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

Using the bond-based peridynamics and the parallel computation with binary decomposition, an adjoint shape design sensitivity analysis(DSA) method is developed for the dynamic crack propagation problems. The peridynamics includes the successive branching of cracks and employs the explicit scheme of time integration. The adjoint variable method is generally not suitable for path-dependent problems but employed since the path of response analysis is readily available. The accuracy of analytical design sensitivity is verified by comparing it with the finite difference one. The finite difference method is susceptible to the amount of design perturbations and could result in inaccurate design sensitivity for highly nonlinear peridynamics problems with respect to the design. It turns out that C1-continuous volume fraction is necessary for the accurate evaluation of shape design sensitivity in peridynamic discretization.

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.

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...

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.

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.

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 optimization for nonlinear buckling of finite-dimensional elastic conservative structures

New adjoint variable method for design sensitivity analysis of distinct eigenvlaues and eigenvectors is presented. In the viewpoint of efficiency for the design sensitivity analysis of eigenvectors especially, the developed adjoint variable method is required to compute adjoint variables from simultaneous linear equations, the so-called adjoint equations, instead of linear combination of eigenvectors. Once we obtain the adjoint variables, design sensitivity analysis of response function that is given in terms of eigenvalues, eigenvectors and design variables can be computed directly. In this way, design sensitivity analysis of eigenvectors can be obtained by using eigenvalues and their corresponding eigenvectors of the mode being differentiated only. To verify the proposed method, numerical examples are demonstrated. This can have considerable impact on computer implementation of the developed method in the design sensitivity analysis of eigenproblem needed for practical applications.

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

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.

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

This report first analysis quasi static power flow for photovoltaic power plant. Then, fault analysis for different symmetrical and asymmetrical fault is done. The system used is IEEE 39 bus New England system that a Photovoltaic farm with 500 MW power is connected to bus 24. Newton-Raphson Power flow method is used with Gauss-Jordan elimination method for matrix inversion in this power flow analysis. Different operation condition including faults are also considered for fault at bus 16. The results are validated with simulation in MATLAB software.