Robust design in geotechnical engineering – an update

This paper presents an update for the robust geotechnical design (RGD) methodology, which seeks an optimal design with respect to design robustness and cost efficiency, while satisfying the safety requirements. In general, the design robustness is achieved if the system response is insensitive to the variation in the uncertain input parameters (called “noise factors”). In other words, a design is considered robust if the system response exhibits little variation, even though there is high variation in the input parameters. Robust design achieves this desirable outcome by carefully adjusting ‘design parameters’ (i.e., the parameters that can be controlled by the designer, such as the geometry and dimensions) without reducing the uncertainty in the noise factors. In this paper, the existing RGD methodology is updated with a gradient-based robustness measure and a simplified procedure for seeking the knee point. The RGD methodology and its simplified version (with new updates) are illustrated with three design examples. The results presented in this paper show that the RGD methodology and its simplified version are effective design tools that considers safety, cost and design robustness simultaneously. The advantages of the simplified RGD approach are discussed.     

Robust geotechnical design of shield-driven tunnels

This paper presents a fuzzy set-based robust geotechnical design (RGD) methodology for the design of shield-driven tunnels. Here, uncertain geotechnical parameters required for analysis of tunnel performance (referred to herein as the structure safety and serviceability performance of tunnel cross section) are represented as fuzzy sets. Given fuzzy input parameters, the performance of a shield-driven tunnel will be uncertain, which is expressed in this study as a fuzzy factor of safety, according to the analysis of vertex method. Then, the fuzzy factor of safety for a given design is used to evaluate the failure probability and design robustness, which are, in turn, employed in the proposed RGD framework. Note that a design is considered robust if the performance of the shield-driven tunnel is insensitive to the variation of its uncertain geotechnical parameters. Within the RGD framework, each candidate design in the design space is analyzed for its safety state (in terms of failure probability), design robustness, and cost. The goal of the RGD of a shield-driven tunnel is to bring the safety state to an acceptable level, while maximizing the robustness and cost efficiency simultaneously. To this end, a multi-objective optimization is performed and a Pareto front is obtained, which provides a trade-off that may be used to select the most preferred design. Through an illustrative case, the effectiveness and significance of this new robust design methodology is demonstrated.     

Robust design of rock slopes with multiple failure modes: modeling uncertainty of estimated parameter statistics with fuzzy number

The variability of shear characteristics of rock discontinuities is often difficult to ascertain. Thus, even with the reliability-based design (RBD) approach, which allows for consideration of the uncertainty of input parameters, the design of a rock slope system may be either cost-inefficient (overdesign) or unsafe (under-design), depending on whether the variation of input parameters is overestimated or underestimated. The uncertainty about the variation of input parameters is a critical issue in a RBD. This paper presents a feasible approach to addressing this problem using robust design concept. First, the uncertainty of the estimated statistics of input parameters (such as rock properties) is represented by fuzzy sets, which requires only the knowledge of lower and upper bounds of the estimated statistics. Then, the robust design concept is implemented to ensure that the final design is insensitive to, or robust against, the uncertainty of the estimated statistics of input parameters. The design methodology is demonstrated with an application to the design of a rock slope system with multiple failure modes. This design methodology, termed robust geotechnical design (RGD), aims to achieve a certain level of design robustness, in addition to meeting safety and cost requirements. In this paper, the RGD framework is realized through a multi-objective optimization, as it involves three requirements, safety, cost, and robustness. The significance of the design methodology is demonstrated with an example of rock slope design.     

Reliability-based robust geotechnical design of spread foundations using multi-objective genetic algorithm

This paper presents a new geotechnical design concept, called robust geotechnical design (RGD). The new design methodology seeks to achieve a certain level of design robustness, in addition to meeting safety and cost requirements. Here, a design is considered robust if the variation in the system response is insensitive to the variation of noise factors such as uncertain soil parameters and construction quality. When multiple objectives are considered, a single best design may not exist, and a trade-off may be necessary. In such a case, a genetic algorithm is adopted for multi-objective optimization and a Pareto Front, which describes a trade-off relationship between cost and robustness at a given safety level, is established. The new design methodology is illustrated with an example of spread foundation design. The significance of the RGD methodology is demonstrated.     

Reliability-based robust design for reinforcement of jointed rock slope

Traditional reliability-based design methodologies often involve selection of design which is of lowest cost and satisfies safety requirements. But, this design is sensitive to variation in statistics of input parameters (noise parameters) and might become unsatisfactory if an underestimation of coefficient of variation of input parameters is made. A relatively new design methodology known as robust geotechnical design (RGD) is applied for the case of reinforcement of rock slope using end-anchored rock bolts. This ensures selection of a cost-effective and safe design for which probability of failure (P_f) of reinforced rock slope is least sensitive to the noise parameters. Reliability-based RGD approach involves evaluation of P_f for each design with different possible noise parameters. Finding P_f for the complex geotechnical structure is computationally expensive, and thus an augmented radial basis function-based response surface is used as a surrogate to the finite element model of rock slope. This response surface, being very efficient, also performs well for a range of values of noise parameters. Later, minimum distance algorithm is applied to obtain a cost-effective and robust design. Finally, a comparison is made in the costs between two robust designs obtained for different target probability of failure for the same rock slope.     

An intelligent multi-objective EPR technique with multi-step model selection for correlations of soil properties

Current multi-objective evolutionary polynomial regression (EPR) methodology has difficulties on decision-making of optimal EPR model. This paper proposes an intelligent multi-objective optimization-based EPR technique with multi-step automatic model selection procedure. A newly developed multi-objective differential evolution algorithm (MODE) is adopted to improve the optimization performance. The proposed EPR process is composed of two stages: (1) intelligent roughing model selection and (2) model delicacy identification. In the first stage, besides two objectives (model accuracy and model complexity), the model robustness measured by robustness ratio is considered as an additional objective in the multi-objective optimization. In the second stage, a new indicator named selection index is proposed and incorporated to find the optimal model. After intelligent roughing selection and delicacy identification, the optimal EPR model is obtained considering the combined effects of correlation coefficient, size of polynomial terms, number of involved variables, robustness ratio and monotonicity. To show the practicality of the proposed EPR technique, three illustrative cases helpful for geotechnical design are presented: (a) modelling of compressibility, (b) modelling of undrained shear strength and (c) modelling of hydraulic conductivity. For each case, a practical formula with better performance in comparison with various existing empirical equations is finally provided. All results demonstrate that the proposed intelligent MODE-based EPR technique is efficient and effective.

     

柱下独立基础稳健性设计与分析

岩土工程随机参数统计特征的不确定性,使得岩土工程可靠度设计存在一定风险。岩土工程稳健性设计能够充分考虑随机参数的不确定性结合结构安全性、稳健性和经济性实现最优设计。针对随机参数统计特征的不确定性对柱下独立基础设计的影响,基于可靠度理论和岩土工程稳健性设计方法,考虑岩土参数、混凝土和钢筋材料力学参数统计特征不确定性的影响,以独立基础几何尺寸作为可控设计参数进行设计分析。将独立基础地基承载力、地基变形、基础结构冲切破坏和基础弯曲破坏4种失效模式视为串联系统,进行多失效模式下的结构体系稳健性设计,分析了多失效模式下结构几何参数与结构体系可靠度的关系。结合稳健性和经济性,进行了独立基础多目标优化设计,确定柱下独立基础设计的最优解。     

Towards an integrated arid zone water management using simulation-based optimisation

For ensuring both optimal sustainable water resources management and long-term planning in a changing arid environment, we propose an integrated Assessment-, Prognoses-, Planning- and Management tool (APPM). The new APPM integrates the complex interactions of the strongly nonlinear meteorological, hydrological and agricultural phenomena, considering the socio-economic aspects. It aims at achieving best possible solutions for water allocation, groundwater storage and withdrawals including saline water management together with a substantial increase of the water use efficiency employing novel optimisation strategies for irrigation control and scheduling. To obtain a robust and fast operation of the water management system, it unites process modeling with artificial intelligence tools and evolutionary optimisation techniques for managing both water quality and quantity. We demonstrate some key components of our methodology by an exemplary application to the south Al-Batinah region in the Sultanate of Oman which is affected by saltwater intrusion into a coastal aquifer due to excessive groundwater withdrawal for irrigated agriculture. We show the effectiveness and functionality of a new simulation-based water management system for the optimisation and evaluation of different irrigation practices, crop pattern and resulting abstraction scenarios. The results of several optimisation runs indicate that due to contradicting objectives, such as profit-oriented agriculture versus aquifer sustainability only a multi-objective optimisation can provide sustainable solutions for the management of the water resources in respect of the environment as well as the socio-economic development.     

Robust optimisation of CO2 sequestration strategies under geological uncertainty using adaptive sparse grid surrogates

Geologic CO₂ sequestration in deep saline aquifers is a promising technique to mitigate the effect of greenhouse gas emissions. Designing optimal CO₂ injection strategy becomes a challenging problem in the presence of geological uncertainty. We propose a surrogate assisted optimisation technique for robust optimisation of CO₂ injection strategies. The surrogate is built using Adaptive Sparse Grid Interpolation (ASGI) to accelerate the optimisation of CO₂ injection rates. The surrogate model is adaptively built with different numbers of evaluation points (simulation runs) in different dimensions to allow automatic refinement in the dimension where added resolution is needed. This technique is referred to as dimensional adaptivity and provides a good balance between the accuracy of the surrogate model and the number of simulation runs to save computational costs. For a robust design, we propose a utility function which comprises the statistical moment of the objective function. Numerical testing of the proposed approach applied to benchmark functions and reservoir models shows the efficiency of the method for the robust optimisation of CO₂ injection strategies under geological uncertainty.     

New approach to milling circuit control—robust inverse Nyquist array design

The aim of this paper is to present an application of robust inverse Nyquist array control design method to a milling circuit in Majdanpek Copper Mine. An existent model of a milling circuit was adopted to respond to robust inverse Nyquist array methodology demand by introducing model uncertainty, and optimal robust controller of milling circuit is designed. The proposed controller is robust for reference signals. The robustness of multivariable feedback systems with designed controller is tested and compared with different, previously designed multivariable controllers. Simulation results, in time domain, demonstrate the preferable performance of the designed robust controller in setpoint tracking. Implementation of the new controller without any investments in new equipment significantly improves performance of the milling process.     

Application of differential evolutionary optimization methodology for parameter structure identification in groundwater modeling

Parameter structure identification is formulated in terms of solving an inverse problem, which allows for a determination of an appropriate level of parameter structure complexity, and the identification of its pattern and the associated parameter values. With the increasing complexity of parameter structure identification in groundwater modeling, demand for robust, fast, and accurate optimizers is on the rise among researchers from groundwater hydrology fields. A novel global optimizer, differential evolution (DE), has been proposed to solve the parameter-structure-identification problem. The Voronoi tessellation is adopted for the automatic parameterization. The stepwise regression method and the error covariance matrix are used to determine the optimal structure complexity. Numerical experiments with a continuous hydraulic conductivity distribution are conducted to demonstrate the proposed methodology. The results indicate that the DE can identify the global optimum effectively and efficiently. A sensitivity analysis of the control parameters and mutation schemes implemented in the DE is employed to examine their influence on the objective function. The comparison between DE and genetic algorithm shows the advantage of DE in terms of robustness and efficiency. The proposed methodology is also applied to a real groundwater system, Pingtung Plain in Taiwan, and the properties of aquifers are successfully identified.     

Determination of partial factors for the verification of the bearing capacity of shallow foundations under open channels

The limit state design method has been introduced into the design criteria for geotechnical structures. The current paper attempts to apply the reliability-based design method, at Level II, to the bearing capacity of the foundations of open channels from the viewpoint of the limit state design. To examine the applicability of the proposed procedure for practical structures, the reliability index is computed for evaluating the stability of the foundations of existing open channels designed by the conventional method. The conventional design procedure makes excessively safe side design. We applied the FORM to the existing open channels designed by the conventional design procedure, and consequently, large values of reliability index, 3 and 5 were obtained for clayey and sandy soils, respectively. Finally, the partial factors for the soil parameters have been determined, corresponding to the target reliability indices β _t =1, 3 and 4.     

Optimal design of groundwater remediation systems using a multi-objective fast harmony search algorithm

A new multi-objective optimization methodology is developed, whereby a multi-objective fast harmony search (MOFHS) is coupled with a groundwater flow and transport model to search for optimal design of groundwater remediation systems under general hydrogeological conditions. The MOFHS incorporates the niche technique into the previously improved fast harmony search and is enhanced by adding the Pareto solution set filter and an elite individual preservation strategy to guarantee uniformity and integrity of the Pareto front of multi-objective optimization problems. Also, the operation library of individual fitness is introduced to improve calculation speed. Moreover, the MOFHS is coupled with the commonly used flow and transport codes MODFLOW and MT3DMS, to search for optimal design of pump-and-treat systems, aiming at minimization of the remediation cost and minimization of the mass remaining in aquifers. Compared with three existing multi-objective optimization methods, including the improved niched Pareto genetic algorithm (INPGA), the non-dominated sorting genetic algorithm II (NSGAII), and the multi-objective harmony search (MOHS), the proposed methodology then demonstrated its applicability and efficiency through a two-dimensional hypothetical test problem and a three-dimensional field problem in Indiana (USA).     

Slope reliability analysis accounting for spatial variation

A practical and transparent procedure is described for implementing a generalized limit equilibrium method via cell-object-oriented constrained optimization in the spreadsheet platform. The formulation allows switching among the Spencer, Bishop simplified and wedge methods on the same template by specifying different side-force inclination and different constraints of optimization. Search for the critical circular or non-circular slip surface is possible. The deterministic procedure is extended probabilistically by implementing the first-order reliability method via constrained optimization of the equivalent dispersion ellipsoid in the original space of the random variables. This procedure is illustrated for an embankment on soft ground, and for a clay slope in southern Norway, both involving spatially correlated soil properties. The effects of autocorrelation distance on the results of reliability analysis are studied. Shear strength anisotropy is modelled via user-created simple function codes in the programming environment of the spreadsheet. The meaning of probability of failure is discussed.     

边坡可靠度分析的支持向量机法

提出了基于支持向量机（SVM）的边坡可靠度分析新算法。该方法采用均匀设计确定样本点,通过一定数量的确定性计算来训练SVM,拟合边坡的功能函数;采用一阶可靠度方法（FORM）和迭代算法优化SVM模型,获得可靠度指标和验算点信息;在SVM模型基础上进一步通过二阶可靠度方法（SORM）和蒙特卡罗模拟（MCS）计算边坡的失稳概率。以两个典型边坡为例,通过与其他方法比较,证明了该方法的准确性和高效性。结果表明：提出的在标准正态空间（U空间）中取样并构建SVM,在原始空间（X空间）中计算功能函数的算法,有效地解决了具有相关非正态分布变量的可靠度分析问题,并且可很容易扩展到SORM的计算。算例结果证明,该方法的精度高于FORM;而效率优于MCS。分析过程中,边坡安全系数计算和可靠度分析相互独立。因此,该方法既适用于具有显式功能函数的简单问题,也适用于需要软件计算安全系数的实际边坡问题。     

System Reliability Assessment for a Rock Tunnel with Multiple Failure Modes

This paper presents a practical procedure for assessing the system reliability of a rock tunnel. Three failure modes, namely, inadequate support capacity, excessive tunnel convergence, and insufficient rockbolt length, are considered and investigated using a deterministic model of ground-support interaction analysis based on the convergence–confinement method (CCM). The failure probability of each failure mode is evaluated from the first-order reliability method (FORM) and the response surface method (RSM) via an iterative procedure. The system failure probability bounds are estimated using the bimodal bounds approach suggested by Ditlevsen (1979), based on the reliability index and design point inferred from the FORM. The proposed approach is illustrated with an example of a circular rock tunnel. The computed system failure probability bounds compare favorably with those generated from Monte Carlo simulations. The results show that the relative importance of different failure modes to the system reliability of the tunnel mainly depends on the timing of support installation relative to the advancing tunnel face. It is also shown that reliability indices based on the second-order reliability method (SORM) can be used to achieve more accurate bounds on the system failure probability for nonlinear limit state surfaces. The system reliability-based design for shotcrete thickness is also demonstrated.     

Enhancement of backtracking search algorithm for identifying soil parameters

In this paper, an enhanced backtracking search algorithm (so-called MBSA-LS) for parameter identification is proposed with two modifications: (a) modifying the mutation of original backtracking search algorithm (BSA) considering the contribution of current best individual for accelerating convergence speed and (b) novelly incorporating an efficient differential evolution (DE) as local search for improving the quality of population. The proposed MBSA-LS is first validated with better performance than the original BSA and some other typical state-of-the-art optimization algorithms on a benchmark of soil parameter identification in terms of effectiveness, efficiency, and robustness. Then, the efficiency of the MBSA-LS is further illustrated by two representative cases: identifying soil parameters from both laboratory tests and field measurements. All comparisons demonstrate that the proposed MBSA-LS algorithm can give accurate results in a short time. Finally, to conveniently solve the problems of parameter identification, a practical tool ErosOpt for parameter identification is developed by integrating the proposed MBSA-LS and some other efficient algorithms for readers to conduct the parameter identification using optimisation algorithms.     

Calibration and assessment of reliability-based serviceability limit state procedures for foundation engineering

This paper uses an existing reliability-based serviceability limit state (RBSLS) procedure to illustrate some of the critical elements in the calibration of RBSLS models and to serve as guide for future calibration work. The impact of copula model selection and therefore correlation structure of bearing pressure-displacement model parameters on reliability is assessed. Then, a framework for evaluating bearing pressure-displacement normalisation techniques is presented as a revision to the existing RBSLS procedure proposed. The reliability of the revised RBSLS procedure is then evaluated by comparing its accuracy to newly conducted, full-scale loading tests of spread footings on aggregate pier-reinforced ground. It is shown that the new RBSLS procedure produces a more accurate estimate of the actual reliability and validates the proposed framework.     

Prediction of California Bearing Ratio from Index Properties of Soils Using Parametric and Non-parametric Models

This work proposes a methodology to obtain from the soils properties the best prediction model for the California bearing ratio index. The methodology proposes three different prediction techniques: (1) the multiple linear regression, a classical parametric technique; and two non-parametric techniques: (2) the local polynomial regression (LPR) and (3) the radial basis network. The LPR is a known statistical method, but in the geotechnical engineering field is not in common use. Besides, although several research works have been published in this field, they do not include a robust procedure for making good comparison between different models. Here, a cross validation method is proposed with this aim. A data set of 96 samples from Peruvian soils is used to illustrate the methodology. To validate the proposed methodology, a data set from the literature is also analyzed.     

Response surface-based robust geotechnical design of supported excavation – spreadsheet-based solution

ABSTRACT

The robust geotechnical design (RGD) approach which involves optimization to obtain a design that is safe, cost-efficient, and robust in the face of uncertainties, can be computationally challenging for complex geotechnical structures. In this study, the RGD approach has become practical by introducing a response surface as a surrogate to finite element- or finite difference-based computer code that is used for analyzing the system, and developing a fast algorithm for the optimization process. For demonstration purposes, a real-world supported excavation project is designed using this modified RGD approach and it is compared with the one designed by a local expert.