Design Sensitivity Analysis and Optimization of Kinematically Driven Systems

Problems of optimal design of mechanisms are formulated in a state space setting that allows treatment of general design objectives and constraints. A constrained multi-element technique is employed for velocity, acceleration, and kineto-static analysis of mechanisms. An adjoint variable technique is employed to compute derivatives with respect to design of general cost and constraint functions involving kinematic, force, and design variables. A generalized steepest descent optimization algorithm is employed, using the design sensitivity analysis methods developed, as the basis for a general purpose kinematic system optimization algorithm. Two optimal design problems are solved to demonstrate effectiveness of the method.

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.

Design sensitivity analysis and optimization for nonlinear buckling of finite-dimensional elastic conservative structures

<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>

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

The computational cost of reliability analysis and optimization of medium/large structural systems is prohibitively high. One of the reasons for this situation is the use of inefficient methods, such as finite differences, for design sensitivity analysis. A significant increase in the efficiency of reliability computations is expected from the integration of reliability methods with the discrete and continuum methods of design sensitivity analysis. So far, only the discrete method has been used in the so-called stochastic finite element method. Integration of reliability computations with the continuum method of design sensitivity analysis is proposed in this paper as a new, powerful approach to the probabilistic design of large structural systems. Basic formulations and solution methods of reliability analysis, design sensitivity analysis of reliability indices, reliability-based optimization, and multiobjective reliability-based optimization are reviewed with focus on the sensitivity informatio...

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.

The objective of this study is to examine two analytical methods for the design sensitivity analysis of optimal control problems and to develop the analyses necessary for the application of the methods to thermal systems. The two methods are the direct differentiation method and the adjoint variable method. Results for these methods are compared with those obtained using central finite difference approximations for design sensitivity analysis. The adjoint variable method has the potential to reduce significantly computational times for design sensitivity analysis while sacrificing only a small amount of accuracy. As the number of functional requiring design sensitivity analysis increases, however, the adjoint variable method becomes less attractive and the direct differentiation method becomes more attractive.

Design sensitivity analysis (DSA) and topology design optimization methods forpiezoelectric resonators are developed. The governing equations for piezoelectric vibrationproblems are the Cauchy’s equation of motion and the charge equation of electrostatics. AMindlin plate theory using higher-order interpolations along the thickness direction that iseffective for the thickness-shear vibration analysis of piezoelectric resonators is used. Anoptimal design problem is formulated to maximize the electromechanical coupling strengthat the mode of interest. Necessary design gradients for eigenvalue problems are computedvery efficiently and accurately using an analytical DSA method with an eigenvectorexpansion concept. A robust mode-tracking method based on thickness-shear strain energyis proposed to precisely identify the thickness-shear mode at high frequencies.Through demonstrative numerical examples, the DSA method developed is verifiedto show excellent efficiency and agreement compared with the finite differencemethod. It is observed that the frequencies are almost invariant even though the sizeand shape of the crystal varies significantly. However, it is observed that theoptimal topology is sensitive to the aspect ratio even if the thicknesses are thesame.

ACOMPUTATIONAL method for design sensitivity analysis of an eigenvalue and an eigenvector of a beam under nonlinear forced vibration is presented in this paper. The nonlinear vibration problem is only analyzed once in the proposed method. The geometric nonlinearity of concern results from the large deflection of a beam. The finite element system equation for nonlinear vibration is symmetric. However, it is found that the equation for computing the design sensitivity of an eigenvector is linear and unsymmetric. A numerical example is included to validate the proposed computational procedure. Contents The nonlinear vibrations studied herein refer to the periodic (though not necessarily harmonic) and stable motions of a nonlinear system. Recently, Hou and Yuan1 addressed the design sensitivity analysis of eigenvalues and eigenvectors of a beam under nonlinear free vibration. In that paper, the longitudinal inertia was not considered in the nonlinear vibration formulation, so that the problem could be expressed in terms of the lateral deflection alone. In the present work the design sensitivity analysis has been extended successfully to a nonlinear forced vibration problem whose formulation includes the in-plane inertia and displacement. The variational formulation2 for the nonlinear forced vibrations of beams under harmonic excitation is given as

Explicit and implicit methods for design sensitivity analysis of nonlinear structures subjected to transient dynamic loads are described. The direct differentiation method is used to calculate the sensitivities. Elastic and elasto-plastic problems undergoing large deformation are treated in the formulation. The explicit method is quite straightforward to implement for analysis as well as design sensitivity analysis: knowing all the quantities at time t, one updates the quantities for the next time point without solution of any linear system of equations, or any iterations. For the implicit method, the effective stiffness matrix for the sensitivity equation is different from that for the analysis phase due to inclusion of large deformation in the formulation. A numerical procedure to integrate the equation is proposed which uses the symmetric effective stiffness matrix from the analysis phase. Although the sensitivity analysis problem is linear, the procedure requires iterations for sensitivity calculations at each time step, as for the analysis phase. The implicit method is implemented using the pre-conditioned conjugate gradient iterative solution scheme. Numerical results for sensitivities indicate the implicit method to be more accurate than the explicit method for transient elastoplastic problems. However, the explicit method is substantially more efficient than the implicit method.

Sensitivity analysis of eigenmodes and dynamic responses for intelligent structures

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

Design sensitivity analysis for transient response of non-viscously damped systems based on direct differentiate method

Isogeometric configuration design sensitivity analysis of geometrically exact shear-deformable beam structures