High Dimensional Model Representation Technique Research Articles

Cable force optimization of cable-stayed bridges: A surrogate model-assisted differential evolution method combined with B-Spline interpolation curves

Cable-stayed bridges have been widely constructed to cross large obstacles in transportation networks and metropolitan cities. As a highly redundant structure, the structural performance of a cable-stayed bridge can be significantly influenced by the cable prestressing forces. Therefore, the determination of optimal cable forces is necessary at the design stage. As computational mechanics matures, numerical computational methods such as high-fidelity finite element analyses facilitate accurate assessment of the bridge state, and evolutionary algorithms have been applied to enhance the global search ability. However, the combination of evolutionary algorithms and finite element model-based evaluations can substantially increase the computational burden, rendering the efficiency of optimization significantly time-consuming for practical engineering projects. To overcome this limitation, this work proposes a new surrogate model-assisted differential evolution method to facilitate the determination of optimal cable forces. Different from the existing methods, the proposed method develops a new individual selection strategy together with an additional archive to manage the diversity of population and further enhance the exploration ability. A new criterion is also proposed for the refinement of the surrogate model in the local search stage to avoid redundant information. Furthermore, B-Spline interpolation curve is integrated within the framework as a high-dimensional model representation technique for the design variables of cable forces. Five classic numerical examples and a practical engineering application validate the performance and advantages of the proposed method.

AN RBF BASED HIGH-DIMENSIONAL MODEL REDUCTION APPROACH IN BUILDING OPTIMIZATION

Building optimization has gained importance with the recent push to create the most economical and efficient buildings possible. As the effects of optimization are a function of the building size, it is crucial to understand and further develop optimization techniques for large-scale building structures. Practical structural optimization of buildings requires the use of a structural analysis software package and an iterative optimization procedure. As a result, finite element (FE) software shall be linked with an optimization solver. It is an expensive process which requires extensive computer coding. Alternative methods are available, including metamodeling methods, which are used to create simple and approximate functions based on complex FE simulations. In this study, the approximate functions are generated using a high-dimensional model representation (HDMR) framework. The HDMR framework is a model reduction approach and is found to be very accurate for different functions. The component functions of HDMR are expressed using augmented radial basis functions (RBFs). To further improve the numerical efficiency of the metamodels and reduce the total required number of structural analyses, a few different HDMR sampling approaches are investigated, including one static approach and two iterative strategies. An existing nonlinear programming (NLP) solver is employed in the design process. To illustrate the proposed approach, a three-dimensional building structure is selected as a numerical example. The numerical optimization is conducted to reduce the torsional response of the building. The proposed optimization method works very well and the results from different HDMR techniques are compared.

A Novel Probabilistic Power Flow Algorithm Based on Principal Component Analysis and High-Dimensional Model Representation Techniques

Because the penetration level of renewable energy sources has increased rapidly in recent years, uncertainty in power system operation is gradually increasing. As an efficient tool for power system analysis under uncertainty, probabilistic power flow (PPF) is becoming increasingly important. The point-estimate method (PEM) is a well-known PPF algorithm. However, two significant defects limit the practical use of this method. One is that the PEM struggles to estimate high-order moments accurately; this defect makes it difficult for the PEM to describe the distribution of non-Gaussian output random variables (ORVs). The other is that the calculation burden is strongly related to the scale of input random variables (IRVs), which makes the PEM difficult to use in large-scale power systems. A novel approach based on principal component analysis (PCA) and high-dimensional model representation (HDMR) is proposed here to overcome the defects of the traditional PEM. PCA is applied to decrease the dimension scale of IRVs and eliminate correlations. HDMR is applied to estimate the moments of ORVs. Because HDMR considers the cooperative effects of IRVs, it has a significantly smaller estimation error for high-order moments in particular. Case studies show that the proposed method can achieve a better performance in terms of accuracy and efficiency than traditional PEM.

Selection of PolSAR Observables for Crop Biophysical Variable Estimation With Global Sensitivity Analysis

The role of global sensitivity analysis (GSA) is to quantify and rank the most influential features for biophysical variable estimation. In this letter, an approximation model, called high-dimensional model representation (HDMR), is utilized to develop a regression method in conjunction with a GSA in the context of determining key input drivers in the estimation of crop biophysical variables from polarimetric synthetic aperture radar data. A multitemporal Radarsat-2 data set is used for the retrieval of three biophysical variables of barley: leaf area index, normalized difference vegetation index, and Biologische Bundesanstalt, Bundessortenamt und CHemische Industrie stage. The HDMR technique is first adopted to estimate a regression model with all available polarimetric features for each biophysical parameter, and sensitivity indices of each feature are then derived to explain the original space with a smaller number of features in which a final regression model is established. To evaluate the applicability of this methodology, root-mean square and coefficient of determination were performed under different amounts of samples. Results highlight that HDMR can be used effectively in biophysical variable estimation for not only reducing computational cost but also for providing a robust regression.

Design of a bio-inspired composite using probabilistic fracture mechanics

Artificial composites inspired from biological materials such as nacre and bone has a staggered structure of hard inclusions surrounded by a soft interface. These composites possess an excellent combination of strength, toughness and stiffness. These properties depends on a critical value of overlap ratio of inclusions, and hence is one of major parameters in the design of bio-inspired composites. Randomness in material and geometrical parameters could exist in such composites. In this work, the randomness in various parameters are considered, and using probabilistic methods, the critical overlap ratio is found out. This is done for two dimensional, and three dimensional inclusions. For the three dimensional case, due to unavailability of the analytical solution, high dimensional model representation technique in conjunction with finite element analysis is done to find out the critical overlap ratio. To determine the probability of failure for two dimensional as well as three dimensional cases, Monte Carlo Simulation is used. Finally, based on the study a factor of safety is proposed for the critical stress intensity factor which could be used in design to account for uncertainties in the various parameters.

An efficient Bayesian uncertainty quantification approach with application to k-ω-γ transition modeling

An efficient Bayesian uncertainty quantification approach is proposed, which combines the adaptive high dimensional model representation technique (HDMR) and stochastic collocation (SC) method based generalized polynomial chaos (gPC) to construct the surrogate for sampling procedure in Bayesian calibration step. Specifically, the adaptive HDMR technique is used to decompose the original high dimensional problems into several lower-dimensional subproblems, which are subsequently solved with the gPC-based SC method. Then the Bayesian calibration and prediction are carried out with the so-constructed surrogate model. A new indicator based on the variance of the corresponding component function is employed to identify the important components of the HDMR, instead of the original one based on the impact on the output mean, as the input parameters that can be well informed in the inverse problem are the ones that the model output is sensitive to. Further, a rigorous convergence study of the approximate posterior to the true posterior is carried out for the proposed approach. Its applications to both a simple mathematical function and a complex fluid dynamic model, i.e. k-ω-γ transition model, are investigated, demonstrating both its efficiency and accuracy. In the application to k-ω-γ transition model, the results show not only a quantified uncertainty overlapping well with the experimental data, but also a great improvement of the match between the prediction mean and the experimental data, which may be due to the further account of the intermittency through the spread of the model parameters.

A new methodology based on covariance and HDMR for global sensitivity analysis

To execute variance based global sensitivity analysis efficiently and purposefully, a computing methodology containing two covariance methods is proposed. The first method estimates sensitivity indices by making full use of a sampling matrix and a re-sampling matrix. The second one is proposed for compensating the systematic error of the first method. Sources of error of the two methods become clear when utilizing high-dimensional model representation technique, then the application scope is obtained and verified by test examples, and it can help researchers choose an appropriate method according to the size of sensitivity index of a given variable. Compared with other sampling-based methods, the new methodology is proved to be efficient and robust by examples.

A stochastic dimension reduction multiscale finite element method for groundwater flow problems in heterogeneous random porous media

Summary In this paper we present a stochastic dimension reduction multiscale finite element method for solving groundwater flow problems in heterogeneous random porous media. The stochastic conductivity is parameterized in a stochastic space using a truncated Karhunen–Loeve expansion with finite random variables. This often results in high stochastic dimensionality due to small correlation length and large variance of the inherent covariance function. In order to treat the high-dimensional stochastic problems efficiently, a truncated high-dimensional model representation (HDMR) technique is proposed that decomposes the high-dimensional stochastic problem into a moderate-dimensional stochastic problem and a few one-dimensional stochastic problems. The derived low-dimensional stochastic problems are solved by incorporating a sparse grid stochastic collocation method (SGSCM) with the proposed HDMR. This leads to a set of uncoupled deterministic problems at the sparse grid collocation points. To capture the small-scale heterogeneities in physical space, a multiscale finite element method (MsFEM) is utilized to solve these deterministic problems. Since the SGSCM and MsFEM are implemented in the different spaces, the integrated approach inherits the advantages from both of the SGSCM and the MsFEM, in which each low-dimensional stochastic problem is decoupled each other at small number of stochastic collocation points and is solved on a coarse spatial mesh. To demonstrate the efficiency and accuracy of the proposed stochastic MsFEM with the HDMR technique, a few numerical experiments are carried out for the transient water flow problems in heterogeneous random media with different correlation lengths and various degrees of spatial variabilities.

Assessment of high dimensional model representation techniques for reliability analysis

This paper presents an assessment of efficient response surface techniques based on the High Dimensional Model Representation (HDMR) and the Factorized High Dimensional Model Representation (FHDMR). The 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 an additive nature, allowing the physical model to be captured by the first few lower order terms. But, if the multiplicative nature of the response function is dominant, then all the right hand side components of the HDMR must be used to be able to obtain the best result. However, if the HDMR requires all components, which means 2 N of them, to get a desired accuracy, making the method very expensive in practice, then the FHDMR can be used. The component functions of the FHDMR are determined by using the component functions of the HDMR. This paper presents the formulation of the FHDMR based response surface approximation of a limit state/performance function which is dominantly multiplicative in nature. It is a given that conventional methods for reliability analysis are computationally very demanding, when applied in conjunction with complex finite element models. This study aims to assess how accurately and efficiently HDMR/FHDMR based response surface techniques can capture complex model output uncertainty. As a part of this effort, the efficacy of the HDMR, which is recently applied to reliability analysis, is also demonstrated. The response surface is constructed using the moving least squares interpolation formula by including constant, first-order, and second-order terms of the HDMR and the FHDMR. Once the response surface form is defined, the failure probability can be obtained by statistical simulation. Results of seven numerical examples involving structural/solid-mechanics/geo-technical engineering problems indicate that the failure probability obtained using the FHDMR based response surface method for a limit state/performance function that is dominantly multiplicative in nature, provides a significant accuracy when compared with the conventional Monte Carlo method, while requiring fewer original model simulations.

Parametric process synthesis for general nonlinear models

This paper presents a new approach towards parametric analysis of MINLP models in the context of process synthesis problems under uncertainty. The approach is based on the idea of High Dimensional Model Representation technique which utilize a reduced number of model runs to build an uncertainty propagation model that expresses the variability of optimal solution in the uncertain space. Based on this idea, a systematic procedure is developed where in the first step the possible changes in the optimal design configurations due to parametric uncertainty are identified. In the next step, the variability of optimal solution with parameter uncertainty for each design is captured. Having obtained a parametric expression of optimal objective for each design, the optimal solution can be determined by comparing the solutions for different designs. The proposed approach provides information about variation of the optimal objective and optimal design configuration over the entire uncertain space. This information can then be judiciously utilized in any decision making depending on specific process requirements. The main advantage of the proposed approach is that it does not depend on the nature or existence of a mathematical model to describe the input-output relationship of the process.

An Efficient Chemical Kinetics Solver Using High Dimensional Model Representation

A high dimensional model representation (HDMR) technique is introduced to capture the input−output behavior of chemical kinetic models. The HDMR expresses the output chemical species concentrations as a rapidly convergent hierarchical correlated function expansion in the input variables. In this paper, the input variables are taken as the species concentrations at time ti and the output is the concentrations at time ti + δ, where δ can be much larger than conventional integration time steps. A specially designed set of model runs is performed to determine the correlated functions making up the HDMR. The resultant HDMR can be used to (i) identify the key input variables acting independently or cooperatively on the output, and (ii) create a high speed fully equivalent operational model (FEOM) serving to replace the original kinetic model and its differential equation solver. A demonstration of the HDMR technique is presented for stratospheric chemical kinetics. The FEOM proved to give accurate and stable chemical concentrations out to long times of many years. In addition, the FEOM was found to be orders of magnitude faster than a conventional stiff equation solver. This computational acceleration should have significance in many chemical kinetic applications.