Hyunkyoo Cho

Modified Bayesian Kriging for Noisy Response Problems for Reliability Analysis

Abstract : This paper develops a new modified Bayesian Kriging (MBKG) surrogate modeling method for problems in which simulation analyses are inherently noisy and thus standard Kriging approaches fail to properly represent the responses. The purpose is to develop a method that can be used to carry out reliability analysis to predict probability of failure. The formulation of the MBKG surrogate modeling method is presented, and the full conditional distributions of the unknown MBKG parameters are presented. Using the full conditional distributions with a Gibbs sampling algorithm, Markov chain Monte Carlo is used to fit the MBKG surrogate model. A sequential sampling method that uses the posterior credible sets for inserting new design of experiment (DoE) sample points is proposed. The sequential sampling method is developed in such a way that the newly added DoE sample points will provide the maximum amount of information possible to the MBKG surrogate model, making it an efficient and effective way to reduce the number of DoE sample points needed. Therefore, the proposed method improves the posterior distribution of the probability of failure efficiently. To demonstrate the developed MBKG and sequential sampling methods, a 2-D mathematical example with added random noise is used. It is shown how, with the use of the sequential sample method, the posterior distribution of the probability of failure converges to capture the true probability of failure. A 3-D multibody dynamics (MBD) engineering block-car example illustrates an application of the new method to a simple engineering example for which standard Kriging methods fail.

Design Sensitivity Method for Sampling-Based RBDO With Varying Standard Deviation

Conventional reliability-based design optimization (RBDO) uses the mean of input random variable as its design variable; and the standard deviation (STD) of the random variable is a fixed constant. However, the constant STD may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is present as a percentage of the mean value. For this kind of design problem, the STD of the input random variable should vary as the corresponding design variable changes. In this paper, a method to calculate the design sensitivity of the probability of failure for RBDO with varying STD is developed. For sampling-based RBDO, which uses Monte Carlo simulation (MCS) for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STD in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with varying STD. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed design sensitivity method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.

Confidence Level Estimation and Design Sensitivity Analysis for Confidence-Based RBDO

In practical engineering problems, often only limited input data are available to generate the input distribution model. The insufficient input data induces uncertainty on the input distribution model, and this uncertainty will cause us to lose confidence in the optimum design obtained using the reliability-based design optimization (RBDO) method. Uncertainty on the input distribution model requires us to consider the reliability analysis output, which is defined as the probability of failure, to follow a probabilistic distribution. This paper proposes a new formulation for the confidence-based RBDO method and design sensitivity analysis of the confidence level. The probability of the reliability analysis output is obtained with consecutive conditional probabilities of input distribution parameters and input distribution types using a Bayesian approach. The approximate conditional probabilities of input distribution parameters and types are suggested under certain assumptions. The Monte Carlo simulation is applied to practically calculate the output distribution, and the copula is used to describe the correlated input distribution types. A confidence-based RBDO problem is formulated using the derived the distribution of output. In this new formulation, the probabilistic constraint is modified to include both the target reliability and the target confidence level. Finally, the sensitivity of the confidence level, which is a new probabilistic constraint, is derived to support an efficient optimization process. Using accurate surrogate models, the proposed method does not require generation of additional surrogate models during the RBDO iteration; it only requires several evaluations of the same surrogate models. Hence, the efficiency of the method is obtained. For the numerical example, the confidence level is calculated and the accuracy of the derived sensitivity is verified when only limited data are available.Copyright

Design Sensitivity Method for Sampling-Based RBDO With Fixed COV

Conventional reliability-based design optimization (RBDO) uses the means of input random variables as its design variables; and the standard deviations (STDEVs) of the random variables are fixed constants. However, the fixed STDEVs may not correctly represent certain RBDO problems well, especially when a specified tolerance of the input random variable is presented as a percentage of the mean value. For this kind of design problem, the coefficients of variations (COVs) of the input random variables should be fixed, which means STDEVs are not fixed. In this paper, a method to calculate the design sensitivity of probability of failure for RBDO with fixed COV is developed. For sampling-based RBDO, which uses Monte Carlo simulation for reliability analysis, the design sensitivity of the probability of failure is derived using a first-order score function. The score function contains the effect of the change in the STDEV in addition to the change in the mean. As copulas are used for the design sensitivity, correlated input random variables also can be used for RBDO with fixed COV. Moreover, the design sensitivity can be calculated efficiently during the evaluation of the probability of failure. Using a mathematical example, the accuracy and efficiency of the developed method are verified. The RBDO result for mathematical and physical problems indicates that the developed method provides accurate design sensitivity in the optimization process.Copyright

Sampling-Based RBDO of Ship Hull Structures Considering Thermo-Elasto-Plastic Residual Deformation

We present a shape optimization method using a sampling-based RBDO method linked with a commercial finite element analysis (FEA) code ANSYS, which is applicable to residual deformation problems of the ship hull structure in welding process. The programming language ANSYS Parametric Design Language (APDL) and shell elements are used for the thermo-elasto-plastic analysis. The shape of the ship hull structure is modeled using the bicubic Ferguson patch and coordinate components of vertices, tangential vectors of boundary curves are selected as design variables. The sensitivity of probabilistic constraint is calculated from the probabilistic sensitivity analysis using the score function and Monte Carlo Simulation (MCS) on the surrogate model constructed by using the Dynamic Kriging (DKG) method. The sequential quadratic programming (SQP) algorithm is used for the optimization. In two numerical examples, the suggested optimization method is applied to practical residual deformation problems in welding ship hull structures, which proves the sampling-based RBDO can be successfully utilized for obtaining a reliable optimum design in highly nonlinear multi-physics problem of thermo-elasto-plasticity.

Reliability-based design optimization of fluid–solid interaction problems

Using a sampling-based reliability-based design optimization method, we present a shape reliability-based design optimization method for coupled fluid–solid interaction problems. For the fluid–solid interaction problem in arbitrary Lagrangian–Eulerian formulation, a coupled variational equation is derived from a steady state Navier–Stokes equation for incompressible flows, an equilibrium equation for geometrically nonlinear solids, and a traction continuity condition at interfaces. The fluid–solid interaction problem is solved using the finite element method and the Newton–Raphson scheme. For the fluid mesh movement, we formulated and solved a pseudo-structural sub-problem. The shape of the solid is modeled using the Non-Uniform Rational B-Spline (NURBS) surface, and the coordinate components of the control points are selected as random design variables. The sensitivity of the probabilistic constraint is calculated using the first-order score functions obtained from the input distributions and from the Monte Carlo simulation on the surrogate model constructed by using the Dynamic Kriging method. The sequential quadratic programming algorithm is used for the optimization. In two numerical examples, the proposed optimization method is applied to the shape design problems of solid structure which is loaded by prescribed fluid flow, and this proves that the sampling-based reliability-based design optimization can be successfully utilized for obtaining a reliable optimum design in highly nonlinear multi-physics problems.

Confidence-Based Method for Reliability-Based Design Optimization

An accurate input probabilistic model is necessary to obtain a trustworthy result in the reliability analysis and the reliability-based design optimization (RBDO). However, the accurate input probabilistic model is not always available. Very often only insufficient input data are available in practical engineering problems. When only the limited input data are provided, uncertainty is induced in the input probabilistic model and this uncertainty propagates to the reliability output which is defined as the probability of failure. Then, the confidence level of the reliability output will decrease. To resolve this problem, the reliability output is considered to have a probability distribution in this paper. The probability of the reliability output is obtained as a combination of consecutive conditional probabilities of input distribution type and parameters using Bayesian approach. The conditional probabilities that are obtained under certain assumptions and Monte Carlo simulation (MCS) method is used to calculate the probability of the reliability output. Using the probability of the reliability output as constraint, a confidence-based RBDO (C-RBDO) problem is formulated. In the new probabilistic constraint of the C-RBDO formulation, two threshold values of the target reliability output and the target confidence level are used. For effective C-RBDO process, the design sensitivity of the new probabilistic constraint is derived. The C-RBDO is performed for a mathematical problem with different numbers of input data and the result shows that C-RBDO optimum designs incorporate appropriate conservativeness according to the given input data.Copyright

Hybrid Reliability Analysis Method for Electromagnetic Design Problems With Non-Gaussian Probabilistic Parameters

This paper proposes an efficient and stable reliability analysis method for reliability-based electromagnetic design problems with non-normal probability distributions of input parameters. The reliability analysis strongly depends on distribution types of random variables since nonlinear transformations between an original random space and a standard normal random space cause additional nonlinearity into the reliability assessment of probabilistic constraint functions. That can lead to numerical inaccuracy and instability in the reliability-based design process, or may fail to have a solution to the probabilistic constraint assessment. To overcome these difficulties, a hybrid mean-value method is introduced to seeking a most probable failure point in the performance measure approach, which is one of the first-order reliability analysis methods. The proposed method is tested with a mathematical model and a loudspeaker design, of which random variables are assumed to follow five different probability distributions case by case.

Hybrid reliability analysis method for electromagnetic design problems with non-Gaussian probabilistic parameters

This paper proposes an efficient and stable reliability analysis method for electromagnetic (EM) design problems with non-normal probability distributions of input parameters. A hybrid search algorithm for a most probable failure point (MPP) is introduced to assess the reliability of highly nonlinear constraint functions. The proposed method is tested with a loudspeaker design, of which random variables follow five different probability distributions case by case.

Development of a Conservative Model Validation Approach for Reliable Analysis

Abstract : Simulation models are approximations of real-world physical systems. Therefore, simulation model validation is necessary for the simulation-based design process to provide reliable products. However, due to the cost of product testing, experimental data in the context of model validation is limited for a given design. When the experimental data is limited, a true output PDF cannot be correctly obtained. Therefore, reliable target output PDF needs to be used to update the simulation model. In this paper, a new model validation approach is proposed to obtain a conservative estimation of the target output PDF for validation of the simulation model in reliability analysis. The proposed method considers the uncertainty induced by insufficient experimental data in estimation of predicted output PDFs by using Bayesian analysis. Then, a target output PDF and a probability of failure are selected from these predicted output PDFs at a user-specified conservativeness level for validation. For validation, the calibration parameter and model bias are optimized to minimize a validation measure of the simulation output PDF and the conservative target output PDF subject to the conservative probability of failure. For the optimization, accurate sensitivity of the validation measure is obtained using the complex variable method (CVM) for sensitivity analysis. As the target output PDF satisfies the user-specified conservativeness level, the validated simulation model provides a conservative representation of the experimental data. A simply supported beam is used to carry out the convergence study and demonstrate that the proposed method establishes a conservatively reliable simulation model.