G. Gary Wang

Adaptive Response Surface Method Using Inherited Latin Hypercube Design Points

This paper addresses the difficulty of the previously developed Adaptive Response Surface Method (ARSM) for high-dimensional design problems. ARSM was developed to search for the global design optimum for computation-intensive design problems. This method utilizes Central Composite Design (CCD), which results in an exponentially increasing number of required design experiments. In addition, ARSM generates a complete new set of CCD points in a gradually reduced design space. These two factors greatly undermine the effciency of ARSM. In this work, Latin Hypercube Design (LHD) is utilized to generate saturated design experiments. Because of the use of LHD, historical design experiments can be inherited in later iterations. As a result, ARSM only requires a limited number of design experiments even for high-dimensional design problems. The improved ARSM is tested using a group of standard test problems and then applied to an engineering design problem. In both testing and design application, significant improvement in the efficiency of ARSM is realized. The improved ARSM demonstrates strong potential to be a practical global optimization tool for computation-intensive design problems. Inheriting LHD points, as a general sampling strategy, can be integrated into other approximation-based design optimization methodologies.

ADAPTIVE RESPONSE SURFACE METHOD - A GLOBAL OPTIMIZATION SCHEME FOR APPROXIMATION-BASED DESIGN PROBLEMS

Abstract For design problems involving computation-intensive analysis or simulation processes, approximation models are usually introduced to reduce computation lime. Most approximation-based optimization methods make step-by-step improvements to the approximation model by adjusting the limits of the design variables. In this work, a new approximation-based optimization method for computation-intensive design problems - the adaptive response surface method(ARSM), is presented. The ARSM creates quadratic approximation models for the computation-intensive design objective function in a gradually reduced design space. The ARSM was designed to avoid being trapped by local optima and to identify the global design optimum with a modest number of objective function evaluations. Extensive tests on the ARSM as a global optimization scheme using benchmark problems, as well as an industrial design application of the method, are presented. Advantages and limitations of the approach are also discussed

Definition and Review of Virtual Prototyping

Virtual Prototyping (VP) technique has been interpreted in many different ways, which causes confusion and misunderstanding among researchers and practitioners. Based on a review of the current related research and application, this paper proposes a definition of VP as well as components of a virtual prototype. VP is then compared with and distinguished from virtual reality (VR), virtual environment (VE), and virtual manufacturing (VM) techniques. Given the proposed definition and review of VP, future VP related research topics are suggested.

Mode-pursuing sampling method for global optimization on expensive black-box functions

The presence of black-box functions in engineering design, which are usually computation-intensive, demands efficient global optimization methods. This article proposes a new global optimization method for black-box functions. The global optimization method is based on a novel mode-pursuing sampling method that systematically generates more sample points in the neighborhood of the function mode while statistically covering the entire search space. Quadratic regression is performed to detect the region containing the global optimum. The sampling and detection process iterates until the global optimum is obtained. Through intensive testing, this method is found to be effective, efficient, robust, and applicable to both continuous and discontinuous functions. It supports simultaneous computation and applies to both unconstrained and constrained optimization problems. Because it does not call any existing global optimization tool, it can be used as a standalone global optimization method for inexpensive problems as well. Limitations of the method are also identified and discussed.

Reliable design space and complete single-loop reliability-based design optimization

Reliability-based design optimization (RBDO) has been intensively studied due to its significance and its conceptual and mathematical complexity. This paper proposes a new method for RBDO on the basis of the concept of reliable design space (RDS), within which any design satisfies the reliability requirements. Therefore, a RBDO problem becomes a simple, deterministic optimization problem constrained by RDS rather than its deterministic feasible space. The RDS is found in this work by using the partial derivatives at the current design point as an approximation of the derivatives at its corresponding most probable point (MPP) on the limit state function. This work completely resolves the double loop in RBDO and turns RBDO into a simple optimization problem. Well-known problems from the literature are selected to illustrate the steps of the approach and for result comparison. Discussions will also be given on the limitation of the proposed method, which is shown to be a common limitation overlooked by the research community on RBDO.

Metamodeling for High Dimensional Simulation-Based Design Problems

Computational tools such as finite element analysis and simulation are widely used in engineering, but they are mostly used for design analysis and validation. If these tools can be integrated for design optimization, it will undoubtedly enhance a manufacturers competitiveness. Such integration, however, faces three main challenges: (1) high computational expense of simulation, (2) the simulation process being a black-box function, and (3) design problems being high dimensional. In the past two decades, metamodeling has been intensively developed to deal with expensive black-box functions, and has achieved success for low dimensional design problems. But when high dimensionality is also present in design, which is often found in practice, there lacks of a practical method to deal with the so-called high dimensional, expensive, and black-box (HEB) problems. This paper proposes the first metamodel of its kind to tackle the HEB problem. This paper integrates the radial basis function with high dimensional model representation into a new model, RBF-HDMR. The developed RBF-HDMR model offers an explicit function expression, and can reveal (1 ) the contribution of each design variable, (2) inherent linearity/nonlinearity with respect to input variables, and (3) correlation relationships among input variables. An accompanying algorithm to construct the RBF - HDMR has also been developed. The model and the algorithm fundamentally change the exponentially growing computation cost to be polynomial. Testing and comparison confirm the efficiency and capability of RBF-HDMR for HEB problems.

Fuzzy clustering based hierarchical metamodeling for design space reduction and optimization

For computation-intensive design problems, metamodeling techniques are commonly used to reduce the computational expense during optimization; however, they often have difficulty or even fail to model an unknown system in a large design space, especially when the number of available samples is limited. This article proposes an intuitive methodology to systematically reduce the design space to a relatively small region. This methodology entails three main elements: (1) constructing metamodels using either response surface or kriging models to capture unknown system behavior in the original large space; (2) calculating many inexpensive points from the obtained metamodel, clustering these points using the fuzzy c-means clustering method, and choosing an attractive cluster and its corresponding reduced design space; (3) progressively generating sample points to construct kriging models and identify the design optimum within the reduced design space. The proposed methodology is illustrated using the well-known six-hump camel back problem, a highly nonlinear constrained optimization problem, and a real design problem. Through comparison with other methods, it is found that the proposed methodology can intuitively capture promising design regions and can efficiently identify the global or near-global design optimum in the presence of highly nonlinear constraints. The effect of using either response surface or kriging models in the original design space is also compared and contrasted. Limitations of the proposed methodology are discussed.

Mode Pursuing Sampling Method for Discrete Variable Optimization on Expensive Black-Box Functions

Based on previously developed Mode Pursuing Sampling (MPS) approach for continuous variables, a variation of MPS for discrete variable global optimization problems on expensive black-box functions is developed in this paper. The proposed method, namely, the discrete variable MPS (D-MPS) method, differs from its continuous variable version not only on sampling in a discrete space, but moreover, on a novel double-sphere strategy. The double-sphere strategy features two hyperspheres whose radii are dynamically enlarged or shrunk in control of, respectively, the degree of “exploration” and “exploitation” in the search of the optimum. Through testing and application to design problems, the proposed D-MPS method demonstrates excellent efficiency and accuracy as compared to the best results in literature on the test problems. The proposed method is believed a promising global optimization strategy for expensive black-box functions with discrete variables. The double-sphere strategy provides an original search control mechanism and has potential to be used in other search algorithms. DOI: 10.1115/1.2803251

Center-based sampling for population-based algorithms

Population-based algorithms, such as Differential Evolution (DE), Particle Swarm Optimization (PSO), Genetic Algorithms (GAs), and Evolutionary Strategies (ES), are commonly used approaches to solve complex problems from science and engineering. They work with a population of candidate solutions. In this paper, a novel center-based sampling is proposed for these algorithms. Reducing the number of function evaluations to tackle with high-dimensional problems is a worthwhile attempt; the center-based sampling can open a new research area in this direction. Our simulation results confirm that this sampling, which can be utilized during population initialization and/or generating successive generations, could be valuable in solving large-scale problems efficiently. Quasi-Oppositional Differential Evolution is briefly discussed as an evidence to support the proposed sampling theory. Furthermore, opposition-based sampling and center-based sampling are compared in this paper. Black-box optimization is considered in this paper and all details about the conducted simulations are provided.

First and second order approximate reliability analysis methods using evidence theory

The first order approximate reliability method (FARM) and second order approximate reliability method (SARM) are formulated based on evidence theory in this paper. The proposed methods can significantly improve the computational efficiency for evidence-theory-based reliability analysis, while generally provide sufficient precision. First, the most probable focal element (MPFE), an important concept as the most probable point (MPP) in probability-theory-based reliability analysis, is searched using a uniformity approach. Subsequently, FARM approximates the limit-state function around the MPFE using the linear Taylor series, while SARM approximates it using the quadratic Taylor series. With the first and second order approximations, the reliability interval composed of the belief measure and the plausibility measure is efficiently obtained for FARM and SARM, respectively. Two simple problems with explicit expressions and one engineering application of vehicle frontal impact are presented to demonstrate the effectiveness of the proposed methods.