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Popular regression methodologies are inapplicable to obtain accurate metamodels for high dimensional practical problems since the computational time increases exponentially as the number of dimensions rises. The purpose of this paper is to use support vector regression with high dimensional model representation (SVR-HDMR) model to obtain accurate metamodels for high dimensional problems with a few sampling points.

High-dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for improving the efficiency of deducing high dimensional input-output system behavior. Support vector regression (SVR) method can approximate the underlying functions with a small subset of sample points. Dividing Rectangles (DIRECT) algorithm is a deterministic sampling method.

This paper proposes a new form of HDMR by integrating the SVR, termed as SVR-HDMR. And an intelligent sampling strategy, namely, DIRECT method, is adopted to improve the efficiency of SVR-HDMR.

Compared to other metamodeling techniques, the accuracy and efficiency of SVR-HDMR were significantly improved. The SVR-HDMR helped engineers understand the essence of underlying problems visually.

The purpose of study is to overcome the error estimation of standard deviation derived from Expected improvement (EI) criterion. Compared with other popular methods, a quantitative model assessment and analysis tool, termed high-dimensional model representation (HDMR), is suggested to be integrated with an EI-assisted sampling strategy.

To predict standard deviation directly, Kriging is imported. Furthermore, to compensate for the underestimation of error in the Kriging predictor, a Pareto frontier (PF)-EI (PFEI) criterion is also suggested. Compared with other surrogate-assisted optimization methods, the distinctive characteristic of HDMR is to disclose the correlations among component functions. If only low correlation terms are considered, the number of function evaluations for HDMR grows only polynomially with the number of input variables and correlative terms.

To validate the suggested method, various nonlinear and high-dimensional mathematical functions are tested. The results show the suggested method is potential for solving complicated real engineering problems.

In this study, the authors hope to integrate superiorities of PFEI and HDMR to improve optimization performance.

To develop a new computational tool for predicting failure probability of structural/mechanical systems subject to random loads, material properties, and geometry.

High dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is a very efficient formulation of the system response, if higher order variable correlations are weak and if the response function is dominantly of additive nature, allowing the physical model to be captured by the first few lower order terms. But, if multiplicative nature of the response function is dominant then all right hand side components of HDMR must be used to be able to obtain the best result. However, if HDMR requires all components, which means 2^N number of components, to get a desired accuracy, making the method very expensive in practice, then factorized HDMR (FHDMR) can be used. The component functions of FHDMR are determined by using the component functions of HDMR. This paper presents the formulation of FHDMR approximation of a multivariate limit state/performance function, which is dominantly of multiplicative nature. Given that conventional methods for reliability analysis are very computationally demanding, when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently HDMR/FHDMR based approximation techniques can capture complex model output uncertainty. As a part of this effort, the efficacy of HDMR, which is recently applied to reliability analysis, is also demonstrated. Response surface is constructed using moving least squares interpolation formula by including constant, first‐order and second‐order terms of HDMR and FHDMR. Once the response surface form is defined, the failure probability can be obtained by statistical simulation.

Results of five numerical examples involving structural/solid‐mechanics/geo‐technical engineering problems indicate that the failure probability obtained using FHDMR approximation for the limit state/performance function of dominantly multiplicative in nature, provides significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.

This is the first time where application of FHDMR concepts is explored in the field of reliability and system safety. Present computational approach is valuable to the practical modeling and design community, where user often suffers from the curse of dimensionality.

Reflow soldering process is an important step of the surface mount technology. The purpose of this paper is to minimize the maximum warpage of shielding frame by controlling reflow soldering control parameters.

Compared with other reflow-related design methods, both time and temperate of each extracted time region are considered. Therefore, the number of design variable is increased. To solve the high-dimensional problem, a surrogate-assisted optimization (SAO) called adaptive Kriging high-dimensional representation model (HDMR) is used.

Therefore, the number of design variable is increased. To solve the high-dimensional problem, a surrogate-assisted optimization (SAO) called HDMR is used. The warpage of shield frame is significantly reduced. Moreover, the correlations of design variables are also disclosed.

Compared with the original Kriging HDMR, the expected improvement (EI) criterion is used and a new projection strategy is suggested to improve the efficiency of optimization method. The application suggests that the adaptive Kriging HDMR has potential capability to solve such complicated engineering problems.

The plain High Dimensional Model Representation (HDMR) method needs Dirac delta type weights to partition the given multivariate data set for modelling an interpolation problem. Dirac delta type weight imposes a different importance level to each node of this set during the partitioning procedure which directly effects the performance of HDMR. The purpose of this paper is to develop a new method by using fluctuation free integration and HDMR methods to obtain optimized weight factors needed for identifying these importance levels for the multivariate data partitioning and modelling procedure.

A common problem in multivariate interpolation problems where the sought function values are given at the nodes of a rectangular prismatic grid is to determine an analytical structure for the function under consideration. As the multivariance of an interpolation problem increases, incompletenesses appear in standard numerical methods and memory limitations in computer‐based applications. To overcome the multivariance problems, it is better to deal with less‐variate structures. HDMR methods which are based on divide‐and‐conquer philosophy can be used for this purpose. This corresponds to multivariate data partitioning in which at most univariate components of the Plain HDMR are taken into consideration. To obtain these components there exist a number of integrals to be evaluated and the Fluctuation Free Integration method is used to obtain the results of these integrals. This new form of HDMR integrated with Fluctuation Free Integration also allows the Dirac delta type weight usage in multivariate data partitioning to be discarded and to optimize the weight factors corresponding to the importance level of each node of the given set.

The method developed in this study is applied to the six numerical examples in which there exist different structures and very encouraging results were obtained. In addition, the new method is compared with the other methods which include Dirac delta type weight function and the obtained results are given in the numerical implementations section.

The authors' new method allows an optimized weight structure in modelling to be determined in the given problem, instead of imposing the use of a certain weight function such as Dirac delta type weight. This allows the HDMR philosophy to have the chance of a flexible weight utilization in multivariate data modelling problems.

This study aims to suggest and develops a global sensitivity analysis-assisted multi-level sequential optimization method for the heat transfer problem.

Compared with other surrogate-assisted optimization methods, the distinctive characteristic of the suggested method is to decompose the original problem into several layers according to the global sensitivity index. The optimization starts with the several most important design variables by the support vector regression-based efficient global optimization method. Then, when the optimization process progresses, the filtered design variables should be involved in optimization one by one or the setting value. Therefore, in each layer, the design space should be reduced according to the previous optimization result. To improve the accuracy of the global sensitivity index, a novel global sensitivity analysis method based on the variance-based method incorporating a random sampling high-dimensional model representation is introduced.

The advantage of this method lies in its capability to solve complicated problems with a limited number of sample points. Moreover, to enhance the reliability of optimum, the support vector regression-based global efficient optimization is used to optimize in each layer.

The developed optimization tool is built by MATLAB and can be integrated by commercial software, such as ABAQUS and COMSOL. Lastly, this tool is integrated with COMSOL and applied to the plant-fin heat sink design. Compared with the initial temperature, the temperature after design is over 49°. Moreover, the relationships among all design variables are also disclosed clearly.

The D-MORPH-HDMR is integrated to obtain the coupling relativities among the design variables efficiently. The suggested method can be decomposed into multiplier layers according to the GSI. The SVR-EGO is used to optimize the sub-problem because of its robustness of modeling.

In the absence of random variables, random variables are generated by the Monte Carlo (MC) simulation method. There are some methods for generating fragility curves with fewer nonlinear analyses. However, the accuracy of these methods is not suitable for all performance levels and peak ground acceleration (PGA) range. This paper aims to present a method through the seismic improvement of the high-dimensional model representation method for generating fragility curves while taking advantage of fewer analyses by choosing the right border points.

In this method, the values of uncertain variables are selected based on the results of the initial analyses, the damage limit of each performance level or according to acceptable limits in the design code. In particular, PGAs are selected based on the general shape of the fragility curve for each performance limit. Also, polynomial response functions are estimated for each accelerogram. To evaluate the accuracy, fragility curves are estimated by different methods for a single degree of freedom system and a reinforced concrete frame.

The results indicated that the proposed method can not only reduce the computational cost but also has a higher accuracy than the other methods, compared with the MC baseline method.

The proposed response functions are more consistent with the actual values and are also congruent with each performance level to increase the accuracy of the fragility curves.

High‐dimensional model representation (HDMR) is a general set of quantitative model assessment and analysis tools for capturing the high‐dimensional relationships between sets of input and output model variables. It is an efficient formulation of the system response, if higher‐order cooperative effects are weak, allowing the physical model to be captured by the lower‐order terms. The paper's aim is to develop a new computational tool for estimating probabilistic sensitivity of structural/mechanical systems subject to random loads, material properties and geometry.

When first‐order HDMR approximation of the original high‐dimensional limit state is not adequate to provide the desired accuracy to the sensitivity analysis, this paper presents an enhanced HDMR (eHDMR) method to represent the higher‐order terms of HDMR expansion by expressions similar to the lower‐order ones with monomial multipliers. The accuracy of the HDMR expansion can be significantly improved using preconditioning with a minimal number of additional input‐output samples without directly invoking the determination of second‐ and higher‐order terms. As a part of this effort, the efficacy of HDMR, which is recently applied to uncertainty analysis, is also demonstrated. The method is based on computing eHDMR approximation of system responses and score functions associated with probability distribution of a random input. Surrogate model is constructed using moving least squares interpolation formula. Once the surrogate model form is defined, both the probabilistic response and its sensitivities can be estimated from a single probabilistic analysis, without requiring the gradients of performance functions.

The results of two numerical examples involving mathematical function and structural/solid‐mechanics problems indicate that the sensitivities obtained using eHDMR approximation provide significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.

This is the first time where application of eHDMR concepts is explored in the stochastic sensitivity analysis. The present computational approach is valuable to the practical modelling and design community.

Global sensitivity can measure the influence of input variables on model responses and is of positive significance for the improvement design of structural systems. This work aims to study the global sensitivity of structural models by combining the active subspace theory and neural network.

This study aims to improve the efficiency of global sensitivity analysis for high-dimensional structural systems, a novel method based on active subspace and surrogate model is proposed. Active subspace can reduce the dimension of input variables, and an adaptive scaling strategy is proposed to improve the accuracy in finding the active subspace. The uncertainty propagation of active variables and model response is performed through the artificial neural network. Then the global sensitivity analysis is carried out.

Several examples are studied by using the Monte Carlo simulation method and the proposed method. Comparison of the results shows that the proposed method has preferable accuracy and low computational cost.

The proposed method provides a practicable tool for the variance-based sensitivity analysis of structural systems. Apart from sensitivity analysis, the method can be also extended for use in other fields relating to uncertainty propagation.

The purpose of this study is to improve the efficiency and accuracy of response surface method (RSM), as well as its robustness.

By introducing cut-high-dimensional representation model (HDMR), the delineation of cross terms and the constitution analysis of component function, a new adaptive RSM is presented for reliability calculation, where a sampling scheme is also proposed to help constructing response surface close to limit-state.

The proposed method has a more feasible process of evaluating undetermined coefficients of each component function than traditional RSM, and performs well in terms of balancing the efficiency and accuracy when compared to the traditional second-order polynomial RSM. Moreover, the proposed method is robust on the parameter in a wide range, indicating that it is able to obtain convergent result in a wide feasible domain of sample points.

This study constructed an adaptive bivariate cut-HDMR by introducing delineation of cross-terms and constitution of univariate component function; and a new sampling technique is proposed.