Radial Basis Function Metamodel Research Articles

Aerodynamic Analysis of Conventional and Boundary Layer Ingesting Propellers

Abstract The boundary layer ingestion (BLI) concept has emerged as a novel technology for reducing aircraft fuel consumption. Several studies designed BLI-fans for aircraft. BLI-propellers, although, have still received little attention, and the choice of open-rotors or ducted propellers is still an open question regarding the best performance. The blade design is also challenging because the BLI-propulsors ingest a nonuniform flow. These aspects emphasize further investigation of unducted and ducted BLI-propulsors and the use of optimization frameworks, coupled with computational fluid dynamics simulations, to design the propeller to adapt to the incoming flow. This paper uses a multi-objective NSGA-II optimization framework, coupled with three-dimensional RANS simulations and radial basis function (RBF) metamodeling, used for the design and optimization of three propeller configurations at cruise conditions: (a) conventional propeller operating in the freestream, (b) unducted BLI-propeller, and (c) ducted BLI-propeller, both ingesting the airframe boundary layer. The optimization results showed a significant increase in chord and a decrease in the blade angles in the BLI configurations, emphasizing that these geometric parameters optimization highly affects the BLI-blade design. The unducted BLI-propeller needs approximately 40% less shaft power than the conventional propeller to generate the same amount of propeller force. The ducted BLI-propeller needs even less power, 47%. The duct contributes to the tip vortex weakening, recovering the swirl, and turning into propeller force, as noticed from 80% of the blade span to the tip. However, the unducted and ducted BLI-configurations presented a higher backward force, 26% and 46%, respectively, compared to the conventional propeller, which can be detrimental and narrow the use of these configurations.

Design of Approximate Explicit Model Predictive Controller Using Parametric Optimization

AbstractThis paper introduces a new technique, called state-parameterized nonlinear programming control (sp-NLPC), for designing a feedback controller that can stabilize intrinsically unstable nonlinear dynamical systems using parametric optimization. Stability-preserving constraints are included in the optimization problem solved offline by the predictive parameterized Pareto genetic algorithm (P3GA), a constrained nonlinear parametric optimization algorithm. The optimal control policy is approximated from P3GA output using radial basis function (RBF) metamodeling. The sp-NLPC technique requires fewer assumptions and is more data-efficient than alternative methods. Two nonlinear systems (single and double inverted pendulums on a cart) are used as benchmarking problems. Performance and computational efficiency are compared to several competing control design techniques. Results show that sp-NLPC outperforms and is more efficient than competing methods. A qualitative investigation on phase plane analysis for the controlled systems is presented to ensure stability. The approximating state-dependent solution for the control input lends itself to applications of control design for control co-design (CCD). Such extensions are discussed as part of future work.

Optimal Operation of Microgrids Based on a Radial Basis Function Metamodel

The optimal operation of a microgrid (MG) is a nonlinear multiconstraint problem. In addition to optimizing the output of different distributed generations (DGs) at the same time, the output of DGs at different times must be coordinated to achieve a balance between power supply and load demand. To solve the optimal operation problem of an MG, an optimal operation model to minimize user electricity cost and MG operation cost is proposed in this article. This model refines the load types and considers the scheduling elasticity of multiple types of home loads of end-users. It also quantifies the scheduling elasticity as the demand response based on the time-of-use price. Furthermore, to solve the problems of a complex solving process and long computation times for the MG optimal operation model, a global optimization algorithm based on a radial basis function (RBF) metamodel is proposed. The novelty of the proposed algorithm is that it adopts Latin hypercube sampling and an RBF to construct a metamodel. The constructed RBF metamodel is used to replace the proposed MG optimal operation model to estimate the function values, thereby avoiding calling the original complex objective function repeatedly. The proposed model and algorithm are tested on the modified IEEE 37-bus and 300-bus systems. The test results show that, the proposed model can effectively reduce the electricity cost of the end-users and enable the MG to obtain more electricity selling revenue. Meanwhile, the proposed algorithm can effectively reduce computational burden and time and has higher search efficiency of the global optimal solution and computational accuracy.

Data-driven global sensitivity indices for the load-carrying capacity of large-diameter stiffened cylindrical shells

The influence of input parameters on the load-carrying capacity of the large-diameter stiffened cylindrical shell has not been satisfactory understood. To obtain the global sensitivity indices for the load-carrying capacity of large-diameter stiffened cylindrical shells, a novel data-driven sensitivity analysis method is presented for the efficient calculation of Sobol’s indices. In the work, the analytical expressions to compute Sobol’s indices are derived in detail based on the Radial Basis Function (RBF) metamodel. Then, the collapse mode of stiffened cylindrical shells with different sections of stringers are simulated, and the sensitivity of the load-carrying performance to geometric parameters are analyzed. The results show that for the considered stiffened cylindrical shell: (1) The coupled flexural-torsional induced deformation becomes the main factor leading to the overall collapse of the large-diameter and heavy-load stiffened cylindrical shells. (2) The flange thickness of stringers has the strongest influence on the load-bearing capacity, followed by the web thickness of stringers. (3) The interaction influences between stringers and other components in stiffened cylindrical shells are insignificant and negligible compared to the individual influence of stringers.

Optimal design of differential mount using nature-inspired optimization methods

Abstract Structural performance and lightweight design are a significant challenge in the automotive industry. Optimization methods are essential tools to overcome this challenge. Recently, nature-inspired optimization methods have been widely used to find optimum design variables for the weight reduction process. The objective of this study is to investigate the best differential mount design using nature-based optimum design techniques for weight reduction. The performances of the nature-based algorithms are tested using convergence speed, solution quality, and robustness to find the best design outlines. In order to examine the structural performance of the differential mount, static analyses are performed using the finite element method. In the first step of the optimization study, a sampling space is generated by the Latin hypercube sampling method. Then the radial basis function metamodeling technique is used to create the surrogate models. Finally, differential mount optimization is performed by using genetic algorithms (GA), particle swarm optimization (PSO), grey wolf optimizer (GWO), moth-flame optimization (MFO), ant lion optimizer (ALO) and dragonfly algorithm (DA), and the results are compared. All methods except PSO gave good and close results. Considering solution quality, robustness and convergence speed data, the best optimization methods were found to be MFO and ALO. As a result of the optimization, the differential mount weight is reduced by 14.6 wt.-% compared to the initial design.

Performance Analysis of Radial Basis Function Metamodels for Predictive Modelling of Laminated Composites.

High-fidelity structural analysis using numerical techniques, such as finite element method (FEM), has become an essential step in design of laminated composite structures. Despite its high accuracy, the computational intensiveness of FEM is its serious drawback. Once trained properly, the metamodels developed with even a small training set of FEM data can be employed to replace the original FEM model. In this paper, an attempt is put forward to investigate the utility of radial basis function (RBF) metamodels in the predictive modelling of laminated composites. The effectiveness of various RBF basis functions is assessed. The role of problem dimensionality on the RBF metamodels is studied while considering a low-dimensional (2-variable) and a high-dimensional (16-variable) problem. The effect of uniformity of training sample points on the performance of RBF metamodels is also explored while considering three different sampling methods, i.e., random sampling, Latin hypercube sampling and Hammersley sampling. It is observed that relying only on the performance metrics, such as cross-validation error that essentially reuses training samples to assess the performance of the metamodels, may lead to ill-informed decisions. The performance of metamodels should also be assessed based on independent test data. It is further revealed that uniformity in training samples would lead towards better trained metamodels.

Multiobjective crashworthiness optimization of graphene type multi-cell tubes under various loading conditions

Nature is an important source of inspiration for researchers to create better designs. In this study, graphene type multi-cell tubes is inspired by graphene due to its strong and lightweight mechanical properties. Peak crushing force (PCF), energy absorption (EA) and crushing force efficiency crashworthiness indicators have been taken into consideration under different loading angles, and the complex proportion assessment (COPRAS) which is a multicriteria decision-making method has been used to determine the best model. The best model is found to be GTMT5 (second-order and third-order hollow cylinders in the graphene type multi-cell tube) by the COPRAS selection method. The multiobjective optimization, whose objective is to minimize PCF and maximize EA, is applied on the GTMT5 using the multiobjective particle swarm optimization and non-dominated sorting genetic algorithm II methods, and the techniques are compared. The optimization study is carried out on the radial basis function metamodels. This study shows that circular structures placed in multi-cell tubes have a significant effect on the crashworthiness performance.

Crashworthiness Analysis for Structural Stability and Dynamics

In this paper, we fabricate human DNA and polar bear inspired thin-walled tube that tends to reduce the strength of decelerating force during impact, while escalating the amount of energy absorbed. The crashworthiness performance under axial impact is investigated using experimental analysis and non-linear finite element analysis (FEA). The investigation is conducted in three phases; the first phase consists of the design and fabrication of a novel bio-inspired tube (BIT) motivated by the most stable human DNA. Twelve BITs are created by filling cylindrical tubes into different positions of the BIT, which was inspired by the microstructural of polar bear hair. The second phase comprises the nonlinear FEA of energy-absorbed ability for different BITs under axial impact loading using LS-DYNA software, and then validated by the Simplified Super Folding Element (SSFE) theorem. In the third phase, Radial Basis Function (RBF) meta-models and Non-dominated Sorting Genetic Algorithm II (NSGA-II) are used for the multi-objective optimization design of BIT-11. The numerical simulation results are compared with the experimental results to confirm the crash behavior and energy absorption (EA) characteristics of the optimal structure over a base one. Based on the results, the suited configuration with required performance in crashworthiness is suggested, which should be incorporated into automobiles for safety consideration of passengers during an impact. The results show an increment of 49% in Specific Energy Absorption (SEA), suggesting the better choice of a particular tube over the base tube.

Comparison of Metamodel Performances on an Electronic Circuit Problem

Aims: Investigation of building and validation of metamodels which of linear regression, simple kriging, ordinary kriging and radial basis function for an electronic circuit problem are the main aim of this study.
 Study Design: An electronic circuit problem was considered to compare the performances of the metamodels. Latin hypercube design was used for experimental design of five input variables of the considered problem.
 Methodology: A training data set consisting of 45 experiments and a validation data set consisting of 500 experiments were obtained using Latin hypercube design. Input variables were used by coded to calculate the spatial distances between observation points more consistently. Then using training data set linear regression, simple kriging, ordinary kriging and radial basis function metamodels were built. And, performance measures were calculated for the validation data set.
 Results: It has been shown that simple kriging which are applied to outputs the differences from the mean, and ordinary kriging metamodels, produce superior solutions compared to the linear regression and radial basis function metamodels for the electronic circuit problem considered in this study. Prediction superiority of SK and OK than RBF on five-dimensional problem is another important result of the study.
 Conclusion: Kriging metamodels are considered to be strong alternatives to the other metamodels for the problems that are considered in this study and have a similar nature. Since the superiority of metamodel methods to each other may vary from problem to problem, it is another important issue to compare their performance by considering more than one method in problem solving stage.

Sequential Radial Basis Function-Based Optimization Method Using Virtual Sample Generation

Abstract To further reduce the computational expense of metamodel-based design optimization (MBDO), a novel sequential radial basis function (RBF)-based optimization method using virtual sample generation (SRBF-VSG) is proposed. Different from the conventional MBDO methods with pure expensive samples, SRBF-VSG employs the virtual sample generation mechanism to improve the optimization efficiency. In the proposed method, a least squares support vector machine (LS-SVM) classifier is trained based on expensive real samples considering the objective and constraint violation. The classifier is used to determine virtual points without evaluating any expensive simulations. The virtual samples are then generated by combining these virtual points and their Kriging responses. Expensive real samples and cheap virtual samples are used to refine the objective RBF metamodel for efficient space exploration. Several numerical benchmarks are tested to demonstrate the optimization capability of SRBF-VSG. The comparison results indicate that SRBF-VSG generally outperforms the competitive MBDO methods in terms of global convergence, efficiency, and robustness, which illustrates the effectiveness of virtual sample generation. Finally, SRBF-VSG is applied to an airfoil aerodynamic optimization problem and a small Earth observation satellite multidisciplinary design optimization problem to demonstrate its practicality for solving real-world optimization problems.

A comparative study on surrogate models for SAEAs

Surrogate model assisted evolutionary algorithms (SAEAs) are metamodel-based strategies usually employed on the optimization of problems that demand a high computational cost to be evaluated. SAEAs employ metamodels, like Kriging and radial basis function (RBF), to speed up convergence towards good quality solutions and to reduce the number of function evaluations. However, investigations concerning the influence of metamodels in SAEAs performance have not been developed yet. In this context, this paper performs an investigative study on commonly adopted metamodels to compare the ordinary Kriging (OK), first-order universal Kriging (UK1), second-order universal Kriging (UK2), blind Kriging (BK) and RBF metamodels performance when embedded into a single-objective SAEA Framework (SAEA/F). The results obtained suggest that the OK metamodel presents a slightly better improvement than the others, although it does not present statistically significant difference in relation to UK1, UK2, and BK. The RBF showed the lowest computational cost, but the worst performance. However, this worse performance is around 2% in relation to the other metamodels. Furthermore, the results show that BK presents the highest computational cost without any significant improvement in solution quality when compared to OK, UK1, and UK2.

Derivative-based global sensitivity measure using radial basis function

Global sensitivity analysis (GSA) is always used to measure the contribution of input variables on the variation of model output. In the fields of GSA, derivative-based global sensitivity measure (DGSM) has gained much attention because of its high efficiency in the identification of insignificant input variables. The commonly used approach to estimate DGSM is the Morris method, which requires a large number of model evaluations to compute the gradient of model output. Therefore, it leads to a great challenge in the computational costs, particularly for expensive problems. To alleviate the intensive computation, an estimator based on Gaussian radial basis function (RBF) metamodel is proposed to compute DGSM. With the aid of RBF, the DGSM can be formulated in an explicit expression of one-dimensional Gaussian integrals. For the case of uniformly and normally distributed variables, the analytical expressions of the integral can be derived while for other cases with general distribution, the numerical approach with the quasi-Monte Carlo method is developed to approximate the integrals. The performance of the proposed method is validated by five numerical examples, which illustrate the proposed RBF-based estimator has the capability to yield desirable accuracy and convergence with a limited number of model evaluations.

A Novel Reliability Analysis Approach With Collaborative Active Learning Strategy-Based Augmented RBF Metamodel

Metamodels in lieu of time-demanding performance functions can accelerate the reliability analysis effectively. In this paper, we propose an efficient collaborative active learning strategy-based augmented radial basis function metamodel (CAL-ARBF), for reliability analysis with implicit and nonlinear performance functions. For generating the suitable samples, a CAL function is first designed to constrain the new samples being generated in sensitivity region, near limit state surface and keep certain distances mutually. Then by adjusting the adjustment coefficient of CAL function, the CAL-ARBF is mathematically modeled and the corresponding reliability analysis theory is developed. The effectiveness of the proposed approach is validated by four numerical samples, including global nonlinear problem, local nonlinear problem, nonlinear oscillator and truss structure. Through comparison of several state-of-the-art methods, the proposed CAL-ARBF is demonstrated to possess the computational advantages in efficiency and accuracy for reliability analysis.

A multi-criteria adaptive sequential sampling method for radial basis function

A multi-criteria adaptive sequential sampling method is proposed for radial basis function metamodel and a new global approximation method is developed in this paper. In this new sampling method, objective, curvature and distance are considered as sampling criteria. With the three criteria, it guarantees that the entire domain will be covered by samples, and more sampling points will be gathered in the peak and valley regions, which is useful for enhance accuracy and efficiency of approximation model. Intensive testing shows that the efficiency of method and accuracy of metamodel are satisfactory by this new global approximation method.

Performance assessment and optimal design of hybrid material bumper for pedestrian lower extremity protection

Lower extremity injuries have exhibited a great proportion in pedestrian-vehicle accidents. This study aims to assess the performance of an innovative aluminum-steel hybrid material bumper for pedestrian lower extremity protection. Finite element (FE) models of the hybrid bumper and its two single-material counterparts were built, validated, and integrated into an automotive front-end structure. The Transport Research Laboratory (TRL) legform model was used to obtain values of the lower extremity injury indicators including the knee-bending angle α, the knee-shearing displacement Disp, and the tibia acceleration Acc, upon impacting on the three types of bumpers. Numerical results showed that the aluminum and the hybrid bumpers result in much less pedestrian lower extremity injuries than the steel bumper due to the lower stiffness and strength of the constitutive material. Besides, the hybrid bumper is superior to its homogeneous counterparts as to the requirements for both pedestrian protection at low-speed and load transfer at high-speed crashes. Moreover, multi-objective optimization designs (MOD) of the three bumpers were performed to minimize the pedestrian's lower extremity injuries, by means of the non-dominated Sorting Genetic Algorithm (NSGA-II) and the radial basis function (RBF) meta-models. The optimal results confirmed the greatest potential of the hybrid bumper for pedestrian lower extremity protection. By virtue of the entropy weight method to determine the weighting factor of each injury indicator, Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) was used to sort the obtained Pareto designs and to select the comprehensive optimal bumpers. The TOPSIS optimum of the hybrid bumper is much closer to the ‘utopian point’, showing again its advantages over the homogeneous counterparts for pedestrian lower extremity protection in pedestrian-vehicle accidents.

Metamodel‐based robust simulation‐optimization assisted optimal design of multiloop integer and fractional‐order PID controller

AbstractClassical proportional‐integral‐derivative (PID) tuning methods such as Ziegler‐Nichols simply increase robustness against disturbances that may arise from the load. Thus, uncertainty in the load parameters has generally not been considered in previous methods, and tuning has been done only in fixed and under certain conditions. Moreover, when one or more load parameters are changed, the best previous results for the PID controller lose their validity and need to be adjusted according to the change in those load parameters. In such case, finding an optimal design for a PID controller might be too inefficient in terms of computational time and cost. This paper aims at proposing a new less time‐consuming method for tuning of a multiloop PID controller for robustness when the output of the model is varying due to the changeability of load parameters in multiple subsystems as a source of variability. Kriging and radial basis function (RBF) metamodels as two common global approximation models together with the Latin hypercube sampling (LHS) method are used to fit input‐output models with the least number of running computer experiments. Robust design terminology in the class of dual response is applied to design a multi‐input multi‐output (MIMO) mathematical programming model under disturbance factors. A MIMO numerical case in tuning a multiloop PID controller, proportional‐integral (PI) controller, and fractional‐order PID (FOPID) controller for speed control of a DC motor is provided as an example to show the flexibility and applicability of the proposed method.

Adaptive multi-fidelity sampling for CFD-based optimisation via radial basis function metamodels

The paper presents a study on four adaptive sampling methods of a multi-fidelity (MF) metamodel, based on stochastic radial basis functions (RBF), for global design optimisation based on expensive CFD computer simulations and adaptive grid refinement. The MF metamodel is built as the sum of a low-fidelity-trained metamodel and an error metamodel, based on the difference between high- and low-fidelity simulations. The MF metamodel is adaptively refined using dynamic sampling criteria, based on the prediction uncertainty in combination with the objective optimum and the computational cost of high- and low-fidelity evaluations. The adaptive sampling methods are demonstrated by four analytical benchmark and two design optimisation problems, pertaining to the resistance reduction of a NACA hydrofoil and a destroyer-type vessel. The performance of the adaptive sampling methods is assessed via objective function convergence.

Metamodel-assisted Kalai-Smorodinsky equilibria to solve a multicriteria optimization problem

The main idea of solving a multicriteria optimization problem (MOP) is to find non-dominated solutions belonging to the Pareto frontier (e.g., a compromise between the criteria). If we consider our criteria as players, we can remark that the obtained solutions are the result of a cooperation of the players to increase their profits which is the same principle of solving a bargaining problem (BP). The Kalai-Smorodinsky (KS) model suggests an attractive solution for the BP called KS equilibria that can be also a MOP solution without having to calculate the Pareto Frontier known to be computationally so expensive. In this paper, we propose an algorithm aimed to rapidly finding the KS solution. The idea is to make a coupling between the KS algorithm and a Radial Basis Function (RBF) metamodel called KS-RBF. In fact, the KS algorithm transforms the MOP into a succession of single objective problems (SOP); for our proposed algorithm, the objective function of each SOP will be replaced by an approximate one using a reliable RBF metamodel (SOP-RBF). The performance of the proposed approach is firstly validated by some well-known mathematical multicriteria problems (Tanaka, Poloni, and ConstMIN problems) by finding a KS solution belonging to the Pareto Frontier; then, we used it to solve a realistic industrial case, namely, shape optimization of the bottom of an aerosol can undergoing non-linear elasto-plastic deformation in order to simultaneously minimize the dome growth (DG) (e.g., displacement of can base) at a proof pressure and maximize the dome reversal pressure (DRP), a critical pressure at which the aerosol can’s bottom loses stability (e.g., initiates buckling). The KS solutions are compared with Pareto Frontier results previously suggested in other papers. The comparison leads us to confirm the performance of the KS-RBF coupling.

AN ENGINEERING RELIABILITY ANALYSIS METHOD BASED ON ADAPTIVE METAMODELS

In practical engineering problems, numerical analyses using the finite element (FE) method or other methods are generally required to evaluate system responses including stresses and deformations. For problems involving expensive FE analyses, it is not efficient or straightforward to directly apply conventional sampling-based or gradient-based reliability analysis approaches. To reduce computational efforts, it is useful to develop efficient and accurate metamodeling techniques to replace the original FE analyses. In this work, an adaptive metamodeling technique and a First-Order Reliability Method (FORM) were integrated. In each adaptive iteration, a compactly supported radial basis function (RBF) was adopted and a metamodel was created to explicitly express a performance function. An alternate FORM was implemented to calculate reliability index of the current iteration. Based on the design point, additional samples were generated and added to the existing sample points to re-generate the metamodel. The accuracy of the RBF metamodel could be improved in the neighborhood of the design point at each iteration. This procedure continued until the convergence of the reliability analysis results was achieved. A numerical example was studied. The proposed adaptive approach worked well and reliability analysis results were found with a reasonable number of iterations.

RBF-GA: An adaptive radial basis function metamodeling with genetic algorithm for structural reliability analysis

Owing to its time-consuming computation of complex structural responses, structural reliability analysis is still one of the challenging tasks. In this work, based on the FORM, the structural reliability analysis problem is transformed into a sequential surrogate constrained optimization problem. Subsequently, a reliability analysis method is developed that an adaptive radial basis function (RBF) metamodeling is adopted to approximate the performance function and the genetic algorithm (GA) is employed to solve the constrained optimization problem. The RBF is constructed on the initial design of experiments (DoE) generated by Latin hypercube sampling (LHS). Based on the metamodel, the GA is adopted to find the “potential” most probable point (MPP) by solving the constrained optimization problem, in which the distances between the found “potential” MPP to the existing DoE are dynamically controlled by a distance constraint. Then, the “potential” MPP is added to the DoE to refine the RBF. Finally, the Monte Carlo simulation (MCS) is employed to estimate the failure probability in terms of the metamodel. The above procedures are repeated until the failure probability converges. Five benchmark problems including small probability and high-dimensional situations are analyzed. The results illustrate the efficiency, accuracy, and robustness of the RBF-GA method.