Sequential optimization and reliability assessment for multidisciplinary systems design

With higher reliability and safety requirements, reliability-based design has been increasingly applied in multidisciplinary design optimization (MDO). A direct integration of reliability-based design and MDO may present tremendous implementation and numerical difficulties. In this work, a methodology of sequential optimization and reliability assessment for MDO is proposed to improve the efficiency of reliability-based MDO. The central idea is to decouple the reliability analysis from MDO with sequential cycles of reliability analysis and deterministic MDO. The reliability analysis is based on the first-order reliability method (FORM). In the proposed method, the reliability analysis and the deterministic MDO use two MDO strategies, the multidisciplinary feasible approach and the individual disciplinary feasible approach. The effectiveness of the proposed method is illustrated with two example problems.     

A cell evolution method for reliability-based design optimization

The reliability-based design optimization (RBDO) presents to be a systematic and powerful approach for process designs under uncertainties. The traditional double-loop methods for solving RBDO problems can be computationally inefficient because the inner reliability analysis loop has to be iteratively performed for each probabilistic constraint. To solve RBDOs in an alternative and more effective way, Deb et al. [1] proposed recently the use of evolutionary algorithms with an incorporated fastPMA. Since the imbedded fastPMA needs the gradient calculations and the initial guesses of the most probable points (MPPs), their proposed algorithm would encounter difficulties in dealing with non-differentiable constraints and the effectiveness could be degraded significantly as the initial guesses are far from the true MPPs. In this paper, a novel population-based evolutionary algorithm, named cell evolution method, is proposed to improve the computational efficiency and effectiveness of solving the RBDO problems. By using the proposed cell evolution method, a family of test cells is generated based on the target reliability index and with these reliability test cells the determination of the MPPs for probabilistic constraints becomes a simple parallel calculation task, without the needs of gradient calculations and any initial guesses. Having determined the MPPs, a modified real-coded genetic algorithm is applied to evolve these cells into a final one that satisfies all the constraints and has the best objective function value for the RBDO. Especially, the nucleus of the final cell contains the reliable solution to the RBDO problem. Illustrative examples are provided to demonstrate the effectiveness and applicability of the proposed cell evolution method in solving RBDOs. Simulation results reveal that the proposed cell evolution method outperforms comparative methods in both the computational efficiency and solution accuracy, especially for multi-modal RBDO problems.     

Reliability-based robust design optimization: A general methodology using genetic algorithm

In this paper, we present an improved general methodology including four stages to design robust and reliable products under uncertainties. First, as the formulation stage, we consider reliability and robustness simultaneously to propose the new formulation of reliability-based robust design optimization (RBRDO) problems. In order to generate reliable and robust Pareto-optimal solutions, the combination of genetic algorithm with reliability assessment loop based on the performance measure approach is applied as the second stage. Next, we develop two criteria to select a solution from obtained Pareto-optimal set to achieve the best possible implementation. Finally, the result verification is performed with Monte Carlo Simulations and also the quality improvement during manufacturing process is considered by identifying and controlling the critical variables. The effectiveness and applicability of this new proposed methodology is demonstrated through a case study.     

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.     

A direct decoupling approach for efficient reliability-based design optimization

This paper develops an efficient methodology to perform reliability-based design optimization (RBDO) by decoupling the optimization and reliability analysis iterations that are nested in traditional formulations. This is achieved by approximating the reliability constraints based on the reliability analysis results. The proposed approach does not use inverse first-order reliability analysis as other existing decoupled approaches, but uses direct reliability analysis. This strategy allows a modular approach and the use of more accurate methods, including Monte-Carlo-simulation (MCS)-based methods for highly nonlinear reliability constraints where first-order reliability approximation may not be accurate. The use of simulation-based methods also enables system-level reliability estimates to be included in the RBDO formulation. The efficiency of the proposed RBDO approach is further improved by identifying the potentially active reliability constraints at the beginning of each reliability analysis. A vehicle side impact problem is used to examine the proposed method, and the results show the usefulness of the proposed method.     

A multi-point reduced-order modeling approach of transient structural dynamics with application to robust design optimization

Predicting the transient response of structures by high-fidelity simulation models within design optimization and uncertainty quantification often leads to unacceptable computational cost. This paper presents a reduced-order modeling (ROM) framework for approximating the transient response of linear elastic structures over a range of design and random parameters. The full-order response is projected onto a lower-dimensional basis spanned by modes computed from a proper orthogonal decomposition (POD) of full-order model simulation results at multiple calibration points. The basis is further enriched by gradients of the POD modes with respect to the design/random parameters. A truncation strategy is proposed to compensate for the increase in basis vectors due to the proposed enrichment strategies. The accuracy, efficiency and robustness of the proposed framework are studied with a two-dimensional model problem. The numerical results suggest that the proposed ROM approach is well suited for large parameter changes and that the number of basis vectors needs to be increased only linearly with the number of design and random parameters to maintain a particular ROM performance. The application of the proposed ROM approach to robust shape optimization demonstrates significant savings in computational cost over using full-order models. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC04-94AL85000.     

Component-oriented decomposition for multidisciplinary design optimization in building design

The potential of Multidisciplinary Design Optimization (MDO) is not sufficiently exploited in current building design practice. I argue that this field of engineering requires a special setup of the optimization model that considers the uniqueness of buildings, and allows the designer to interact with the optimization in order to assess qualities of aesthetics, expression, and building function. For this reason, the approach applies a performance optimization based on resource consumption extended by preference criteria. Furthermore, building design-specific components serve for the decomposition and an interactive way of working. The component scheme follows the Industry Foundation Classes (IFC) as a common Building Information Model (BIM) standard in order to allow a seamless integration into an interactive CAD working process in the future. A representative case study dealing with a frame-based hall design serves to illustrate these considerations. An N-Square diagram or Design Structure Matrix (DSM) represents the system of components with the disciplinary dependencies and workflow of the analysis. The application of a Multiobjective Genetic Algorithm (MOGA) leads to demonstrable results.     

Design space optimization using design space adjustment and refinement

To deal with large-scale problems that often occur in industry, the authors propose design space optimization with design space adjustment and refinement. In topology optimization, a design space is specified by the number of design variables, and their layout or configuration. The proposed procedure has two efficient algorithms for adjusting and refining design space. First, the design space can be adjusted in terms of design space expansion and reduction. This capability is evolutionary because the design domain expands or reduces wherever necessary. Second, the design space can be refined uniformly or selectively wherever and whenever necessary, ensuring a target resolution with fewer elements, especially for selective refinement. Accordingly, the proposed procedure can handle large-scale problems by solving a sequence of smaller problems. Two examples show the efficiency of the proposed approach.     

Imparting desired attributes in structural design by means of multi-objective optimization

Commonly available optimization methods typically produce a single optimal design as a constrained minimum of a particular objective function. However, in engineering design practice it is quite often important to explore as much of the design space as possible, with respect to many attributes, to discover what behaviors are possible and not possible within the initially adopted design concept. This paper shows that the very simple method of the sum of weighted objectives is useful for such exploration. By geometrical argument it is demonstrated that if every weighting coefficient is allowed to change its magnitude and its sign then the method returns a set of designs that are all feasible, diverse in their attributes, and include the Pareto and non-Pareto solutions, at least for convex cases. Numerical examples in the paper include the case of an aircraft wing structural box with thousands of degrees of freedom and constraints, and over 100 design variables, whose attributes are structural mass, volume, displacement, and frequency. The weighted coefficients method is inherently suitable for parallel, coarse-grained implementation that enables exploration of the design space in the elapsed time of a single structural optimization.     

Design optimization and sensitivity analysis on buckling stability of piezoelectric truss structures

The design optimization of buckling behavior is studied for piezoelectric intelligent truss structures. First, on the basis of mechanical–electric coupling equation and considering electric load and mechanical loads together, the finite element model of piezoelectric trusses has been built up. Then, the computational formula has been derived for the design sensitivities of critical buckling load factor of the structure with respect to size and shape design variables. The electric voltage is taken as a new kind of design variable and the calculation method of critical load buckling factor with respect to the electric voltage variables is proposed. Particularly, the variations of the loads and pre-buckling inner forces with design variables have been accounted for. Finally, the sequential linear programming algorithm is employed to solve the optimization problem, and a new method of controlling structural buckling stability by optimizing the voltages of piezoelectric active bars is proposed. Numerical examples given in the paper have demonstrated the effectiveness of the methods presented.     

Tree decomposition and discrete optimization problems: A survey

The paper considers tree decomposition methods as applied to discrete optimization and presents relevant mathematical results. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 102–118, July–August 2007.     

A crossing sensitivity filter for structural topology optimization with chamfering,rounding, and checkerboard-free patterns

Chamfering and rounding are the most common features of structures and mechanisms. Introducing these features is another way to make material usage more efficient by lowering or avoiding stress con centrations and making the structure even stronger. In this brief note, a new crossing sensitivity filter is proposed for structural topology optimization with chamfering and rounding. This method also ensures the final optimal solution without checkerboard patterns or mesh-dependency. It can also partially prevent one-node hinges. Numerical examples that include both design of stiffest/strongest structures and synthesis of compliant mechanisms are provided.     

Efficient structural optimization using reanalysis and sensitivity reanalysis

Optimization of large-scale structures using conventional formulations often involves much computational effort. Repeated solutions of the analysis and sensitivity analysis equations usually require most of this effort. The computational cost may become prohibitive in large-scale structures having complex analysis models. To alleviate this difficulty, various procedures are integrated in this study into a general optimization approach. The approach is suitable for different classes of response types and optimization methods, including linear and non-linear response; static and dynamic response; direct and gradient optimization methods. Combined approximations are used for reanalysis and repeated sensitivity analysis. The advantage is that the efficiency of local approximations and the improved quality of global approximations are combined to obtain effective solution procedures. Approximate reanalysis and finite-difference sensitivity reanalysis are considered for each intermediate design during the solution process. Reductions in the computational effort may reach several orders of magnitude. Typical numerical examples show that the results achieved by the approach presented are similar to those obtained by exact reanalysis and sensitivity analysis.     

Crashworthiness optimization of helicopter subfloor based on decomposition and global approximation

This work deals with the development of an optimization procedure under crashworthiness requirements applied to a typical helicopter subfloor. The difficulties due to the nonlinear design space and structural behaviour are overcome by developing an optimization procedure based on decomposition, where the structure to be optimized is converted into a set of smaller and linked substructures. To evaluate the response of each substructure, global approximation strategies, based on neural networks, are used. Size variables (dimensions, thickness) and geometrical variables (element number and position) are considered in order to maximize the global crashworthiness performance. The energy absorbed per unit mass by the subfloor is chosen as objective function and acceleration constraints are considered. Genetic algorithms are used to find the optimal configuration. The optimization allowed an increase in the crush force efficiency of 12% and a decrease in the subfloor mass of 4%. A significant CPU time saving was also obtained.     

Integration of topology and shape optimization for design of structural components

This paper presents an integrated approach that supports the topology optimization and CAD-based shape optimization. The main contribution of the paper is using the geometric reconstruction technique that is mathematically sound and error bounded for creating solid models of the topologically optimized structures with smooth geometric boundary. This geometric reconstruction method extends the integration to 3-D applications. In addition, commercial Computer-Aided Design (CAD), finite element analysis (FEA), optimization, and application software tools are incorporated to support the integrated optimization process. The integration is carried out by first converting the geometry of the topologically optimized structure into smooth and parametric B-spline curves and surfaces. The B-spline curves and surfaces are then imported into a parametric CAD environment to build solid models of the structure. The control point movements of the B-spline curves or surfaces are defined as design variables for shape optimization, in which CAD-based design velocity field computations, design sensitivity analysis (DSA), and nonlinear programming are performed. Both 2-D plane stress and 3-D solid examples are presented to demonstrate the proposed approach. Received January 27, 2000 Communicated by J. Sobieski     

An efficient decomposition and dual-stage multi-objective optimization method for water distribution systems with multiple supply sources

This paper proposes an efficient decomposition and dual-stage multi-objective optimization (DDMO) method for designing water distribution systems with multiple supply sources (WDS-MSSs). Three phases are involved in the proposed DDMO approach. In Phase 1, an optimal source partitioning cut-set is identified for a WDS-MSS, allowing the entire WDS-MSS to be decomposed into sub-networks. Then in Phase 2 a non-dominated sorting genetic algorithm (NSGA-II) is employed to optimize the sub-networks separately, thereby producing an optimal front for each sub-network. Finally in Phase 3, another NSGA-II implementation is used to drive the combined sub-network front (an approximate optimal front) towards the Pareto front for the original complete WDS-MSS. Four WDS-MSSs are used to demonstrate the effectiveness of the proposed approach. Results obtained show that the proposed DDMO significantly outperforms the NSGA-II that optimizes the entire network as a whole in terms of efficiently finding good quality optimal fronts.     

Design parameterization and tool integration for CAD-based mechanism optimization

This paper presents an open and integrated tool environment that enables engineers to effectively search, in a CAD solid model form, for a mechanism design with optimal kinematic and dynamic performance. In order to demonstrate the feasibility of such an environment, design parameterization that supports capturing design intents in product solid models must be available, and advanced modeling, simulation, and optimization technologies implemented in engineering software tools must be incorporated. In this paper, the design parameterization capabilities developed previously have been applied to support design optimization of engineering products, including a High Mobility Multi-purpose Wheeled Vehicle (HMMWV). In the proposed environment, Pro/ENGINEER and SolidWorks are supported for product model representation, DADS (Dynamic Analysis and Design System) is employed for dynamic simulation of mechanical systems including ground vehicles, and DOT (Design Optimization Tool) is included for a batch mode design optimization. In addition to the commercial tools, a number of software modules have been implemented to support the integration; e.g., interface modules for data retrieval, and model update modules for updating CAD and simulation models in accordance with design changes. Note that in this research, the overall finite difference method has been adopted to support design sensitivity analysis.     

航空产品研发过程实时动态分解与优化配置

为了实现航空制造企业产品研发过程实时动态分解与优化配置,实现过程体系的优化,提出了基于活动及其实现过程（ProA）的动态分解方法。详细讨论了ProA分解工作流程序、基于DSM的子ProA的确定、ProA分解参数的优化配置及监测与更改流程。     

On the enhancement of computation and exploration of discretization approaches for meshless shape design sensitivity analysis

A new implementation of Reproducing Kernel Particle Method (RKPM) is proposed to enhance the process of shape design sensitivity analysis (DSA). The acceleration process is accomplished by expressing RKPM shape functions and their derivatives explicitly in terms of kernel function moments. In addition, two different discretization approaches are explored elaborately, which emanate from discretizing design sensitivity equation using the direct differentiation method. Comparison of these two approaches is made, and the equivalence of these two superficially different approaches is demonstrated through two elastostatics problems. The effectiveness of the enhanced RKPM is also verified by comparison of consumption of computer time between the classical method and the improved method.