特性間の競合・協調関係の解明に基づく最適設計における設計感度解析の適用

<p>The primary objective of this study is to develop new computational methods for robust design optimization (RDO) and reliability-based design optimization (RBDO) of high-dimensional, complex engineering systems. Four major research directions, all anchored in polynomial dimensional decomposition (PDD), have been defined to meet the objective. They involve: (1) development of new sensitivity analysis methods for RDO and RBDO; (2) development of novel optimization methods for solving RDO problems; (3) development of novel optimization methods for solving RBDO problems; and (4) development of a novel scheme and formulation to solve stochastic design optimization problems with both distributional and structural design parameters.</p> <p>The major achievements are as follows. Firstly, three new computational methods were developed for calculating design sensitivities of statistical moments and reliability of high-dimensional complex systems subject to random inputs. The first method represents a novel integration of PDD of a multivariate stochastic response function and score functions, leading to analytical expressions of design sensitivities of the first two moments. The second and third methods, relevant to probability distribution or reliability analysis, exploit two distinct combinations built on PDD: the PDD-SPA method, entailing the saddlepoint approximation (SPA) and score functions; and the PDD-MCS method, utilizing the embedded Monte Carlo simulation (MCS) of the PDD approximation and score functions. For all three methods developed, both the statistical moments or failure probabilities and their design sensitivities are both determined concurrently from a single stochastic analysis or simulation. Secondly, four new methods were developed for RDO of complex engineering systems. The methods involve PDD of a high-dimensional stochastic response for statistical moment analysis, a novel integration of PDD and score functions for calculating the second-moment sensitivities with respect to the design variables, and standard gradient-based optimization algorithms. The methods, depending on how statistical moment and sensitivity analyses are dovetailed with an optimization algorithm, encompass direct, single-step, sequential, and multi-point single-step design processes. Thirdly, two new methods were developed for RBDO of complex engineering systems. The methods involve an adaptive-sparse polynomial dimensional decomposition (AS-PDD) of a high-dimensional stochastic response for reliability analysis, a novel integration of AS-PDD and score functions for calculating the sensitivities of the failure probability with respect to design variables, and standard gradient-based optimization algorithms, resulting in a multi-point, single-step design process. The two methods, depending on how the failure probability and its design sensitivities are evaluated, exploit two distinct combinations built on AS-PDD: the AS-PDD-SPA method, entailing SPA and score functions; and the AS-PDD-MCS method, utilizing the embedded MCS of the AS-PDD approximation and score functions. In addition, a new method, named as the augmented PDD method, was developed for RDO and RBDO subject to mixed design variables, comprising both distributional and structural design variables. The method comprises a new augmented PDD of a high-dimensional stochastic response for statistical moment and reliability analyses; an integration of the augmented PDD, score functions, and finite-difference approximation for calculating the sensitivities of the first two moments and the failure probability with respect to distributional and structural design variables; and standard gradient-based optimization algorithms, leading to a multi-point, single-step design process.</p> <p>The innovative formulations of statistical moment and reliability analysis, design sensitivity analysis, and optimization algorithms have achieved not only highly accurate but also computationally efficient design solutions. Therefore, these new methods are capable of performing industrial-scale design optimization with numerous design variables.</p>

This paper describes a method for structural optimization considering the elastoplastic structural behaviour. The proposed method consists of three parts : elastoplastic incremental structural analysis, elastoplastic sensitivity analysis, and optimization. The structural analysis and sensitivity analysis are performed using the incremental finite element method. For structural optimization, the Sequential Unconstrained Minimization Technique (SUMT) is used.In the optimization process, reanalysis is performed using the elastoplastic sensitivities. To get rational reanalysis solutions, some limitations should be imposed on the magnitude of changes in design variables. The maximum allowable changes in design variables are automatically evaluated by applying repeated sensitivity analysis during the optimization process.After verifying the accuracy of the reanalysis solution based on repeated sensitivity analysis, a three-bar and a ten-bar plane trusses are analyzed taking the cross-sectional areas as design variables. Comparison of the results shows that the proposed method gives very good results.

Three methods for design sensitivity such as numerical differentiation, analytical method and semianalytical method have been developed for the last three decades. Although analytical design sensitivity analysis is exact, it is hard to implement for practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable for most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate because its computational cost depends on the number of design variables and large numerical errors can be included especially in nonlinear design sensitivity analysis. Thus semi-analytical method is more suitable for complicated design problems. Moreover semianalytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure for the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and computational technique is proposed, which evaluates the pseudo-load for design sensitivity analysis easily by using the design variation of corresponding internal nodal forces. Errors in semi-analytical design sensitivity analysis are examined and numerical examples are illustrated to confirm the reduction of numerical error considerably.

α = magnitude of the change in design variables β = new design variable (Vanderplaats’ formulation) e = relative error λ = buckling load factor σ = normal stress c = positive constant (Vanderplaats’ formulation) F = objective function g = design constraint i = design variable index j = active constraint index p = parameter ∆p = parameter increase r = inactive constraint index S = vector of feasible directions w = vertical displacement X = vector of design variables X = vector of optimum design variables Xaprox = vector of approximate optimum design variables

In this paper, a novel optimal shape design method is proposed using the finite-difference time-domain (FDTD) method and the design sensitivity analysis to obtain broad-band characteristics of microwave devices. In shape design problem, the nodes that describe the shape of geometry to be optimized are taken as design variables. The design sensitivity is evaluated using the adjoint variable equation that is obtained from a terminal-value problem. The adjoint equation can be solved by the FDTD technique with the backward time scheme. With this method, a Ka-band unilateral fin line is tested to show validity.

A shape design sensitivity analysis (DSA) and optimization of structural transient dynamics are proposed for the finite deformation elastoplastic materials under impact with a rigid surface. A shape variation of the structure is considered using the material derivative approach in continuum mechanics. Hyperelasticitybased multiplicatively decomposed elastoplasticity is used for the constitutive model. The implicit Newmark time integration scheme is used for the structural dynamics. The design sensitivity equation is solved at each converged time step with the same tangent stiffness matrix as response analysis without iteration. The cost of sensitivity computation is more efficient than the cost of response analysis for the implicit time integration method. The efficiency and the accuracy of the proposed method are shown through the design optimization of a vehicle bumper. *Communicated by D. Tortorelli.

Design sensitivity analysis and topology optimization of displacement–loaded non-linear structures

Recent developments of the modal-based aeroelastic optimization approach facilitated efficient treatment of static-aeroelastic and stress constraints. The reduced-size models use low-frequency normal modes of the baseline structure as fixed generalized coordinates throughout the optimization process. The modal approach is extended to allow the expansion of the modal basis by the addition of static modes generated in previous design steps. Being based on modal perturbations stored in the modal data base before the optimization process starts, the basis expansion accelerates the convergences to the optimal design. Numerical examples with realistic fighter- aircraft models demonstrate practical applications with CPU speed-up factors of ~10, compared with the regular discrete-coordinate approach, with negligible loss of accuracy. HE desire for efficient procedures for optimal design of com- plex structures motivated the development of reduced-size op- timization schemes in which calculations of stability and response parameters and their sensitivity to changes in the design variables are based on a set of low-frequency vibration modes of a base- line structure. The modal approach is especially attractive in multi- disciplinary cases in which the excitation loads are affected by the structural response, such as in aeroelastic and control-augmented systems. The ASTROS code1 was developed to provide a multidisciplinary analysis and design capability for aerospace structures. The con- sidered disciplines include structural analysis, aeroelastic analysis, and some features of the interaction with the control system. As in other commonly used analysis and optimization schemes,24 the dynamic response and the stability features are treated by the modal approach, but the static-aeroelastic and stress disciplines are treated by the discrete approach. The use of modal coordinates in static-aeroelastic analysis was shown in Ref. 5 to be extremely efficient and of good accuracy in application to realistic aircraft models. Structural optimization with static-aeroelastic constraints was later performed with the base- line normal modes serving as a fixed set of generalized coordinates throughout the entire optimization run.6 The fixed-modal-basis ap- proach was shown in Ref. 6 to be of good accuracy with design variable changes of less than ~20%. Larger changes required the update of the full finite-element model such that the optimization process could be resumed with new modal coordinates. Even when the modal approach was successfully applied in multidisciplinary optimization with static- and dynamic-aeroelastic constraints,7 it was not ready yet for application to stress constraints. The main reason for not using the modal approach for stresses was that it might yield erroneous results in cases of concentrated loads and in cases of local structural changes. Recent developments of the

We study the use of Hessian interaction terms to quickly identify design variables that reduce variability of system performance. To start we quantify the uncertainty and compute the variance decomposition to determine noise variables that contribute most, all at an initial design. Minimizing the uncertainty is next sought, though probabilistic optimization becomes computationally difficult, whether by including distribution parameters as an objective function or through robust design of experiments. Instead, we consider determining the more easily computed Hessian interaction matrix terms of the variance-contributing noise variables and the variables of any proposed design change. We also relate the Hessian term coefficients to subtractions in Sobol indices and reduction in response variance. Design variable changes that can reduce variability are thereby identified quickly as those with large Hessian terms against noise variables. Furthermore, the Jacobian terms of these design changes can indicate which design variables can shift the mean response, to maintain a desired nominal performance target. Using a combination of easily computed Hessian and Jacobian terms, design changes can be proposed to reduce variability while maintaining a targeted nominal. Lastly, we then recompute the uncertainty and variance decomposition at the more robust design configuration to verify the reduction in variability. This workflow therefore makes use of UQ/SA methods and computes design changes that reduce uncertainty with a minimal 4 runs per design change. An example is shown on a Stirling engine design where the top four variance-contributing tolerances are matched with two design changes identified through Hessian terms, and a new design found with 20% less variance.

In this paper, continuum-based sizing and configuration design sensitivity analysis (DSA) and optimization methods using the established transient dynamic analysis code DYNA3D are developed. For transient dynamic analysis, the explicit central difference method is used. For the structural domain, the updated Lagrangian formulation is used since it is appropriate for the path dependent problem. The first-order variations of the kinetic energy form, strain energy form, load form, and kinematic and structural responses with respect to the sizing and configuration design variables are derived. For the updated Lagrangian formulation, the design velocity field that defines the mapping between the initial and perturbed designs is updated at each updated configuration. The direct differentiation method is used in this paper for DSA. Numerical implementation of sizing DSA, configuration DSA, and optimization of a vehicle frame and A-pillar models has been carried out using the nonlinear finite element analysis (FEA) code DYNA3D.

A state space design sensitivity analysis and optimization method is presented in which problems of optimal design of machines are formulated in a setting that allows treatment of general design objectives and constraints. Synthesis of machines to perform both kinematic and kinetic functions is considered. A Cartesian coordinate formulation is employed for position, velocity, acceleration, and force analysis. An adjoint variable technique is employed to compute derivatives with respect to design of general cost and constraint functions that involve kinematic, force, and design variables. Linearization (or sequential quadratic programming) and gradient projection optimization algorithms are employed, using the design sensitivity analysis method developed, for design optimization. Four optimal design problems are solved to demonstrate use of the method.

Topology design optimization of structures at high frequencies using power flow analysis

A continuum-based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non-linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non-linear dynamic problems, such as crashworthiness design. The first-order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path-dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate-based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.

Improved semi-analytic sensitivity analysis combined with a higher order approximation scheme in the framework of adjoint variable method

<div class="htmlview paragraph">A procedure is described for rapidly correlating finite element models to modal test results using a combination of design sensitivity and optimization methods. The design sensitivity coefficients estimate the effect of many possible model changes or design variables. Optimization methods are used to select optimum changes which minimize the error between test and analysis results with minimum change to the finite element model. Weighting factors provide flexibility for correlating selected modes or design variables. The procedure is demonstrated using a spacecraft model with up to 50% error in initial properties.</div>