Polynomial genetic programming for response surface modeling Part 2: adaptive approximate models with probabilistic optimization problems

This is the second in a series of papers. The first deals with polynomial genetic programming (PGP) adopting the directional derivative-based smoothing (DDBS) method, while in this paper, an adaptive approximate model (AAM) based on PGP is presented with the partial interpolation strategy (PIS). The AAM is sequentially modified in such a way that the quality of fitting in the region of interest where an optimum point may exist can be gradually enhanced, and accordingly the size of the learning set is gradually enlarged. If the AAM uses a smooth high-order polynomial with an interpolative capability, it becomes more and more difficult for PGP to obtain smooth polynomials, whose size should be larger than or equal to the number of the samples, because the order of the polynomial becomes unnecessarily high according to the increase in its size. The PIS can avoid this problem by selecting samples belonging to the region of interest and interpolating only those samples. Other samples are treated as elements of the extended data set (EDS). Also, the PGP system adopts a multiple-population approach in order to simultaneously handle several constraints. The PGP system with the variable-fidelity response surface method is applied to reliability-based optimization (RBO) problems in order to significantly cut the high computational cost of RBO. The AAMs based on PGP are responsible for fitting probabilistic constraints and the cost function while the variable-fidelity response surface method is responsible for fitting limit state equations. Three numerical examples are presented to show the performance of the AAM based on PGP.     

Reliability-based design optimization of aeroelastic structures

Aeroelastic phenomena are most often either ignored or roughly approximated when uncertainties are considered in the design optimization process of structures subject to aerodynamic loading, affecting the quality of the optimization results. Therefore, a design methodology is proposed that combines reliability-based design optimization and high-fidelity aeroelastic simulations for the analysis and design of aeroelastic structures. To account for uncertainties in design and operating conditions, a first-order reliability method (FORM) is employed to approximate the system reliability. To limit model uncertainties while accounting for the effects of given uncertainties, a high-fidelity nonlinear aeroelastic simulation method is used. The structure is modelled by a finite element method, and the aerodynamic loads are predicted by a finite volume discretization of a nonlinear Euler flow. The usefulness of the employed reliability analysis in both describing the effects of uncertainties on a particular design and as a design tool in the optimization process is illustrated. Though computationally more expensive than a deterministic optimum, due to the necessity of solving additional optimization problems for reliability analysis within each step of the broader design optimization procedure, a reliability-based optimum is shown to be an improved design. Conventional deterministic aeroelastic tailoring, which exploits the aeroelastic nature of the structure to enhance performance, is shown to often produce designs that are sensitive to variations in system or operational parameters.     

Extensions of design potential concept for reliability-based design optimization to nonsmooth and extreme cases

The reliability-based design optimization (RBDO) can be described by the design potential concept in a unified system space, where the probabilistic constraint is identified by the design potential surface of the reliability target that is obtained analytically from the first-order reliability method (FORM). This paper extends the design potential concept to treat nonsmooth probabilistic constraints and extreme case design in RBDO. In addition, refinement of the design potential surface, which yields better optimum design, can be obtained using more accurate second-order reliability method (SORM). By integrating performance probability analysis into the iterative design optimization process, the design potential concept leads to a very effective design potential method (DPM) for robust system parameter design. It can also be applied effectively to extreme case design (ECD) by directly representing a probabilistic constraint in terms of the system performance function. Received July 25, 2000     

Parameter optimization in multiquadric response surface approximations

Multiquadric (MQ) response surface approximation uses the function C_i|X-X_i|²+h^1/2 to interpolate a given set of data. The performance of MQ approximation depends on the shift parameter h. Efficient methods of computing the optimal shift parameter based on the leave-one-out cross validation technique are presented in this paper. We also proved that the condition number of the MQ coefficient matrix is an increasing function of the shift parameter h. Two numerical examples are included to illustrate the proposed formulation.     

Two-level optimization of airframe structures using response surface approximation

In this paper, a two-level optimization approach is developed for the preliminary and conceptual design of airframe structures. The preliminary design, involving a single objective multidisciplinary optimization, constitutes the lower level where ASTROS (Automated STRuctural Optimization System) is employed for multidisciplinary optimization. The conceptual design, which is carried out at the upper level, aims mainly at configuration design. The multiple objectives are incorporated as a single objective function by using the K-S function formulation. The objective function and constraints at the upper level are modelled through response surface approximation. During the upper level optimization process, the branch and bound method is applied for solving the problem with discrete design variables. The proposed strategy is demonstrated by the optimization of an Intermediate Complexity Wing (ICW) model. Received June 23, 1999     

Reliability-based structural optimization of frame structures for multiple failure criteria using topology optimization techniques

Topology optimization methods using discrete elements such as frame elements can provide useful insights into the underlying mechanics principles of products; however, the majority of such optimizations are performed under deterministic conditions. To avoid performance reductions due to later-stage environmental changes, variations of several design parameters are considered during the topology optimization. This paper concerns a reliability-based topology optimization method for frame structures that considers uncertainties in applied loads and nonstructural mass at the early conceptual design stage. The effects that multiple criteria, namely, stiffness and eigenfrequency, have upon system reliability are evaluated by regarding them as a series system, where mode reliabilities can be evaluated using first-order reliability methods. Through numerical calculations, reliability-based topology designs of typical two- or three-dimensional frames are obtained. The importance of considering uncertainties is then demonstrated by comparing the results obtained by the proposed method with deterministic optimal designs.     

基于响应面法的三维炮尾结构设计优化

为满足某炮尾结构布置的特殊需要,在输弹槽贯穿托弹板的开放式炮尾结构基础上,重新设计传力结构;采用基于响应面法（Response Surface Method,RSM）的多目标遗传算法NSGA-II寻找齿形传力结构的最优参数,通过编写Python脚本控制Abaqus内核实现自动前处理,对其进行有限元分析,并基于iSight实现多目标三维模型设计优化.该方法摒弃传统的二维优化三维验证的理念,将多目标遗传算法与RSM结合起来,在iSight中直接进行三维模型设计优化,可节省计算时间、提高计算效率、改善设计水平.     

Multiobjective optimization of complex antenna structures using response surface models

A very efficient multiobjective (MO) design technique for complex antenna structures involving a large number of design parameters is presented. This design technique, multiobjective‐fractional factorial design (MO‐FFD), is very different from conventional Pareto‐based MO algorithms, which take a great deal of effort to balance the trade‐off between all the design specifications. By performing one single combination of simulations, all the response surface models of design goals are simultaneously built, and Derringer's desirability functions are readily applied to these models so that the optimum structure is obtained. Compared to classical MO algorithms such as Strength Pareto Evolutionary Algorithm 2, nondominated sorting particle swarm optimizer, and cultural MO particle swarm optimization, MO‐FFD yields more desirable performances yet the required number of simulations is reduced by 97%. This article thoroughly illustrates the mathematical development of MO‐FFD, deriving a novel application of ultrawideband (UWB) antennas because of its MO optimization capability. More explicitly, MO‐FFD overcomes all the design challenges of dual band‐notched UWB antennas including desired impedance characteristics, enhanced fidelity factors, and uniform peak gains over the passband, which are what conventional Pareto‐based algorithms cannot attain. The measured results show that all the performance criteria are met; especially, the time‐domain signal distortion is minimized. © 2015 Wiley Periodicals, Inc. Int J RF and Microwave CAE 26:62–71, 2016.     

Buckling design optimization of complex built-up structures with shape and size variables

The design optimization of buckling behaviour is studied for complex built-up structures composed of various kinds of elements and implemented within JIFEX95, a general-purpose software for finite element analysis and design optimization. The direct and adjoint methods of sensitivity analysis for critical buckling loads are presented with detailed computational procedures. Particularly, the variations of prebuckling stresses and external loads have been accounted for. The design model and solution methods presented in this paper are available for both shape and size optimization, and buckling optimization can also be combined with static, frequency and dynamic response optimization. The numerical examples show the applications of the buckling optimization method and the effectiveness of the methods and the program of this paper. Received February 23, 1999     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.     

Interval optimization of dynamic response for structures with interval parameters

This paper presents an interval optimization method for the dynamic response of structures with interval parameters. The matrices of structures with interval parameters are given. Combining the interval extension of function with the perturbation theory of dynamic response, the method for interval dynamic response analysis is derived. The interval optimization problem is transformed into a corresponding deterministic one. Because the mean values and the uncertainties of the interval parameters can be elected as the design variables, more information of the optimization results can be obtained by the present method than that obtained by the deterministic one. The present method is implemented for a truss structure and a frame structure. The numerical results show that the method is effective.     

Reinforcement layout and sizing optimization of composite submarine sail structures

A topology optimization approach that makes use of nonlinear design variable-to-sizing relationship is presented. A finite element (FE) model is used to describe the loaded structure, but unlike the microstructure approach, the decision is whether an element in the continuum should have maximum or minimum cross-sectional dimension while its material density and moduli are held constant. This approach is applied to reinforcement layout optimization of a very large and geometrically complex Composite Advanced Sail (CAS) structure under an asymmetric wave slap loading condition. A high-complexity model in the form of multilayered shell and a low-complexity model in the form of stiffened shell are developed for the layout optimization of the CAS and solved for minimum strain energy. The effects of constraints such as buckling instability on optimal placement of internal stiffeners are also explored. Based on the results of the layout optimization, a new FE model of the CAS is developed and optimized for minimum weight. Depending upon the degree of variability in skin thickness, the results show a weight saving of up to 19% over the original model.     

An optimal design of magnetic actuators using topology optimization and the response surface method

The magnetic actuator is a device that transforms electric energy to mechanical energy. By mechanical energy transformation, some part of the electric energy creates force, and the other part is stored within the ferrous material. An actuator with improved magnetic force can be designed by reducing the stored energy within the ferrous material at the core or the armature. Topology optimization based on the homogenization design method (HDM) is used for the initial design by determining the porous hole size of each element created. The homogenized magnetic permeability is applied in calculation of the magnetic energy stored. The magnetic energy is calculated by finite element analysis and the sensitivity is calculated mathematically by determining the effects of the magnetic energy according to the permeability change at each element. Repeating the process of the porous hole size determination by the sequential linear programming (SLP), eventually leads to a design of an actuator that makes the most improved magnetic force within the limited volume. The initial actuator model derived from topology optimization uses parameter optimization for detail designs. In parameter optimization design, the response surface method (RSM) based on the central composite design is used to obtain a clear final design.     

汽车薄壁冲压件焊点的拓扑优化

汽车薄壁冲压件的焊点数量是汽车制造成本的决定性因素,为了降低成本,焊点的拓扑优化逐渐成为人们关注的热点。基于ICM（Independent Continuous Mapping,即独立、连续、映射）方法,将焊点有无的离散拓扑优化问题转化为[0,1]区间上的连续优化问题;建立了以结构刚度最大为目标、结构强度为约束的连续拓扑优化模型;采用K-S函数将多目标和多约束问题转化为单目标和单约束问题;运用响应面（RSM）方法将位移和应力转化为设计变量的显式函数,采用序列二次规划（SQP）方法求解优化模型。为了提高求解效率,对优化问题的可并行性进行了分析,搭建并行环境,用C++和Fortran语言开发了焊点优化问题的并行程序。工程实例表明,优化算法和程序是可靠、有效的。     

Multi-criteria shape optimization of a funnel in cathode ray tubes using a response surface model

One purpose of simulation describing the behaviors of structures is to optimize the performances within specific functional requirements and customers needs with respect to the design variables. For reduction of the volume of cathode ray tubes, the design of the glass geometry, especially funnel geometry, is essential while maintaining the internal vacuum pressure of the cathode ray tube. In order to describe the three-dimensional geometry of the funnel in cathode ray tubes, a higher-order response surface model is employed in the simulation model instead of non-uniform rational B-splines (NURBS) or Bezier curves because the response surface model is more robust for understanding the geometry change in finite element analysis. We formulate the design problem as a multi-criteria optimization because minimization of both volume and maximum stress is required. Using the response surface model of the geometry of the funnel and sequential quadratic programming within the process integration framework, the shape optimization of a funnel is successfully performed and the maximum stress level of the funnel is decreased by almost half.     

响应面法优化PVA/PVDF复合膜纯化IPA的操作条件

常压下异丙醇与水形成共沸物,应用精馏法生产工艺能耗较大,本文采用自制的PVA/PVDF复合膜蒸汽渗透分离异丙醇(IPA)-水混合溶液,并应用响应面法确定最优的操作条件。首先,制备了理论交联度为8%的马来酸酐交联的PVA/PVDF复合膜:然后根据Box-Behnken中心组合实验设计原理,以膜的渗透通量和选择性为响应值,操作温度、IPA入膜浓度、膜渗透侧压力3个操作条件为影响因素进行实验:最后对实验结果进行响应面分析,得到了优化的操作条件并通过实验进行验证。综合考虑膜的渗透通量和选择性这2个响应值,求得实验所采用的PVA/PVDF复合膜在蒸汽渗透分离IPA水体系中的最优操作条件是:操作温度为95℃,IPA入膜浓度为82.5 wt.%,膜渗透侧压力为1 kPa,此时的渗透通量和选择性分别达到了1.57 kg/(m~2·h)和38.90。在此最佳操作条件下对模型预测值进行实验验证,膜的渗透通量和选择性的预测值与实验值的相对误差分别为5.7%和9.2%。     

混合响应面法对汽车吸能部件优化关键技术*

结合sequential和successful的响应面方法以及Kriging插值技术,针对高维优化问题,建立一种混合响应面优化方法.首先采用sequential响应面方法基于初始化变量进行优化,采用Kriging插值方法对样本点和优化过程中形成的优化点重新进行响应面构造,以确定优化范围内新的初始值并将约束减小到一定范围;随后采用successful响应面对更新的设计变量初始值和约束范围进行优化.如此循环,直至得到最优解.采用该方法对汽车前纵梁中板的厚度问题进行优化,取得了不错的效果.     

Topology and parameter optimization of a foaming jig reinforcement structure by the response surface method

A dilemma in the foaming of inner polyurethane (PU) pieces for household refrigerators is that of keeping the production costs down without adversely affecting the dimension precision. One way to do this is to reduce the electric power consumption spent running the polyurethane foaming line in which a number of heavy foaming jigs are continuously circulating, by optimally designing the reinforcement structure of the jig. In this paper, topology optimization and parameter optimization utilizing the response surface method (RSM) are applied for the optimum design of the jig reinforcement structure, in order to minimize the total jig weight while securing the dimension precision of the foamed urethane case. Both the reliability of the approximated response surfaces and the validity of the proposed optimization procedure are verified through illustrative optimization experiments. In addition, it is confirmed that the proposed procedure provides us with an optimum reinforcement structure which remarkably reduces the total jig weight.