Metamodeling for High Dimensional Simulation-Based Design Problems

2010 ◽  
Vol 132 (5) ◽  
Author(s):  
Songqing Shan ◽  
G. Gary Wang
Keyword(s):  
Practical Method ◽  
Black Box ◽  
High Dimensional ◽  
High Dimensional Model Representation ◽  
Element Analysis ◽  
Design Problems ◽  
Function Expression ◽  
Input Variables ◽  
Dimensional Simulation ◽  
Low Dimensional

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 manufacturer’s 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.

Turning Black-Box Into White Functions

2010 ◽  
Author(s):  
Songqing Shan ◽  
G. Gary Wang
Keyword(s):  
Functional Form ◽  
Black Box ◽  
High Dimensional ◽  
High Dimensional Model Representation ◽  
Significant Variable ◽  
Dimensional Model ◽  
Adaptive Process ◽  
Order Form ◽  
Effectiveness And Efficiency ◽  
Test Result

Modeling of high dimensional expensive black-box (HEB) functions is challenging. A recently developed method, radial basis function-based high dimensional model representation (RBF-HDMR), has been found promising. This work extends RBF-HDMR to enhance its modeling capability beyond the current second order form and “uncover” black-box functions so that not only a more accurate metamodel is obtained, but also key information of the function can be gained and thus the black-box function can be turned “white.” The key information that can be gained includes 1) functional form, 2) (non)linearity with respect to each variable, 3) variable correlations. The resultant model can be used for applications such as sensitivity analysis, visualization, and optimization. The RBF-HDMR exploration is based on identifying the existence of certain variable correlations through derived theorems. The adaptive process of exploration and modeling reveals the black-box functions till all significant variable correlations are found. The black-box functional form is then represented by a structure matrix that can manifest all orders of correlated behavior of variables. The proposed approach is tested with theoretical and practical examples. The test result demonstrates the effectiveness and efficiency of the proposed approach.

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

2015 ◽  
Vol 137 (2) ◽  
Author(s):  
George H. Cheng ◽  
Adel Younis ◽  
Kambiz Haji Hajikolaei ◽  
G. Gary Wang
Keyword(s):  
Global Optimization ◽  
Optimization Problems ◽  
Trust Region ◽  
Standard Test ◽  
Black Box ◽  
High Dimensional ◽  
Efficient Global Optimization ◽  
Test Problems ◽  
Design Problems ◽  
Design Variables

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.

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

2012 ◽  
Author(s):  
Kambiz Haji Hajikolaei ◽  
G. Gary Wang
Keyword(s):  
Orthonormal Basis ◽  
Black Box ◽  
Basis Functions ◽  
High Dimensional ◽  
High Dimensional Model Representation ◽  
Design Practice ◽  
Test Functions ◽  
Sample Data ◽  
Sample Points ◽  
Relative Errors

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.

Turning Black-Box Functions Into White Functions

2011 ◽  
Vol 133 (3) ◽  
Author(s):  
Songqing Shan ◽  
G. Gary Wang
Keyword(s):  
Functional Form ◽  
Black Box ◽  
High Dimensional ◽  
High Dimensional Model Representation ◽  
Significant Variable ◽  
Test Results ◽  
Dimensional Model ◽  
Adaptive Process ◽  
Modeling Process ◽  
Effectiveness And Efficiency

A recently developed metamodel, radial basis function-based high-dimensional model representation (RBF-HDMR), shows promise as a metamodel for high-dimensional expensive black-box functions. This work extends the modeling capability of RBF-HDMR from the current second-order form to any higher order. More importantly, the modeling process “uncovers” black-box functions so that not only is a more accurate metamodel obtained, but also key information about the function can be gained and thus the black-box function can be turned “white.” The key information that can be gained includes: (1) functional form, (2) (non)linearity with respect to each variable, and (3) variable correlations. The black-box “uncovering” process is based on identifying the existence of certain variable correlations through two derived theorems. The adaptive process of exploration and modeling reveals the black-box functions until all significant variable correlations are found. The black-box functional form is then represented by a structure matrix that can manifest all orders of correlated behavior of the variables. The resultant metamodel and its revealed inner structure lend themselves well to applications such as sensitivity analysis, decomposition, visualization, and optimization. The proposed approach is tested with theoretical and practical examples. The test results demonstrate the effectiveness and efficiency of the proposed approach.

An Emulator-Based Prediction of Dynamic Stiffness for Redundant Parallel Kinematic Mechanisms

2015 ◽  
Vol 8 (2) ◽  
Author(s):  
Mario Luces ◽  
Pinar Boyraz ◽  
Masih Mahmoodi ◽  
Farhad Keramati ◽  
James K. Mills ◽  
...  
Keyword(s):  
Dynamic Stiffness ◽  
Training Data ◽  
Analytical Models ◽  
High Dimensional ◽  
Computationally Efficient ◽  
Element Analysis ◽  
Data Set ◽  
Parallel Kinematic ◽  
Low Dimensional ◽  
Mls Approximation

The accuracy of a parallel kinematic mechanism (PKM) is directly related to its dynamic stiffness, which in turn is configuration dependent. For PKMs with kinematic redundancy, configurations with higher stiffness can be chosen during motion-trajectory planning for optimal performance. Herein, dynamic stiffness refers to the deformation of the mechanism structure, subject to dynamic loads of changing frequency. The stiffness-optimization problem has two computational constraints: (i) calculation of the dynamic stiffness of any considered PKM configuration, at a given task-space location, and (ii) searching for the PKM configuration with the highest stiffness at this location. Due to the lack of available analytical models, herein, the former subproblem is addressed via a novel effective emulator to provide a computationally efficient approximation of the high-dimensional dynamic-stiffness function suitable for optimization. The proposed method for emulator development identifies the mechanism's structural modes in order to breakdown the high-dimensional stiffness function into multiple functions of lower dimension. Despite their computational efficiency, however, emulators approximating high-dimensional functions are often difficult to develop and implement due to the large amount of data required to train the emulator. Reducing the dimensionality of the approximation function would, thus, result in a smaller training data set. In turn, the smaller training data set can be obtained accurately via finite-element analysis (FEA). Moving least-squares (MLS) approximation is proposed herein to compute the low-dimensional functions for stiffness approximation. Via extensive simulations, some of which are described herein, it is demonstrated that the proposed emulator can predict the dynamic stiffness of a PKM at any given configuration with high accuracy and low computational expense, making it quite suitable for most high-precision applications. For example, our results show that the proposed methodology can choose configurations along given trajectories within a few percentage points of the optimal ones.

scholarly journals Improving High-dimensional Simulation-driven Optimization

2020 ◽  
Vol 17 ◽  
Keyword(s):  
Performance Analysis ◽  
Real World ◽  
Prediction Accuracy ◽  
Engineering Application ◽  
High Dimensional ◽  
Test Functions ◽  
Computationally Intensive ◽  
Dimensional Simulation ◽  
Low Dimensional ◽  
Extensive Performance

Computer simulations are being extensively used as a partial substitute for real-world experiments. Such simulations are often computationally intensive and hence metamodels are used to approximate them and to yield estimated output values more economically. While this setup can work well in low dimensional problems it can struggle in high-dimensional ones due to poor metamodel prediction accuracy. As such this study examines the application of dimensionality-reduction procedures during the search so that a simplified problems is formulated which is easier to solve and which could yield a better solution of the original one. An extensive performance analysis with both mathematical test functions and an engineering application shows the effectiveness of the proposed approach.

Development of Adaptive RBF-HDMR Model for Approximating High Dimensional Problems

2009 ◽  
Author(s):  
Songqing Shan ◽  
G. Gary Wang
Keyword(s):  
Curse Of Dimensionality ◽  
Black Box ◽  
Divide And Conquer ◽  
High Dimensional ◽  
High Dimensional Model Representation ◽  
Dimensional Model ◽  
Adaptive Modeling ◽  
Computationally Expensive ◽  
New Form ◽  
Modeling Strategy

Modeling or approximating high dimensional, computationally-expensive, black-box problems faces an exponentially increasing difficulty, the “curse-of-dimensionality”. This paper proposes a new form of high-dimensional model representation (HDMR) by integrating the radial basis function (RBF). The developed model, called RBF-HDMR, naturally explores and exploits the linearity/nonlinearity and correlation relationships among variables of the underlying function that is unknown or computationally expensive. This work also derives a lemma that supports the divide-and-conquer and adaptive modeling strategy of RBF-HDMR. RBF-HDMR circumvents or alleviates the “curse-of-dimensionality” by means of its explicit hierarchical structure, adaptive modeling strategy tailored to inherent variable relation, sample reuse, and a divide-and-conquer space-filling sampling algorithm. Multiple mathematical examples of a wide scope of dimensionalities are given to illustrate the modeling principle, procedure, efficiency, and accuracy of RBF-HDMR.

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