Improving variable-fidelity modelling by exploring global design space and radial basis function networks for aerofoil design

The global variable-fidelity modelling (GVFM) method presented in this article extends the original variable-complexity modelling (VCM) algorithm that uses a low-fidelity and scaling function to approximate a high-fidelity function for efficiently solving design-optimization problems. GVFM uses the design of experiments to sample values of high- and low-fidelity functions to explore global design space and to initialize a scaling function using the radial basis function (RBF) network. This approach makes it possible to remove high-fidelity-gradient evaluation from the process, which makes GVFM more efficient than VCM for high-dimensional design problems. The proposed algorithm converges with 65% fewer high-fidelity function calls for a one-dimensional problem than VCM and approximately 80% fewer for a two-dimensional numerical problem. The GVFM method is applied for the design optimization of transonic and subsonic aerofoils. Both aerofoil design problems show design improvement with a reasonable number of high- and low-fidelity function evaluations.     

An Overview of First-Order Model Management for Engineering Optimization

First-order approximation/model management optimization (AMMO) is a rigorous methodology for solving high-fidelity optimization problems with minimal expense in high-fidelity function and derivative evaluation. AMMO is a general approach that is applicable to any derivative based optimization algorithm and any combination of high-fidelity and low-fidelity models. This paper gives an overview of the principles that underlie AMMO and puts the method in perspective with other similarly motivated methods. AMMO is first illustrated by an example of a scheme for solving bound-constrained optimization problems. The principles can be easily extrapolated to other optimization algorithms. The applicability to general models is demonstrated on two recent computational studies of aerodynamic optimization with AMMO. One study considers variable-resolution models, where the high-fidelity model is provided by solutions on a fine mesh, while the corresponding low-fidelity model is computed by solving the same differential equations on a coarser mesh. The second study uses variable-fidelity physics models, with the high-fidelity model provided by the Navier-Stokes equations and the low-fidelity model—by the Euler equations. Both studies show promising savings in terms of high-fidelity function and derivative evaluations. The overview serves to introduce the reader to the general concept of AMMO and to illustrate the basic principles with current computational results.     

Construction of nested maximin designs based on successive local enumeration and modified novel global harmony search algorithm

Engineering design often involves different types of simulation, which results in expensive computational costs. Variable fidelity approximation-based design optimization approaches can realize effective simulation and efficiency optimization of the design space using approximation models with different levels of fidelity and have been widely used in different fields. As the foundations of variable fidelity approximation models, the selection of sample points of variable-fidelity approximation, called nested designs, is essential. In this article a novel nested maximin Latin hypercube design is constructed based on successive local enumeration and a modified novel global harmony search algorithm. In the proposed nested designs, successive local enumeration is employed to select sample points for a low-fidelity model, whereas the modified novel global harmony search algorithm is employed to select sample points for a high-fidelity model. A comparative study with multiple criteria and an engineering application are employed to verify the efficiency of the proposed nested designs approach.     

A multi-objective variable-fidelity optimization method for genetic algorithms

A novel variable-fidelity optimization (VFO) scheme is presented for multi-objective genetic algorithms. The technique uses a low- and high-fidelity version of the objective function with a Kriging scaling model to interpolate between them. The Kriging model is constructed online through a fixed updating schedule. Results for three standard genetic algorithm test cases and a two-objective stiffened panel optimization problem are presented. For the stiffened panel problem, statistical analysis of four performance metrics are used to compare the Pareto fronts between the VFO method, full high-fidelity optimizer runs, and Pareto fronts developed by enumeration. The fixed updating approach is shown to reduce the number of high-fidelity calls significantly while approximating the Pareto front in an efficient manner.     

基于序贯层次Kriging模型的微型飞行器机身结构设计优化

在基于仿真模型的工程设计优化中,采用高精度、高成本的分析模型会导致计算量大,采用低精度、低成本的分析模型会导致设计优化结果的可信度低,难以满足实际工程的要求。为了有效平衡高精度与低成本之间的矛盾关系,通过建立序贯层次Kriging模型融合高/低精度数据,采用大量低成本、低精度的样本点反映高精度分析模型的变化趋势,并采用少量高成本、高精度的样本点对低精度分析模型进行校正,以实现对优化目标的高精度预测。为了避免层次Kriging模型误差对优化结果的影响,将层次Kriging模型与遗传算法相结合,根据6σ设计准则计算每一代最优解的预测区间,具有较大预测区间的当前最优解即为新的高精度样本点。同时,在优化过程中序贯更新层次Kriging模型,提高最优解附近的层次Kriging模型的预测精度,从而保证设计结果的可靠性。将所提出的方法应用于微型飞行器机身结构的设计优化中,以验证该方法的有效性和优越性。采用具有不同单元数的网格模型分别作为低精度分析模型和高精度分析模型,利用最优拉丁超立方设计分别选取60个低精度样本点和20个高精度样本点建立初始层次Kriging模型,采用本文方法求解并与直接采用高精度仿真模型求解的结果进行比较。结果表明,所提出的方法能够有效利用高/低精度样本点处的信息,建立高精度的层次Kriging模型;本文方法仅需要少量的计算成本就能求得近似最优解,有效提高了设计效率,为类似的结构设计优化问题提供了参考。     

Fast simulation-driven design of microwave structures using improved variable-fidelity optimization technique

An improved variable-fidelity optimization algorithm for the simulation-driven design of microwave structures is presented. It exploits a set of electromagnetic-based models of increasing discretization density. These models are sequentially optimized with the optimum of the ‘coarser’ model being the initial design for the ‘finer’ one. The found optimum is further refined using a response surface approximation model constructed from the coarse-discretization simulation data. In this work, the computational efficiency of the variable-fidelity algorithm is enhanced by employing a novel algorithm for optimizing the coarse-discretization models. This allows reduction of the overall design time by up to 50% compared to the previous version. The presented technique is particularly suitable for problems where simulation-driven design is the only option, for example, ultra wideband and dielectric resonator antennas. Operation of the presented approach is demonstrated using two examples of antennas and a microstrip filter. In all cases, the optimal design is obtained at a low computational cost corresponding to a few high-fidelity simulations of the structure.     

Sparse hierarchical solvers with guaranteed convergence

Solving sparse linear systems from discretized partial differential equations is challenging. Direct solvers have, in many cases, quadratic complexity (depending on geometry), while iterative solvers require problem dependent preconditioners to be robust and efficient. Approximate factorization preconditioners such as incomplete LU factorization provide cheap approximations to the system matrix. However, even a highly accurate preconditioner may have deteriorating performance when the condition number of the system matrix increases. By increasing the accuracy on low-frequency errors, we propose a novel hierarchical solver with improved robustness with respect to the condition number of the linear system. This solver retains the linear computational cost and memory footprint of the original algorithm.     

An adaptive sampling method for variable-fidelity surrogate models using improved hierarchical kriging

Variable-fidelity (VF) modelling methods have been widely used in complex engineering system design to mitigate the computational burden. Building a VF model generally includes two parts: design of experiments and metamodel construction. In this article, an adaptive sampling method based on improved hierarchical kriging (ASM-IHK) is proposed to refine the improved VF model. First, an improved hierarchical kriging model is developed as the metamodel, in which the low-fidelity model is varied through a polynomial response surface function to capture the characteristics of a high-fidelity model. Secondly, to reduce local approximation errors, an active learning strategy based on a sequential sampling method is introduced to make full use of the already required information on the current sampling points and to guide the sampling process of the high-fidelity model. Finally, two numerical examples and the modelling of the aerodynamic coefficient for an aircraft are provided to demonstrate the approximation capability of the proposed approach, as well as three other metamodelling methods and two sequential sampling methods. The results show that ASM-IHK provides a more accurate metamodel at the same simulation cost, which is very important in metamodel-based engineering design problems.     

基于广义协同高斯过程模型的结构不确定性量化解析方法

结构不确定性量化是定量参数不确定性传递到结构响应的不确定性大小。传统的蒙特卡洛法需要进行大量的数值计算,耗时较高,难以应用于大型复杂结构的不确定性量化。代理模型方法是基于少量训练样本建立的近似数学模型,可代替原始物理模型进行不确定性量化以提高计算效率。针对高精度样本计算成本高而低精度样本计算精度低的问题,该文提出了整合高、低精度训练样本的广义协同高斯过程模型。基于该模型框架推导了结构响应均值和方差的解析表达式,实现了结构不确定性的量化解析。采用三个空间结构算例来验证结构不确定性量化解析方法的准确性,并与传统的蒙特卡洛法、协同高斯过程模型和高斯过程模型的计算结果对比,结果表明所提方法在计算精度和效率方面均具有优势。     

An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization

Response surface methods based on kriging and radial basis function (RBF) interpolation have been successfully applied to solve expensive, i.e. computationally costly, global black-box nonconvex optimization problems. In this paper we describe extensions of these methods to handle linear, nonlinear, and integer constraints. In particular, algorithms for standard RBF and the new adaptive RBF (ARBF) are described. Note, however, while the objective function may be expensive, we assume that any nonlinear constraints are either inexpensive or are incorporated into the objective function via penalty terms. Test results are presented on standard test problems, both nonconvex problems with linear and nonlinear constraints, and mixed-integer nonlinear problems (MINLP). Solvers in the TOMLAB Optimization Environment () have been compared, specifically the three deterministic derivative-free solvers rbfSolve, ARBFMIP and EGO with three derivative-based mixed-integer nonlinear solvers, OQNLP, MINLPBB and MISQP, as well as the GENO solver implementing a stochastic genetic algorithm. Results show that the deterministic derivative-free methods compare well with the derivative-based ones, but the stochastic genetic algorithm solver is several orders of magnitude too slow for practical use. When the objective function for the test problems is costly to evaluate, the performance of the ARBF algorithm proves to be superior.     

A new bi-fidelity model reduction method for Bayesian inverse problems

This work presents a new bi-fidelity model reduction approach to the inverse problem under the framework of Bayesian inference. A low-rank approximation is introduced to the solution of the corresponding forward problem and admits a variable-separation form in terms of stochastic basis functions and physical basis functions. The calculation of stochastic basis functions is computationally predominant for the low-rank expression. To significantly improve the efficiency of constructing the low-rank approximation, we propose a bi-fidelity model reduction based on a novel variable-separation method, where a low-fidelity model is used to compute the stochastic basis functions and a high-fidelity model is used to compute the physical basis functions. The low-fidelity model has lower accuracy but efficient to evaluate compared with the high-fidelity model; it accelerates the derivative of recursive formulation for the stochastic basis functions. The high-fidelity model is computed in parallel for a few samples scattered in the stochastic space when we construct the high-fidelity physical basis functions. The required number of forward model simulations in constructing the basis functions is very limited. The bi-fidelity model can be constructed efficiently while retaining good accuracy simultaneously. In the proposed approach, both the stochastic basis functions and physical basis functions are calculated using the model information. This implies that a few basis functions may accurately represent the model solution in high-dimensional stochastic spaces. The bi-fidelity model reduction is applied to Bayesian inverse problems to accelerate posterior exploration. A few numerical examples in time-fractional derivative diffusion models are carried out to identify the smooth field and channel-structured field in porous media in the framework of Bayesian inverse problems.     

Flower Pollination Heuristics for Nonlinear Active Noise Control Systems

Abstract In this paper, a novel design of the flower pollination algorithm is presented for model identification problems in nonlinear active noise control systems. The recently introduced flower pollination based heuristics is implemented to minimize the mean squared error based merit/cost function representing the scenarios of active noise control system with linear/nonlinear and primary/secondary paths based on the sinusoidal signal, random and complex random signals as noise interferences. The flower pollination heuristics based active noise controllers are formulated through exploitation of nonlinear filtering with Volterra series. The comparative study on statistical observations in terms of accuracy, convergence and complexity measures demonstrates that the proposed meta-heuristic of flower pollination algorithm is reliable, accurate, stable as well as robust for active noise control system. The accuracy of the proposed nature inspired computing of flower pollination is in good agreement with the state of the art counterpart solvers based on variants of genetic algorithms, particle swarm optimization, backtracking search optimization algorithm, fireworks optimization algorithm along with their memetic combination with local search methodologies. Moreover, the central tendency and variation based statistical indices further validate the consistency and reliability of the proposed scheme mimic the mathematical model for the process of flower pollination systems.     

On the effect of numerical noise in approximate optimization of forming processes using numerical simulations

The coupling of Finite Element (FE) simulations with approximate optimization techniques is becoming increasingly popular in forming industry. By doing so, it is implicitly assumed that the optimization objective and possible constraints are smooth functions of the design variables and, in case of robust optimization, design and noise variables. However, non-linear FE simulations are known to introduce numerical noise caused by the discrete nature of the simulation algorithms, e.g. errors caused by re-meshing, time-step adjustments or contact algorithms. The subsequent usage of metamodels based on such noisy data reduces the prediction quality of the optimization routine and is known to even magnify the numerical errors. This work provides an approach to handle noisy numerical data in approximate optimization of forming processes, covering several fundamental research questions in dealing with numerical noise. First, the deteriorating effect of numerical noise on the prediction quality of several well-known metamodeling techniques is demonstrated using an analytical test function. Next, numerical noise is quantified and its effect is minimized by the application of local approximation and regularization techniques. A general approximate optimization strategy is subsequently presented and coupling with a sequential update algorithm is proposed. The strategy is demonstrated by the sequential deterministic and robust optimization of 2 industrial metal forming processes i.e. a V-bending application and a cup-stretching application. Although numerical noise is often neglected in practice, both applications in this work show that the general awareness of its presence is highly important to increase the overall accuracy of optimization results.     

An adaptive metamodel-based global optimization algorithm for black-box type problems

In this article, an adaptive metamodel-based global optimization (AMGO) algorithm is presented to solve unconstrained black-box problems. In the AMGO algorithm, a type of hybrid model composed of kriging and augmented radial basis function (RBF) is used as the surrogate model. The weight factors of hybrid model are adaptively selected in the optimization process. To balance the local and global search, a sub-optimization problem is constructed during each iteration to determine the new iterative points. As numerical experiments, six standard two-dimensional test functions are selected to show the distributions of iterative points. The AMGO algorithm is also tested on seven well-known benchmark optimization problems and contrasted with three representative metamodel-based optimization methods: efficient global optimization (EGO), GutmannRBF and hybrid and adaptive metamodel (HAM). The test results demonstrate the efficiency and robustness of the proposed method. The AMGO algorithm is finally applied to the structural design of the import and export chamber of a cycloid gear pump, achieving satisfactory results.     

Structural damage detection using improved particle swarm optimization

An approach based on an improved particle swarm optimization (PSO) algorithm is proposed for structural damage detection in this study. A disturbance is introduced in the evolution process to avoid the occurrence of premature. The present algorithm focuses on the mutation of global or individual best known positions to guide the swarm to escape from the local minimum. The feasibility and robustness of the modified PSO are verified by three different structures, including a beam, a truss and a plate. The results show that the method is efficient and effective for structural damage identification when measurement noise is considered.     

改进的 QPSO 算法在自融资投资组合中的应用

针对量子粒子群优化 (Quantum Particle Swarm Optimization, QPSO) 算法的缺陷,提出了一种基于 L$\acute{\rm e}$vy 飞行策略和混合概率分布的改进量子粒子群优化 (Hybrid Quantum Particle Swarm Optimization, HQPSO) 算法。在算法的设计中,借助 L$\acute{\rm e}$vy 飞行策略对粒子位置的迭代公式进行更新,用于改善算法的局部收敛精度,增强其全局探索能力。另外,考虑到迭代后期的早熟问题,在势阱模型中引入了指数分布和正态分布相结合的混合概率分布,帮助算法及时逃离局部最优。基于 16 个基准函数的测试结果表明,HQPSO 算法在收敛精度和鲁棒性上比其他几种算法表现更好。最后,将改进的 QPSO 算法应用到自融资投资组合模型的求解中,其数值结果与差分进化、粒子群优化算法和量子粒子群优化算法相比,HQPSO 算法展现出更好的可比性和优越性。     

An efficient algorithm for modelling progressive damage accumulation in disordered materials

This paper presents an efficient algorithm for the simulation of progressive fracture in disordered quasi‐brittle materials using discrete lattice networks. The main computational bottleneck involved in modelling the fracture simulations using large discrete lattice networks stems from the fact that a new large set of linear equations needs to be solved every time a lattice bond is broken. Using the present algorithm, the computational complexity of solving the new set of linear equations after breaking a bond reduces to a simple triangular solves (forward elimination and backward substitution) using the already Cholesky factored matrix. This algorithm using the direct sparse solver is faster than the Fourier accelerated iterative solvers such as the preconditioned conjugate gradient (PCG) solvers, and eliminates the critical slowing down associated with the iterative solvers that is especially severe close to the percolation critical points. Numerical results using random resistor networks for modelling the fracture and damage evolution in disordered materials substantiate the efficiency of the present algorithm. In particular, the proposed algorithm is especially advantageous for fracture simulations wherein ensemble averaging of numerical results is necessary to obtain a realistic lattice system response. Copyright © 2005 John Wiley & Sons, Ltd.     

A SBS-BD based solver for domain decomposition in BE methods

In boundary element methods (BEM), subregioning may be needed either to model complex solids (with cracks, stiffeners, layers, inclusions, etc.) or simply to decompose a problem by computational reasons (e.g. for parallelization). Since the development of the first BEM codes, many attempts have been made to efficiently devise generic boundary-element subregioning techniques. Crucial points are how to profit from the sparsity of the global matrix, and how to deal with traction discontinuities. In this work, the most fundamental steps for efficiently devising reliable and efficient subregioning algorithms are discussed. The subregion-by-subregion (SBS) algorithm and the preconditioning of the embedded Krylov solver are addressed. Besides the BiCG solver, the BiCGSTAB(l) is newly incorporated into the BE-SBS code. The 3D microstructural analysis of carbon-nanotube-reinforced composites (CNT composites) is considered to verify the performance of the algorithm. Numerical results showing the efficiency of the preconditioned solvers studied are presented.