Cameron J. Turner

Multidimensional sequential sampling for NURBs-based metamodel development

Adaptive design of experiments approaches are intended to overcome the limitations of a priori experimental design by adapting to the results of prior runs so that subsequent runs yield more significant information. Such approaches are valuable in engineering applications with metamodels, where efficiently collecting a dataset to define an unknown function is important. While a variety of approaches have been proposed, most techniques are limited to sampling for only one phenomenon at a time. We propose a multicriteria optimization approach that effectively simultaneously samples for multiple phenomena. In addition to determining the next sequential sampling point, such an algorithm also can be formulated to support conclusions about the adequacy of the experiment through the use of convergence criteria. A multicriteria adaptive sequential sampling algorithm, along with convergence metrics, is defined and demonstrated on five trial problems of engineering interest. The results of these five problems demonstrate that a multicriteria sequential sampling approach is a useful engineering tool for modeling engineering design spaces using NURBs-based metamodels.

Selecting an Appropriate Metamodel: The Case for NURBs Metamodels

Metamodels are becoming increasingly popular for representing unknown black box functions. Several metamodel classes exist, including response surfaces and spline-based models, kriging and radial basis function models, and neural networks. For an inexperienced user, selecting an appropriate metamodel is difficult due to a limited understanding of the advantages and disadvantages of each metamodel type. This paper reviews several major metamodeling techniques with respect to their advantages and disadvantages and compares several significant metamodel types for use as a black box metamodeling tool. The results make a strong case for using Non-Uniform Rational B-spline (NURBs) HyPerModels as a generic metamodeling tool.Copyright

Global optimization of NURBs-based metamodels

The emergence of metamodels as approximate objective function representations offers the ability to ‘design’ metamodels with favourable optimization characteristics without compromising the accurate representational capabilities of arbitrary function topologies and modalities. With non-uniform rational B-splines (NURBs) as a metamodel basis, favourable optimization properties can be obtained which allow the intelligent selection of starting points for multistart optimization algorithms and which constrain optimization searches to metamodel regions containing the global metamodel optimum. In this article NURBs-based metamodels are used to define an optimization algorithm (HyPerOp) which guarantees the discovery of the global metamodel optimum with known computational effort. Emphasis is placed on demonstrating how NURBs’ properties contribute to a favourable objective function approximation. Through a large non-linear optimization trial problem set, the claim that HyPerOp is guaranteed to find the global metamodel optimum is demonstrated and the performance of HyPerOp with respect to random multistart approaches is evaluated.

N-Dimensional Nonuniform Rational B-Splines for Metamodeling

Nonuniform rational B-splines (NURBs) have unique properties that make them attractive for engineering metamodeling applications. NURBs are known to accurately model many different continuous curve and surface topologies in one- and two-variate spaces. However, engineering metamodels of the design space often require hypervariate representations of multidimensional outputs. In essence, design space metamodels are hyper-dimensional constructs with a dimensionality determined by their input and output variables. To use NURBs as the basis for a metamodel in a hyperdimensional space, traditional geometric fitting techniques must be adapted to hypervariate and hyperdimensional spaces composed of both continuous and discontinuous variable types. In this paper, we describe the necessary adaptations for the development of a NURBs-based metamodel called a hyperdimensional performance model or HyPerModel. HyPerModels are capable of accurately and reliably modeling nonlinear hyperdimensional objects defined by both continuous and discontinuous variables of a wide variety of topologies, such as those that define typical engineering design spaces. We demonstrate this ability by successfully generating accurate HyPerModels of ten trial functions laying the foundation for future work with N-dimensional NURBs in design space applications.

Metamodel defined multidimensional embedded sequential sampling criteria.

Collecting data to characterize an unknown space presents a series of challenges. Where in the space should data be collected? What regions are more valuable than others to sample? When have sufficient samples been acquired to characterize the space with some level of confidence? Sequential sampling techniques offer an approach to answering these questions by intelligently sampling an unknown space. Sampling decisions are made with criteria intended to preferentially search the space for desirable features. However, N-dimensional applications need efficient and effective criteria. This paper discusses the evolution of several such criteria based on an understanding of the behaviors of existing criteria, and desired criteria properties. The resulting criteria are evaluated with a variety of planar functions, and preliminary results for higher dimensional applications are also presented. In addition, a set of convergence criteria, intended to evaluate the effectiveness of further sampling are implemented. Using these sampling criteria, an effective metamodel representation of the unknown space can be generated at reasonable sampling costs. Furthermore, the use of convergence criteria allows conclusions to be drawn about the level of confidence in the metamodel, and forms the basis for evaluating the adequacy of the original sampling budget.Copyright

Robust Optimization With NURBS HyPerModels

Engineering systems are generally susceptible to parameter uncertainties that influence real-time system performance and long-term system reliability. However, designers and engineers must design system solutions that are both optimal and dependable. Robust design techniques and robust optimization methods in particular, have emerged as promising methodologies to address the problem of dealing with parameter uncertainties. This research advances a robust optimization approach that exploits gradient information embedded in proximate NURBs control point clusters that are inherent in NURBs metamodel design space representations. The proximate control point clusters embody the target sensitivity profile and therefore include robust optimal solutions, thus enabling selective optimization within regions associated with the clusters. This robust optimization framework has been implemented and is demonstrated on unconstrained robust optimization problems from two test functions and a constrained robust optimization problem from a practical engineering design problem.© 2008 ASME

An Introduction to VitalS

VitalS is a system or methodology for evaluating the health or state of a variety of systems. Important or critical components of a system are identified and assigned weights based on how critical they are. Data are collected for each of these parameters, and the resulting data are analyzed using the VitalS to produce an indication of the system’s health or state. Component problems are readily identified using this technique. Various examples of VitalS are given in this paper. Examples included are medical, automobile, and avionic systems. The discussions includes introduction, utility and application of VitalS.Copyright

Mixed Integer Optimization With NURBs HyPerModels

The challenge of determining the best design in a multimodal design space with multiple local optimal solutions often challenges the best available optimization techniques. By casting the objective function of the optimization problem in the form of a Non-Uniform Rational B-spline (NURBs) metamodel, known as a HyPerModel, significant optimization advantages can be achieved, including the ability to efficiently find the global metamodel optimum solution with less computational expense than traditional approaches. This optimization strategy, defined by the HyPerOp algorithm, uses the underlying structure of a HyPerModel to intelligently select starting points for optimization runs and to identify regions of the design space that do not contain locations for the global metamodel optimum location. This paper describes the application of the HyPerOp algorithm to mixed integer programming problems and demonstrates its use with two example applications. The algorithm works with design spaces composed of continuous and integer design variables and provides a complementary approach for improved optimization capabilities.

Modeling Design Spaces With Discontinuous Variables Using NURBs HyPerModels

The vast majority of metamodeling demonstrations focuses on problems composed of continuous variables. However, important engineering design problems often include one or more discontinuous variables that require special attention. Previous work demonstrated the ability of Non-Uniform Rational B-spline HyPerModels to represent highly nonlinear functions composed of continuous variables. With minor modifications those capabilities can be extended to include functions defined by combinations of discontinuous input and output variables of different types, including discrete integer variables, feasibility variables and membership functions. Examples are used to demonstrate these modeling capabilities including applications developed from real engineering design problems such as the optimal positioning of a construction site crane and the optimal lay-up of a composite material I-beam.Copyright

Fault Detection With NURBs-Based Metamodels

Developing the ability for a system to self-monitor its condition is a desirable feature in many modern engineering systems. This capability facilitates a maintenance-as-needed rather than a maintenance-as-scheduled paradigm, offering potential efficiency improvements and corresponding cost savings. By using continuously updated Non-Uniform Rational B-spline (NURBs) metamodels of system performance to monitor the system condition, the onset of incipient faults can be detected by comparison to a self-generated as-built system metamodel, providing a basis for determining off-normal operating conditions. This capability is demonstrated for three distinct fault conditions prevalent in brushless DC motors. The results show that this technique can be used to develop an as-built system metamodel, develop a current system model during system operation, and detect the presence of an incipient fault condition despite the compensation provided by a feedback control system.© 2006 ASME