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Journal Article

Reliability-Based Design Optimization with Model Bias and Data Uncertainty

2013-04-08
2013-01-1384
Reliability-based design optimization (RBDO) has been widely used to obtain a reliable design via an existing CAE model considering the variations of input variables. However, most RBDO approaches do not consider the CAE model bias and uncertainty, which may largely affect the reliability assessment of the final design and result in risky design decisions. In this paper, the Gaussian Process Modeling (GPM) approach is applied to statistically correct the model discrepancy which is represented as a bias function, and to quantify model uncertainty based on collected data from either real tests or high-fidelity CAE simulations. After the corrected model is validated by extra sets of test data, it is integrated into the RBDO formulation to obtain a reliable solution that meets the overall reliability targets while considering both model and parameter uncertainties.
Journal Article

On Stochastic Model Interpolation and Extrapolation Methods for Vehicle Design

2013-04-08
2013-01-1386
Finite Element (FE) models are widely used in automotive for vehicle design. Even with increasing speed of computers, the simulation of high fidelity FE models is still too time-consuming to perform direct design optimization. As a result, response surface models (RSMs) are commonly used as surrogates of the FE models to reduce the turn-around time. However, RSM may introduce additional sources of uncertainty, such as model bias, and so on. The uncertainty and model bias will affect the trustworthiness of design decisions in design processes. This calls for the development of stochastic model interpolation and extrapolation methods that can address the discrepancy between the RSM and the FE results, and provide prediction intervals of model responses under uncertainty.
Journal Article

An Ensemble Approach for Model Bias Prediction

2013-04-08
2013-01-1387
Model validation is a process of determining the degree to which a model is an accurate representation of the real world from the perspective of the intended uses of the model. In reliability based design, the intended use of the model is to identify an optimal design with the minimum cost function while satisfying all reliability constraints. It is pivotal that computational models should be validated before conducting the reliability based design. This paper presents an ensemble approach for model bias prediction in order to correct predictions of computational models. The basic idea is to first characterize the model bias of computational models, then correct the model prediction by adding the characterized model bias. The ensemble approach is composed of two prediction mechanisms: 1) response surface of model bias, and 2) Copula modeling of a series of relationships between design variables and the model bias, between model prediction and the model bias.
Journal Article

Sampling-Based RBDO Using Score Function with Re-Weighting Scheme

2013-04-08
2013-01-0377
Sampling-based methods are general but time consuming for solving a Reliability-Based Design Optimization (RBDO) problem. In order to alleviate the computation burden, score function together with the Monte Carlo method was used to compute the stochastic sensitivities of reliability functions. In literature, re-weighting schemes were shown to converge faster than the regular Monte Carlo method. In this paper, a reweighting scheme together with score function is employed to perform sampling-based stochastic sensitivity analysis to improve the computational efficiency and accuracy. An analytical example is used to show the advantages of the proposed method. Comparisons to the conventional methods are made and discussed. Two RBDO problems are solved to demonstrate the use of the proposed method.
Technical Paper

A Study of Model Validation Method for Dynamic Systems

2010-04-12
2010-01-0419
This paper presents an enhanced Bayesian based model validation method together with probabilistic principal component analysis (PPCA). The PPCA is employed to address multivariate correlation and to reduce the dimensionality of the multivariate functional responses. The Bayesian hypothesis testing is used to quantitatively assess the quality of a multivariate dynamic system. Unlike the previous approach, the differences between test and CAE results are used for dimension reduction though PPCA and then to assess the model validity. In addition, physics-based thresholds are defined and transformed to the PPCA space for Bayesian hypothesis testing. This new approach resolves some critical drawbacks of the previous method and provides desirable properties of a validation method, e.g., symmetry. A dynamic system with multiple functional responses is used to demonstrate this new approach.
Technical Paper

Comparative Benchmark Studies of Response Surface Model-Based Optimization and Direct Multidisciplinary Design Optimization

2014-04-01
2014-01-0400
Response Surface Model (RSM)-based optimization is widely used in engineering design. The major strength of RSM-based optimization is its short computational time. The expensive real simulation models are replaced with fast surrogate models. However, this method may have some difficulties to reach the full potential due to the errors between RSM and the real simulations. RSM's accuracy is limited by the insufficient number of Design of Experiments (DOE) points and the inherent randomness of DOE. With recent developments in advanced optimization algorithms and High Performance Computing (HPC) capability, Direct Multidisciplinary Design Optimization (DMDO) receives more attention as a promising future optimization strategy. Advanced optimization algorithm reduces the number of function evaluations, and HPC cut down the computational turnaround time of function evaluations through fully utilizing parallel computation.
Technical Paper

A Modified Particle Swarm Optimization Algorithm with Design of Experiment Technique and a Perturbation Process

2015-04-14
2015-01-0422
Particle swarm optimization (PSO) is a relatively new stochastic optimization algorithm and has gained much attention in recent years because of its fast convergence speed and strong optimization ability. However, PSO suffers from premature convergence problem for quick losing of diversity. That is to say, if no particle discovers a new superiority position than its previous best location, PSO algorithm will fall into stagnation and output local optimum result. In order to improve the diversity of basic PSO, design of experiment technique is used to initialize the particle swarm in consideration of its space-filling property which guarantees covering the design space comprehensively. And the optimization procedure of PSO is divided into two stages, optimization stage and improving stage. In the optimization stage, the basic PSO initialized by Optimal Latin hypercube technique is conducted.
Journal Article

A Bayesian Inference based Model Interpolation and Extrapolation

2012-04-16
2012-01-0223
Model validation is a process to assess the validity and predictive capabilities of a computer model by comparing simulation results with test data for its intended use of the model. One of the key difficulties for model validation is to evaluate the quality of a computer model at different test configurations in design space, and interpolate or extrapolate the evaluation results to untested new design configurations. In this paper, an integrated model interpolation and extrapolation framework based on Bayesian inference and Response Surface Models (RSM) is proposed to validate the designs both within and outside of the original design space. Bayesian inference is first applied to quantify the distributions' hyper-parameters of the bias between test and CAE data in the validation domain. Then, the hyper-parameters are extrapolated from the design configurations to untested new design. They are then followed by the prediction interval of responses at the new design points.
Technical Paper

An Effective Optimization Strategy for Structural Weight Reduction

2010-04-12
2010-01-0647
Multidisciplinary design optimization (MDO) methods are commonly used for weight reduction in automotive industry. The design variables for MDO are often selected based on critical parts, which usually are close to optimal after many design iterations. As a result, the real weight reduction benefit may not be fully realized due to poor selection of design parameters. In addition, most applications require running design of experiments (DOE) to explore the full design space and to build response surfaces for optimization. This approach is often too costly if too many design variables are simultaneously considered. In this research, an alternative approach to address these issues is presented. It includes two optimization phases. The first phase uses critical parts for design iterations and the second phase use non-critical for weight reduction. A vehicle body structure is used to demonstrate the proposed strategy to show its effectiveness.
Technical Paper

Auto-Correlation of an Occupant Restraint System Model Using a Bayesian Validation Metric

2009-04-20
2009-01-1402
Computer Aided Engineering (CAE) has become a vital tool for product development in automotive industry. Various computer models for occupant restraint systems are developed. The models simulate the vehicle interior, restraint system, and occupants in different crash scenarios. In order to improve the efficiency during the product development process, the model quality and its predictive capabilities must be ensured. In this research, an objective model validation metric is developed to evaluate the model validity and its predictive capabilities when multiple occupant injury responses are simultaneously compared with test curves. This validation metric is based on the probabilistic principal component analysis method and Bayesian statistics approach for multivariate model assessment. It first quantifies the uncertainties in both test and simulation results, extracts key features, and then evaluates the model quality.
Journal Article

Bayesian Probabilistic PCA Approach for Model Validation of Dynamic Systems

2009-04-20
2009-01-1404
In the automobile industry, the reliability and predictive capabilities of computer models for a dynamic system need to be assessed quantitatively. Quantitative validation allows engineers to assess and improve model reliability and quality objectively and ultimately lead to potential reduction in the number of prototypes built and tests. A good metric, which is essential in model validation, requires considering uncertainties in both testing and computer modeling. In addition, it needs to be able to compare multiple responses simultaneously, as multiple quantities are often encountered at different spatial and temporal points of a dynamic system. In this paper, a state-of-the-art validation technology is developed for multivariate complex dynamic systems by exploiting a probabilistic principal component analysis method and Bayesian statistics approach.
Journal Article

A Comparative Benchmark Study of using Different Multi-Objective Optimization Algorithms for Restraint System Design

2014-04-01
2014-01-0564
Vehicle restraint system design is a difficult optimization problem to solve because (1) the nature of the problem is highly nonlinear, non-convex, noisy, and discontinuous; (2) there are large numbers of discrete and continuous design variables; (3) a design has to meet safety performance requirements for multiple crash modes simultaneously, hence there are a large number of design constraints. Based on the above knowledge of the problem, it is understandable why design of experiment (DOE) does not produce a high-percentage of feasible solutions, and it is difficult for response surface methods (RSM) to capture the true landscape of the problem. Furthermore, in order to keep the restraint system more robust, the complexity of restraint system content needs to be minimized in addition to minimizing the relative risk score to achieve New Car Assessment Program (NCAP) 5-star rating.
Technical Paper

Enhanced Error Assessment of Response Time Histories (EEARTH) Metric and Calibration Process

2011-04-12
2011-01-0245
Computer Aided Engineering (CAE) has become a vital tool for product development in automotive industry. Increasing computer models are developed to simulate vehicle crashworthiness, dynamic, and fuel efficiency. Before applying these models for product development, model validation needs to be conducted to assess the validity of the models. However, one of the key difficulties for model validation of dynamic systems is that most of the responses are functional responses, such as time history curves. This calls for the development of an objective metric which can evaluate the differences of both the time history and the key features, such as phase shift, magnitude, and slope between test and CAE curves. One of the promising metrics is Error Assessment of Response Time Histories (EARTH), which was recently developed. Three independent error measures that associated with physically meaningful characteristics (phase, magnitude, and slope) were proposed.
Technical Paper

A Model Validation Approach for Various Design Configurations with Insufficient Experimental Data for Model Accuracy Check

2012-04-16
2012-01-0228
Analytical models (math or computer simulation models) are typically built on the basis of many assumptions and simplifications and hence model prediction could be inaccurate in intended applications. Model validation is thus critical to quantify and improve the degree of accuracy of these models. So far, little work considers model validation for various design configurations so that model prediction is accurate in the intended design space. Furthermore, there is a lack of effective approaches that can be used to quantify model accuracy considering different number of experimental data. To overcome these limitations, objective of this paper is to develop a model validation approach for various design configurations with a reference metric for model accuracy check considering different number of experimental data.
Journal Article

Towards Optimization of Multi-material Structure: Metamodeling of Mixed-Variable Problems

2016-04-05
2016-01-0302
In structural design optimization, it is challenging to determine the optimal dimensions and material for each component simultaneously. Material selection of each part is always formulated as a categorical design variable in structural optimization problems. However, it is difficult to solve such mixed-variable problems using the metamodelbased strategy, because the prediction accuracy of metamodels deteriorates significantly when categorical variables exist. This paper investigates two different strategies of mixed-variable metamodeling: the “feature separating” strategy and the “all-in-one” strategy. A supervised learning-enhanced cokriging method is proposed, which fuses multi-fidelity information to predict new designs’ responses. The proposed method is compared with several existing mixed-variable metamodeling methods to understand their pros and cons. These methods include Neural Network (NN) regression, Classification and Regression Tree (CART) and Gaussian Process (GP).
Journal Article

A Copula-Based Approach for Model Bias Characterization

2014-04-01
2014-01-0735
Available methodologies for model bias identification are mainly regression-based approaches, such as Gaussian process, Bayesian inference-based models and so on. Accuracy and efficiency of these methodologies may degrade for characterizing the model bias when more system inputs are considered in the prediction model due to the curse of dimensionality for regression-based approaches. This paper proposes a copula-based approach for model bias identification without suffering the curse of dimensionality. The main idea is to build general statistical relationships between the model bias and the model prediction including all system inputs using copulas so that possible model bias distributions can be effectively identified at any new design configurations of the system. Two engineering case studies whose dimensionalities range from medium to high will be employed to demonstrate the effectiveness of the copula-based approach.
Journal Article

A Stochastic Bias Corrected Response Surface Method and its Application to Reliability-Based Design Optimization

2014-04-01
2014-01-0731
In vehicle design, response surface model (RSM) is commonly used as a surrogate of the high fidelity Finite Element (FE) model to reduce the computational time and improve the efficiency of design process. However, RSM introduces additional sources of uncertainty, such as model bias, which largely affect the reliability and robustness of the prediction results. The bias of RSM need to be addressed before the model is ready for extrapolation and design optimization. This paper further investigates the Bayesian inference based model extrapolation method which is previously proposed by the authors, and provides a systematic and integrated stochastic bias corrected model extrapolation and robustness design process under uncertainty. A real world vehicle design example is used to demonstrate the validity of the proposed method.
Journal Article

Research on Validation Metrics for Multiple Dynamic Response Comparison under Uncertainty

2015-04-14
2015-01-0443
Computer programs and models are playing an increasing role in simulating vehicle crashworthiness, dynamic, and fuel efficiency. To maximize the effectiveness of these models, the validity and predictive capabilities of these models need to be assessed quantitatively. For a successful implementation of Computer Aided Engineering (CAE) models as an integrated part of the current vehicle development process, it is necessary to develop objective validation metric that has the desirable metric properties to quantify the discrepancy between multiple tests and simulation results. However, most of the outputs of dynamic systems are multiple functional responses, such as time history series. This calls for the development of an objective metric that can evaluate the differences of the multiple time histories as well as the key features under uncertainty.
Journal Article

Development of a Comprehensive Validation Method for Dynamic Systems and Its Application on Vehicle Design

2015-04-14
2015-01-0452
Simulation based design optimization has become the common practice in automotive product development. Increasing computer models are developed to simulate various dynamic systems. Before applying these models for product development, model validation needs to be conducted to assess their validity. In model validation, for the purpose of obtaining results successfully, it is vital to select or develop appropriate metrics for specific applications. For dynamic systems, one of the key obstacles of model validation is that most of the responses are functional, such as time history curves. This calls for the development of a metric that can evaluate the differences in terms of phase shift, magnitude and shape, which requires information from both time and frequency domain. And by representing time histories in frequency domain, more intuitive information can be obtained, such as magnitude-frequency and phase-frequency characteristics.
Journal Article

Validation Metric for Dynamic System Responses under Uncertainty

2015-04-14
2015-01-0453
To date, model validation metric is prominently designed for non-dynamic model responses. Though metrics for dynamic responses are also available, they are specifically designed for the vehicle impact application and uncertainties are not considered in the metric. This paper proposes the validation metric for general dynamic system responses under uncertainty. The metric makes use of the popular U-pooling approach and extends it for dynamic responses. Furthermore, shape deviation metric was proposed to be included in the validation metric with the capability of considering multiple dynamic test data. One vehicle impact model is presented to demonstrate the proposed validation metric.
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