Univariate Dimension Reduction Method Research Articles

Reliability and sensitivity analyses of porous functionally graded graphene platelet reinforced composite plate

The performance of composite is seriously influenced by numerous uncertain factors, in which the uncertainty analysis plays a vital role in composite materials analysis by providing a comprehensive understanding of uncertain factors and their influence on design and performance. In this study, uncertain behaviors of porous functionally graded graphene platelets reinforced composite (FG-GPLRC) plates are first investigated for assessing the safety under bending deformation. The mechanical model for static analysis of the porous FG-GPLRC plate is formulated by Reddy’s higher-order shear deformation theory, and the bending responses are derived by Hamilton’s principle and the Navier method. Material properties and loads are deemed as random variables, and the performance function is determined by the Tsai-Wu criterion. The univariate dimension reduction method and maximum entropy method are employed to estimate the reliability of porous FG-GPLRC plates with diverse distribution patterns. Additionally, the analytical sensitivities of failure probability with respect to material parameters are derived. The analysis results illustrate that the existence of uncertain factors significantly influences the performance and reliability of composites, and the structural reliability can be improved by adding appropriate pores and graphene platelets at the edge layer of the composite plate.

Topology optimization of truss structure under load uncertainty with gradient‑free proportional topology optimization method

This study proposes a method for truss topology optimization (TO) that incorporates uncertain static and dynamic loads based on the efficient gradient‑free proportional topology optimization (PTO) method. The uncertain loads are modeled using probabilistic parameters and the objective function is constructed as the mean structural compliance with the constraint is imposed to the structural material volume. The univariate dimension reduction method and Gauss-type (UDRG) quadrature are used to quantify and propagate the load uncertainty to estimate the objective function. By the UDRG method, the TO of truss structures under uncertain loads can be transformed into a simple deterministic problem. To obviate the necessity for sensitivity analysis, the PTO method with comparable accuracy and efficiency as the gradient-based method is used. The proposed methodology is corroborated through multiple illustrative examples, which also serve to elucidate the ramifications of load uncertainty on optimized truss designs.

Soft Monte Carlo Simulation for imprecise probability estimation: A dimension reduction-based approach

This paper proposes an efficient solution for solving hybrid reliability problems involving random and interval variables. To meet this aim, using the soft Monte Carlo (SMC) method, a solution is proposed that breaks the random variables space into local 1-D coordinates and then, considers 1-D coordinate as an additional dimension of interval variables. Accordingly, using an optimization in increased interval variables space, the upper and lower bounds of failure probability for each 1-D problem are estimated. In addition, the total failure probabilities are presented as the mathematical expectation of the obtained probability bounds for 1-D coordinates. Then, it is shown that this approach is fit for application of univariate dimension reduction method to reduce the function calls of analysis in the optimization phase. This approach is validated by solving benchmark reliability problems as well as the application of the proposed method for solving real world engineering problems investigated by solving hybrid reliability analysis of reinforced concrete columns. It is shown that the proposed approach efficiently approximates the failure probability bound of problems with moderate nonlinear limit state functions with high accuracy.

System-Level Robust Design Optimization of a Permanent Magnet Motor Under Design Parameter Uncertainties

In the presence of design parameter uncertainties, a system-level robust design optimization method for a permanent magnet motor is proposed to enhance the dynamic and steady performances of a whole motor drive system. To achieve the goal, the influence of individual motor parameters on transient system responses is first investigated with the method of Taguchi experimental planning. The temperature-dependent material property of each motor component is then examined around an operating temperature. A conventionally customized motor drive system is fully simulated based on the finite element method. Finally, it is optimized by the univariate dimension reduction method to ensure the robustness of system performances against the manufacturing tolerance and operating temperature fluctuation.

High-Dimensional Uncertainty Quantification in Electrical Impedance Tomography Forward Problem Based on Deep Neural Network

In electrical impedance tomography (EIT), the uncertainty of conductivity distribution may cause the uncertainty in the forward calculation and further affect the inverse problem. In this paper, an improved univariate dimension reduction method based on deep neural network (DNN-UDR) is proposed for the high-dimensional uncertainty quantification in EIT forward problem. Firstly, DNN is studied to build a substitute model for EIT forward problem in order to solve the high-dimensional problem. Three normalized circular finite element models are established with random uniform conductivity distribution. Then UDR is used to analyze and quantify the uncertainty in the simulation with the form of probability. Compared with Monte Carlo simulation (MCS), the probability distribution of voltage is fitted, and the quantification indicators such as mean, variance, variation coefficient and covariance, are also consistent. On the other hand, with the increase of parameter dimensions, DNN-UDR accelerates the computations obviously. This indicates that DNN-UDR is effective and has high structural stability, accurate prediction results and high computational efficiency.

Research on the Application of Uncertainty Quantification (UQ) Method in High-Voltage (HV) Cable Fault Location

In HV cable fault location technology, line parameter uncertainty has an impact on the location criterion and affects the fault location result. Therefore, it is of great significance to study the uncertainty quantification of line parameters. In this paper, an impedance-based fault location criterion was used for an uncertainty study. Three kinds of uncertainty factors, namely the sheath resistivity per unit length, the equivalent grounding resistance on both sides, and the length of the cable section, were taken as random input variables without interaction. They were subject to random uniform distribution within a 50% amplitude variation. The relevant statistical information, such as the mean value, standard deviation and probability distribution, of the normal operation and fault state were calculated using the Monte Carlo simulation (MCS) method, the polynomial chaos expansion (PCE) method, and the univariate dimension reduction method (UDRM), respectively. Thus, the influence of uncertain factors on fault location was analyzed, and the calculation results of the three uncertainty quantification methods compared. The results indicate that: (1) UQ methods are effective for simulation analysis of fault locations, and UDRM has certain application prospects for HV fault location in practice; (2) the quantification results of the MCS, PCE, and UDRM were very close, while the mean convergence rate was significantly higher for the UDRM; (3) compared with the MCS, PCE, and UDRM, the PCE and UDRM had higher accuracy, and MCS and UDRM required less running time.

Robust Design Optimization of a Permanent Magnet Motor from the System-Level Design Perspective

From the system-level design perspective, a robust design optimization method for a permanent magnetic motor is proposed to enhance the dynamic performance of a drive system while maintaining its steady performance. To achieve the goal, an elaborate numerical model for the whole drive system is first constructed by incorporating a control circuit simulator into a finite element analysis tool, and then the influence of motor parameter variations on transient system responses is investigated by means of the method of Taguchi experimental planning. A conventionally customized motor is optimized by the univariate dimension reduction method to ensure the robustness of system performances against manufacturing tolerances. Finally, a comparative performance analysis between two motor drive systems is provided to demonstrate the validity of the proposed method.

Bayesian reliability-based robust design optimization of mechanical systems under both aleatory and epistemic uncertainties

Uncertainties can be divided into two general categories: aleatory and epistemic. Conventional reliability-based robust design optimization approaches, which disregard epistemic uncertainties due to lack of knowledge about the physical nature of systems, have previously been developed. To overcome this weakness, unlike previous methods, a Bayesian reliability-based robust design optimization method is proposed in the presence of both aleatory and epistemic uncertainties. The proposed formulation is presented as a multi-objective optimization problem. The univariate dimension reduction method is used to approximate the mean and variance of the design function. The non-dominated sorting genetic algorithm-II is used to solve the multi-objective optimization problem. To find the final optimum design from the Pareto front, Shannon’s entropy-based technique for order of preference by similarity to ideal solution (TOPSIS) algorithm is applied. Finally, to demonstrate the applicability of the proposed method, two case studies are considered and the results are compared and discussed.

Long-term fatigue analysis of mooring lines considering wind-sea and swell waves using the Univariate Dimension-Reduction Method

In offshore structures analysis, one of the greatest challenges is the high computational cost of evaluating long-term waves-induced fatigue damage. Complication lies on the large number of required numerical stochastic simulations to compute the long-term multi-dimensional integral associated to the problem. In addition, when effects of wind-sea and swell waves are simultaneously considered as independent phenomena, the order of the damage integral increases, which leads to an even greater amount of numerical simulations. Aiming at reducing the number of short-term stochastic simulations needed in order to evaluate the multi-dimensional fatigue damage integral, this work investigates the Univariate Dimension-Reduction Method (UDRM) as an efficient numerical approach. In this method the original random variables, which represent the environmental conditions parameters, are transformed into standard Gaussian variables and, in this reduced space, an additive decomposition to represent the short-term fatigue damage by a simpler function is employed. Due to this transformation, the problem of simultaneous independent wind and swell waves that previously could only be solved by one four-dimensional integral can now be solved by four one-dimensional integrals. These one-dimensional integrals can be solved by appropriate quadrature techniques, such as the one based on the Gauss-Hermite quadrature. Final result is a significantly reduced number of numerical analyses required for a good estimation of the long-term fatigue damage, allowing its use for practical cases. To evaluate the methodology performance, two case studies involving mooring systems of FPSOs (a spread-moored and another turret-moored) located offshore Brazil in approximately 1800 m (∼6000 ft) water depth are presented. Fatigue damages of the mooring lines are individually calculated by UDRM and results are compared to those obtained by the traditional approach of estimation of fatigue damage considering environmental data of a large enough number of short-term conditions. It is shown that, even with less than 2% of the total short-term numerical simulations required by the traditional approach, UDRM leads to very suitable fatigue damages results, i.e., with differences lower than 9% in the first case study and 24% in the second one. The accuracy of UDRM due to statistical modeling of environmental parameters is also investigated.

Robust Topology Optimization of Graphene Platelets Reinforced Functionally Graded Materials Considering Hybrid Bounded Uncertainties

Abstract An efficient scheme for the robust topology optimization considering hybrid bounded uncertainties (RTOHBU) is proposed for the graphene platelets (GPLs) reinforced functionally graded materials (FGMs). By introducing the concept of the layer-wise FGMs, the properties of the GPLs reinforced FGMs are calculated based on the Halpin-Tsai micromechanics model. The practical boundedness of probabilistic variables is naturally ensured by utilizing a generalized Beta distribution in constructing the robust topology optimization model. To address the issue of lacking the information of critical loads in existing topology optimization approaches considering hybrid uncertainties, a gradient-attributed search is carried out at first based on the hypothesis of linear elasticity to determine the critical loads leading to the worst structural performance. Subsequently, the statistical characteristics of the objective structural performance under such critical loads are efficiently evaluated by integrating the univariate dimension reduction method and the Gauss–Laguerre quadrature, the accuracy of which is verified by the comparison analyses utilizing the results of Monte Carlo simulation as references. Furthermore, a novel realization vector set is constructed for the bounded probabilistic uncertainties to parallelize the sensitivity analysis and accelerate the optimization process. All the proposed innovations are integrated into the robust topology optimization scheme, the effectiveness and efficiency of which are verified by both numerical and realistic engineering examples.

Wellhead fatigue analysis considering the effect of wind-sea and swell waves by using Univariate Dimension Reduction Method

In recent years, the observation of dynamic motions in wellhead structures during drilling operations turned wellhead fatigue into a great concern for the oil industry. As stated by DNV in its Recommended Practice E104 (DNVGL, 2018), the fatigue damage evaluation demands an accurate representation of environmental loads acting on the whole drilling structure, including rig, drilling riser and BOP/LMRP. In terms of computational costs, this representation can make the fatigue analysis very expensive. A high number of loading cases may be needed to take into account the statistical independence between environmental parameters, mainly when wind-sea and swell waves are taken into account. To solve this issue, this paper proposes the use of the Univariate Dimension Reduction Method (UDRM) to compute the fatigue damage in wellhead structures considering the effects of both types of waves. The main advantage of this procedure is that it replaces the solution of the four-fold integral associated to the problem by the solution of four one-dimensional integrals, which drastically reduces the number of numerical simulations required in the fatigue assessment. The method is applied to two study cases considering the environmental conditions of the Brazilian coast. In the first, the drilling unit is a semi-submersible platform, while in the second case a ship-based drilling unit is considered. The obtained results indicate a substantial reduction in computational costs without loss of accuracy of UDRM when compared to a Monte Carlo Simulation-based approach.

A new multivariate quadrature rule for calculating statistical moments of stochastic response

As a derivative-free algorithm, the multivariate quadrature rule has been widely used to calculate statistical moments of the output of a system with uncertain inputs. In this paper, by using Rosenblatt transformation and copula method, the system inputs are related to an independent standard normal random vector U; then, a multivariate polynomial model is introduced to relate the system outputs to U, whereby a set of moment matching equations are derived to define quadrature weights and points. After reviewing the tensor product (TP) based quadrature rule and sparse grid method (SGM), the univariate dimension reduction (UDR) method is reformulated by Kronecker product. Following this routine, a new multivariate quadrature rule is derived by combining a Hadamard matrix based quadrature rule and discrete sine transformation matrix (DSTM). Compared with TP and SGM, the proposed quadrature rule can significantly alleviate the curse of dimensionality, its computational burden increases linearly with respect to the number of uncertain inputs. Besides, the proposed algorithms can match moment matching equations neglected by the UDR method, and thus perform more robustly in the uncertainty quantification problem. Finally, numerical examples are performed to check the proposed quadrature rule.

Reliability Sensitivity Analysis Method for Mechanical Components

Based on the univariate dimension-reduction method (UDRM), Edgeworth series, and sensitivity analysis, a new method for reliability sensitivity analysis of mechanical components is proposed. The univariate dimension-reduction method is applied to calculate the response origin moments and their sensitivity with respect to distribution parameters (e.g., mean and standard deviation) of fundamental input random variables. Edgeworth series is used to estimate failure probability of mechanical components by using first few response central moments. The analytic formula of reliability sensitivity can be derived by calculating partial derivative of the failure probability P f with respect to distribution parameters of basic random variables. The nonnormal random parameters need not to be transformed into equivalent normal ones. Three numerical examples are employed to illustrate the accuracy and efficiency of the proposed method by comparing the failure probability and reliability sensitivity results obtained by the proposed method with those obtained by Monte Carlo simulation (MCS).

Proficiency of statistical moment-based methods for analysis of positional accuracy reliability of industrial robots

General presence of uncertainties in geometrical parameters of industrial robots, such as link length, distance between two connecting rods, joint rotation angle and torsional angle, leads to deviations from the specified trajectory of robotic end-effector. It is of practical significance to analyze the positional accuracy reliability for industrial robots in terms of these uncertainties. Among the existing analysis methods, statistical moment-based methods are highly prioritized in evaluating the positional accuracy reliability for industrial robots due to the high accuracy and good computing efficiency. In this study, three different statistical moment-based methods, namely the sparse grid numerical integration (SGNI) method, the point estimation method (PEM) and the univariate dimension reduction method (UDRM), are applied to quantitatively evaluate the positional accuracy reliability of industrial robots. The kinematics model of industrial robots is established through the Denavit-Hartenberg method. The aforementioned three methods are then programmed to calculate the first-four order moments of the established kinematics model. The industrial robots’ positional accuracy reliability is calculated using the SGNI, PEM and UDRM under specified threshold and compared with that from Monte Carlo simulation (MCS) method. Comparison of results shows that the SGNI method performs best in terms of computational accuracy and the PEM exhibits the highest computational efficiency among the three candidate methods.

An improved adaptive bivariate dimension-reduction method for efficient statistical moment and reliability evaluations

Statistical moment estimations of the performance function with the aim of balancing accuracy and efficiency still remains a challenge for moment-based structural reliability analysis. In this paper, an improved adaptive bivariate dimension-reduction method in terms of vectors (i-VBDRM) is proposed for efficient statistical moments and reliability evaluations. In the proposed method, the delineation of cross terms of two-dimensional functions involved in the bivariate dimension-reduction method (BDRM) is first implemented, where the random variables are classified into several sub-vectors after the non-normal to normal transformation. Then, the explicit expressions of the moments of multiple component sub-vector functions are formulated, where a novel cubature formula is proposed and the Gauss-Hermite quadrature is employed to evaluate the involved two- and one-dimensional Gaussian-weighted integrals. In that regard, the first-four central moments can be calculated with accuracy and efficiency. Then, the probability density function of the performance function is rebuilt by a flexible distribution model called the shifted generalized lognormal distribution based on the first-four central moments evaluated by the proposed method. To demonstrate the efficacy of the proposed method, four numerical examples are presented, where some other forms of BDRM, univariate dimension-reduction method and the crude Monte Carlo simulation are performed for comparisons. The results show that the proposed method can keep the trade-off of precision and efficiency for both the statistical moment assessments and structural reliability analysis.

Hybrid multiplicative dimension reduction method for uncertainty analysis of engineering structures

For uncertainty analysis of high-dimensional complex engineering problems, this article proposes a hybrid multiplicative dimension reduction method based on the existent multiplicative dimension reduction method. It uses the multiplicative dimension reduction method to approximate the original high-dimensional performance function which is sufficiently smooth and has a small high-order derivative as the product of a series of one-dimensional functions, and then uses this approximation to calculate the statistical moments of the function. Then the variance-based global sensitivity index is employed to identify the important variables, and the identified important variables are subjected to bivariate decomposition approximation. Combined with the univariate multiplicative dimension reduction method, the hybrid decomposition approximation is obtained. Compared with the existing method, the proposed method is more accurate than the univariate decomposition approximation when used for uncertainty analysis of engineering models and needs less computational efforts than the bivariate decomposition. In the end, a numerical example and two engineering applications are tested to verify the effectiveness of the proposed method.

Sequential optimization and moment-based method for efficient probabilistic design

To increase the range of applicability of decoupling strategies for reliability-based design optimization (RBDO), a sequential optimization and moment-based reliability assessment (SOMRA) is proposed. In this approach, a moment method based on the univariate dimension reduction method (UDRM) and probability density function (PDF) estimation is employed. Meanwhile, a corresponding mathematical model and a PDF-based method of calculating the shifting scalar are developed to decouple the reliability assessment from the optimization process. The shifting scalar is corrected according to the nonlinear degree of the limit state surface of the performance function before reconstructing the mathematical model for the next iteration of optimization. This approach uses statistical moments to check whether the constraints are active, and rather than assessing the reliability and calculating the shifting scalars for all constraints, only the active constraints are considered for the PDF estimation and shifting scalar calculation. Three numerical examples and an automobile crashworthiness lightweight design problem are presented to demonstrate the effectiveness of the proposed method.

Maximum Entropy Method-Based Reliability Analysis With Correlated Input Variables via Hybrid Dimension-Reduction Method

The estimation of the statistical moments is widely involved in the industrial application, whose accuracy affects the reliability analysis result considerably. In this study, a novel hybrid dimension-reduction method based on the Nataf transformation is proposed to calculate the statistical moments of the performance function with correlated input variables. Nataf transformation is intrinsically the Gaussian copula, which is commonly used to transform the correlated input variables into independent ones. To calculate the numerical integration of the univariate component function in the proposed method, a normalized moment-based quadrature rule is employed. According to the statistical moments obtained by the proposed method, the probability density function of the performance function can be recovered accurately via maximum entropy method. Six examples are tested to illustrate the accuracy and efficiency of the proposed method, compared with that of Monte Carlo simulation, the conventional univariate dimension-reduction method, and the bivariate dimension-reduction method. It is confirmed that the proposed method achieves a good tradeoff between accuracy and efficiency for structural reliability analysis with correlated input variables.

A novel approach to uncertainty analysis using methods of hybrid dimension reduction and improved maximum entropy

Methods of uncertainty analysis based on statistical moments are more convenient than methods that use a Taylor series expansion because the moments methods require neither an iteration process to locate the most probable point nor the computation of derivatives of the performance function. However, existing moments estimation methods are either computationally expensive (e.g., the full factorial numerical integration method) or produce large errors (e.g., the univariate dimension-reduction method). In this paper, a hybrid dimension-reduction method taking account of interactions among variables is presented for estimating the probability moments of the system performance function. In this method, a contribution-degree analysis with finite changes is implemented to identify the relative importance of the input variables on the output. Then, an approximate performance function is generated with the hybrid dimension-reduction method that is based on the results of contribution-degree analysis. Finally, the statistical moments of the performance function can be calculated from the approximate performance function. Once the probability moments are obtained, an improved maximum entropy method is used to generate the probability density function of the performance function. The uncertainty analysis can be implemented by using the approximation probability density function. Five illustrative numerical examples are presented, and different methods are compared in those examples. The statistical moments estimation results reveal that the proposed moments estimation method can dramatically improve efficiency and also guarantee accuracy. Compared with the other probability density function approximation methods, our improved maximum entropy method, using more statistical moments, is more accurate and robust.

A time-variant extreme-value event evolution method for time-variant reliability analysis

Abstract In this paper, we propose a time-variant extreme-value event evolution method (TEEM). The time-evolution process of extreme-value event is firstly proposed in this paper. And by solving it, we can obtain the time-variant reliability of arbitrary time interval and arbitrary failure threshold. In this method, the random process in limit-state function is firstly expanded by an improved orthogonal series expansion method (iOSE). Second, we introduce the idea of extreme-value event to describe the time-variant reliability problem. And by discretizing the time domain, we can obtain a series of extreme-value events. The moments of extreme-value event in every discrete time interval will be solved by the integration of Broyden–Fletcher–Goldfarb–Shanno (BFGS) method and univariate dimension reduction method (UDRM). Third, a time-dependent polynomial chaos expansion method (t-PCE) is proposed to simulate the extreme-value event's time-evolution process, and it will be simulated as a function in terms of a standard normal variable and time. Finally, Monte Carlo simulation (MCS) is adopted to sample the standard normal variable to obtain the time-variant reliability of arbitrary failure threshold and time interval. Three numerical examples are investigated to demonstrate the effectiveness of the proposed methods.