Over the recent decades, Topology Optimization (TO) has become an important tool in the design and analysis of mechanical structures. Although structural TO is already used in many industrial ...applications, it needs much more investigation in the context of vehicle crashworthiness. Indeed, crashworthiness optimization problems present strong nonlinearities and discontinuities, and gradient-based methods are of limited use. The aim of this work is to present an in-depth analysis of the novel Kriging-Assisted Level Set Method (KG-LSM) for TO. It is based on an adaptive optimization strategy using the Kriging surrogate model and a modified version of the Expected Improvement (EI) as the update criterion, which allows for embedding opportune constraints. The adopted representation using Moving Morphable Components (MMCs) significantly reduces the dimensionality of the problem, enabling an efficient use of surrogate-based optimization techniques. A minimum compliance cantilever beam test case of different dimensionalities is used to validate the presented strategy, as well as identify its potential and limits. The method is then applied to a 2D crash test case, involving a cylindrical pole impact on a rectangular beam fixed at both ends. Compared to the state-of-the-art Covariance Matrix Adaptation Evolution Strategy (CMA-ES), the KG-LSM optimization algorithm demonstrates to be efficient in terms of convergence speed and performance of the optimized designs.
•An active failure-pursuing Kriging modeling method is proposed for time-dependent reliability analysis.•An equivalent stochastic process transformation is developed to form a uniform high ...probability density region.•An active failure-pursing strategy is proposed to identify the most valuable samples.•Correlation-based screening and space partition are applied to represent the sensitive local regions.
Some time-dependent reliability analysis methods use surrogate models to approximate the implicit limit state functions of complex systems. However, the performance of these methods is usually affected by the situations that the used models are not accurate and some samples have no significant contribution to the accuracy improvement. To construct a more suitable model for reliability analysis, this work proposes an active failure-pursuing Kriging modeling method to identify the most valuable samples for improving the accuracy of the predicted failure probability. On the one hand, a global predicted failure probability error index calculated through the real-time reliability result is proposed to pursue the sensitive sample and the corresponding local region that is most likely to maximize the improvement of the accuracy of the reliability result. A fault-tolerant scheme is further applied to ensure the accuracy of the failure-pursuing process. On the other hand, the correlation-based screening and space partition strategy is developed to describe the local regions and avoid the clustering of samples. In each iteration, the Kriging model is updated with the exploitation of new sample from the local regions around the sensitive samples. Additionally, an equivalent stochastic process transformation is developed to form a uniform high probability density sampling space. The results of three cases demonstrate the efficiency, accuracy and stability of the proposed method.
Macro-scale computations of shocked particulate flows require closure laws that model the exchange of momentum/energy between the fluid and particle phases. Closure laws are constructed in this work ...in the form of surrogate models derived from highly resolved mesoscale computations of shock-particle interactions. The mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of Mach Number and particle volume fraction. Two Kriging-based methods, viz. the Dynamic Kriging Method (DKG) and the Modified Bayesian Kriging Method (MBKG) are evaluated for their ability to construct surrogate models with sparse data; i.e. using the least number of mesoscale simulations. It is shown that if the input data is noise-free, the DKG method converges monotonically; convergence is less robust in the presence of noise. The MBKG method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments. This work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows.
•The Tellus and LUCAS databases were used to explore soil spatial variability.•Elevation was the most influential covariate to model target soil properties.•Spatial maps produced from kriging ...techniques showed high accuracy.•Generated maps would be useful for site-specific management options evaluation.•A strong spatial structure was found for the fractal parameters.
Understanding how topography affects the distribution of soil properties is essential in the management of landscape hydrology and establishment of sustainable soil management practices. This study investigated the impact of topography on the variation in particle size distribution, coarse fragments, and soil bulk density using different interpolation techniques and fractal analysis. It also evaluated the performance of various interpolation techniques in predicting and characterizing the distribution of soil properties. The study was conducted using data from 620 samples extracted from the Tellus and LUCAS databases in Eglinton and Castlederg counties, Northern Ireland. Terrain attributes were obtained at a 30 × 30 m resolution using a global digital elevation model (GDEM) reintroduced to the Universal Transverse Mercator (UTM) projection. Interpolation analyses were conducted using inverse distance weighting (IDW), ordinary kriging (OK), block kriging (BK) and co-kriging (CK). Among the terrain attributes, elevation was the most influential covariate for CK. In addition, fractal analysis was conducted to assess the self-similarity of the soil properties. Prediction accuracy of the interpolation methods was evaluated using the Nash-Sutcliffe efficiency, mean absolute error, index of agreement, and Pearson correlation coefficient. Spatial maps produced from the kriging techniques showed high accuracy in the prediction of soil particle size distribution and bulk density. The use of elevation as an auxiliary variable was successful in producing accurate soil property distribution maps with CK. The fractal parameters showed that the soil properties had short range spatial variability, anti-persistent nature, and strong spatial structure. Additionally, the fractal dimension was strongly correlated with sand, silt and clay contents and bulk density, and weakly correlated with the coarse fragments.
The present paper discusses a new methodology to assess stress-cycle fatigue design using meta-modelling. Research on its application is presented for offshore wind turbine towers. Kriging models are ...used to surrogate the complex time-domain stress-cycle fatigue assessment that demands multiple evaluations in the design phase.
The presented development highlights the importance of having a notion of improvement for the problem of meta-modelling. Literature shows that, when meta-modelling complex engineering problems, the idea of improvement is not always considered. To tackle the problem of assessing fatigue in the design phase, a learning criterion is introduced. It has the particularity of relating to the physical description of the stress-cycle fatigue. Results of the proposed criterion are compared with meta-modelling using a Latin hypercube sampling, and with the standard design approach that bins environmental conditions for fatigue design calculations. A full 1 year validation sample is used to study convergence.
As surrogate models for stress-cycle fatigue, Kriging models significantly decrease the efforts of the design procedure. Results showed that computational efforts can be reduced consistently by a factor of 5–8 without compromising accuracy. This may correspond to a reduction of up to approximately 85% of the effort needed to assess stress-cycle fatigue in the design phase.
To conclude, it is important to highlight that the methodology presented has a universal character. It can be implemented to reduce computational time or assess the probabilistic response with only one requirement, the definition of a single representative indicator. In the case of fatigue, the short-term stress-cycle damage rate was considered.
We propose a new multi-fidelity Gaussian process regression (GPR) approach for prediction of random fields based on observations of surrogate models or hierarchies of surrogate models. Our method ...builds upon recent work on recursive Bayesian techniques, in particular recursive co-kriging, and extends it to vector-valued fields and various types of covariances, including separable and non-separable ones. The framework we propose is general and can be used to perform uncertainty propagation and quantification in model-based simulations, multi-fidelity data fusion, and surrogate-based optimization. We demonstrate the effectiveness of the proposed recursive GPR techniques through various examples. Specifically, we study the stochastic Burgers equation and the stochastic Oberbeck–Boussinesq equations describing natural convection within a square enclosure. In both cases we find that the standard deviation of the Gaussian predictors as well as the absolute errors relative to benchmark stochastic solutions are very small, suggesting that the proposed multi-fidelity GPR approaches can yield highly accurate results.
•An active learning method combining Kriging and Subset Simulation (AK–SS) is proposed.•AK–SS takes advantages of Subset Simulation and the Kriging metamodel.•The proposed method is applied to ...several benchmark functions and a tunnel lining structure.•AK–SS is shown to be more efficient than the other methods in the literature.•AK–SS can deal with small probability problems with time-consuming function evaluations.
With complex performance functions and time-demanding computation of structural responses, the estimation of small failure probabilities is a challenging problem in engineering. Although Subset Simulation (SS) is a powerful tool for small probabilities, the computation amount is still large for time-consuming numerical procedures. Metamodelling is an important approach to increase the computational efficiency for engineering problems, however, a larger set of sample points is required for higher accuracy. This is a time-consuming task when the performance function needs to be numerically evaluated. To address this issue, AK–SS: an active learning method combining Kriging model and SS is proposed in this paper. The efficiency of this new method relies upon the advantages of SS in evaluating small failure probabilities and the Kriging model with active learning and updating characteristic for approximating the true performance function. The proposed method is applied to several benchmark functions in the literature, and to the reliability analysis of a shield tunnel, which requires finite element analysis. The results demonstrated that as compared to the other approaches in literature, AK–SS can provide accurate solutions more efficiently, making it a promising approach for structural reliability analyses involving small failure probabilities, high-dimensional performance functions, and time-consuming simulation codes in practical engineering.
•Two accuracy measures of Kriging for reliability analysis are proposed.•The joint PDF of performance values at untried points is derived.•The proposed accuracy measures are validated by four ...benchmark examples.
The Kriging model for structural reliability analysis applications has attracted much attention within recent years. Several Kriging-based strategies of design of experiments are constructed for structural reliability analysis procedure. However, the quantitative accuracy measure of Kriging is still undesirable. Two accuracy measures are introduced. The first one is further studied through derivation and proposed to quantify accuracy of the Kriging-based estimate of the limit state. This paper treats the target failure probability as a variable with epistemic randomness. The second measure is innovatively defined as the standard deviation of the target failure probability. Combining with Chebyshev's inequality, the second measure is available to construct a lower upper bound of the error of the failure-probability estimate. The joint distribution of performance values at untried points of a given Kriging model is derived and proved, which is indeed essential for computing the innovative accuracy measure because it takes the correlation between performance values of untried points into account. Monte Carlo simulation is employed to compute them with acceptable computational cost. To validate the accuracy measures, four benchmark examples are studied. Results demonstrate the availability of them.
A variable-fidelity method can remarkably improve the efficiency of a design optimization based on a high-fidelity and expensive numerical simulation, with assistance of lower-fidelity and cheaper ...simulation(s). However, most existing works only incorporate “two” levels of fidelity, and thus efficiency improvement is very limited. In order to reduce the number of high-fidelity simulations as many as possible, there is a strong need to extend it to three or more fidelities. This article proposes a novel variable-fidelity optimization approach with application to aerodynamic design. Its key ingredient is the theory and algorithm of a Multi-level Hierarchical Kriging (MHK), which is referred to as a surrogate model that can incorporate simulation data with arbitrary levels of fidelity. The high-fidelity model is defined as a CFD simulation using a fine grid and the lower-fidelity models are defined as the same CFD model but with coarser grids, which are determined through a grid convergence study. First, sampling shapes are selected for each level of fidelity via technique of Design of Experiments (DoE). Then, CFD simulations are conducted and the output data of varying fidelity is used to build initial MHK models for objective (e.g. CD) and constraint (e.g. CL, Cm) functions. Next, new samples are selected through infill-sampling criteria and the surrogate models are repetitively updated until a global optimum is found. The proposed method is validated by analytical test cases and applied to aerodynamic shape optimization of a NACA0012 airfoil and an ONERA M6 wing in transonic flows. The results confirm that the proposed method can significantly improve the optimization efficiency and apparently outperforms the existing single-fidelity or two-level-fidelity method.
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