Non-probabilistic Interval Analysis Research Articles

Consideration of uncertainty in damage detection using interval analysis wavelet without baseline data

ABSTRACT The effectiveness of vibration-based damage detection (VBDD) method has been demonstrated by researchers to provide reliable results. However, the existence of uncertainties in measurement and modelling data hinders the accuracy of results obtained from VBDD. Researches have yielded favourable results by integrating probabilistic method. Despite these successes, the probabilistic method faces the problem of obtaining an unbiased probabilistic distribution of uncertainties. Furthermore, the probabilistic method involves long and complex computations. In dealing with these problems, the nonprobabilistic method that requires no assumptions of the uncertainties distribution was proposed. It involves estimating only the upper and lower bounds of the uncertain parameter. However, the success of the nonprobabilistic method is shortened by its reliance on baseline (undamaged) data that is often not available for existing structures. In this study, a nonprobabilistic interval analysis wavelet (NIAW) method to consider uncertainties in damage identification without using baseline healthy data is proposed. The proposed method is demonstrated by using a plate structure and applying the symmetrical properties of the plate structure. The wavelet coefficient of the plate mode shape is divided along the line of symmetry to obtain wavelet coefficients WL and WR , and the bounds (upper and lower) of WL and WR are estimated. The PoDE and wavelet coefficient increment factor (WCIF) are estimated to obtain damage identity by using the bounds of WL and WR . The product of PoDE and WCIF provides the value of DMI which indicates the level of damage severity. This method is demonstrated using numerical models of a steel plate. The results show that the proposed method accurately identifies damage when noise-contaminated mode shape data is applied.

Non-probabilistic method to consider uncertainties in structural damage identification based on Hybrid Jaya and Tree Seeds Algorithm

This paper proposes a novel non-probabilistic structural damage identification approach by developing a hybrid swarm intelligence technique based on Jaya and Tree Seeds Algorithm (TSA), taking into account the high-level uncertainties in the measurements and finite element modelling. The damage in structure is simulated as reduction of elemental stiffness, and structural damage identification is formulated as an optimization problem. To overcome the challenge for structural damage identification with a limited number of measurement data, an objective function based on the modal data and sparse regularization technique is defined. To make the optimization algorithm more powerful and robust, a hybridization of the K-means clustering based Jaya and TSA is proposed. Jaya algorithm is taken as the core in the hybridization. The clustering strategy is employed to replace solutions with low-quality objective values in the Jaya algorithm. Then the search strategy of the TSA is introduced into the best-so-far solution of each cycle. The proposed hybridization algorithm is termed as “ C-Jaya-TSA”. To enhance the capacity of the proposed algorithm to consider uncertainties, a non-probabilistic method is also integrated to calculate the interval bound (lower and upper bounds) of the elemental stiffness changes by using the interval analysis method. To better quantify the structural damage extents, Damage Measure Index (DMI) values are introduced for representing structural damage states. The DMI value can be viewed as a combination of deterministic stiffness reduction and the Possibility of Damage Existence (PoDE). Numerical benchmark functions, numerical studies and experimental investigations are conducted to verify the accuracy and performance of the proposed method. The identification results show that the developed C-Jaya-TSA integrated with the non-probabilistic interval analysis method is a promising tool to accurately identify the structural damage, even high-level uncertainties exist.

Sensitivity and uncertainty analysis of interfacial effect in SHPB tests for concrete-like materials

The Split Hopkinson Pressure Bar (SHPB) equipment has been widely used to measure the dynamic properties of concrete-like materials at high strain rate between 10s-1 and 103s-1. As we know, a crucial factor that influences the accuracy of SHPB tests is the interfacial effect between specimen and bars. In our work, interfacial effect was investigated and summarized into three types of parameters, i.e., interfacial friction coefficient, specimen diameter and specimen non-parallelism. A series of concrete-like (cement mortar) specimens with two different strengths were tested at three different strain rates on SHPB equipment, and the uncertainty of the interfacial effect was transformed into intervals of these three types of parameters. A numerical model was established in ABAQUS and verified to study the interfacial effect, then the sensitivity analysis of three types of parameters was undertaken to figure out a sensitivity ranking using the full factorial design method. The sensitivity analysis provides an effective measure to promote the experimental accuracy and it is meaningful for understanding the interfacial effect in SHPB tests. Moreover, uncertainty analysis was conducted to give an interval of dynamic response of SHPB tests based on a non-probabilistic interval analysis method. The uncertainty analysis can provide an effective estimation of SHPB experimental results considering the uncertainty of interfacial effect.

Tolerance Analysis of Antenna Array Pattern and Array Synthesis in the Presence of Excitation Errors

This paper analyzes array pattern tolerance against excitation errors. The nonprobabilistic interval analysis algorithm is used for tolerance analysis of the nonideal uniform linear array in this work. Toward this purpose, corresponding interval models of the power pattern functions are established, respectively, with the consideration of the amplitude errors, phase errors, or both simultaneously, in antenna arrays. The tolerance for the amplitude-phase error of the main function parameters including the beamwidth, sidelobe level, and the directivity is simulated by computer according to the indicators and the actual requirements. Accordingly, the worst admissible performance of an array can be evaluated, which may provide theoretical reference for optimal antenna array design. As for the problem of array synthesis in the presence of various array errors, interval analysis-convex programming (IA-CP) is presented. Simulation results show that the proposed IA-CP based synthesis technique is robust for the amplitude and phase errors.

Interval prediction of responses for uncertain multidisciplinary system

Considering that numerous sample data points are required in the probabilistic method, a non-probabilistic interval analysis method can be an alternative when the information is insufficient. In the paper, new strategies, which are iterative algorithm based interval uncertainty analysis methods (IA-IUAMs), are developed to acquire the bounds of the responses in multidisciplinary system. Two iterative processes, Jacobi iteration and Seidel iteration, are applied in the new methods respectively. The Jacobi iteration based interval uncertainty analysis method (JI-IUAM) utilizes the strategy of concurrent subsystem analysis to improve computational efficiency while the Seidel iteration based interval uncertainty analysis method (SI-IUAM) can accelerate convergence by utilizing the newest information. Both IA-IUAMs are able to evaluate the bounds of responses accurately and quickly. The presented methods are compared with general sensitivity analysis based interval uncertainty analysis method (SIUAM) and conventional Monte Carlo simulation approach (MCS). The validity and efficiency of the new methods are demonstrated by two numerical examples and two engineering examples. Results show that, on the one hand, IA-IUAMs are more efficient than MCS by avoiding hundreds of system analyses, on the other hand, IA-IUAMs are more accurate and have a wider range of application than SIUAM by avoiding linear approximation and global sensitivity calculation.

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.

Probabilistic and Nonprobabilistic Sensitivity Analyses of Uncertain Parameters

Parameter sensitivity analyses have been widely applied to industrial problems for evaluating parameter significance, effects on responses, uncertainty influence, and so forth. In the interest of simple implementation and computational efficiency, this study has developed two sensitivity analysis methods corresponding to the situations with or without sufficient probability information. The probabilistic method is established with the aid of the stochastic response surface and the mathematical derivation proves that the coefficients of first-order items embody the parameter main effects on the response. Simultaneously, a nonprobabilistic interval analysis based method is brought forward for the circumstance when the parameter probability distributions are unknown. The two methods have been verified against a numerical beam example with their accuracy compared to that of a traditional variance-based method. The analysis results have demonstrated the reliability and accuracy of the developed methods. And their suitability for different situations has also been discussed.

Structural damage measure index based on non-probabilistic reliability model

Abstract Uncertainties in the structural model and measurement data affect structural condition assessment in practice. As the probabilistic information of these uncertainties lacks, the non-probabilistic interval analysis framework is developed to quantify the interval of the structural element stiffness parameters. According to the interval intersection of the element stiffness parameters in the undamaged and damaged states, the possibility of damage existence is defined based on the reliability theory. A damage measure index is then proposed as the product of the nominal stiffness reduction and the defined possibility of damage existence. This new index simultaneously reflects the damage severity and possibility of damage at each structural component. Numerical and experimental examples are presented to illustrate the validity and applicability of the method. The results show that the proposed method can improve the accuracy of damage diagnosis compared with the deterministic damage identification method.

Non-probabilistic interval analysis method for dynamic response analysis of nonlinear systems with uncertainty

Effects of uncertainties on the dynamic response of the nonlinear vibration systems with general form are investigated. Based on interval mathematics, modeling the uncertain parameters as interval numbers, a non-probabilistic interval analysis method, which estimates the range of the nonlinear dynamic response with the help of Taylor series expansion, is presented, where the partial derivatives of the dynamic response with respect to uncertain parameters are considered to be interval numbers. The sensitivity matrices of dynamic response with the uncertain parameters are derived. For the presented method, only the bounds on uncertain parameters are needed, instead of probabilistic density distribution or statistical quantities. Numerical examples are used to illustrate the validity and feasibility of the presented method.

Solution theorems for the standard eigenvalue problem of structures with uncertain-but-bounded parameters

Generalized eigenvalue problems from the modal analysis are often converted to the standard eigenvalue problems. In this paper, it evaluates the upper and lower bounds on the eigenvalues of the standard eigenvalue problem of structures subject to severely deficient information about the structural parameters. Here, we focus on non-probabilistic interval analysis models of uncertainty, which are adapted to the case of severe lack of information on uncertainty. Non-probabilistic, interval analysis method in which uncertainties are defined by interval numbers appears as an alternative to the classical probabilistic models. For the standard eigenvalue problem of structures with uncertain-but-bounded parameters, the vertex solution theorem, the positive semi-definite solution theorem and the parameter decomposition solution theorem for the standard eigenvalue problem are presented, and compared with Deif's solution theorem in numerical examples. It is shown that, for the upper and lower bounds on the eigenvalues of the standard eigenvalue problem with uncertain-but-bounded parameters, the presented vertex solution theorem is unconditional, and the positive semi-definite solution theorem and the parameter decomposition solution theorem have less limitary conditions compared with Deif's solution theorem. The effectiveness of the vertex solution theorem, the positive semi-definite solution theorem and the parameter decomposition solution theorem are illustrated by numerical examples

Comparison of dynamic response of structures with uncertain-but-bounded parameters using non-probabilistic interval analysis method and probabilistic approach

Now by combining the finite element analysis and interval mathematics, a new, non-probabilistic, set-theoretical models, that is interval analysis method is being developed in scientific and engineering communities to predict the variability or uncertainty resulting from the unavoidable scatter in structural parameters and the external excitations as an alternative to the classical probabilistic approaches. Interval analysis methods of uncertainty were developed for modeling uncertain parameters of structures, in which bounds on the magnitude of uncertain parameters are only required, not necessarily knowing the probabilistic distribution densities. Instead of conventional optimization studies, where the minimum possible response is sought, here an uncertainty modeling is developed as an anti-optimization problem of finding the least favorable response and the most favorable response under the constraints within the set-theoretical description. In this study, we presented the non-probabilistic interval analysis method for the dynamical response of structures with uncertain-but-bounded parameters. Under the condition of the interval vector, which contains the uncertain-but-bounded parameters, determined from probabilistic statistical information or stochastic sample test, through comparing between the non-probabilistic interval analysis method and the probabilistic approach in the mathematical proof and the numerical examples, we can see that the region of the dynamical response of structures with uncertain-but-bounded parameters obtained by the interval analysis method contains that produced by the probabilistic approach. In other words, the width of the maximum or upper and minimum or lower bounds on the dynamical responses yielded by the probabilistic approach is tighter than those produced by the interval analysis method. This kind of results is coincident with the meaning of the probabilistic theory and interval mathematics.