Response Surface Approximation Models Research Articles

Multi-objective design optimization of antenna structures using sequential domain patching with automated patch size determination

ABSTRACTIn this article, a simple yet efficient and reliable technique for fully automated multi-objective design optimization of antenna structures using sequential domain patching (SDP) is discussed. The optimization procedure according to SDP is a two-step process: (i) obtaining the initial set of Pareto-optimal designs representing the best possible trade-offs between considered conflicting objectives, and (ii) Pareto set refinement for yielding the optimal designs at the high-fidelity electromagnetic (EM) simulation model level. For the sake of computational efficiency, the first step is realized at the level of a low-fidelity (coarse-discretization) EM model by sequential construction and relocation of small design space segments (patches) in order to create a path connecting the extreme Pareto front designs obtained beforehand. The second stage involves response correction techniques and local response surface approximation models constructed by reusing EM simulation data acquired in the first step. A major contribution of this work is an automated procedure for determining the patch dimensions. It allows for appropriate selection of the number of patches for each geometry variable so as to ensure reliability of the optimization process while maintaining its low cost. The importance of this procedure is demonstrated by comparing it with uniform patch dimensions.

Optimized Reduction of Unsteady Radial Forces in a Singlechannel Pump for Wastewater Treatment

A single-channel pump for wastewater treatment was optimized to reduce unsteady radial force sources caused by impeller-volute interactions. The steady and unsteady Reynolds- averaged Navier-Stokes equations using the shear-stress transport turbulence model were discretized by finite volume approximations and solved on tetrahedral grids to analyze the flow in the single-channel pump. The sweep area of radial force during one revolution and the distance of the sweep-area center of mass from the origin were selected as the objective functions; the two design variables were related to the internal flow cross-sectional area of the volute. These objective functions were integrated into one objective function by applying the weighting factor for optimization. Latin hypercube sampling was employed to generate twelve design points within the design space. A response-surface approximation model was constructed as a surrogate model for the objectives, based on the objective function values at the generated design points. The optimized results showed considerable reduction in the unsteady radial force sources in the optimum design, relative to those of the reference design.

Three-objective optimization of a centrifugal pump with double volute to minimize radial thrust at off-design conditions

Optimization of a centrifugal pump with double volute was performed using surrogate modeling and a multi-objective genetic algorithm to minimize the radial thrust at off-design conditions with minimal loss in hydraulic efficiency. Both steady and unsteady numerical analyses were conducted using three-dimensional Reynolds-averaged Navier–Stokes equations with a shear stress transport turbulent model. The three objective functions are hydraulic efficiency at the design flow rate, and radial thrust coefficient at 70% and 120% of the design flow rate. In addition, three geometric parameters related to the surface curve and the locations of both ends of the rib structure were selected as the design variables for optimization. The Latin hypercube sampling method was used to choose the design points in the design space. The three objective functions were approximated using response surface approximation models. Pareto-optimal solutions representing the trade-off between the objective functions were obtained using a genetic algorithm. Representative optimal designs show that the optimized double volutes significantly reduce both the radial force and the amplitude of the radial force fluctuation with small loss in hydraulic efficiency.

Response features and circuit decomposition for accelerated EM‐driven design of compact impedance matching transformers

ABSTRACTIn this letter, a procedure for low‐cost simulation‐driven design optimization of compact impedance matching transformers is presented. The proposed methodology exploits a bottom‐up design with the design requirements for the entire transformer circuit being translated into specifications for its building blocks, here, compact microstrip resonant cells. These elementary cells are optimized using response features which permits considerable numerical savings compared to conventional approach. Subsequently, the entire circuit is fine‐tuned using local response surface approximation models and response correction methods. A three‐section 50‐to‐100 Ω transformer is designed for demonstration purposes. Its optimized design is obtained at the computational cost corresponding to only 31 evaluations of the high‐fidelity electromagnetic transformer model, despite large number of geometry parameters to be adjusted (here, 15). © 2016 Wiley Periodicals, Inc. Microwave Opt Technol Lett 58:2130–2133, 2016

Rapid Multiobjective Antenna Design Using Point-By-Point Pareto Set Identification and Local Surrogate Models

Antenna design is inherently a multicriterial problem. Determination of the best possible tradeoffs between conflicting objectives (a so-called Pareto front), such as reflection response, gain, and antenna size, is indispensable from the designer’s point of view, yet challenging when high-fidelity electromagnetic (EM) simulations are utilized for performance evaluation. Here, a novel and computationally efficient methodology for multiobjective optimization of antenna structures is presented. In our approach, the tradeoff designs are obtained by moving along the Pareto front and identifying the subsequent Pareto-optimal solutions using surrogate-based optimization techniques. Computational efficiency of the process is achieved by employing coarse-discretization EM simulations and local response surface approximation (RSA) models. The proposed approach is demonstrated using a compact ultrawideband (UWB) monopole antenna with a representation of the Pareto front obtained at the cost corresponding to just a few dozen of evaluations of the high-fidelity EM antenna model. Experimental validation is also provided.

Efficient multi-objective optimization of a boot-shaped rib in a cooling channel

Multi-objective optimization of a boot-shaped rib in a cooling channel was conducted to assess the trade-off between two conflicting objectives, heat transfer performance and pressure drop. Numerical analysis using three-dimensional Reynolds-averaged Navier–Stokes equations, surrogate modelling for function approximation, and a multi-objective genetic algorithm were used for the optimization. Four geometric variables (the ratios of the tip width to rib width, rib width to rib height, rib height to channel height, and tip height to rib height) were chosen as the design variables. To characterize the heat transfer performance and pressure drop, the area-averaged Nusselt number at the rib-roughened surface and the friction factor were selected as the objective functions. For modelling turbulence, a low-Re k–ω turbulence model was used. The Latin hypercube sampling method was used to select the design points in the design space. Two different cases were considered. In the first case, the objective functions were formulated using response surface approximation models while the second case involved kriging models. The Pareto-optimal fronts obtained using the multi-objective genetic algorithm for the two cases were compared using convergence and spread performance criterion. Based on the performance analysis results, a new approach to obtain the best Pareto-optimal front is presented. Further, the flow dynamics and heat transfer characteristics of some representative Pareto-optimal solutions of the best Pareto-optimal front were used to analyze the variations of the objective functions with the design variables.

Fast multi‐objective surrogate‐assisted design of multi‐parameter antenna structures through rotational design space reduction

A technique for fast multi-objective optimisation of antennas is introduced. The core of the proposed methodology is a reliable initial estimation of the design space subset that contains a set of Pareto optimal solutions, i.e. those representing the best possible trade-offs between the conflicting objectives (such as the antenna size and its electrical performance parameters). A fast response surface approximation (RSA) surrogate is subsequently constructed in a reduced search space using sampled coarse-discretisation electromagnetic (EM) simulation data. Owing to the authors’ reduction approach, the surrogate model construction is computationally feasible even when the number of antenna parameters is high. The RSA model is optimised using a multi-objective evolutionary algorithm to yield an initial approximation of the Pareto set. The latter is further refined (to obtain its representation at the high-fidelity EM antenna model level). The approach is illustrated using two design cases. A comparison with previously published methods, as well as experimental validation, is also provided.

Expedited constrained multi-objective aerodynamic shape optimization by means of physics-based surrogates

In the paper, computationally efficient constrained multi-objective design optimization of transonic airfoil profiles is considered. Our methodology focuses on fixed-lift design aimed at finding the best possible trade-offs between the two objectives: minimization of the drag coefficient and maximization of the pitching moment. The algorithm presented here exploits the surrogate-based optimization principle, variable-fidelity computational fluid dynamics (CFD) models, as well as auxiliary data-driven surrogates (here, using Kriging). In order to permit computationally feasible construction of the Kriging models, initial design space reduction is also utilized. The design process has three major stages: (i) identification of the extreme points of the Pareto front through single-objective optimization (one objective at a time), (ii) construction of the Kriging model and initial Pareto front generation using multi-objective evolutionary algorithm (MOEA), and (iii) Pareto front refinement using response correction techniques and local response surface approximation (RSA) models. For the sake of computational efficiency, stages (i) and (ii) are realized at the level of coarse-discretization CFD model. The RSA models are also utilized to predict the angle of attack necessary to achieve the target lift coefficient, which considerably reduces the CFD simulation effort involved in the design process. Two design case studies are considered involving B-spline-parameterized airfoil shapes with 8 and 12 design variables. The 10-element Pareto front representations are obtained at the cost corresponding to just over two hundred of high-fidelity CFD model evaluations. This cost is not only considerably lower (up to two orders of magnitude) than the cost of direct high-fidelity model optimization using metaheuristics but, more importantly, renders multi-objective optimization of aerodynamic components computationally tractable even at the level of accurate CFD models.

Strategies for computationally feasible multi-objective simulation-driven design of compact RF/microwave components

Purpose – Strategies for accelerated multi-objective optimization of compact microwave and RF structures are investigated, including the possibility of exploiting surrogate modeling techniques for electromagnetic (EM)-driven design speedup for such components. The paper aims to discuss these issues. Design/methodology/approach – Two algorithmic frameworks are described that are based on fast response surface approximation models, structure decomposition, and Pareto front refinement. Numerical case studies are provided demonstrating feasibility of solving real-world problems involving multi-objective optimization of miniaturized microwave passives and expensive EM-simulation models of such structures. Findings – It is possible, through appropriate combination of the surrogate modeling techniques and response correction methods, to identify the set of alternative designs representing the best possible trade-offs between conflicting design objectives in a realistic time frame corresponding to a few dozen of high-fidelity EM simulations of the respective structures. Research limitations/implications – The present study sets a direction for further studied on expedited optimization of computationally expensive simulation models for miniaturized microwave components. Originality/value – The proposed algorithmic framework proved useful for fast design of microwave structures, which is extremely challenging when using conventional methods. To the authors’ knowledge, this is one of the first attempts to surrogate-assisted multi-objective optimization of compact components at the EM-simulation level.

Scalability of surrogate-assisted multi-objective optimization of antenna structures exploiting variable-fidelity electromagnetic simulation models

Multi-objective optimization of antenna structures is a challenging task owing to the high computational cost of evaluating the design objectives as well as the large number of adjustable parameters. Design speed-up can be achieved by means of surrogate-based optimization techniques. In particular, a combination of variable-fidelity electromagnetic (EM) simulations, design space reduction techniques, response surface approximation models and design refinement methods permits identification of the Pareto-optimal set of designs within a reasonable timeframe. Here, a study concerning the scalability of surrogate-assisted multi-objective antenna design is carried out based on a set of benchmark problems, with the dimensionality of the design space ranging from six to 24 and a CPU cost of the EM antenna model from 10 to 20 min per simulation. Numerical results indicate that the computational overhead of the design process increases more or less quadratically with the number of adjustable geometric parameters of the antenna structure at hand, which is a promising result from the point of view of handling even more complex problems.

Rapid design and size reduction of microwave couplers using variable-fidelity EM-driven optimization

In this article, fast electromagnetic (EM) simulation-driven design optimization of compact microwave couplers is addressed. The main focus is on explicit reduction of the circuit footprint. Our methodology relies on the penalty function approach, which allows us to minimize the circuit area while ensuring equal power split between the output ports and providing a sufficient bandwidth with respect to the return loss and isolation around the operating frequency. Computational efficiency of the design process is achieved by exploiting variable-fidelity EM simulations, local response surface approximation models, as well as suitable response correction techniques for design tuning. The technique described in this work is demonstrated using two examples of compact rat-race couplers. The size-reduction-oriented designs are compared with performance-oriented ones to illustrate available design trade-offs. Final design solutions of the former case illustrate ∼92% of miniaturization for both coupler examples (with corresponding fractional bandwidths of 16%). Alternative design solutions pertaining to the latter case show a lesser size reduction (∼90% for both examples), but present a much wider bandwidths (∼25% for both couplers). The overall computational cost of the design procedure corresponds to about 20 and 10 high-fidelity coupler simulations for the first and second design example, respectively. Numerical results are also validated experimentally. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:27–35, 2016.

Multi-objective optimization of solar air heater with obstacles on absorber plate

A multi-objective optimization of a solar air heater with obstacles on an absorber plate is performed for maximum heat transfer and minimum pressure loss. In this work, shape optimization is carried out in conjunction with three-dimensional Reynolds-averaged Navier–Stokes analysis and two basic surrogate models: the response surface approximation and the Kriging models. Three geometric variables (the ratio of the obstacle height to the height of the duct; the ratio of the transverse pitch to the base length of the obstacle; and the angle of attack) were used as design variables for the optimization. The average Nusselt number and friction factor were used to define the two objective functions. The Latin hypercube sampling method was used to select the design points in the design space. A hybrid multi-objective genetic algorithm coupled with the surrogate model was used to find the Pareto-optimal solutions. The representative Pareto-optimal solutions were selected to study the trade-off between the two objectives. The response surface approximation model leads to a better set of non-dominated solutions over a wide range of functional space than the Kriging model. The optimization results show that the objective functions are significantly affected by the design variables, and the constructed surrogate models show good prediction accuracies for the objective functions. A performance factor was used to study the thermal–hydraulic performance of the Pareto-optimal solutions.

Optimal shape design of multi-element trawl-doors using local surrogate models

Trawl-doors have a large influence on the fuel consumption of fishing vessels. Design and optimization of trawl-doors using computational models are a key factor in minimizing the fuel consumption. This paper presents an optimization algorithm for the shape design of trawl-door shapes using computational fluid dynamic (CFD) models. Accurate CFD models are computationally expensive. Therefore, the direct use of traditional optimization algorithms, which often require a large number of evaluations, may be prohibitive. The proposed approach is iterative and uses low-order local response surface approximation models of the expensive CFD model, constructed in each iteration, to reduce the number of evaluations. The algorithm is applied to the design of a multi-element trawl-doors, involving up to four design variables controlling the angle of attack and the slat and flap position and orientation. The results show that a satisfactory design can be obtained at the cost of a few iterations of the algorithm.

Fast Multiobjective Optimization of Narrowband Antennas Using RSA Models and Design Space Reduction

A computationally efficient technique for multiobjective design optimization of narrowband antennas is presented. In our approach, the corrected low-fidelity antenna model [obtained through coarse-discretization electromagnetic (EM) simulations] is enhanced using frequency scaling and response correction, sampled, and utilized to obtain a fast response surface approximation (RSA) antenna surrogate. The RSA model is constructed in the reduced design space. The initial set of Pareto-optimal designs is then utilized to confine the design space further by identifying designs that satisfy minimum requirements with respect to antenna reflection response. The updated RSA model set up in the confined space is subsequently optimized to yield the final Pareto set at the low computational cost.

Design Optimization of a Centrifugal Fan with Splitter Blades

AbstractMulti-objective optimization of a centrifugal fan with additionally installed splitter blades was performed to simultaneously maximize the efficiency and pressure rise using three-dimensional Reynolds-averaged Navier-Stokes equations and hybrid multi-objective evolutionary algorithm. Two design variables defining the location of splitter, and the height ratio between inlet and outlet of impeller were selected for the optimization. In addition, the aerodynamic characteristics of the centrifugal fan were investigated with the variation of design variables in the design space. Latin hypercube sampling was used to select the training points, and response surface approximation models were constructed as surrogate models of the objective functions. With the optimization, both the efficiency and pressure rise of the centrifugal fan with splitter blades were improved considerably compared to the reference model.

Efficient Multi-Objective Simulation-Driven Antenna Design Using Co-Kriging

A methodology for fast multi-objective antenna optimization is presented. Our approach is based on response surface approximation (RSA) modeling and variable-fidelity electromagnetic (EM) simulations. In the design process, a computationally cheap RSA surrogate model constructed from sampled coarse-discretization EM antenna simulations is optimized using a multi-objective evolutionary algorithm. The initially determined Pareto optimal set representing the best possible trade-offs between conflicting design objectives is then iteratively refined. In each iteration, a limited number of high-fidelity EM model responses are incorporated into the RSA model using co-kriging. The enhanced RSA model is subsequently re-optimized to yield the refined Pareto set. Combination of low- and high-fidelity simulations as well as co-kriging results in the low overall optimization cost. The proposed approach is validated using two UWB antenna examples.

Inverse airfoil design using variable-resolution models and shape-preserving response prediction

The paper presents a computationally efficient surrogate-based optimization algorithm for the inverse design of transonic airfoils. Our approach replaces the direct optimization of an accurate, but computationally expensive, high-fidelity airfoil model by an iterative re-optimization of two different surrogate models. Initially, for a few design iterations, a corrected physics-based low-fidelity model is employed, which is subsequently replaced by a response surface approximation model. The low-fidelity model is based on the same governing fluid flow equations as the high-fidelity one, but uses coarser mesh resolution and relaxed convergence criteria. The shape-preserving response prediction (SPRP) technique is utilized to predict the high-fidelity model response, here, the airfoil pressure distribution. In this prediction process, SPRP employs the actual changes of the low-fidelity model response (or the surface approximation model) due to the design variable adjustments. The SPRP algorithm is embedded into the trust region framework to ensure good convergence properties of the optimization procedure. Our algorithm is applied to constrained inverse airfoil design in inviscid transonic flow. A comparison with the basic version of the optimization algorithm, exploiting only a physics-based low-fidelity model, is carried out. While the performance of both versions is similar with respect to their ability to match the target pressure distribution, the improved algorithm offers substantial design cost savings, from 25 to 72 percent, depending on the test case.

Low-cost EM-simulation-driven Multi-objective Optimization of Antennas

A surrogate-based method for efficient multi-objective antenna optimization is presented. Our technique exploits response surface approximation (RSA) model constructed from sampled low-fidelity antenna model (here, obtained through coarse-discretization EM simulation). The RSA model enables fast determination of the best available trade-offs between conflicting design goals. A low-cost RSA model construction is possible through initial reduction of the design space. Optimization of the RSA model has been carried out using a multi-objective evolutionary algorithm (MOEA). Additional response correction techniques have been subsequently applied to improve selected designs at the high-fidelity EM antenna model level. The refined designs constitute the final Pareto set representation. The proposed approach has been validated using an ultra-wideband (UWB) monocone and a planar Yagi-Uda antenna.

Multiobjective Optimization of a Counterrotating Type Pump-Turbine Unit Operated at Turbine Mode

A multiobjective optimization for improving the turbine output and efficiency of a counterrotating type pump-turbine unit operated at turbine mode was carried out in this work. The blade geometry of both the runners was optimized using a hybrid multiobjective evolutionary algorithm coupled with a surrogate model. Three-dimensional Reynolds-averaged Navier-Stokes equations with the shear stress transport turbulence model were discretized by finite volume approximations and solved on hexahedral grids to analyze the flow in the pump-turbine unit. As major hydrodynamic performance parameters, the turbine output and efficiency were selected as objective functions with two design variables related to the hub profiles of both the runner blades. These objectives were numerically assessed at twelve design points selected by Latin hypercube sampling in the design space. Response surface approximation models for the objectives were constructed based on the objective function values at the design points. A fast nondominated sorting genetic algorithm for the local search coupled with the response surface approximation models was applied to determine the global Pareto-optimal solutions. The trade-off between the two objectives was determined and described with respect to the Pareto-optimal solutions. The results of this work showed that the turbine outputs and efficiencies of optimized pump-turbine units were simultaneously improved in comparison to the reference unit.

Surrogate Assisted Design Optimization of an Air Turbine

Surrogates are cheaper to evaluate and assist in designing systems with lesser time. On the other hand, the surrogates are problem dependent and they need evaluation for each problem to find a suitable surrogate. The Kriging variants such as ordinary, universal, and blind along with commonly used response surface approximation (RSA) model were used in the present problem, to optimize the performance of an air impulse turbine used for ocean wave energy harvesting by CFD analysis. A three-level full factorial design was employed to find sample points in the design space for two design variables. A Reynolds-averaged Navier Stokes solver was used to evaluate the objective function responses, and these responses along with the design variables were used to construct the Kriging variants and RSA functions. A hybrid genetic algorithm was used to find the optimal point in the design space. It was found that the best optimal design was produced by the universal Kriging while the blind Kriging produced the worst. The present approach is suggested for renewable energy application.