Adaptive Response Surface Method Research Articles

An optimization framework for the design of LMUs used in offshore platform float-over installation

Leg mating units (LMU), consisting of steels and hyperelastic materials, are utilized to protect the structures during offshore platform float-over installation. Initial LMU design usually does not satisfy required performance since it is geometrically adjusted to fit given requirements based on previous design. To meet design criteria, design optimization can be applied to the initial design. However, a direct implementation of an accurate local search algorithm may cause heavy computations with the risk of falling into a local optimum. It can be also challenging to find an accurate optimum with efficient global search algorithms due to the characteristics of LMU problems. A small perturbation in nonlinear material can have a significant impact on LMU performance. Therefore, in this research, to find an optimum LMU design efficiently and accurately, an optimization framework that combines global and local search algorithms is implemented. In the framework, an efficient meta-model of optimal prognosis based evolutionary algorithm is first applied. The optimal design obtained from the first step, premature but near the optimum, is set as a new starting point for the sequential optimization where accurate polynomial based local adaptive response surface method is utilized. The robustness of the proposed framework is shown by the development of an LMU suitable for float-over installation of a 20,000-ton topside over an eight-legged fixed offshore jacket platform.

Development of an additively-manufactured functionally-graded expandable implant via the application of the adaptive response surface method: feasibility study on intramedullary humerus nail

The paper introduces the concept of the rational design of a deployable humeral intramedullary nail plug based on a honeycomb structure used for the surgical treatment of humeral shaft fractures. The concept serves for to restore the axial alignment of bone fragments and to maintain stability via bone-nail friction and locking screws. The design nail plug was gained by optimisation process the Latin Hypercube Sampling Design algorithm and Multi-Objective Genetic Algorithm. It was shown that we can use statistical shape function combined by 3 D printing for designing of a new rationally designed implants.

Optimization of automotive battery pack casing based on equilibrium response surface model and multi-objective particle swarm algorithm

Lightweight research based on battery pack structural strength can improve the endurance and safety of electric vehicles. Based on the adaptive response surface and multi-objective particle swarm optimization algorithm, this paper proposes an optimization design method for lightweight of battery pack shell. The thickness of the battery pack shell is the optimization parameter. The stress, strain, and frequency of the battery pack shell under typical working conditions are used as boundary conditions. The response surface model is established according to the criterion of cross terms in adaptive response surface method, and the multi-objective particle swarm optimization algorithm is used for iterative solution. The optimization results show that the maximum stress of the battery pack is reduced to the appropriate range, the first-order frequency is increased by 41% to reduce resonance, the maximum deformation is reduced from 2.7 to 1.12 mm, and the total mass is reduced by 26.8%. The battery pack optimization design method proposed in this paper can achieve lightweighting while meeting safety performance.

Multi-objective optimization of a multi-cellular aluminum extruded crush rail subjected to dynamic axial and oblique impact loading conditions

This work presents a multi-objective shape optimization study of a multi-cellular aluminum extrusion subjected to dynamic axial and oblique (20° angled) impact loading conditions. Finite element models are developed, validated with experimental data, parameterized, and used to exhaust the design space using a full factorial design of experiments. An analysis of different energy absorption characteristics, such as axial and oblique mean crush force, crush efficiency, and oblique impact coefficient, are presented. A multi-objective optimization problem is defined to generate new solutions that dominate a baseline profile in all aspects of performance, which produced a geometry with up to 16% improvements in energy absorption. The parametric study is used to define the “true” Pareto front, which is the set of non-dominated designs and the solution to the optimization problem. The response surface methodology (RSM) is used to perform the multi-objective optimization analysis. The traditional RSM approach is compared to an adaptive surrogate-assisted response surface method (ASA-RSM) in their speed and ability to identify the true Pareto front correctly. Using the traditional metamodel constraint, the traditional RSM approach could only identify 18% of the true Pareto front using up to 8% of the design domain. After relaxing the constraint, the traditional RSM approach still required approximately 63% of the entire domain to identify 67% of the true Pareto front accurately. In comparison, the ASA-RSM approach was able to identify 84.5% of the true Pareto front using 42% of the entire domain. In some instances, the ASA-RSM approach correctly identified 100% of the true Pareto front by using as little as 18% of the domain. However, the ASA-RSM approach’s peak performance is dependent on the initial sampling and resulting metamodel. This study provides a comprehensive understanding of the performance gains of multi-objective optimization in the application of structural crashworthiness considering various loading conditions.

A new response surface method based on the adaptive bivariate cut-HDMR

PurposeThe purpose of this study is to improve the efficiency and accuracy of response surface method (RSM), as well as its robustness.Design/methodology/approachBy introducing cut-high-dimensional representation model (HDMR), the delineation of cross terms and the constitution analysis of component function, a new adaptive RSM is presented for reliability calculation, where a sampling scheme is also proposed to help constructing response surface close to limit-state.FindingsThe proposed method has a more feasible process of evaluating undetermined coefficients of each component function than traditional RSM, and performs well in terms of balancing the efficiency and accuracy when compared to the traditional second-order polynomial RSM. Moreover, the proposed method is robust on the parameter in a wide range, indicating that it is able to obtain convergent result in a wide feasible domain of sample points.Originality/valueThis study constructed an adaptive bivariate cut-HDMR by introducing delineation of cross-terms and constitution of univariate component function; and a new sampling technique is proposed.

Uncertainty Quantification of the Crosstalk in Multiconductor Transmission Lines via Degree Adaptive Stochastic Response Surface Method

The degree adaptive stochastic response surface method is applied to analyze statistically the crosstalk in multiconductor transmission lines (MTLs). The coefficient of polynomial chaos expansion (PCE) is obtained based on the least angle regression. The truncation degree of PCE is iterated using the degree adaptive truncation algorithm, and the optimal proxy model of the crosstalk of the original MTLs that satisfies the actual error requirements is calculated. The statistical properties of crosstalk in MTLs (such as mean, standard deviation, skewness, kurtosis, and probability density distribution) are obtained. The failure probability of the electromagnetic compatibility in the MTLs system is considered. The global sensitivity indices of crosstalk-related factors are analyzed. Finally, the proposed method is proved to be effective compared with the conventional Monte Carlo method. The uncertainty quantification of crosstalk in MTLs can be calculated efficiently and accurately.

ANN Based Design Parameter Estimation for Structural Systems

Estimation of the probability of failure of multi-dimensional structural systems is expensive from the computation perspective. To decrease the burden of computation, one can use simple approximation methods like Surrogate models, Kriging model, Support vector machine, Artificial neural network, and more based on the suitability for the problems. In Surrogate or Response surface modeling, the limit state function of any system is suitably approximated by making use of known mathematical models like polynomials, exponentials, etc. During the construction of surrogates, variables in the model should be well known prior to the approximation. In practical consideration, the design parameters are the unknowns that need to be evaluated before reliability-based design. Inverse Response surface procedure is proposed in the paper to address the above-mentioned issue. The procedure developed is the combination of adaptive Response surface method with appropriate experimental design i.e. Halton low discrepancy sequence sampling technique for evaluating the probability of failure or reliability index and an Artificial neural network is utilised as an inverse reliability procedure for design optimisation. The method gives an accurate result and the efficiency is increased for the same number of iterations in comparison to the work of David Lehky and Martina Somodikova [1] with Latin hypercube sampling as experimental design.

An efficient hybrid sequential approximate optimization method for problems with computationally expensive objective and constraints

To improve the computational cost and approximate optimization accuracy of solving engineering optimization problems with both expensive objective and constraints, a hybrid sequential approximate optimization (HSAO) method is proposed. First, the radial basis function neural network (RBFNN) and response surface method (RSM) approximate models fit the optimization objectives and constraints, respectively. Second, the fitted model searches for the optimal solution via global optimization algorithm, and then, the optimal solution is used as a newly added sample point for updating approximation model. The region of the optimal solution is iteratively updated until the optimal solution converges. The performance of the HSAO method is tested on numerical test functions. Compared with the traditional approximate model, adaptive RSM, and adaptive RBFNN approximate optimization, the results indicate that the proposed HSAO method is higher in search ability and efficiency. Finally, the method is applied in S-shaped tube crashworthiness optimization, and has achieved desired results.

Adaptive anisotropic response surface method based on univariate dimension-reduction model and its high-order revision

PurposeBased on univariate dimension-reduction model, this study aims to propose an adaptive anisotropic response surface method (ARSM) and its high-order revision (HARSM) to balance the accuracy and efficiency for response surface method (RSM).Design/methodology/approachFirst, judgment criteria for the constitution of a univariate function are derived mathematically, together with the practical implementation. Second, by combining separate polynomial approximation of each component function of univariate dimension-reduction model with its constitution analysis, the anisotropic ARSM is proposed. Third, the high-order revision for component functions is introduced to improve the accuracy of ARSM, namely, HARSM, in which the revision is also anisotropic. Finally, several examples are investigated to verify the accuracy, efficiency and convergence of the proposed methods, and the influence of parameters on the proposed methods is also performed.FindingsThe criteria for constitution analysis are appropriate and practical. Obtaining the undetermined coefficients for every component functions is easier than the existing RSMs. The existence of special component functions is useful to improve the efficiency of the ARSM. HARSM is helpful for improving accuracy significantly and it is more robust than ARSM and the existing quadratic polynomial RSMs and linear RSM. ARSM and HARSM can achieve appropriate balance between precision and efficiency.Originality/valueThe constitution of univariate function can be determined adaptively and the nonlinearity of different variables in the response surface can be treated in an anisotropic way.

Efficient aero-structure coupled wing optimization using decomposition and adaptive metamodeling techniques

This paper proposes an efficient decomposition-based optimization framework using adaptive metamodeling for expensive aero-structure coupled wing optimization problems. First, high-fidelity aero-structure coupled analysis method is presented through spatial interpolations of distributed aerodynamic loads and structural deformation. The proposed optimization framework decomposes the original complex coupled optimization problem into 2-D airfoil optimization (i.e., Stage-I) and 3-D wing optimization (i.e., Stage-II) to alleviate the expensive computational costs. In Stage-II, a two-level optimization strategy is tailored to further decompose the 3-D wing optimization into system-level and subsystem-level for dimension reduction. An adaptive response surface method using intelligent space exploration strategy is used to perform optimization tasks involving CFD simulations (i.e., 2-D airfoil optimization and system-level optimization in Stage-II), while sequential quadratic programming is employed to optimize the massive subsystem structure sizing variables. The effectiveness of two-level optimization strategy is validated on a numerical testing problem and a simplified wing optimization case. Finally, the developed models and proposed methods are successfully applied to aero-structure coupled optimization of a high aspect ratio wing. The optimization results demonstrate the effectiveness of the developed aero-structural analysis models and efficiency of the proposed optimization methods.

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.

An adaptive order response surface method for structural reliability analysis

Purpose Balancing accuracy and efficiency is an important evaluation index of response surface method. The purpose of this paper is to propose an adaptive order response surface method (AORSM) based on univariate decomposition model (UDM). Design/methodology/approach First, the nonlinearity of the univariate function can be judged by evaluating the goodness of fit and the error of curve fit rationally. Second, combining UDM with the order analysis of separate component polynomial, an easy-to-implement AORSM is proposed. Finally, several examples involving mathematical functions and structural engineering problems are studied in detail. Findings With the proposed AORSM, the orders of component functions in the original response surface can be determined adaptively and the results of those cases in this paper indicate that the proposed method performs good accuracy, efficiency and robustness. Research limitations/implications Because just the cases with single failure mode and single MPP are studied in this paper, the application in multi-failure mode and multi-MPP cases need to be investigated in the coming work. Originality/value The nonlinearity of the univariate in the response surface can be determined adaptively and the undetermined coefficients of each component function are obtained separately, which reduces the computation dramatically.

Research on the Lightweight Design of Body-Side Structure Based on Crashworthiness Requirements

Crashworthiness is the most significant variable during lightweight design of vehicle structures. However, crashworthiness studies using the single substructure-based method are limited due to the negligence of interactions among substructures. Thus, a whole structure-based study was conducted for the lightweight design of a body-side structure. In this study, a full finite element model was firstly created and then modified into a simplified model for structural improvements, where the major load-carrying subassemblies were improved from the perspectives of crashworthiness and manufacturing costs. Finally, sensitivity analyses were conducted to further optimize the strength distribution, based on which an adaptive response surface method was employed for thickness optimization of the structure. It is found that through the structural improvements and optimizations, the weight of the structure was significantly reduced even when its crashworthiness was improved. This indicates that the whole structure-based method is effective for lightweight design of vehicle structures.

Efficient Robust Design Optimization of a Stacker–Reclaimer Structure Under Uncertainty

A stacker–reclaimer structure (SRS) is a massive structure used for bulk material exploration. Performance of SRS is sensitive to the effect of uncertainty which may lead to catastrophic failure consequences. Thus, in this paper a comparatively new robust design optimization (RDO) approach for design of SRS is explored. The involved parameter of SRS e.g., material loading, incrustation, normal digging, etc., may not have well-defined probability density functions and can be expressed as uncertain but bounded (UBB) type parameters. Hence, RDO is explored for probabilistic as well as UBB cases. Solution of such RDO problem in full simulation approach would require extensive computational time. Hence, response surface method (RSM) based metamodeling approach has been adopted here to alleviate computational burden. Also, as conventional least squares method (LSM) based RSM may be a source of error, this study adopts a comparatively new moving LSM (MLSM) based adaptive RSM in RDO. The RDO results depict that UBB type uncertainty is more critical than the probabilistic case. The proposed MLSM based RDO approach yields reasonably accurate design solutions in a computationally efficient way. The proposed MLSM based RDO approach yields design solutions which ensures safe performance of SRS even in the presence of uncertainty.

Adaptive improved response surface method for reliability-based design optimization

ABSTRACTReliability-based design optimization (RBDO) requires the evaluation of probabilistic constraints (or reliability), which can be very time consuming. Therefore, a practical solution for efficient reliability analysis is needed. The response surface method (RSM) and dimension reduction (DR) are two well-known approximation methods that construct the probabilistic limit state functions for reliability analysis. This article proposes a new RSM-based approximation approach, named the adaptive improved response surface method (AIRSM), which uses the moving least-squares method in conjunction with a new weight function. AIRSM is tested with two simplified designs of experiments: saturated design and central composite design. Its performance on reliability analysis is compared with DR in terms of efficiency and accuracy in multiple RBDO test problems.

Optimal design of sandwich panels with layered-gradient aluminum foam cores under air-blast loading

The dynamic response and design optimization of clamped sandwich panels comprising two aluminum alloy face-sheets and a layered-gradient closed-cell aluminum foam core subjected to air-blast loading were examined in this study. The numerical approach was first validated by blast test results of sandwich panels with monolithic aluminum foam cores, and then the dynamic responses of layered-gradient core sandwich panels were briefly discussed in terms of deflection response and energy absorption. Two surrogate model methods (i.e., response surface method – RSM, and radial basis function – RBF) were adopted to construct objective response functions, and the single-objective adaptive response surface method (ARSM) and multi-objective genetic algorithm (MOGA) were used for the defined optimization problem. The optimization results show the trade-off relationships among the maximum energy absorption, minimum structural mass and minimum deflection, and the advantage of “Pareto front” in such design circumstances. Furthermore, the applicability and accuracy of RSM and RBF agent models in the multi-objective design optimization (MDO) of sandwich panels with layered-gradient foam cores under air-blast loading were also compared and revealed.

An adaptive ANN‐based inverse response surface method

AbstractThe paper deals with an adaptive artificial neural network‐based inverse response surface method utilized when performing reliability‐based design optimization of complex structural systems. Since calculating their reliability indicators (failure probabilities or reliability indices) is usually a time‐consuming task, the utilization of approximation methods with a view to reducing the computational effort to an acceptable level is an appropriate solution. A popular approximation method is the response surface method, where the limit state function is approximated using a suitable surrogate model. In this case, an artificial neural network is utilized. Construction of a response surface requires all variables of stochastic model to be known in advance. However, during the structural design, which is an inverse task, there are design parameters which are subject of reliability‐based design optimization procedure and thus not known at the start of the process. For such cases, an adaptive inverse response surface procedure is proposed. The procedure is based on a coupling of the adaptive response surface method and the artificial neural network‐based inverse reliability method. The validity and accuracy of the method is tested using example with explicit nonlinear limit state function. Obtained results as well as important aspects of the method are discussed.

Seismic reliability analysis of reinforced concrete bridge pier using efficient response surface method–based simulation

Seismic reliability analysis of bridge structures during and succeeding an earthquake event is of significant importance. The more accurate and robust approach of seismic reliability analysis is based on direct Monte Carlo simulation technique. But it is computationally challenging due to the requirement of large number of nonlinear time history analyses. The response surface method–based metamodeling approach is a viable alternative in such situation. This study explores the advantage of moving least squares method–based adaptive response surface method compared to the usually applied least squares method–based response surface method for improved seismic reliability analysis of multi-span bridge pier. The nonlinear time history analyses of the bridge pier are performed in the OpenSees with fibre sections considering a ground motion bin corresponding to the specified hazard level of the bridge site. The seismic reliability analysis results obtained by the usual least squares method and the proposed moving least squares method–based response surface method are compared with that of obtained by more accurate direct Monte Carlo simulation technique to elucidate the effectiveness of the proposed approach.

Analysis and parameters optimization of an expanding energy-absorbing structure for a rail vehicle coupler

This paper describes the energy absorption and crashworthiness optimization of an expanding structure under axial loading, which has been applied to a rail vehicle coupling device. An experiment was set up to study the energy absorption characteristics of the coupling device. Based on the structure of the device, a finite element (FE) model was established and validated by experimental results. The valid FE model was used to predict the responses of the expanding structure. A full factorial design and a central composite design of experiments (DOE) were used to take samples for the variables (friction coefficient (μ), thickness of deformable tube (T) and slope angle of conical mandrel （α）). Based on these samples, an approximation model was built with the moving least squares method (MLSM). Main effects analysis was performed with the full factorial design results and it was found that T had the most significant effect on the average force (Favg), while α most influenced the specific energy absorption (SEA). Considering Favg, fracture and buckling of the structure as constraints, parameters optimization was carried out using the adaptive response surface method (ARSM) to gain a higher SEA. Finally, the optimum parameters (u = 0.25, T = 5.5mm, α = 34.5°) with the SEA value of 34.8kJ/kg was obtained. The value of SEA increased by 45.55% compared with the initial results.

An innovative adaptive sparse response surface method for structural reliability analysis

In the scope of infrastructure risk assessment, structural reliability analysis leads to a challenging problem in order to deal with conflicting objectives: accurate estimation of failure probabilities and computational efficiency. Since the application of classical reliability methods is limited and often leads to a prohibitive computational cost, metamodeling techniques (e.g. polynomial chaos, kriging, response surface methods (RSM), etc.) have been widely used. Nevertheless, existing RSM present limitations handling with highly non-linear limit states, large-scale problems and approximation error. To overcome these problems, this paper describes a cutting-edge response surface algorithm covering the following issues: (i) dimensionality reduction by a variable screening procedure; (ii) definition of a promising search domain; (iii) initial experimental design based on an optimized space-filling scheme; (iv) model selection according to a stepwise regression procedure; (v) model validation by a cross-validation approach; (vi) model fitting using a double weighted regression technique; (vii) sequential sampling scheme by exploring a defined region of interest; (viii) confidence interval of reliability estimates based on a bootstrapping technique. With the aim of proving its efficiency, a wide collection of six illustration examples, concerning both analytical and FE-based problems, was selected. By benchmarking obtained results with literature findings, proposed method not only outperforms existing RSM, but also provides a powerful alternative to the use of other metamodeling techniques.