An adaptive Dendrite-HDMR metamodeling technique for high dimensional problems

High dimensional model representation (HDMR), decomposing the high-dimensional problem into summands of different order component terms, has been widely researched to work out the dilemma of “curse-of-dimensionality” when using surrogate techniques to approximate high-dimensional problems in engineering design. However, the available one-metamodel-based HDMRs usually encounter the predicament of prediction uncertainty, while current multi-metamodels-based HDMRs cannot provide simple explicit expressions for black-box problems, and have high computational complexity in terms of constructing the model by the explored points and predicting the responses of unobserved locations. Therefore, aimed at such problems, a new stand-alone HDMR metamodeling technique, termed as Dendrite-HDMR, is proposed in this study based on the hierarchical Cut-HDMR and the white-box machine learning algorithm, Dendrite Net. The proposed Dendrite-HDMR not only provides succinct and explicit expressions in the form of Taylor expansion, but also has relatively higher accuracy and stronger stability for most mathematical functions than other classical HDMRs with the assistance of the proposed adaptive sampling strategy, named KKMC, in which k-means clustering algorithm, k-Nearest Neighbor classification algorithm and the maximum curvature information of the provided expression are utilized to sample new points to refine the model. Finally, the Dendrite-HDMR technique is applied to solve the design optimization problem of the solid launch vehicle propulsion system with the purpose of improving the impulse-weight ratio, which represents the design level of the propulsion system.

Implementation and Performance Evaluation of a Bivariate Cut-HDMR Metamodel for Semiconductor Packaging Design Problems with a Large Number of Input Variables

A metamodeling technique based on Bivariate Cut High Dimensional Model Representation (Bivariate Cut HDMR) is implemented for a semiconductor packaging design problem with 10 design variables. Bivariate Cut-HDMR constructs a metamodel by considering only up to second-order interactions. The implementation uses three uniformly distributed sample points (s = 3) with quadratic spline interpolation to construct the component functions of Bivariate Cut-HDMR, which can be used to make a direct comparison with a metamodel based on Central Composite Design (CCD). The performance of Bivariate Cut-HDMR is evaluated by two well-known error metrics: R-squared and Relative Average Absolute Error (RAAE). The results are compared with the performance of CCD. Bivariate Cut HDMR does not compromise the accuracy compared to CCD, although the former uses only one-fifth of sample points (201 sample points) required by the latter (1045 sample points). The sampling schemes and the predictions of cut-planes and boundary-planes are discussed to explain possible reasons for the outstanding performance of Bivariate Cut HDMR.

Introduction to Metamodeling for Reducing Computational Burden of Advanced Analyses with Health Economic Models: A Structured Overview of Metamodeling Methods in a 6-Step Application Process

Metamodels can be used to reduce the computational burden associated with computationally demanding analyses of simulation models, although applications within health economics are still scarce. Besides a lack of awareness of their potential within health economics, the absence of guidance on the conceivably complex and time-consuming process of developing and validating metamodels may contribute to their limited uptake. To address these issues, this article introduces metamodeling to the wider health economic audience and presents a process for applying metamodeling in this context, including suitable methods and directions for their selection and use. General (i.e., non–health economic specific) metamodeling literature, clinical prediction modeling literature, and a previously published literature review were exploited to consolidate a process and to identify candidate metamodeling methods. Methods were considered applicable to health economics if they are able to account for mixed (i.e., continuous and discrete) input parameters and continuous outcomes. Six steps were identified as relevant for applying metamodeling methods within health economics: 1) the identification of a suitable metamodeling technique, 2) simulation of data sets according to a design of experiments, 3) fitting of the metamodel, 4) assessment of metamodel performance, 5) conducting the required analysis using the metamodel, and 6) verification of the results. Different methods are discussed to support each step, including their characteristics, directions for use, key references, and relevant R and Python packages. To address challenges regarding metamodeling methods selection, a first guide was developed toward using metamodels to reduce the computational burden of analyses of health economic models. This guidance may increase applications of metamodeling in health economics, enabling increased use of state-of-the-art analyses (e.g., value of information analysis) with computationally burdensome simulation models.

AN ENGINEERING RELIABILITY ANALYSIS METHOD BASED ON ADAPTIVE METAMODELS

In practical engineering problems, numerical analyses using the finite element (FE) method or other methods are generally required to evaluate system responses including stresses and deformations. For problems involving expensive FE analyses, it is not efficient or straightforward to directly apply conventional sampling-based or gradient-based reliability analysis approaches. To reduce computational efforts, it is useful to develop efficient and accurate metamodeling techniques to replace the original FE analyses. In this work, an adaptive metamodeling technique and a First-Order Reliability Method (FORM) were integrated. In each adaptive iteration, a compactly supported radial basis function (RBF) was adopted and a metamodel was created to explicitly express a performance function. An alternate FORM was implemented to calculate reliability index of the current iteration. Based on the design point, additional samples were generated and added to the existing sample points to re-generate the metamodel. The accuracy of the RBF metamodel could be improved in the neighborhood of the design point at each iteration. This procedure continued until the convergence of the reliability analysis results was achieved. A numerical example was studied. The proposed adaptive approach worked well and reliability analysis results were found with a reasonable number of iterations.

Optimal Parallel Machine Allocation Problem in IC Packaging Using IC-PSO: An Empirical Study

We model and apply a stochastic-simulation-based methodology to optimize the machine allocation of a flexible flow shop (FFS) dedicated to integrated circuit (IC) packaging. This contrasts with most previous research on non-deterministic FFS problems wherein stochastic simulation is mostly used to estimate throughput, cycle time, delay cost, or some other measure(s) in order to compare the performances of already-existing heuristic-based algorithms. The methodology applied in this research, called progressive simulation metamodeling for IC Packaging (IC-PSO), while rooted in the traditional metamodeling technique known as Response Surface Methodology (RSM), contrasts with RSM in that it is equipped with well-designed mechanisms to ensure an ever-increasing solution quality in an attempt to achieve the desirable optimality. The computational efficiency that IC-PSO affords IC packaging companies is demonstrated via a numerical study. Meanwhile, an empirical study based on real data was conducted to validate the viability of the proposed methodology in real settings.

Enhanced Gaussian Process Metamodeling and Collaborative Optimization for Vehicle Suspension Design Optimization

Dynamic stability is a key performance metric of motor vehicles and has a direct impact on passenger experience and customer satisfaction. The desired vehicle dynamics behavior can be achieved by optimizing the design of vehicle suspensions. Two challenges are associated with this design optimization task. The first one arises from the large number (e.g., 40 or 50) of design variables in modern suspension systems. Such multitude of variables not only makes it expensive to build a training dataset for metamodeling purposes, but also renders accurate surrogate modeling extremely difficult. The second challenge is a lack of guideline for choosing a proper multidisciplinary design optimization (MDO) method for a single MDO problem such as one for vehicle suspension design. In this paper, an enhanced Gaussian process (GP) metamodeling technique is developed and several versions of the collaborative optimization (CO) method are compared via a vehicle suspension design problem. In our enhanced GP modeling method, the model parameters are efficiently estimated using the smoothing effect of the so-called nugget parameter to reduce the search space. In addition, various versions of the CO method are studied where the enhanced collaborative optimization (ECO) method is found to perform the best. A simplified ECO formulation is also investigated to provide insights for future engineering applications.