Kambiz Haji Hajikolaei

Trust Region Based Mode Pursuing Sampling Method for Global Optimization of High Dimensional Design Problems

Mode pursuing sampling (MPS) was developed as a global optimization algorithm for design optimization problems involving expensive black box functions. MPS has been found to be effective and efficient for design problems of low dimensionality, i.e., the number of design variables is less than 10. This work integrates the concept of trust regions into the MPS framework to create a new algorithm, trust region based mode pursuing sampling (TRMPS2), with the aim of dramatically improving performance and efficiency for high dimensional problems. TRMPS2 is benchmarked against genetic algorithm (GA), dividing rectangles (DIRECT), efficient global optimization (EGO), and MPS using a suite of standard test problems and an engineering design problem. The results show that TRMPS2 performs better on average than GA, DIRECT, EGO, and MPS for high dimensional, expensive, and black box (HEB) problems. [DOI: 10.1115/1.4029219]

Broadening the Frequency Bandwidth of a Tire-Embedded Piezoelectric-Based Energy Harvesting System Using Coupled Linear Resonating Structure

The efficiency of single-degree-of-freedom (SDOF) vibration-based energy harvesters significantly drops when the resonance frequency of the harvester is different from that of the ambient vibration. In this study, a novel piezoelectric-based energy harvesting mechanism is introduced for rotary motion applications, which can generate power over a broad range of angular velocities of the wheel. The proposed design, which comprises a coupled spring-mass system attached to a PZT beam, has the advantage that it can easily be tuned in an off-line position by simply changing the tip mass and/or spring stiffness. A theoretical and experimental study is undertaken to check the performance of the proposed design for the range of speeds typical of commercial tires. It is shown that by tuning the resonance frequency of the mass-spring system the design can significantly increase the frequency bandwidth of the energy harvester.

High Dimensional Model Representation With Principal Component Analysis

In engineering design, spending excessive amount of time on physical experiments or expensive simulations makes the design costly and lengthy. This issue exacerbates when the design problem has a large number of inputs, or of high dimension. High dimensional model representation (HDMR) is one powerful method in approximating high dimensional, expensive, black-box (HEB) problems. One existing HDMR implementation, random sampling HDMR (RS-HDMR), can build an HDMR model from random sample points with a linear combination of basis functions. The most critical issue in RS-HDMR is that calculating the coefficients for the basis functions includes integrals that are approximated by Monte Carlo summations, which are error prone with limited samples and especially with nonuniform sampling. In this paper, a new approach based on principal component analysis (PCA), called PCA-HDMR, is proposed for finding the coefficients that provide the best linear combination of the bases with minimum error and without using any integral. Several benchmark problems of different dimensionalities and one engineering problem are modeled using the method and the results are compared with RS-HDMR results. In all problems with both uniform and nonuniform sampling, PCA-HDMR built more accurate models than RS-HDMR for a given set of sample points. [DOI: 10.1115/1.4025491]

Modification of DIRECT for high-dimensional design problems

DIviding RECTangles (DIRECT), as a well-known derivative-free global optimization method, has been found to be effective and efficient for low-dimensional problems. When facing high-dimensional black-box problems, however, DIRECTs performance deteriorates. This work proposes a series of modifications to DIRECT for high-dimensional problems (dimensionality d>10). The principal idea is to increase the convergence speed by breaking its single initialization-to-convergence approach into several more intricate steps. Specifically, starting with the entire feasible area, the search domain will shrink gradually and adaptively to the region enclosing the potential optimum. Several stopping criteria have been introduced to avoid premature convergence. A diversification subroutine has also been developed to prevent the algorithm from being trapped in local minima. The proposed approach is benchmarked using nine standard high-dimensional test functions and one black-box engineering problem. All these tests show a significant efficiency improvement over the original DIRECT for high-dimensional design problems.

Adaptive Orthonormal Basis Functions for High Dimensional Metamodeling With Existing Sample Points

High Dimensional Model Representation (HDMR) is a tool for generating an approximation of an input-output model for a multivariate function. It can be used to model a black-box function for metamodel-based optimization. Recently the authors’ team has developed a radial basis function based HDMR (RBF-HDMR) model that can efficiently model a high dimensional black-box function and, moreover, to uncover inner variable structures of the black-box function. This approach, however, requests a complete new, although optimized, set of sample points, as dictated by the methodology, while in engineering design practice one often has many existing sample data. How to utilize the existing data to efficiently construct a HDMR model is the focus of this paper. We first identify the Random-Sampling HDMR (RS-HDMR), which uses orthonormal basis functions as HDMR component functions and existing sample points can be used to calculate the coefficients of the basis functions. One of the important issues related to the RS-HDMR is that in theory the basis functions are obtained based on the continuous integrations related to the orthonormality conditions. In practice, however, the integrations are approximated by Monte Carlo summation and thus the basis functions may not satisfy the orthonormality conditions. In this paper, we propose new and adaptive orthonormal basis functions with respect to a given set of sample points for RS-HDMR approximation. RS-HDMR models are built for different test functions using the standard and new adaptive basis functions for different number of sample points. The relative errors for both models are calculated and compared. The results show that the models that are built using the new basis functions are more accurate.Copyright

Decomposition for large-scale global optimization based on quantified variable correlations uncovered by metamodelling

The recently developed Radial Basis Function High-Dimensional Model Representation (RBF-HDMR) yields qualitative information about which variables are correlated. Such qualitative information is only applicable for a few problems that can be completely decomposed. This work develops a strategy to quantify the variable correlations so that decomposition can be fully supported for a wider range of problems. A simple optimization scheme is also proposed to systematically solve the decomposed subproblems, instead of solving the original undecomposed problem. The proposed decomposition–optimization strategy is compared to the direct optimization case without decomposition, for four categories of problems with different decomposability levels. The results show that except for the category of non-decomposable problems in which all variable correlations are strong, the proposed methodology is effective and has similar accuracy to the case of solving the original undecomposed problems, and it finds the optimum with a lower number of function evaluations.

Designing scalable product families by the radial basis function–high-dimensional model representation metamodelling technique

Product family design is cost-efficient for achieving the best trade-off between commonalization and diversification. However, for computationally intensive design functions which are viewed as black boxes, the family design would be challenging. A two-stage platform configuration method with generalized commonality is proposed for a scale-based family with unknown platform configuration. Unconventional sensitivity analysis and information on variation in the individual variants’ optimal design are used for platform configuration design. Metamodelling is employed to provide the sensitivity and variable correlation information, leading to significant savings in function calls. A family of universal electric motors is designed for product performance and the efficiency of this method is studied. The impact of the employed parameters is also analysed. Then, the proposed method is modified for obtaining higher commonality. The proposed method is shown to yield design solutions with better objective function values, allowable performance loss and higher commonality than the previously developed methods in the literature.

Design Optimization on "White-Box" Uncovered by Metamodeling

In the area of Multidisciplinary Design Optimization (MDO), a majority of problems involve so called high-dimensional, expensive, black-box (HEB) functions, such as complex finite element analyses or computational fluid dynamics simulations. A new metamodeling approach, the radial-basis function-high dimensional model representation (RBF-HDMR) method, was recently developed for HEB problems. RBF-HDMR adaptively models a HEB problem according to the problem’s intrinsic (non)linearity, variable correlations, and variable structures. Therefore in a sense it is able to turn a “black-box” function in a “white-box.” This work explores the application of RBF-HDMR in the context of optimization. The model is first applied to uncover the variable structure and correlations, based on which the HEB problem is then decomposed to sub-problems. Optimization is then applied to those sub-problems. This simple strategy is then compared with direct optimization without decomposition. From the tests, the pros and cons of the strategy will be discussed.

Surrogate-assisted Self-accelerated Particle Swarm Optimization

Surrogate-assisted self-accelerated particle swarm optimization (SASA-PSO) is a major modification of an original PSO which uses all previously evaluated particles aiming to increase the computational efficiency. A newly in-house developed metamodeling approach named high dimensional model representation with principal component analysis (PCAHDMR), which was specifically established for so called high-dimensional, expensive, blackbox (HEB) problems, is used to approximate a function using all particles calculated during the optimization process. Then, based on the minimum of the constructed metamodel, a term called “metamodeling acceleration” is added to the velocity update formula in the original PSO algorithm. The proposed optimization algorithm performance is investigated using several benchmark problems with different number of variables and the results are also compared with original PSO results. Preliminary results show a considerable performance improvement in terms of number of function evaluations as well as achieved global optimum specifically for high-dimensional problems.

Visual HDMR Model Refinement Through Iterative Interaction

In engineering design, time-consuming simulations may be needed to find the input-output relationship of a system. High Dimensional Model Representation (HDMR) alleviates the need for intensive simulation by approximating the system’s design space with a surrogate model. Although HDMR can provide an overview, specific regions of interest to the designer may require higher accuracy. This paper presents a tool to visualize and interactively improve HDMR accuracy in specified regions of the design space. Regions of the HDMR are selected by iterative brushing in two-dimensional scatterplot planes. Once a region is chosen, designers may concentrate sampling within its bounds to improve the model locally. Regions can be also improved by modeling the error with a localized radial basis function (RBF) metamodel. The effect of local refinement was further evaluated with localized performance metrics. Testing of the tool shows that it can effectively display and improve HDMR models in regions of interest, if there are variables which have a dominating influence on the output.Copyright