Sensitivity analysis of eigenmodes and dynamic responses for intelligent structures

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.

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

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...

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.

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.

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.

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

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

A parameter optimization method based on sensitivity analysis is presented for the structural optimization of variable section slender manipulator. Structure mechanism of a polishing robot is introduced firstly, and its stiffness model is established. Then, a design sensitivity analysis method and a sequential liner programming (SLP) strategy are proposed. In the beginning of the optimization, the design sensitivity analysis method can be used to select the sensitive design variables which can make the optimized results more efficient and accurate. And then, it can be used to improve the convergence during the process of the optimization. The design sensitivities are calculated using the finite difference method. The search for the final optimal structure is performed using the SLP method. Simulation results show that the structure optimization method is effective to enhance the stiffness of the manipulator, no matter when the manipulator suffers constant force or variable force. This work lays a theoretical foundation for the structural optimization for such manipulators.

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.

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

Direct differentiation method for response sensitivity analysis of a bounding surface plasticity soil model

Design sensitivity analysis of dynamic responses for a BLDC motor with mechanical and electromagnetic interactions

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.

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