierarchy.

DeLoach, Richard↗