Current Design Point Research Articles

An interval sequential linear programming for nonlinear robust optimization problems

In this paper, interval sequential linear programming (ISLP) is proposed to solve nonlinear robust optimization (RO). The main idea of the programming is to transform the uncertain optimization into several possibility-sensitivity analyses and deterministic linear optimization problems that are sequentially solved. At each cycle, a possibility-sensitivity analysis method is proposed to obtain the approximate partial derivatives of the uncertain constraints at the current design point, based on which a deterministic linear optimization model is constructed and the design point is updated by solving the linear optimization. Moreover, an iterative mechanism is created to adaptively update the design space and improve the convergence rate. Finally, two numerical examples and two practical engineering problems are applied to verify the accuracy and efficiency of the proposed method.

Reliability-based multi-scale design optimization of composite frames considering structural compliance and manufacturing constraints

The paper proposes an efficient methodology for concurrent reliability-based multi-scale design optimization (RBMDO) of composite frames to minimize structural cost subjecting to compliance constraint. Two types of variables are systematically considered in RBMDO, which are deterministic design variables of the frame components, the discrete fiber winding angles at the two geometrical scales, and random parameters of material properties and loading conditions in both magnitude and direction. To overcome the difficulty of highly nonlinear compliance constraint when using fiber winding angles as design variables and improve efficiency and accuracy of RBMDO of composite frames, the improved single loop and single vector (SLSV) approach based on modified chaos control (MCC) scheme, which is abbreviated hereafter as SLSV-MCC, is proposed, and sensitivities at the current design point are utilized to further increase accuracy of the proposed SLSV-MCC. Six types of specific manufacturing constraints are explicitly considered in the proposed RBMDO to reduce the risk of local failure in the laminated composite. The deterministic multi-scale design optimization (DMDO) model is also presented and utilized for comparison to distinguish differences between deterministic and reliability-based optimization results. Efficiency and accuracy of the proposed SLSV-MCC are compared with the first-order reliability method (FORM) and conventional SLSV approach. Meanwhile, the Monte Carlo simulation (MCS) method is further utilized to validate the accuracy of the proposed RBMDO. The discrete material optimization (DMO) approach is utilized to couple two geometrical scales: macroscopic topology and microscopic material selection. Capabilities of the proposed RBMDO are demonstrated by optimization of 2D and 3D composite frames. Numerical study reveals that the uncertainties in material properties and loading conditions will lead to different macroscopic sizing and topology configurations for deterministic and reliability-based solutions.

Iterative reliable design space approach for efficient reliability-based design optimization

Reliability-based design optimization has gained much attention in many engineering design problems with the consideration of uncertainties. Nevertheless, the application is limited by the huge computational cost for the repeated reliability analysis in the optimization process. To address this issue, an iterative reliable design space approach is proposed with less number of reliability analysis required. Specifically, a sequential optimization strategy is proposed to identify the iterative reliable design space and perform the equivalent deterministic optimization in a serial loop. Thus, the identification of the reliable design space, which is time-consuming, is eliminated from the equivalent probabilistic constraints. Furthermore, an improved shifting vector strategy is put forward for the identification. In this strategy, an approximate shifting vector is first constructed utilizing the partial derivatives at the current design point, and it is then modified based on an increment of the shifting vector and a correction provided by the true reliability analysis. Besides, several mathematical examples and engineering problems are tested to validate the performance of the proposed method. In conclusion, the proposed method is able to deal with different kinds of nonlinear problems with relatively high accuracy and efficiency.

An adaptive local range sampling method for reliability-based design optimization using support vector machine and Kriging model

Kriging model is an effective method to overcome huge computational cost for reliability-based design optimization (RBDO) problems. However, the results of RBDO usually depend on constraint boundaries within the local range that contains the RBDO optimum. Determining this local range and building adaptive response surfaces within it can avoid selecting samples in unrelated areas. In this research, a new RBDO process is proposed. In the first phase, Kriging models of constraints are built based on Latin Hypercube sampling method, and updated by two new samples in each iteration. One of these two samples is selected based on SVM and mean squared error to make sure it is located near constraint boundaries. Another one is the deterministic optimum point (DOP) of current Kriging models, which is obtained based on the deterministic optimization and specifies the direction to the RBDO optimum. And the RBDO design point is obtained by SORA. When consecutive RBDO design points are close enough to each other, the local range is determined based on the current RBDO design point and the current DOP. In the second phase, new samples are located on constraint boundaries within the local range to refine Kriging models. The location and the size of the local range is adaptively defined by the RBDO design point and the DOP during each iteration. Several optimization examples are selected to test the computation capability of the proposed method. The results indicate that the new method is more efficient and more accurate.

Rule-based spacecraft design space exploration and sensitivity analysis

This second in a series of three papers [1,2] describes the combination of an analytical model with a rule-based approach. The abstract design vocabulary is assembled by the a sequence of rules, making up a design language. After describing the subsystems in the first paper, in this paper the modeling of the FireSat payload is the starting point. The activity diagrams to integrate the payload and the subsystems lead to a generic satellite design cycle which produces a Design Graph containing all relevant information. From the integrated model in the Design Graph, different analyses of the complex system design are derived. The visualization of the constraint network containing all the variables of the design, gives a glimpse on the complexity of the example. Then the design space for the satellite is shown by plotting the mass of the system over different initial requirements. This analysis is completed by a second plot incorporating the design decisions that cannot be reflected analytically but are stored in concrete design rules. Finally a sensitivity analysis of the full equation system of the satellite is presented. The heat map used to visualize the derivative values is a mean to display all influences on the current design point in one graph.

A local sampling method with variable radius for RBDO using Kriging

Purpose– Kriging model has been widely adopted to reduce the high computational costs of simulations in Reliability-based design optimization (RBDO). To construct the Kriging model accurately and efficiently in the region of significance, a local sampling method with variable radius (LSVR) is proposed. The paper aims to discuss these issues.Design/methodology/approach– In LSVR, the sequential sampling points are mainly selected within the local region around the current design point. The size of the local region is adaptively defined according to the target reliability and the nonlinearity of the probabilistic constraint. Every probabilistic constraint has its own local region instead of all constraints sharing one local region. In the local sampling region, the points located on the constraint boundary and the points with high uncertainty are considered simultaneously.Findings– The computational capability of the proposed method is demonstrated using two mathematical problems, a reducer design and a box girder design of a super heavy machine tool. The comparison results show that the proposed method is very efficient and accurate.Originality/value– The main contribution of this paper lies in: a new local sampling region computational criterion is proposed for Kriging. The originality of this paper is using expected feasible function (EFF) criterion and the shortest distance to the existing sample points instead of the other types of sequential sampling criterion to deal with the low efficiency problem.

An efficient framework for the reliability-based design optimization of large-scale uncertain and stochastic linear systems

This paper is focused on the development of an efficient reliability-based design optimization algorithm for solving problems posed on uncertain linear dynamic systems characterized by large design variable vectors and driven by non-stationary stochastic excitation. The interest in such problems lies in the desire to define a new generation of tools that can efficiently solve practical problems, such as the design of high-rise buildings in seismic zones, characterized by numerous free parameters in a rigorously probabilistic setting. To this end a novel decoupling approach is developed based on defining and solving a limited sequence of deterministic optimization sub-problems. In particular, each sub-problem is formulated from information pertaining to a single simulation carried out exclusively in the current design point. This characteristic drastically limits the number of simulations necessary to find a solution to the original problem while making the proposed approach practically insensitive to the size of the design variable vector. To demonstrate the efficiency and strong convergence properties of the proposed approach, the structural system of a high-rise building defined by over three hundred free parameters is optimized under non-stationary stochastic earthquake excitation.

Analytical sensitivity analysis of eigenvalues and lightweight design of composite laminated beams

The paper describes an optimisation technique for the lightweight design of composite laminated beams with a big ratio of span to thickness. The optimisation model for the lightweight design is to find the fibre volume fractions to minimise the mass of the composite laminated beams under the eigenvalue (or frequency) constraints. The analytical sensitivity of the eigenvalues, with respect to the fibre volume fractions, is formulated using the Euler–Bernoulli beam theory. The analytical formulae for sensitivity analysis are suitable for the different boundary conditions (pinned–pinned, fixed–fixed, fixed–free and fixed–pinned). The eigenvalues are linearised using Taylor’s series so that optimisation model is converted to a linear programming problem. The iterative computational procedure is proposed to solve the optimisation model and eliminate the Taylor’s series approximation since the current design point maybe not near the optimum design. Finally, the lightweight designs of the composite laminated beams with different boundary conditions are performed. The merits of the proposed method and lightweight designs obtained by proposed method are discussed.

샘플링 기반 민감도를 이용한 국부 유도 가열용 코일의 최적 설계

본 논문에서는 전자기 결합 문제에 대한 효율적인 최적 설계를 위하여 샘플링 기반 민감도법을 제시하였다. 전자기 해석과 열/구조 해석 사이의 결합 특성에 무관하게 최적해를 도출하기 위하여 블랙박스 개념에 기초한 설계기법이 적용되었다. 현 설계점을 중심으로 초입방체 국부 영역에서 크리깅 근사모델을 생성하고 근사모델의 기저함수의 미분을 통하여 설계민감도 값을 도출하였다. 제안된 설계기법은 샘플링 기반 설계기법과 민감도 기반 설계기법의 장점만을 취한 혼합형 설계기법으로 열/구조 특성과 강성 결합된 전자기 문제의 최적 설계에도 적용이 가능하다. 제안된 방법은 비선형성이 강한 수학 예제 및 과도상태 해석이 요구되는 국부 유도가열용 코일 설계에 적용하여 그 정확성과 타당성을 검증하였다. This paper proposes a sampling-based sensitivity method for dealing with electromagnetic coupled design problems effectively. The black-box modeling technique is basically applied to obtain an optimum regardless of how strong the electromagnetic, thermal and structural analyses are coupled with each other. To achieve this, Kriging surrogate models are produced in a hyper-cubic local window with the center of a current design point. Then design sensitivity values are extracted from the differentiation of basis functions which consist of the models. The proposed method falls under a hybrid optimization method which takes advantages of the sampling-based and the sensitivity-based methods. Owing to the aforementioned feature, the method can be applied even to electromagnetic problems of which the material properties are strongly coupled with thermal or structural outputs. To examine the accuracy and validity of the proposed method, a strongly nonlinear mathematical example and a coil design problem for local induction heating are tested.

Simulation studies for a space-based CO2 lidar mission

We report results of initial space mission simulation studies for a laser-based, atmospheric CO2 sounder, which are based on real-time carbon cycle process modelling and data analysis. The mission concept corresponds to the Active Sensing of CO2 Emissions over Nights, Days and Seasons (ASCENDS) recommended by the US National Academy of Sciences’ Decadal Survey. As a pre-requisite for meaningful quantitative evaluation, we employ a CO2 model that has representative spatial and temporal gradients across a wide range of scales. In addition, a relatively complete description of the atmospheric and surface state is obtained from meteorological data assimilation and satellite measurements. We use radiative transfer calculations, an instrument model with representative errors and a simple retrieval approach to quantify errors in ‘measured’ CO2 distributions, which are a function of mission and instrument design specifications along with the atmospheric/surface state. Uncertainty estimates based on the current instrument design point indicate that a CO2 laser sounder can provide data consistent with ASCENDS requirements and will significantly enhance our ability to address carbon cycle science questions. Test of a dawn/dusk orbit deployment, however, shows that diurnal differences in CO2 column abundance, indicative of plant photosynthesis and respiration fluxes, will be difficult to detect.

Reliable design space and complete single-loop reliability-based design optimization

Reliability-based design optimization (RBDO) has been intensively studied due to its significance and its conceptual and mathematical complexity. This paper proposes a new method for RBDO on the basis of the concept of reliable design space (RDS), within which any design satisfies the reliability requirements. Therefore, a RBDO problem becomes a simple, deterministic optimization problem constrained by RDS rather than its deterministic feasible space. The RDS is found in this work by using the partial derivatives at the current design point as an approximation of the derivatives at its corresponding most probable point (MPP) on the limit state function. This work completely resolves the double loop in RBDO and turns RBDO into a simple optimization problem. Well-known problems from the literature are selected to illustrate the steps of the approach and for result comparison. Discussions will also be given on the limitation of the proposed method, which is shown to be a common limitation overlooked by the research community on RBDO.

Further study on efficiency of sequential approximate programming for probabilistic structural design optimization

Compared to the traditional deterministic optimization based on safety factors, the probabilistic structural design optimization (PSDO) is considered to be a more rational design philosophy because of reasonable account of uncertainties in material properties, loading, boundary condition and geometry, and even mathematical representation of the system model. However, it is well known that the computation for PSDO can be prohibitive when the associated function evaluation is expensive. As a result, many approximate PSDO methods have been developed in recent literatures. In previous works, we developed two sequential approximate programming (SAP) strategies for PSDO based on reliability index approach (RIA) and performance measure approach (PMA). In PMA with SAP, a sequence of approximate programming of PSDO was formulated and solved before the final optimum was located. In each subprogramming, rather than relying on direct linear Taylor expansion of the probabilistic performance measure (PPM), we developed a formulation for approximate PPM at the current design point and used its linearization instead. The approximate PPM and its sensitivity were obtained by approximating the optimality conditions in the vicinity of the minimum performance target point (MPTP). The present paper further elaborates the SAP for PMA. In addition to detailed description of the algorithm, we present error analysis and show that in the ɛ-vicinity of optimum design and corresponding MPTP, the difference between the Taylor expansion of PPM and the linear expansion of approximate PPM is of higher order of ɛ. Four examples are optimized by six algorithms appearing in recent literatures for efficiency comparison. The effect of target reliability index and statistical distribution of random variables on the comparison is discussed. The third example shows that PMA with SAP performs well even for the problem for which reliability index calculation by first order reliability method (FORM) fails. Finally, the fourth example with 144 probabilistic constraints is shown to demonstrate the effectiveness of PMA with SAP. All example results illustrate that with the algorithm PMA with SAP, we get concurrent convergence of both design optimization and probabilistic performance measure calculation, which agrees well with the error analysis.

A sequential approximate programming strategy for reliability-based structural optimization

Although reliability-based structural optimization (RBSO) is recognized as a rational structural design philosophy that is more advantageous to deterministic optimization, most common RBSO is based on straightforward two-level approach connecting algorithms of reliability calculation and that of design optimization. This is achieved usually with an outer loop for optimization of design variables and an inner loop for reliability analysis. A number of algorithms have been proposed to reduce the computational cost of such optimizations, such as performance measure approach, semi-infinite programming, and mono-level approach. Herein the sequential approximate programming approach, which is well known in structural optimization, is extended as an efficient methodology to solve RBSO problems. In this approach, the optimum design is obtained by solving a sequence of sub-programming problems that usually consist of an approximate objective function subjected to a set of approximate constraint functions. In each sub-programming, rather than direct Taylor expansion of reliability constraints, a new formulation is introduced for approximate reliability constraints at the current design point and its linearization. The approximate reliability index and its sensitivity are obtained from a recurrence formula based on the optimality conditions for the most probable failure point (MPP). It is shown that the approximate MPP, a key component of RBSO problems, is concurrently improved during each sub-programming solution step. Through analytical models and comparative studies over complex examples, it is illustrated that our approach is efficient and that a linearized reliability index is a good approximation of the accurate reliability index. These unique features and the concurrent convergence of design optimization and reliability calculation are demonstrated with several numerical examples.

ARIES-AT magnet systems

This report presents a conceptual design of the magnet systems for an advanced tokamak fusion reactor (ARIES-AT). The main focus of the paper is to anticipate and extrapolate the current state-of-the-art in high temperature superconductors and coil design, and apply them to an advanced commercial fusion reactor concept. The current design point is described and supported with a preliminary structural analysis and a discussion of the merits, performance, and economics of high temperature vs. low temperature superconductors in an advanced fusion reactor design.

Improved global–local simulated annealing formulation for solving non-smooth engineering optimization problems

This paper is concerned with a novel optimization algorithm that implements an enhanced formulation of simulated annealing (SA). The new algorithm is denoted as ISA (improved simulated annealing) in the rest of the paper. ISA includes a two-level random search: “global annealing” where all design variables are perturbed simultaneously and “local annealing” where design variables are perturbed one at a time. The improvement with respect to classical SA is in the fact that trial designs are generated always taking care to choose directions along which the cost function may improve. To this purpose, cost function sensitivities are computed in order to properly choose the size of each random perturbation. In addition, the optimization problem is linearized about the current design point if the optimizer ends up in an infeasible region or there is no significant reduction in cost even though the cost function gradient is not close to zero. The linearization is controlled by a trust region model. The optimization algorithm continuously shifts from global to local annealing based on the current best record at the beginning of each cooling cycle. Finally, the cooling schedule is automatically adjusted within ISA based on the convergence behavior. In this work, the ISA algorithm is successfully utilized to solve complicated optimization problems which exhibit non-smooth/non-convex behavior: (i) the large-scale (200 design variables and 3500 constraints) weight minimization of a 200 bar truss under five independent loading conditions; (ii) the configuration optimization of a cantilevered bar truss with 45 elements and 81 design variables; (iii) an example of reverse engineering where in-plane elastic properties of an eight-ply woven composite laminate are to be determined. The performance of ISA is compared to that of classical SA, gradient based optimization codes recently published in literature and commercial software. The results obtained in this study indicate that ISA is a very efficient optimization code. In fact, ISA was much faster than classical SA. The present code allowed about 300 kg weight saving in the 200 bar truss case and about 80 kg in the cantilevered bar truss case. In addition, the residual error on elastic constants in the material identification problem was less than 3%.

이산설계공간에서 직교배열표를 이용한 순차적 알고리듬의 국부해

Structural optimization has been carried out in continuous design space or in discrete design space. Generally, available designs are discrete in design practice. However, the methods for discrete variables are extremely expensive in computational cost. An iterative optimization algorithm is proposed for design in a discrete space, which is called a sequential algorithm using orthogonal arrays (SOA). We demonstrate verifying the fact that a local optimum solution can be obtained from the process with this algorithm. The local optimum solution is defined in a discrete design space. Then the search space, which is a set of candidate values of each design variables formed by the neighborhood of a current design point, is defined. It is verified that a local optimum solution can be found by sequentially moving the search space. The SOA algorithm has been applied to problems such as truss type structures. Then it is confirmed that a local solution can be obtained by using the SOA algorithm.

Development of an efficient algorithm for global optimization by simplex elimination

An efficient multi-start algorithm for global optimization is developed by introducing multi-dimensional simplexes as new expression units of attraction regions. The region elimination method generally consists of making a set of eliminated regions called attraction regions, checking adjacency between the current design point and the attraction region, and quitting local optimization for the attracted design points. The efficiency of the elimination method is considerably enhanced by supplementing general simplexes and their neighborhoods to conventional units of attraction regions of points and lines. To show the effectiveness of the proposed algorithm, mathematical problems from the literature are solved and the results are compared with several well-known multi-start algorithms. The present algorithm produces the global optimum in all problems more efficiently than the variants of the multi-start method. Several types of truss, frame, and composite material structures are optimized as engineering applications. Many local optima are found and the differences among the local optima are not negligibly small. These results suggest that an efficient and reliable global optimizer is strongly required in some fields of engineering optimization.

Weight reduction of aluminum disc wheels under fatigue constraints using a sequential neural network approximation method

This paper describes a weight reduction problem of aluminum disc wheels under cornering fatigue constraints. It is a special structural optimization problem because of the existence of the implicit fatigue constraint. A sequential neural network approximation method is presented to solve this type of discrete–variable engineering optimization problems. First a back-propagation neural network is trained to simulate the feasible domain formed by the implicit constraints using just a few training data. A search algorithm then searches for the “optimal point” in the feasible domain simulated by the neural network. This new design point is checked against the true implicit constraints to see whether it is feasible, and the new training data is then added to the training set. This process continues in an iterative manner until we get the same design point repeatedly and no new training point is generated. In each iteration, only one evaluation of the implicit constraints is needed to see whether the current design point is feasible. No precise function value or sensitivity calculation is required.

A Sequential Approximation Method Using Neural Networks for Nonlinear Discrete-Variable Optimization with Implicit Constraints.

This paper presents a sequential approximation method that combines a back-propagation neural network with a search algorithm for nonlinear discrete-variable engineering optimization problems with implicit constraints. This is an iteration process. A back-propagation neural network is trained to simulate the feasible domain formed by the implicit constraints. A search algorithm then searches for the “optimal point” in the feasible domain simulated by the neural network. This new design point is checked against the true constraints to see whether it is feasible, and is added to the training set. Then the neural network is trained again. With more design points in the training set, information about the whole search domain is accumulated to progressively form a better approximation for the feasible domain. This iteration process continues until the approximate model insists the same “optimal” point in consecutive iterations. In each iteration, only one evaluation of the implicit constraints is needed to see whether the current design point is feasible. No precise function value or sensitivity calculation is required. Several engineering design examples are used to demonstrate the practicality of this approach.

Finite elements, genetic algorithms and [formula omitted]-splines: a combined technique for shape optimization

Shape optimization consist in changing the boundary shape of the structure to find the optimal design by verifying the prescribed constraints. Generally, this problem has been solved by replacing the originally design problem by a sequence of explicit subproblems. They are constructed by means of local approximation concepts based on function values and derivatives at the current design point. The aim of this paper is to propose a different approach to solve bidimensional shape optimization problems by using Genetic Algorithms. Boundary β-splines curves with an automatic mesh generation are used to obtain the analytical model and finally, the finite element method is used to estimate the fitness function. The versatility and flexibility of the proposed approach is tested and discussed in two numerical examples, showing that the technique is able to deal with real engineering problems.