Non-probabilistic Convex Model Research Articles

Efficient non-probabilistic parallel model updating based on analytical correlation propagation formula and derivative-aware deep neural network metamodel

Non-probabilistic convex models are powerful tools for structural model updating with uncertain‑but-bounded parameters. However, existing non-probabilistic model updating (NPMU) methods often struggle with detecting parameter correlation due to limited prior information. Worth still, the unique core steps of NPMU, involving nested inner layer forward uncertainty propagation and outer layer inverse parameter updating, present challenges in efficiency and convergence. In response to these challenges, a novel and flexible NPMU scheme is introduced, integrating analytical correlation propagation and parallel interval bounds prediction to enable automatic detection of parameter correlations. In the forward uncertainty propagation phase, a linear coordinate transformation is applied to map the original parameter space to a standard hypercube space, simplifying correlation-involved bounds prediction into conventional interval bounds prediction. Moreover, an analytical correlation propagation formula is derived using a second-order response approximation to sidestep the complexities of geometry-based correlation calculations. To expedite forward propagation, a derivative-aware neural network model is employed to replace the physical solver, facilitating improved fitting capabilities and automatic differentiation, including the calculation of Jacobian and Hessian matrices essential for correlation propagation. The neural network's inherent parallelism accelerates interval bounds prediction through parallel computation of samples. In the inverse parameter updating phase, the block coordinate descent algorithm is embraced to narrow the search space and boost convergence capabilities, while the perturbation method is utilized to determine the optimal starting point for optimization. Two numerical examples illustrate the efficacy of the proposed method in updating structural models while considering correlations.

Uncertainty and reliability analyses of adhesive assembly to the center-of-mass drift of float components based on the non-probabilistic interval model

Multi-source and strong uncertainties of the adhesive assembly process for high-precision inertial devices are critical issues that result in the current low assembly accuracy and poor assembly consistency. This paper addresses the resulting uncertainty in the center-of-mass drift of the float components of a high-precision inertial device. An interval-based non-probabilistic convex model is used to characterize the uncertainty of the bonding defect parameters of the adhesive assembly process, and the uncertainty transferring from those parameters to the center-of-mass drift is constructed by using a surrogate model and numerical simulations. A reliability analysis is carried out to understand the effect of the uncertainties of three types of bonding defects on the center-of-mass drift, and the sensitivities of the center-of-mass drift to the bonding defects are also found. The results show that the reliability index of the center-of-mass drift obtained under the initial parameter intervals is 0.689, indicating the center-of-mass drift at the risk of being out of the accuracy requirement. By adjusting the parameter intervals of the bonding defects to stabilize the center-of-mass of the assembled components, the reliability index is then improved to greater than 1.

Novel bootstrap-based ellipsoidal convex model for non-probabilistic reliability-based design optimization with insufficient input data

Performing reliability-based design optimization with insufficient input data is a significant and challenging issue as accurately quantifying epistemic uncertainty can be difficult in such scenarios. Non-probabilistic models can marginally alleviate the need for a large sample size by constructing the boundaries of uncertainty parameters. However, most of the existing non-probabilistic models account for only the available or known samples. For cases with insufficient data, the compact uncertainty domain may result in considerably hazardous results, particularly for the problems that require high reliability. Therefore, a novel bootstrap-based ellipsoidal convex model (BECM) is proposed herein to account for both known and unknown data. Herein, the bootstrap method is introduced to non-probabilistic convex models for the first time to quantify the ellipsoidal shape uncertainty caused by insufficient input data. Further, the proposed BECM is also integrated into non-probabilistic reliability-based design optimization. The entire framework of uncertainty quantification and reliability optimization design for insufficient input data is established. Marginally conservative results can finally be achieved in a rational and objective manner by partially sacrificing the ellipsoid volume. Comprehensive comparisons and detailed discussions on aspects, including the problem complexity, initial sample size, and distribution type, are also conducted. The numerical results show that the BECM displays the best performance in terms of accuracy and robustness. It can achieve an approximate balance between being conservative and taking risks. The significant advantages are also illustrated by an engineering example of linear buckling analysis of stiffened cylindrical shells.

Imprecise P-Box Sensitivity Analysis of an Aero-Engine Combustor Performance Simulation Model Considering Correlated Variables

Uncertainties are widely present in the design and simulation of aero-engine combustion systems. Common non-probabilistic convex models are only capable of processing independent or correlated uncertainty variables, while conventional precise probabilistic sensitivity analysis based on ideal conditions also fails due to the presence of uncertainties. Given the above-described problem, an imprecise p-box sensitivity analysis method is proposed in this study in accordance with a multi-dimensional parallelepiped model, comprising independent and correlated variables in a unified framework to effectively address complex hybrid uncertainty problems where the two variables co-exist. The concepts of the correlation angle and correlation coefficient of any two parameters are defined. A multi-dimensional parallelepiped model is built as the uncertainty domain based on the marginal intervals and correlation characteristics of all parameters. The correlated variables in the initial parameter space are converted into independent variables in the affine space by introducing an affine coordinate system. Significant and minor variables are filtered out through imprecise sensitivity analysis using pinching methods based on p-box characterization. The feasibility and accuracy of the method are verified based on the analysis of the numerical example and the outlet temperature distribution factor. As indicated by the results, the coupling between the variables can be significantly characterized using a multi-dimensional parallelepiped model, and a notable difference exists in the sensitivity ranking compared with considering only the independence of the variables, in which input parameters (e.g., inlet and outlet pressure, density, and reference flow rate) are highly sensitive to changes in the outlet temperature distribution factor. Furthermore, the structural parameters of the flame cylinder exert a secondary effect.

Dynamic analysis of Jeffcott rotor under uncertainty based on Chebyshev convex method

Rotor system is often affected by various uncertain factors during work, which may affect the normal working process of the rotor system. In this paper, aiming at the uncertainty of the Jeffcott rotor system, a non-probabilistic convex model is constructed to describe the uncertain parameters. By introducing the non-embedded Chebyshev expansion function, the Chebyshev Convex Method (CCM) is proposed. Then, the CCM is employed to calculate the rotor response when the mass and bearing stiffness are respectively uncertain parameters. Meanwhile, by changing the mass of the rotor disc and the bearing support position on the Bently RK4 rotor rig, the Jeffcott rotor with uncertain parameters is simulated. The numerical and experimental results show that the proposed method is valid and effective for quantifying the uncertain parameters of the Jeffcott rotor system. In addition, the Jeffcott rotor response with multiple uncertain parameters is discussed in detail. The results show that compared with the parameters related to the material properties of the rotor such as mass and density, when the parameters related to the bearing support such as stiffness and damping are uncertain, it will have a greater impact on the response of the rotor system.

Life-cycle reliability design optimization of high-power DC electromagnetic devices based on time-dependent non-probabilistic convex model process

The life-cycle reliability of high-power DC electromagnetic devices due to multi-source heterogeneous uncertainties in the design, manufacturing, loads, degradation and cumulative damages has become increasingly prominent, life-cycle reliability optimization has attracted extensive attention. The design optimization strategy based on reliability, which combines the static/time-independent hypothesis and random theory, is inapplicable to life-cycle design optimization. Therefore, a time-dependent uncertainty analysis and life-cycle quality reliability optimization method are proposed in this study. Based on the information entropy transformation method, the fuzzy uncertainty is transformed into a probability density function of equivalent random variables, and intervals for the random variables are determined in the standard normal space. In this way, the heterogeneous uncertainty is unified as an interval type. Moreover, the time-dependent characteristics of the uncertainty parameters are transformed into a time-dependent of the characteristics through ellipsoidal model process. A life-cycle expression method based on an orthonormal basis is proposed to form a life-cycle time-dependent non-probabilistic convex model process. Therefore, the analytical expression of out-crossing rate in which there are many non-probabilistic convex model process parameters are derived and the corresponding upper and lower bounds of out-crossing rate can be obtained. Then, a life-cycle reliability design optimization model is constructed and the optimal solution is determined from intelligence algorithm. The effectiveness of proposed method was verified by a high-power DC electromagnetic relay in renewable energy systems.

Dynamic reliability and variance-based global sensitivity analysis of pipes conveying fluid with both random and convex variables

Purpose This paper aims to solve the problem that pipes conveying fluid are faced with severe reliability failures under the complicated working environment. Design/methodology/approach This paper proposes a dynamic reliability and variance-based global sensitivity analysis (GSA) strategy with non-probabilistic convex model for pipes conveying fluid based on the first passage principle failure mechanism. To illustrate the influence of input uncertainty on output uncertainty of non-probability, the main index and the total index of variance-based GSA analysis are used. Furthermore, considering the efficiency of traditional simulation method, an active learning Kriging surrogate model is introduced to estimate the dynamic reliability and GSA indices of the structure system under random vibration. Findings The variance-based GSA analysis can measure the effect of input variables of convex model on the dynamic reliability, which provides useful reference and guidance for the design and optimization of pipes conveying fluid. For designers, the rankings and values of main and total indices have essential guiding role in engineering practice. Originality/value The effectiveness of the proposed method to calculate the dynamic reliability and sensitivity of pipes conveying fluid while ensuring the calculation accuracy and efficiency in the meantime.

A direct-integration-based structural reliability analysis method using non-probabilistic convex model

In practical structural reliability analysis, there is not only random uncertainty but also fuzzy uncertainty. Aiming at the fuzzy reliability of structure, a novel fuzzy reliability method is proposed based on direct integration method and ellipsoidal convex model. Firstly, the decomposition of fuzzy mathematics principle is used to convert fuzzy reliability model into non-probabilistic reliability model, in which fuzzy variables are converted into interval variables. The upper and lower bounds of interval variables are determined by the possibility distribution function on the membership value. Secondly, multidimensional ellipsoid convex models are constructed to quantify the uncertainty because of the complexity of non-probabilistic reliability. Finally, sigmoid function with adjustable parameter is introduced to direct integration method for approximating the step function, and then direct integration method is used to solve the fuzzy reliability. Numerical examples are investigated to demonstrate the effectiveness of the present method, which provides a feasible way for the structural fuzzy reliability analysis.

Non-probabilistic convex model theory to obtain failure shear stress of simulated lunar soil under interval uncertainties

The properties of the simulated lunar soil, which are of great importance to guide the investigation on the lunar due to the rarity and preciousness of the lunar soil, are extremely complex such that their precise probability distributions are hard to obtain but their bounds are easy to gain. In this study, an innovative non-random analysis method is developed to capture the ultimate soil shear stress based on the non-probabilistic convex model and Mohr–Coulomb failure criterion. Specifically, we extend the traditional Mohr–Coulomb failure criterion, which is usually in deterministic case, to an uncertain case by using a symmetrical positive definite matrix to characterize the correlation of the uncertain parameters related to the soil parameters. To obtain the upper and lower bounds of failure shear stress of the simulated lunar soil, the problem of calculating the stress bounds is converted into two optimization problems, which are solved by using Lagrangian method. Considering whether the time is involved or not in the uncertainties of the soil parameters, both time-invariant and time-variant interval uncertainties problems can be tackled by the proposed method. The effectiveness and accuracy of the proposed method is evidenced by numerical examples and data retrieved from the experimental results exactly between the upper and lower bounds. It indicates that the proposed method can provide important guidance for the design and optimization of lunar soil sampling mechanisms.

Discussions on non-probabilistic convex modelling for uncertain problems

• A unified framework for construction of the non-probabilistic convex models is proposed. • A multidimensional ellipsoid model and five multidimensional parallelepiped models are discussed. • A theoretic evaluation criterion for validity verification for convex modelling methods is proposed. • Two assessment indexes for convex models in practical applications are proposed. Non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain-but-bounded parameters, which is very effective for structural uncertainty analysis with limited or poor-quality experimental data. To overcome the complexity and diversity of the formulations of current convex models, in this paper, a unified framework for construction of the non-probabilistic convex models is proposed. By introducing the correlation analysis technique, the mathematical expression of a convex model can be conveniently formulated once the correlation matrix of the uncertain parameters is created. More importantly, from the theoretic analysis level, an evaluation criterion for convex modelling methods is proposed, which can be regarded as a test standard for validity verification of subsequent newly proposed convex modelling methods. And from the practical application level, two model assessment indexes are proposed, by which the adaptabilities of different convex models to a specific uncertain problem with given experimental samples can be estimated. Four numerical examples are investigated to demonstrate the effectiveness of the present study.

Super parametric convex model and its application for non-probabilistic reliability-based design optimization

• A general hyper parametric convex model is created. • The minimum volume method is developed to select the type of hyper parametric convex model properly. • Nominal value and advanced nominal value methods are proposed simultaneously to perform the non-probabilistic analysis effectively. • Effective non-probabilistic reliability-based design optimization approach is developed based on advanced nominal value method. In this study, we attempt to propose a new super parametric convex model by giving the mathematical definition, in which an effective minimum volume method is constructed to give a reasonable enveloping of limited experimental samples by selecting a proper super parameter. Two novel reliability calculation algorithms, including nominal value method and advanced nominal value method, are proposed to evaluate the non-probabilistic reliability index. To investigate the influence of non-probabilistic convex model type on non-probabilistic reliability-based design optimization, an effective approach based on advanced nominal value method is further developed. Four examples, including two numerical examples and two engineering applications, are tested to demonstrate the superiority of the proposed non-probabilistic reliability analysis and optimization technique.

Study on the Non-Probabilistic Interval Damage of High Temperature Furnace Tube

Uncertainty of parameters like material mechanical properties are traditionally described by probabilistic model, which requires detailed statistical information of parameters limited to obtain in engineering application. In this paper, the problem is solved by using the non-probabilistic convex model that only needs the upper and lower bounds of variables instead of abundant statistical information, which greatly reduces the demand of data and avoids unsafe results caused by the scatter of data. Based on the non-probabilistic convex model, the non-probabilistic interval analysis method is introduced to calculate the creep damage of HK40 high temperature furnace tube and to discuss the influence of operation conditions on creep damage with operation pressure and operation temperature regarded as interval variables. Consequently, the effect of outer wall temperature on creep damage interval is greater at the beginning of furnace tube operation stage. With the increase of running time, the effect of inner wall temperature on creep damage interval is greater and greater, and at the same time greater than that of creep damage interval when the outer wall temperature considered as interval variables. When the operating pressure is treated as interval variable, the upper and lower bounds of creep damage are nearly in a straight line that indicates the effect of pressure change on damage is little. Meanwhile, the life of high temperature furnace tube is predicted according to the definition of non-probabilistic interval index, and the analysis results are compared with the results of probabilistic model. The analysis results sufficiently demonstrate that the non-probabilistic approach presented in this paper is viable.

An improved multidimensional parallelepiped non-probabilistic model for structural uncertainty analysis

The non-probabilistic convex model utilizes a convex set to quantify the uncertainty domain of uncertain parameters. Different with “interval model” and “ellipsoid model”, the parallelepiped convex model can include the dependent and independent interval variables in a unified framework to deal with the complex “multi-source uncertainty” problems. Based on the existing multidimensional parallelepiped (MP) model, this paper proposed an improved MP model for uncertainty quantification. The correlation coefficient between interval variables in this improved MP model is redefined and an explicit expression describing the uncertainty domain of the interval variables is derived based on the correlation matrix. Through matrix transformation, the parallelepiped-shaped uncertainty domain can be projected into a box. The improved MP model is then applied to the uncertainty propagation analysis and reliability analysis of structures. Several numerical examples are investigated to demonstrate the effectiveness of this model.

An efficient Kriging method for global sensitivity of structural reliability analysis with non-probabilistic convex model

The convex model which only needs to know the variation bound of the uncertainty domain is competent to deal with the reliability analysis for the engineering problems lacking sufficient information. However, compared with the researches in solving non-probabilistic reliability index with convex model, the non-probabilistic reliability sensitivity analysis is less available. In this article, the moment-independent global sensitivity analysis of the basic variable based on convex model is performed for investigating the effect of non-probabilistic variable of the structure or system on the dangerous degree in reliability engineering. The proposed sensitivity index inherits the advantages of the traditional moment-independent global sensitivity index. For the problems of which the computational cost of the Monte Carlo simulation is too high, an active learning Kriging solution is established to solve the global sensitivity index. Several examples are adopted to illustrate the correctness of this global reliability sensitivity describing the effect on the reliability of the structure system of the convex model variable and the applicability and feasibility of the active learning Kriging–based solution.

UNCERTAIN BUCKLING AND RELIABILITY ANALYSIS OF THE PIEZOELECTRIC FUNCTIONALLY GRADED CYLINDRICAL SHELLS BASED ON THE NONPROBABILISTIC CONVEX MODEL

The uncertain buckling and reliability of the laminated piezoelectric functionally graded material (FGM) cylindrical shells subjected to axially compressed loads are investigated in this research. Considering the shear deformation, the buckling governing equations of the piezoelectric FGM cylindrical shells are derived on the basis of Donnell assumptions. And then the nonprobabilistic convex model is introduced to predict the uncertain buckling loads of the piezoelectric FGM cylindrical shells resulting from the unavoidable scatter in structural parameters. Finally, the reliability degree of the structures is obtained by computing the ratio of the multidimensional volume falling into the reliability domain to the one of the whole convex model. Numerical results indicate that uncertainties in structural parameters have significant effects on the critical buckling loads and reliability of the piezoelectric FGM cylindrical shells.

Some Important Issues on First-Order Reliability Analysis With Nonprobabilistic Convex Models

Compared with the probability model, the convex model approach only requires the bound information on the uncertainty, and can make it possible to conduct the reliability analysis for many complex engineering problems with limited samples. Presently, by introducing the well-established techniques in probability-based reliability analysis, some methods have been successfully developed for convex model reliability. This paper aims to reveal some different phenomena and furthermore some severe paradoxes when extending the widely used first-order reliability method (FORM) into the convex model problems, and whereby provide some useful suggestions and guidelines for convex-model-based reliability analysis. Two FORM-type approximations, namely, the mean-value method and the design-point method, are formulated to efficiently compute the nonprobabilistic reliability index. A comparison is then conducted between these two methods, and some important phenomena different from the traditional FORMs are summarized. The nonprobabilistic reliability index is also extended to treat the system reliability, and some unexpected paradoxes are found through two numerical examples.

Non-probabilistic convex model process: A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems

In this paper, we propose a method for time-variant uncertainty analysis, namely, the “non-probabilistic convex model process”, which provides an effective mathematical tool for the analysis of structural dynamic uncertainty when lacking relevant information. In the convex model process, we express the variables at any time with intervals and establish the corresponding auto-covariance function and correlation coefficient function to depict the correlation between variables at different times. We also define several important characteristic parameters for the uni- and bi-dimensional convex model processes, including the mid-value function, variance function, auto-covariance function, and cross-covariance function; we provide the definition for the stationary convex model process and its ergodicity. Then, by combining the convex model process with the first-passage failure mechanism, we propose a non-probabilistic analysis model of structural dynamic reliability and formulate the solving algorithm based on Monte Carlo simulation. Finally, through the analysis of numerical examples, we verify the effectiveness of the convex model process and the model of dynamic reliability analysis proposed in this paper.

A non-probabilistic structural reliability analysis method based on a multidimensional parallelepiped convex model

Compared with a probability model, a non-probabilistic convex model only requires a small number of experimental samples to discern the uncertainty parameter bounds instead of the exact probability distribution. Therefore, it can be used for uncertainty analysis of many complex structures lacking experimental samples. Based on the multidimensional parallelepiped convex model, we propose a new method for non-probabilistic structural reliability analysis in which marginal intervals are used to express scattering levels for the parameters, and relevant angles are used to express the correlations between uncertain variables. Using an affine coordinate transformation, the multidimensional parallelepiped uncertainty domain and the limit-state function are transformed to a standard parameter space, and a non-probabilistic reliability index is used to measure the structural reliability. Finally, the method proposed herein was applied to several numerical examples.

Structural reliability analysis using non-probabilistic convex model

By using the convex model approach, the bounds of the uncertain parameters are only required rather than their precise probability distributions, based on which it can be made possible to conduct the reliability analysis for many complex engineering problems with limited information. In this paper, a non-probabilistic reliability model is given for structures with convex model uncertainty, which is defined as a ratio of the multidimensional volume falling into the reliability domain to the one of the whole convex model. A Monte Carlo simulation is then formulated for the reliability model to obtain a reference solution. A first order approximation method (FOAM) is proposed to solve the reliability model based on a linear approximation of the failure surface. A second order approximation method (SOAM) is further formulated to improve the reliability analysis precision for limit-state functions with relatively strong nonlinearity. By introducing the suggested reliability model, a convex-model-based system reliability method is also formulated. Eight numerical examples are provided to demonstrate the effectiveness of the present methods.

Correlation analysis of non-probabilistic convex model and corresponding structural reliability technique

Compared with the probability approach, the non-probabilistic convex model only requires a small amount of samples to obtain the variation bounds of the imprecise parameters, and whereby makes the reliability analysis very convenient and economical. In this paper, we attempt to propose and create a correlation analysis technique mathematically for the non-probabilistic convex model, and based on it develop an effective method to construct the multidimensional ellipsoids on the uncertainty. A marginal convex model is defined to describe the variation range of each uncertain parameter, and a covariance is defined to represent the correlation degree of two uncertain parameters. For a multidimensional problem, the covariance matrix and correlation matrix can be created through all marginal convex models and covariances, based on which the required ellipsoid on the uncertainty can be conveniently achieved. By combining the correlation analysis technique and the reliability index approach, a non-probabilistic reliability analysis method is also developed for uncertain structures. Six numerical examples are presented to demonstrate the effectiveness of the present method.