Design sensitivity analysis and optimization of nonlinear structural dynamics using DYNA3D

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.

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

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...

Using the direct differentiation method, a design sensitivity analysis method for time-dependent incompressible fluids is developed. The fluid behavior is described as the motion of particles involved by the SPH method. In the SPH projection method, instead of changing the fluid density, incompressibility is enforced by the pressure Poisson equation derived from pressure projection, which enable to use larger time steps. In spite of the additional pressure Poisson equation, the computational cost for the design sensitivity is not expensive since the factorized system matrix of pressure Poisson equation can be utilized. Aforementioned computational efficiency is very beneficial for the design sensitivity computation required for every time step in explicit time integration and updated Lagrangian schemes, for which an update scheme of design velocity field is developed using the velocity sensitivity. Through demonstrative numerical examples, the developed DSA method turns out to be efficient and shows excellent agreement with finite differencing.

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

Design sensitivity analysis of sheet metal forming processes with a direct differentiation method

Design Sensitivity Analysis (DSA) is a procedure for obtaining derivatives of design objectives, such as mass and strength with respect to some design parameters, such as size and shape. In the DSA methodology, we take an existing technique for computing objectives from parameters and systematically differentiate it so that the derivatives of the objectives can be computed directly, rather than employing some sort of differencing scheme to approximate derivatives. In this paper, we apply this technique to a non-linear finite element program, Radioss, and begin looking at the properties of the DSA methodology for crashworthiness simulations. It is shown that some major computational advantages can be gained by employing DSA for these kinds of problems. Besides cost savings, there are many cases where numerical differentiation results in erroneous derivatives due to numerical noise in crash simulations. It is shown that the DSA method avoids these difficulties, thus, making it quite important for use in support of design activities.

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

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

Sensitivity analysis of eigenmodes and dynamic responses for intelligent structures

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

Design sensitivity analysis of dynamic response of nonviscously damped systems