Convex Set Model Research Articles

Dynamic failure assessment analysis of 15MnTi steel supporting structure with a circumferential surface crack under impact loads

Supporting structures, whose strength have great influence on the safe operation of the whole equipment, are widely used in various ship equipment. In this paper, dynamic failure assessment analysis is carried out to discuss the structural integrity of the cracked supporting structure under impact loads. Considering the dynamic response characteristics of structures, the dynamic fracture toughness determined by the loading rate of dynamic J-integrals is used to calculate the failure assessment point (FAP). Due to the widespread uncertainties in practical engineering, such as crack size and mechanical properties of materials, the failure assessment results will also have uncertainty. To analyze the structural integrity affected by uncertain variables, the polygonal convex set (PCS) model is introduced to the failure assessment diagram (FAD) to propose a non-probabilistic failure assessment method. The FAP is extended to a failure assessment region, and structural integrity can be analyzed by the non-probabilistic reliability degree. Compared with the traditional interval model, the non-probabilistic reliability degree calculated by PCS model is closer to the probabilistic reliability degree, which avoids premature judgement of structural failure. For the 15MnTi supporting structure with a circumferential semi-elliptical surface crack under impact loads, the critical depth obtained by non-probabilistic methods decrease by 10% compared with deterministic results, and the failure mechanism is summarized as plastic collapse resulting from the excessive net section stress. In the absence of statistical data on crack size or mechanical properties, the non-probabilistic failure assessment method can be used as an effective alternative to the probabilistic method.

A dynamic Gaussian process surrogate model based on the grasshopper optimization algorithm for non-probabilistic reliability analysis of complex structures

In practical complex engineering structures, the performance function (PF) for reliability analysis is commonly implicit and highly nonlinear. Commonly used surrogate models are appropriate for structural reliability analysis with an implicit PF. However, these methods require hundreds of PF values, which are time-consuming to obtain by adopting numerical analysis, such as finite element analysis (FEA). Therefore, since non-probabilistic reliability analysis does not require a large number of samples, it has great development potential. In this paper, a dynamic Gaussian process (GP) surrogate model based on the grasshopper optimization algorithm (DGP-GOA) is proposed for the non-probabilistic reliability analysis. First, with the help of the scale factor of the convex set model, the non-probabilistic reliability analysis problem is transformed into an unconstrained optimization problem. Second, the DGP-GOA fits the PF by constructing a GP surrogate model with a small dataset. Third, the GOA is used to search for the global optimal solution to obtain a non-probabilistic reliability index. Then, a dynamic retraining strategy is proposed to improve the fitting accuracy and efficiency. The results demonstrate that the proposed method is highly applicable to the non-probabilistic structural reliability analysis of complex engineering structures.

Efficient Inverse Method for Structural Identification Considering Modeling and Response Uncertainties

The inverse problem analysis method provides an effective way for the structural parameter identification. However, uncertainties wildly exist in the practical engineering inverse problems. Due to the coupling of multi-source uncertainties in the measured responses and the modeling parameters, the traditional inverse method under the deterministic framework faces the challenges in solving mechanism and computing cost. In this paper, an uncertain inverse method based on convex model and dimension reduction decomposition is proposed to realize the interval identification of unknown structural parameters according to the uncertain measured responses and modeling parameters. Firstly, the polygonal convex set model is established to quantify the epistemic uncertainties of modeling parameters. Afterwards, a space collocation method based on dimension reduction decomposition is proposed to transform the inverse problem considering multi-source uncertainties into a few interval inverse problems considering response uncertainty. The transformed interval inverse problem involves the two-layer solving process including interval propagation and optimization updating. In order to solve the interval inverse problems considering response uncertainty, an efficient interval inverse method based on the high dimensional model representation and affine algorithm is further developed. Through the coupling of the above two strategies, the proposed uncertain inverse method avoids the time-consuming multi-layer nested calculation procedure, and then effectively realizes the uncertainty identification of unknown structural parameters. Finally, two engineering examples are provided to verify the effectiveness of the proposed uncertain inverse method.

Nonprobabilistic Reliability Solution Method of Slope Convex Set Model

Nonprobabilistic Reliability Solution Method of Slope Convex Set Model

A time-variant reliability analysis framework for selective laser melting fabricated lattice structures with probability and convex hybrid models

ABSTRACT We propose a time-variant reliability analysis framework to quantitatively predict the lifetime of the lattice structures fabricated by selective laser melting (SLM), including confirming hybrid uncertainties, establishing a hybrid model, and proposing an efficient time-variant reliability method. We first design and manufacture a representative and complex L-shaped body-centred cubic (BCC) lattice structure utilising the SLM method, followed by morphology and microstructure observations to indicate the necessity of accounting for material uncertainty. Further considering loading fluctuation, we develop an effective time-variant reliability analysis method utilising the mixed probability and convex set model. One benchmark numerical example has been employed to shed a light on the high computational efficiency and acceptable computational accuracy of the developed time-variant reliability method. Finally, the proposed framework is performed to a real L-shaped BCC lattice structure to predict its lifetime, finding that the failure probability after ten years can reach more than 40 times the initial design.

A data-driven deep learning pipeline for quantitative susceptibility mapping (QSM)

PurposeThis study developed a data-driven optimization to improve the accuracy of deep learning QSM quantification. MethodsThe proposed deep learning QSM pipeline consisted of two projections onto convex set (POCS) models designed to decouple trainable network components with the spherical mean value (SMV) filters and dipole kernel in the data-driven optimization. They were a background field removal network (named POCSnet1) and a dipole inversion network (named POCSnet2). Both POCSnet1 and POCSnet2 were the unrolled V-Net with iterative data-driven optimization to enforce the data fidelity. For training POCSnet1, we simulated phantom data with random geometric shapes as the background susceptibility sources. For training POCSnet2, we used geometric shapes to mimic the QSM. The evaluation was performed on synthetic data, a public COSMOS (N = 1), and clinical data from a Parkinson's disease cohort (N = 71) and small-vessel disease cohort (N = 26). For comparison, DLL2, FINE, and autoQSM, were implemented and tested under the same experimental setting. ResultsOn COSMOS, results from POCSnet1 were more similar to that of the V-SHARP method with NRMSE = 23.7% and SSIM = 0.995, compared with the NRMSE = 62.7% and SSIM = 0.975 for SHARQnet, a naïve V-Net model. On COSMOS, the NRMSE and HFEN for POCSnet2 were 58.1% and 56.7%; while for DLL2, FINE, and autoQSM, they were 62.0% and 61.2%, 69.8% and 67.5%, and 87.5% and 85.3%, respectively. On the Parkinson's disease cohort, our results were consistent with those obtained from VSHARP+STAR-QSM with biases <3% and outperformed the SHARQnet+DeepQSM that had biases of 7% to 10%. The sensitivity of cerebral microbleed detection using our pipeline was 100%, compared with 92% by SHARQnet+DeepQSM. ConclusionData-driven optimization improved the accuracy of QSM quantification compared with that of naïve V-Net models.

Time-dependent reliability-based optimization for structural-topological configuration design under convex-bounded uncertain modeling

In this work, a novel convexity-oriented time-dependent reliability-based topology optimization (CTRBTO) framework is investigated with overall consideration of universal uncertainties and time-varying natures in configuration design. For uncertain factors, the initial static ones are quantified by the convex set model and nodal dynamic responses are then expressed by the convex process model, where both the boundary rules and time-dependency properties are revealed by the full-dimensional convex-set collocation theorem. Unlike the original deterministic constraints in topology optimization schemes, a new convex time-dependent reliability (CTR) index is defined to give a reasonable failure judgment of local dynamic stiffness and impel the overall CTRBTO strategy. In addition, the gradient-based iterative algorithm is utilized to guarantee the computational robustness and the CTR-driven design sensitivities are explicitly analyzed by the Lagrange multiplier method. Several numerical examples are used to illustrate the effectiveness of the proposed method, and numerical results reflect the significance of this study to a certain extent.

A new method for calculating failure probability of landslide based on ANN and a convex set model

Calculating the failure probability of a landslide is important in engineering related to geological process and geomorphological evolution. Strength parameters of a soil (i.e., cohesion and internal friction angle) are regarded as uncertain-but-bounded parameters. In this study, a new method is proposed for computing the landslide failure probability based on a convex set model and artificial neural network (ANN). In the new method, ANN is used to determine the limit state function of landslide stability, and the failure probability is determined using a simple iterative algorithm. The new method was applied to calculate the failure probability of the Gufenping landslide in Nanjiang, Sichuan, China. The results calculated using a Monte Carlo simulation (MCS) method confirmed that the new method accurately and quickly obtains the failure probability of a landslide. Additionally, compared with the two-dimensional calculation method, the one-dimensional analysis method overestimates the failure probability of the landslide. The results of single factor and global sensitivity analysis indicate that the average of internal friction angle is the main factor affecting the stability of the landslide. It is easy to calculate failure probability of landslides using the novel method than using the conventional methods.

Long-term safety assessment of large-scale arch dam based on non-probabilistic reliability analysis

Long-term service status of a large-scale arch dam is affected by variations of loads and material properties with randomness and fuzziness. This study aims to resolve the problems of a variety of uncertain variables that coexist in the reliability assessment of arch dams. Firstly, a time-varying non-probabilistic reliability (N-PR) solution method of the arch dams is proposed based on the convex set model. Subsequently, non-probabilistic target reliability indexes are put forward according to the defined probabilistic reliability indexes during long-term operation of the arch dams. Afterwards, a fuzzy prediction index is formulated based on the mixed statistical model and the Info-gap model under the most unfavorable reservoir water depth, so as to timely find defects and avoid the potential failure of the arch dams. Numerical simulation on the Jinping I arch dam is conducted to validate the proposed methods. The possible failure modes and the long-term N-PR of the world’s highest arch dam is appraised, and the fuzzy prediction indexes are concluded. Research results can provide a feasible way for large-scale arch dam reliability and safety assessment.

Uncertainty‐oriented double‐scale topology optimization with macroreliability limitation and micromanufacturing control

AbstractThis article proposes an uncertainty‐oriented double‐scale topology optimization method considering macroreliability limitation and micromanufacturing control. The procedure combines the homogenization method to solve the equivalent elastic property of the microstructure, the feature distance theory to evaluate reliability, a density projection for manufacturability control, and the solid isotropic material with penalization model to suppress intermediate density. To coupling the micro–macro scale, the equivalent elastic property of the cell microstructure is evaluated by the numerical homogenization method and then endowed to the macroelement for finite element analysis. Uncertainty factors existed in optimization parameters are evaluated by the interval convex set model. By utilizing the interval parameter vertex method and the feature distance theory, the reliability of displacement constraint in the optimization model is evaluated and constrained. In terms of length scale control, the microdensity design variables are projected by a threshold function to obtain a clear microstructure that satisfies the preset minimum length scale constraint. Finally, three numerical examples are presented to illustrate the effect of micromanufacturing control, as well as the necessity of considering parameter uncertainty.

Non-probabilistic polygonal convex set model for structural uncertainty quantification

In this paper, a general polygonal convex set model and clustering polygonal convex set model are proposed for more reasonably quantifying non-probabilistic uncertainties. Firstly, through the principal component analysis of uncertain samples, a new interval model based on principal component analysis is constructed to characterize the correlation of uncertain parameters. Then, the polygonal convex set model is further constructed by combining the traditional interval model and the interval model based on principal component analysis. Because the polygonal convex set model more compactly and adaptively envelopes all the uncertain samples using the irregular boundaries, it is very suitable for quantification analysis of high-dimensional uncertain problems. In addition, as the polygonal convex set model has the linear boundaries, the classical simplex optimization method is properly adopted to effectively solve the corresponding uncertainty propagation problems. In order to handle the complex problems with large uncertainties, the clustering polygonal convex set model is further established by combing a several sub polygonal convex set models based on cluster analysis. Because the linear approximation of performance function is suitable in the local range of each sub polygonal convex set model, the simplex optimization method can still provide an effective propagation results by solving each sub model. Finally, three numerical examples are investigated to illustrate the feasibility and effectiveness of the proposed two non-probabilistic models on structural uncertainty quantification.

Non-probabilistic Integrated Reliability Analysis of Structures with Fuzzy Interval Uncertainties using the Adaptive GPR-RS Method

Due to its weak dependence on the quantity of variable samples, the non-probabilistic reliability analysis method based on the convex set model is applicable to practical problems in structural engineering with inherent uncertainties. However, when dealing with the black-box limit-state function issues in practical complex structural engineering, the traditional quadratic polynomial response surface (QP-RS) method has the problem of insufficient precision in approximating a highly nonlinear function. Meanwhile, fixing the limits of interval variables is trickier in case of scant samples and meager statistical information. To remedy the above deficiencies, this paper introduces a reasonable integrated reliability analysis approach. First, an adaptive Gaussian process regression response surface (GPR-RS) method that dynamically improves the fitted accuracy near the design point of the black-box limit-state function is formulated. Furthermore, the integrated reliability index with consideration of fuzzy interval uncertainties is presented. Three validation and two application examples are employed, which have justified the approach as a more reasonable assessor of practical complex structural reliability with safer results.

A global sensitivity analysis method based on ANOVA with convex set model and its state-dependent parameter solution

Convex set model is most widely applied around nonprobabilistic uncertainty description. This paper combines the convex model with global sensitivity analysis theory of variance, and then proposes an index based on convex set model and variance-based global sensitivity analysis method to illustrate the effect of the nonprobability variables on the dangerous degree. The proposed index consists of two parts, including the main and total indices. The main index can quantitatively reflect the effect of uncertainties of input variables on the variance of output response, and the total index reflects the influence of interaction with other variables in addition to the individual influence of input variables. Furthermore, an efficient state-dependent parameter solution for solving the variance-based global sensitivity analysis of nonprobabilistic convex uncertainty is given in this paper. The state-dependent parameter solution not only greatly improves the efficiency but also guarantees the computational accuracy, and the times of performance functions evaluation decrease from [Formula: see text] in single-loop Monte Carlo solution to 2048 in the state-dependent parameter method. Finally, three numerical examples and a finite element example are used to verify the feasibility and rationality of the proposed method.

Extending SORA method for reliability-based design optimization using probability and convex set mixed models

In many practical applications, probabilistic and bounded uncertainties often arise simultaneously, and these uncertainties can be described by using probability and convex set models. However, the computing cost becomes unacceptable when directly solving the reliability-based design optimization (RBDO) problem with these uncertainties involved. To address this issue, in this study, a sequential sampling strategy by extending classical sequential optimization and reliability assessment (SORA) method for RBDO is developed. The proposed strategy can successively select sample points to update the surrogate model at each step of the optimization process. New samples for reliability constraints are mainly chosen from the local region around the approximate minimum performance target point (MPTP) and worst-case point (WCP). Typical design examples, including one engineering application, are investigated to demonstrate the efficiency and accuracy of the proposed method.

Structural eigenvalue analysis under the constraint of a fuzzy convex set model

In small-sample problems, determining and controlling the errors of ordinary rigid convex set models are difficult. Therefore, a new uncertainty model called the fuzzy convex set (FCS) model is built and investigated in detail. An approach was developed to analyze the fuzzy properties of the structural eigenvalues with FCS constraints. Through this method, the approximate possibility distribution of the structural eigenvalue can be obtained. Furthermore, based on the symmetric F-programming theory, the conditional maximum and minimum values for the structural eigenvalue are presented, which can serve as non-fuzzy quantitative indicators for fuzzy problems. A practical application is provided to demonstrate the practicability and effectiveness of the proposed methods.

Structural analysis with alternative uncertainty models: From data to safety measures

When adequate empirical data on uncertain variables is lacking, non-probabilistic approaches to quantify uncertainties become appropriate. This study discusses such situations in the context of structural safety assessment. The problem of developing convex function and fuzzy set models for uncertain variables based on limited data and subsequent application in structural safety assessment is considered. Strategies to develop convex set models for limited data based on super-ellipsoids with minimum volume and Nataf’s transformation based method are proposed. These models are shown to be fairly general (for instance, approximations to interval based models emerge as special cases). Furthermore, the proposed convex functions are mapped to a unit multi-dimensional sphere. This enables the evaluation of a unified measure of safety, defined as the shortest distance from the origin to the limit surface in the transformed standard space, akin to the notion used in defining the Hasofer–Lind reliability index. Also discussed are issues related to safety assessment when mixed uncertainty modeling approach is used. Illustrative examples include safety assessment of an inelastic frame with uncertain properties.

Optimum design of lead-rubber bearing system with uncertainty parameters

In this study, a non-stationary random earthquake Clough-Penzien model is used to describe earthquake ground motion. Using stochastic direct integration in combination with an equivalent linear method, a solution is established to describe the non-stationary response of lead-rubber bearing (LRB) system to a stochastic earthquake. Two parameters are used to develop an optimization method for bearing design: the post-yielding stiffness and the normalized yield strength of the isolation bearing. Using the minimization of the maximum energy response level of the upper structure subjected to an earthquake as an objective function, and with the constraints that the bearing failure probability is no more than 5% and the second shape factor of the bearing is less than 5, a calculation method for the two optimal design parameters is presented. In this optimization process, the radial basis function (RBF) response surface was applied, instead of the implicit objective function and constraints, and a sequential quadratic programming (SQP) algorithm was used to solve the optimization problems. By considering the uncertainties of the structural parameters and seismic ground motion input parameters for the optimization of the bearing design, convex set models (such as the interval model and ellipsoidal model) are used to describe the uncertainty parameters. Subsequently, the optimal bearing design parameters were expanded at their median values into first-order Taylor series expansions, and then, the Lagrange multipliers method was used to determine the upper and lower boundaries of the parameters. Moreover, using a calculation example, the impacts of site soil parameters, such as input peak ground acceleration, bearing diameter and rubber shore hardness on the optimization parameters, are investigated.

A non-probabilistic robust reliability method for analysis and design optimization of structures with uncertain-but-bounded parameters

Uncertainty is inherent and unavoidable in almost all engineering systems. Reliability issues stem rightly from the existence of all sorts of uncertainties. In reliability modeling, an appropriate way for dealing with uncertainties and a corresponding reasonable index system for measuring the reliability are essential. The main purpose of this paper is to present a generalized non-probabilistic reliability methodology for analysis and design of structures with bounded uncertainties. The input uncertain parameters are treated as interval variables, and the two representative convex-set models, hyper-rectangle model and hyper-ellipsoid model, are adopted to describe bounded uncertainties. A non-probabilistic reliability measuring system is developed, in which dimensionless non-probabilistic reliability indices are defined in different situations by adopting a similar way as the traditional probabilistic reliability method for structures. The measuring system of reliability is explained in detail and is illustrated schematically. Based on these, a non-probabilistic reliability methodology for structures with uncertain-but-bounded parameters is presented systematically. The fundamental problem of how to construct a convex-set model is briefly discussed by examples and numerical experiments. Four other numerical examples are provided to illustrate the effectiveness and feasibility of the presented method.

A non-probabilistic model of structural reliability based on fuzzy convex set model

In small sample size problems, it is difficult to judge and control the errors of the ordinary rigid convex model. In view of this, a new uncertainty model named fuzzy convex set model was built, whose properties were systematically researched. Some typical models were established by introducing a new parameter named fuzzy extending parameter into the ordinary convex models. A comprehensive and integral reliability model was presented on the basis of the new uncertainty model. The Gauss-Legendre integral formula was used for the reliability calculation. Two examples were given and showed the rationality and feasibility of the proposed uncertainty and reliability models.

Bound Direct Displacement-Based Seismic Design of Buildings

Limitations of probabilistic model appear when information is insufficient. Convex set model, which much less information is needed, is employed to model uncertainties of target displacement, peak acceleration and predominant period of ground motion. A new bound direct displacement-based seismic design method is proposed. The interval of seismic design results is obtained by considering the uncertainties of design variables and the relative risk of any design scheme within the interval is also simultaneously provided. The results of a four-storey RC building show that the base shear and beam sectional area by Xue's design scheme are close to the lower bound of the bound design method, and the column sectional area and the weight of steel bar are just over the middle value to the upper bound. The sensitivity analysis indicates that the base shear is more sensitive to target displacement.