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This paper describes a bilateral filter that is being proposed as a coding tool for the Versatile Video Codec (VVC). The filter acts as a loop filter in parallel with the sample-adaptive offset (SAO) filter. Both the proposed filter and SAO act on the same input samples, each filter produces an offset, and these offsets are then added to the input sample to produce an output sample that, after clipping, goes to the next stage. The method has been implemented and tested according to the common test conditions in VVC test model version 5.0. For the all-intra configuration, we report a BD rate figure of -0.4% with an encoder run time increase of 6% and a decoder run time increase of 4%. For the random access configuration, the BD rate figure is -0.5% with an encoder run time increase of 2% and a decoder run time increase of 2%.

In reliability-based robust design optimization (RBRDO) formulation, the product quality loss function is minimized subject to probabilistic constraints. Since the quality loss function is expressed in terms of the first two statistical moments, mean and variance, three methods have been recently proposed to accurately and efficiently estimate the moments: the univariate dimension reduction method (DRM), performance moment integration (PMI) method, and percentile difference method (PDM). In this paper, a reliability-based robust design optimization method is developed using DRM and compared to PMI and PDM for accuracy and efficiency. The numerical results show that DRM is effective when the number of random variables is small, whereas PMI is more effective when the number of random variables is relatively large.

The reliability-based design optimization (RBDO) using performance measure approach for problems with correlated input variables requires a transformation from the correlated input random variables into independent standard normal variables. For the transformation with correlated input variables, the two most representative transformations, the Rosenblatt and Nataf transformations, are investigated. The Rosenblatt transformation requires a joint cumulative distribution function (CDF). Thus, the Rosenblatt transformation can be used only if the joint CDF is given or input variables are independent. In the Nataf transformation, the joint CDF is approximated using the Gaussian copula, marginal CDFs, and covariance of the input correlated variables. Using the generated CDF, the correlated input variables are transformed into correlated normal variables and then the correlated normal variables are transformed into independent standard normal variables through a linear transformation. Thus, the Nataf transformation can accurately estimates joint normal and some lognormal CDFs of the input variable that cover broad engineering applications. This paper develops a PMA-based RBDO method for problems with correlated random input variables using the Gaussian copula. Several numerical examples show that the correlated random input variables significantly affect RBDO results.

Structural analysis and design optimization have recently been extended to consider various uncertainties. If the statistical data for the uncertainties are sufficient to construct the input distribution function, the uncertainties can be treated as random variables and RBDO is used; otherwise, the uncertainties can be treated as fuzzy variables and PBDO is used. However, many structural design problems include both uncertainties with sufficient data and uncertainties with insufficient data. For these problems, RBDO will yield an unreliable design since the distribution functions of uncertainties are not believable. On the other hand, treating the random variables as fuzzy variables and invoking PBDO may yield too conservative design with a higher optimum cost. This paper proposes a new design formulation using the performance measure approach (PMA). For the inverse analysis, this paper proposes a new most probable/possible point (MPPP) search method called maximal failure search (MFS), which is an integration of the enhanced hybrid mean value method (HMV+) and maximal possibility search (MPS) method. Some mathematical and physical examples are used to demonstrate the proposed inverse analysis method and design formulation.

DOI: 10.2514/1.16546 Structural analysis and design optimization have recently been extended to the stochastic approach to consider variousuncertainties.However,inareaswhereitisnotpossibletoproduceaccurate statistical information forinput data, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, because improper modeling of uncertainty could cause a greater degree of statistical uncertainty than those of physical uncertainty. For systems with insufficient information for input data, possibility-based (or fuzzy set) methods have recently been introduced in structural analysis and design optimization. Using possibility methods, the extended fuzzy operations are much simpler than of random variables. Possibility-based design optimization will provide more conservative designs than those from probability methods, andwill provide a system level of failure possibility automatically. This paper proposes a new formulation of possibility-based design optimization using the performance measure approach. For the inverse possibility analysis, the maximal possibility search method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the multilevel-cut method. Two mathematical examples, including a nonmonotonic response and a physical example of vehicle side impact, are used to demonstrate the proposed maximal possibility search method and possibility-based design optimization.

In practical engineering applications, there exist two different types of uncertainties: aleatory and epistemic uncertainties. This study attempts to develop a robust design optimization with epistemic uncertainty. For epistemic uncertainties, a possibility-based design optimization improves the failure rate, while a robust design optimization minimizes the product quality loss. In general, product quality loss is described using the first two statistical moments for aleatory uncertainty: mean and standard deviation. However, there is no metric for product quality loss defined when having epistemic uncertainty. This paper first proposes a new metric for product quality loss with epistemic uncertainty, and then a possibility-based robust design optimization. For numerical efficiency and stability, an enriched performance measure approach is employed for possibility-based robust design optimization, and the maximal possibility search is used for a possibility analysis. Three different types of robust objectives are considered for possibility-based robust design optimization: smaller-the-better type (S-Type), larger-the-better type (L-Type), and nominal-the-better type (N-Type). Examples are used to demonstrate the effectiveness of possibility-based robust design optimization using the proposed metric for product quality loss with epistemic uncertainty.

The reliability based design optimization (RBDO) method is prevailing in stochastic structural design optimization by assuming the amount of input data is sufficient enough to create accurate input statistical distribution. If the sufficient input data cannot be generated due to limitations in technical and/or facility resources, the possibility-based design optimization (PBDO) method can be used to obtain reliable designs by utilizing membership functions for epistemic uncertainties. For RBDO, the performance measure approach (PMA) is well established and accepted by many investigators. It is found that the same PMA is a very much desirable approach also for the PBDO problems. In many industry design problems, we have to deal with uncertainties with sufficient data and uncertainties with insufficient data simultaneously. For these design problems, it is not desirable to use RBDO since it could lead to an unreliable optimum design. This paper proposes to use PBDO for design optimization for such problems. In order to treat uncertainties as fuzzy variables, several methods for membership function generation are proposed. As less detailed information is available for the input data, the membership function that provides more conservative optimum design should be selected. For uncertainties with sufficient data, the membership function that yields the least conservative optimum design is proposed by using the possibility-probability consistency theory and the least conservative condition. The proposed approach for design problems with mixed type input uncertainties is applied to some example problems to demonstrate feasibility of the approach. It is shown that the proposed approach provides conservative optimum design.

An enriched performance measure approach is presented for reliability-based design optimization to substantially improve computational efficiency when applied to large-scale applications. In the enriched performance measure approach, four improvements are made over the original performance measure approach: as a way to launch reliability-based design optimization at a deterministic optimum design, as a new enhanced hybrid-mean value method, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some reliability-based design optimization iterations. In reliability-based design optimization, a computational burden on the feasibility check of constraints can be significantly reduced by using a mean value first-order method and by carrying out the refined reliability analysis using the enhanced hybrid-mean value method for e-active and violated constraints. The enhanced hybrid-mean value method is developed to handle nonlinear and/or nonmonotonic constraints in reliability analysis. The fast reliability analysis method is proposed to efficiently evaluate probabilistic constraints under the condition of design closeness. Moreover, two numerical examples are provided to compare the enriched performance measure approach to existing reliability-based design optimization methods from a numerical efficiency and stability point of view.

This paper proposes an adaptive probability analysis method that can effectively generate the probability distribution of the output performance function by identifying the propagation of input uncertainty to output uncertainty. The method is based on an enhanced hybrid mean value (HMV+) analysis in the performance measure approach (PMA) for numerical stability and efficiency in search of the most probable point (MPP). The HMV+ method improves numerical stability and efficiency especially for highly nonlinear output performance functions by providing steady convergent behavior in the MPP search. The proposed adaptive probability analysis method approximates the MPP locus, and then adaptively refines this locus using an a posteriori error estimator. Using the fact that probability levels can be easily set a priori in PMA, the MPP locus is approximated using the interpolated moving least-squares method. For refinement of the approximated MPP locus, additional probability levels are adaptively determined through an a posteriori error estimator. The adaptive probability analysis method will determine the minimum number of necessary probability levels, while ensuring accuracy of the approximated MPP locus. Several examples are used to show the effectiveness of the proposed adaptive probability analysis method using the enhanced HMV+ method.

In this paper, we derive gradient recovery type a posteriori error estimate for the finite element approximation of elliptic equations. We show that a posteriori error estimate provide both upper and lower bounds for the discretization error on the non-uniform meshes. Moreover, it is proved that a posteriori error estimate is also asymptotically exact on the uniform meshes if the solution is smooth enough. The numerical results demonstrating the theoretical results are also presented in this paper.

To obtain reliable designs, aleatory and epistemic uncertainties are considered recently in the structural analysis and design optimization. The reliability based de-sign optimization (RBDO) method [1] is used when the amount of input data is sufficient enough to create accurate statistical distribution. On the other hand, when the sufficient input data are not available due to limitations in time, human, and facility resources, the optimum design may not be reliable if RBDO method is used. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method can be used for structural analysis and possibility based design optimization (PBDO) [2]. However, in many industry design problems, we may have to deal with design problems that involve with the mixed input statistical random and fuzzy variables simultaneously. For these problems, RBDO may yield unreliable optimum designs because of insufficient data. On the other hand, treating the random variables as fuzzy variables and invoking PBDO to solve the mixed design variable problem may yield too conservative designs with higher optimum costs. This paper proposes a new mixed variable design optimization (MVDO) problem based on the performance measure approach (PMA) [1]. To evaluate the possibilistic constraint in MVDO, a sub-optimization problem for inverse analysis is carried out using a hyper-cylinder domain. To solve this sub-problem efficiently and effectively, a new numerical algorithm, maximum failure search (MFS) method, is proposed in this paper by combining the enhanced hybrid mean value (HMV+) method [3] for the inverse reliability analysis in RBDO and the maximal possibility search (MPS) method [2] for the inverse possibility analysis in PBDO. Some mathematical examples are used to demonstrate the efficiency and effectiveness of the proposed numerical MFS method. Some physical design examples are used to compare the proposed MVDO results with RBDO and PBDO results.

In a gradient-based design optimization, it is necessary to know sensitivities of the constraint with respect to the design variables. In a reliability-based design optimization (RBDO), the constraint is evaluated at the most probable point (MPP) and called the probabilistic constraint, thus it requires the sensitivities of the probabilistic constraints at MPP. This paper presents the rigorous analytic derivation of the sensitivities of the probabilistic constraint at MPP for both First Order Reliability Method (FORM)-based Performance Measure Approach (PMA) and Dimension Reduction Method (DRM)-based PMA. Numerical examples are used to demonstrate that the analytic sensitivities agree very well with the sensitivities obtained from the finite difference method (FDM). However, since the sensitivity calculation at the true DRM-based MPP requires the second-order derivatives and additional MPP search, the sensitivity derivation at the approximated DRM-based MPP, which does not require the second-order derivatives and additional MPP search to find the DRM-based MPP, is proposed in this paper. A convergence study illustrates that the sensitivity at the approximated DRM-based MPP converges to the sensitivity at the true DRM-based MPP as the design approaches the optimum design. Hence, the sensitivity at the approximated DRM-based MPP is proposed to be used for the DRM-based RBDO to enhance the efficiency of the optimization.Copyright © 2008 by ASME

For RBDO problems with correlated input variables, it is necessary to obtain the input joint distribution (CDF, cumulative distribution function). Then Rosenblatt transformation is used to transform the correlated input variables into the independent standard normal variables for the purpose of inverse reliability analysis. However, in practical industry RBDO problems, often only the marginal CDFs and paired samples are available from limited experimental data. In this paper, a copula, which is a link between a joint CDF and marginal CDFs, is proposed to generate an input joint CDF from these marginal CDFs and paired samples. To identify the right copula from limited data, Bayesian method is proposed to use in this paper. Using Bayesian method, the number of samples required to properly identify the right copula is investigated for different types of copulas and for different correlation coefficients. A real industry problem is used to show how a copula can be identified from the limited experimental data.Copyright © 2008 by ASME

Reliability-based design optimization (RBDO), which includes design optimization in design space and inverse reliability analysis in standard normal space, has been recently developed under the assumption that all input variables are independent because it is difficult to construct a joint probability distribution function (PDF) of input variables with limited data such as the marginal PDF and covariance matrix. However, since in real applications, it is common that some of the input variables are correlated, the RBDO results might contain a significant error if the correlation between input variables for RBDO is not considered. In this paper, Rosenblatt and Nataf transformations, which are the most representative transformation methods and have been widely used in the reliability analysis, have been studied and compared in terms of applicability to RBDO with correlated input variables. It is identified that Nataf transformation is one of copulas and more applicable than Rosenblatt transformation. Using numerical examples, it is also shown that the correlation of input variables significantly affects the RBDO results.

Whereas the robust design concept has been well established in the probability theory, it has not been developed in the possibility theory. For problems where accurate statistical information for input data is not available, a possibility-based (or fuzzy set) robust design concept is proposed in this paper by investigating the similarity between the membership function of the fuzzy variable and the cumulative distribution function of the random variable. Based on the probability-possibility consistency principle, a random variable that corresponds to the fuzzy variable is introduced in this paper in order to define the robust design concept for the fuzzy variable. For the system with input fuzzy variables, the robustness measure of the output performance is computed using the performance measure integration (PMI) method, while the integration points are obtained from the inverse possibility analysis by using the maximal possibility search method with interpolation (MPS). For the system with mixed random and fuzzy input variables, the robustness measure of the output performance is computed using PMI method, with the integration points obtained from the inverse mixed analysis by using the maximal failure search method (MFS). A new mixed (random and fuzzy) variable robust design optimization (MVRDO) method is proposed and several numerical examples are used to verify the robust design concept in the possibility theory and the MVRDO formulation.Copyright © 2007 by ASME

For the performance measure approach (PMA) of RBDO, a transformation between the input random variables and the standard normal random variables is necessary to carry out the inverse reliability analysis. For reliability analysis, Rosenblatt and Nataf transformations are commonly used. In many industrial RBDO problems, the input random variables are correlated. However, often only limited information such as the marginal distribution and covariance could be practically obtained, and the input joint probability distribution function (PDF) is very difficult to obtain. Thus, in literature, most RBDO methods assume all input random variables are independent. However, in this paper, it is found that the RBDO results can be significantly different when the input variables are correlated. Thus, various transformation methods are investigated for development of a RBDO method for problems with correlated input variables. It is found that Rosenblatt transformation is impractical for problems with correlated input variables due to difficulty of constructing a joint PDF from the marginal distributions and covariance. On the other hand, Nataf transformation can construct the joint CDF using the marginal distributions and covariance, and thus applicable to problems with correlated random input variables. The joint CDF is Nataf model, which is called a Gaussian copula in the copula family. Since the Gaussian copula can describe a wide range of the correlation coefficient, Nataf transformation can be widely used for various types of correlated input variables. In this paper, Nataf transformation is used to develop a RBDO method for design problems with correlated random input variables. Numerical examples are used to demonstrate the proposed method. Also, it is shown that the correlated random input variables significantly affect the RBDO results.

There are two commonly used reliability analysis methods of analytical methods: linear approximation - First Order Reliability Method (FORM), and quadratic approximation - Second Order Reliability Method (SORM), of the performance functions. The reliability analysis using FORM could be acceptable for mildly nonlinear performance functions, whereas the reliability analysis using SORM is usually necessary for highly nonlinear performance functions of multi-variables. Even though the reliability analysis using SORM may be accurate, it is not desirable to use SORM for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is very difficult to develop. This paper proposes a method that can be used for multi-dimensional highly nonlinear systems to yield very accurate probability of failure calculation without requiring the second order sensitivities. For this purpose, the univariate dimension reduction method (DRM) is used. A three-step computational process is proposed to carry out the inverse reliability analysis: constraint shift, reliability index (β) update, and the most probable point (MPP) approximation method. Using the three steps, a new DRM-based MPP is obtained, which computes the probability of failure of the performance function more accurately than FORM and more efficiently than SORM.Copyright © 2007 by ASME

Structural analysis and design optimization have recently been extended to stochastic approach to take various uncertainties into account. However in areas whwew it is not possible to produce accurate statistical information, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, since improper modeling of uncertainty could cause greater degree of statistical uncertainty than those of physical uncertainty. For uncertainty with insufficient information, possibility-based (or fuzzy set) methods have recently been introduced in stochastic structural analysis and design optimization. The main advantage of the fuzzy analysis is that it preserves the intrinsic random nature of physical variables through their membership functions and, when used for evaluation of designs, yields more conservative design than those from the probabilistic methods. There are two computational aspects in the fuzzy analysis compared to the probability analysis. First, the input fuzzy variables can be defined easier than the input random variables when no or very limited statistical data are available. Secondly, extended fuzzy operations are much simpler than those required to use probability, especially when a number of variables are involved. For possibility-based design optimization (PBDO), like for reliability-based design optimization (RBDO), the performance measure approach (PMA) with an inverse fuzzy analysis is more appropriate than other approaches, such as a possibility index approach. This paper proposes a new formulation of PBDO using PMA. It is also found that PBDO is more desirable than RBDO, since PBDO can inherently handle a system level possibility unlike RBDO. For the inverse fuzzy analysis, the maximal possibility search (MPS) method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the

This paper presents an enriched performance measure approach (PMA+) for reliability-based design optimization (RBDO) to substantially improve computational efficiency when applied to large-scale applications. Three aspects of PMA+ are presented: as a way to launch RBDO at a deterministic optimum design, as an efficient probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. It is found that deterministic design optimization helps improve numerical efficiency by reducing some RBDO iterations. Unlike deterministic design optimization, a significant computational burden is imposed on the feasibility check of constraints in the RBDO process due to the costs of a reliability analysis. Such difficulties can be effectively resolved by using a mean value (MV) first-order method with an allowable accuracy for the purpose of feasibility identification, and by carrying out the refined reliability analysis using the enhanced hybrid mean value (HMV+) first-order method for e-active and violate constraints in the RBDO process. In addition, the fast reliability analysis method is proposed by reusing some of the information obtained at the previous RBDO iteration to efficiently evaluate probabilistic constraints at the current design iteration under the condition of design closeness. Other RBDO methods have recently been developed to enhance numerical efficiency of RBDO. Thus, the PMA+ is compared to existing RBDO methods from a numerical efficiency and stability point of view. For a numerical understanding of the RBDO process, two numerical examples are provided, including a large-scale multi-crash application.