Response surface estimation and refinement in collaborative optimization

The use of response surface estimation in collaborative optimization, an architecture for large-scale multidisciplinary design is described. Collaborative optimization preserves the autonomy of individual disciplines while providing a mechanism for coordinating the overall design problem and progressing toward improved designs. Collaborative optimization is a two-level optimization architecture, with discipline-specific optimizations free to specify local designs, and a global optimization that ensures that all of the discipline designs eventually agree on a single value for those variables that are shared in common. Results demonstrate how response surface models of subproblem optimization results improve the performance of collaborative optimization. The utility of response surface estimation in collaborative optimization depends on the generation of inexpensive accurate response surface models and the refinement of these models over several fitting cycles. Special properties of the subproblem optimization formulation are exploited to reduce the number of required subproblem optimizations to develop a quadratic model from O(n 2 ) to O(n/2). Response surface refinement is performed using ideas from trust region methods. Results for the combined approaches are compared through the design optimization of a tailless unmanned air vehicle in 44 design variables.

Response surface methodology (RSM) is a widely used method for simulation optimization. Its strategy is to explore small subregions of the decision space in succession instead of attempting to explore the entire decision space in a single attempt. This method is especially suitable for complex stochastic systems where little knowledge is available. Although RSM is popular in practice, its current applications in simulation optimization treat simulation experiments the same as real experiments. However, the unique properties of simulation experiments make traditional RSM inappropriate in two important aspects: (1) It is not automated; human involvement is required at each step of the search process; (2) RSM is a heuristic procedure without convergence guarantee; the quality of the final solution cannot be quantified. We propose the stochastic trust-region response-surface method (STRONG) for simulation optimization in attempts to solve these problems. STRONG combines RSM with the classic trust-region method developed for deterministic optimization to eliminate the need for human intervention and to achieve the desired convergence properties. The numerical study shows that STRONG can outperform the existing methodologies, especially for problems that have grossly noisy response surfaces, and its computational advantage becomes more obvious when the dimension of the problem increases.

This research falls into the gap between applied statistics and numerical optimization in a specific topic—Ridge Analysis (RA). This article proposes using the trust-region (TR) methods in numerical optimization to solve the RA problem, arising from the literature of response surface methodology (RSM) in applied statistics, where its goal is to help engineers for ‘process improvement’ to find the better response value of the predicted response function within the boundary of experimentation. In the field of numerical optimization, as the family of TR approaches always exhibits excellent mathematical properties during optimization steps, thus the algorithm presented in this study guarantees global optima for the RA problem. Two examples found in the RSM literature are included to illustrate the algorithm, demonstrating its capability of locating better operating conditions than existing computing methods and pointing out particular circumstances (termed the ‘hard case’) where the classical RA procedure fails. An important application to the response modeling problem arising from the philosophy of Taguchi's quality engineering illustrates the hard case. Finally, the utility of the presented TR algorithm is demonstrated through a sequential framework with iterative updates of the TR model under local approximation provided that the predicted response model is a high-order or even non-polynomial function.

A unified response surface framework for the interval and stochastic finite element analysis of structures with uncertain parameters

Effect of Variable Selection on Multidisciplinary Design Optimization: a Flight Vehicle Example

PurposeBecause of the high computational efficiency, response surface method (RSM) has been widely used in structural reliability analysis. However, for a highly nonlinear limit state function (LSF), the approximate accuracy of the failure probability mainly depends on the design point, and the result is that the response surface function composed of initial experimental points rarely fits the LSF exactly. The inaccurate design points usually cause some errors in the traditional RSM. The purpose of this paper is to present a hybrid method combining adaptive moving experimental points strategy and RSM, describing a new response surface using downhill simplex algorithm (DSA-RSM).Design/methodology/approachIn DSA-RSM, the operation mechanism principle of the basic DSA, in which local descending vectors are automatically generated, was studied. Then, the search strategy of the basic DSA was changed and the RSM approximate model was reconstructed by combining the direct search advantage of DSA with the reliability mechanism of response surface analysis.FindingsThe computational power of the proposed method is demonstrated by solving four structural reliability problems, including the actual engineering problem of a car collision. Compared to specific structural reliability analysis methods, the approach of modified DSA interpolation response surface for structural reliability has a good convergent capability and computational accuracy.Originality/valueThis paper proposes a new RSM technology based on proxy model to complete the reliability analysis. The originality of this paper is to present an improved RSM that adjusts the position of the experimental points judiciously by using the DSA principle to make the fitted response surface closer to the actual limit state surface.

Response Surface Methodology (RSM) is a metamodel- based optimization method. Its strategy is to explore small subregions of the parameter space in succession instead of attempting to explore the entire parameter space directly. This method has been widely used in simulation optimization. However, RSM has two significant shortcomings: Firstly, it is not automated. Human involvements are usually required in the search process. Secondly, RSM is heuristic without convergence guarantee. This paper proposes Stochastic Trust Region Gradient-Free Method (STRONG) for simulation optimization with continuous decision variables to solve these two problems. STRONG combines the traditional RSM framework with the trust region method for deterministic optimization to achieve convergence property and eliminate the requirement of human involvement. Combined with appropriate experimental designs and specifically efficient screening experiments, STRONG has the potential of solving high-dimensional problems efficiently.

In allusion to the transmission of a new type of truck with three shafts four levels, it's Collaborative Optimization framework including mathematical model is set up. Through taking multidisciplinary optimization software iSIGHT as optimization design platform to integrate disciplinary analysis tool ANSYS and Response Surface Method, Collaborative Optimization of transmission is completed and satisfying optimization scheme is obtained, where Multi-Island Genetic Algorithm is adopted for system-level optimization and Sequential Quadratic Programming is adopted for subsystem-level optimization. The optimization method used in the paper provides important references for optimization design of other complex systems.

Uncertainties in multidisciplinary design optimization (MDO) have a significant influence on the whole design process of engineering systems. The most probable point (MPP) based reliability analysis is an approach that utilizes the safety index β to measure the effect of uncertainties. Collaborative optimization (CO) is a two-level optimization method specially created for large-scale distributed-analysis applications. Simulated annealing-based collaborative optimization (SA—CO) is one of the improved forms of CO that overcomes the difficulty of convergence given the existing of highly nonlinear consistency constraints. By combining the MPP-based reliability analysis method with SA—CO, we present a new collaborative reliability analysis method under the environment of MDO to deal with uncertainties existing in MDO, that is, MPP—SA—CO. Demonstrated by two typical examples, the proposed method inherits the advantages of CO. Also, accurate and efficient results are obtained by employing simulated annealing algorithmic as the system level optimizer and it features response surface instead of disciplinary optimization.

This paper presents a new improved collaborative optimization (CO) model that provides solution capabilities for multiobjective multidisciplinary optimization problems. Reasons that cause computational difficulties in CO algorithm are firstly analyzed. Then a new system level objective function is advised to minimize relative value between the collaborative objective function and single disciplinary objective function. And it eliminates the effect of dimensions and magnitude orders among objectives. A new subsystem level objective function is developed that includes the disciplinary objective function and the consistency constraint. A new CO framework which is more suitable for multilevel distributed design is advised. In this CO framework, the system level optimizer does not only independently invoke the subdisciplinary analysis tools, but also invoke its subdiscipline optimizer. The improved CO model proposed in this work is demonstrated with two examples. The results of examples show the improved CO is not only feasible, reliable and efficient, but also well suitable to solve the multiobjective optimization problems in multidisciplinary design environment.

Simulation optimization refers to the iterative procedure in search of the optimal parameter when the objective function can only be evaluated by stochastic simulation. STRONG (Stochastic Trust Region Response Surface Convergent Method) is a newly developed design-of-experiments based simulation optimization method. It incorporates the idea of trust region method (TRM) for deterministic optimization into the traditional response surface methodology (RSM) to eliminate the human intervention required by RSM and to achieve the desired convergence. In the earlier paper, we proved the convergence of STRONG and demonstrated its computational efficiency. The original STRONG assumes the stochastic response follows a normal distribution. This paper relaxes the normal assumption and develops a framework called STRONG-X which is applicable for generally distributed additive noise with bounded variance. The convergence of STRONG-X can be proved and the generality of STRONG-X makes it an appealing method.

To improve the carrying capacity and reduce the weight of telescopic boom structure in a truck crane, a Collaborative optimization (CO) approach was applied to solve the problems of strength, stiffness and local stability in the telescopic boom structure. First, the complex optimization problem of the telescopic boom structure was decomposed into two-level optimizations: the system level and two subsystem levels for strength and local stability. Second, the underside curve of the boom’s cross-section was constructed by the Non-uniform rational B-Splines (NURBS) curve. 3D parametric solid model and the parametric finite element analysis model for the strength and the local stability were then established. Third, the mathematical models of the strength and local stability for the subsystem levels, and the system level were optimized, respectively. The adaptive relaxation factor algorithm and the penalty function approach were applied to improve the efficiency of CO. Next, the CO process which integrates the ANSYS package with ISIGHT platform was implemented. The optimal results show that the carrying capacity of the telescopic boom structure can be significantly improved and its weight efficiently is reduced. Finally, with the comparison of the stress values obtained from both the experimental test and the theoretical computation, highly coincident results could be obtained to verify the reliability of CO of a telescopic boom.

Uncertainty quantification on reserves estimation is a fundamental task for project validation on offshore green fields and also for the selection of a development plan that is robust to those uncertainties. To test the robustness of several development plans is then fundamental to be able to evaluate the impact of this uncertainty with low computational cost. Structural, geological and dynamic uncertainties are integrated in a new workflow based on a set of embedded response surfaces. The first step consists in building an experimental design with only structural and geological parameters to evaluate the impact of such uncertainties on static responses such as oil initially in place. In the second step, this experimental design is completed with additional dynamic parameters. The novelty of the static and dynamic design merging is to incorporate the previously obtained static response surface in addition to the uncertain parameters. By embedding the static response in the variables we obtain a response surface on the reserves more regular and more predictive. In this workflow, mainly applied to green fields, the impact of local geological uncertainty is usually of second order respect to other global uncertainties and it is taken into account as random noise whose variance may depend on the static response. The method is applied to a real green field with complex multi-level heterogeneities and it is compared to more standard approaches such as Monte Carlo. We manage to obtain a very good response surface quality for the static response (oil in place) that is then embedded in the reserves response surface. The final accuracy of this response surface is sensibly increased respect to a classical approach considering only static and/or dynamic parameters at minor costs in terms of reservoir simulations. The new approach reduces sensibly the number of reservoir simulations needed for an accurate uncertainty and risk assessment when applied to select the more robust development plan selected from a set of possible candidates. It also provides useful sensitivity analysis measures to guide possible data acquisition campaign in order to reduce risks and take better decisions.

In this research a Collaborative Optimization (CO) approach for multidisciplinary systems design is used to develop a decision based design framework for non-deterministic optimization. To date CO strategies have been developed for use in application to deterministic systems design problems. In this research the decision based design (DBD) framework proposed by Hazelrigg [1,2] is modified for use in a collaborative optimization framework. The Hazelrigg framework as originally proposed provides a single level optimization strategy that combines engineering decisions with business decisions in a single level optimization. By transforming this framework for use in collaborative optimization one can decompose the business and engineering decision making processes. In the new multilevel framework of Decision Based Collaborative Optimization (DBCO) the business decisions are made at the system level. These business decisions result in a set of engineering performance targets that disciplinary engineering design teams seek to satisfy as part of subspace optimizations. The Decision Based Collaborative Optimization framework more accurately models the existing relationship between business and engineering in multidisciplinary systems design.

Bi-level optimization strategy collaborative optimization (CO) strategy is used for bridge system performance-based design. Optimization design process is subdivided into two levels: system level and discipline level. For system level, target function is formulated with the total direct construction cost. For discipline level, In terms of load types and structural seismic performance levels, bridge system design process is decomposed into three disciplines: carrying capacity discipline, E1 earthquake discipline and E2 earthquake discipline. Constraints to every discipline are formulated in terms of corresponding codes demands respectively. In order to examine the feasibility and efficiency of CO, both CO and mono-discipline optimization strategy taking Sequential Quadratic Programming as optimizer are applied to an example of a three span simple supported bridge. The results show that CO has high computing efficiency when achieving the same optimization results, especially for design process involving time-consuming disciplinary analysis process.