Adjoint design sensitivity analysis and optimization of nonlinear structures using geometrical mapping approach

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 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.

This paper presents the implementation of the boundary element method to shape sensitivity analysis of elastic structures with stress concentrators. An elastic body which contains a number of voids (internal boundaries), playing the role of stress concentrators, is considered. We are interested in calculating the first order sensitivity of shape–dependent functionals with respect to the shape variation of the body domain. This task is accomplished using the adjoint variable method. As it has been shown by Dems and Mróz [10], for a basic transformation (i.e. a translation, a rotation or a scale change) of the body, the sensitivity of the considered functional takes the form of a path-independent integral (PII) whose integrand depends on the primary and adjoint state fields, along an arbitrary path (curve in 2D, surface in 3D), enclosing the transformed stress concentrator (void). It is very important for numerical computations, because we can compute this integral along the path placed far from the stress concentrator to eliminate the negative influence of stress concentrations on the accuracy of calculations. The boundary element method (BEM) is used to solve both primary and adjoint problems. Some important cases of adjoint problems related to functionals are analyzed in the paper. A thorough numerical verification of the proposed method is performed in this work. The presented method of sensitivity analysis is utilised in gradient-based optimization and identification problems. Numerical examples of optimization and identification are shown in this paper.

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

Three methods for design sensitivity analysis such as finite difference method(FDM), direct differentiation method(DDM) and adjoint variable method(AVM) are well known. FDM and DDM for design sensitivity analysis cost too much when the number of design variables is too large. An AVM is required to compute adjoint variables from the simultaneous linear system equation, the so-called adjoint equation. Because the adjoint equation is independent of the number of design variables, an AVM is efficient for when number of design variables is too large. In this study, AVM has been extended to the eigenproblem of damped systems whose eigenvlaues and eigenvectors are complex numbers. Moreover, this method is implemented into a commercial finite element analysis program by means of the semi-analytical method to show applicability of the developed method into practical structural problems. The proposed method is compared with FDM and verified its accuracy for analytical and practical cases.

A numerical method of structural dynamic shape optimization is presented by using reproducing kernel particle method (RKPM), by which the mesh distortion that exists in shape optimal method based on finite element can be eliminated completely and the optimal model for structural dynamic optimization design is built. The discreteness-based design sensitivity analysis in both natural frequency and dynamic response is proposed by using direct differentiation method and discrete derivatives on the basis of structural dynamic analysis, in which the penalty method is employed into imposing the essential boundary conditions, and the derivatives of shape functions with respect to design variables are derived. The algorithm of dynamic sensitivity analysis is testified by numerical example, and the numerical results obtained are in good agreement with those obtained using semi-analytical method and global finite differences method. Finally, by integrating the algorithm mentioned based on RKPM with parameterized descriptive method of boundary shape, two examples for structural dynamic shape optimization are performed.

A continuum-based design sensitivity analysis method for hyperelastic structures is presented. Analysis is performed using a meshless method, called the reproducing kernel particle method. Yeoh’ s energy density function is used to describe thehyperelastic structural behavior. The meshless method eliminates mesh distortion orentanglement encountered in using e nite element analysis for large deformation structural analysis and structural shape design optimization. Both the adjoint variable and direct differentiation methods are developed for material and shape design variables. An ine nitely long rubber tube, a two-dimensional rubber band, and an engine mount are examples used to demonstrate the feasibility and accuracy of the method.

A direct differentiation method and an adjoint variable method are proposed for the efficient semi-analytical evaluation of the sensitivities of multibody systems formulated in a matrix Lie group framework. These methods rely on the linearization of the equations of motion and/or of the time integration procedure. The simpler structure of the equations of motion in the Lie group formalism appears as an advantage for that purpose. Lie bracket contributions and the non-linearity of the exponential map need to be taken into account in the sensitivity algorithms. Nevertheless, essential characteristics of formulations of the direct differentiation method and the adjoint variable method on linear spaces are recovered. Some implementation issues are discussed and two relevant examples illustrate the properties of these methods.

Wind resistant size optimization of geometrically nonlinear lattice structures using a modified optimality criterion method

Recent developments in the sensitivity analysis for nonlinear structural mechanics and dynamics problems are reviewed. The activities are grouped into two general categories, namely, computational strategies and facilities for evaluating the sensitivity coefficients, and new applications. Brief description is given of the activities pertaining to the first category. More detailed discussion is presented for two recent applications: a) hierarchical sensitivity coefficients of the various response quantities of sandwich panels with composite face sheets with respect to the different laminate, layer and micromechanical characteristics of the face sheets and core; and b) sensitivity analysis of the large strain response of viscoplastic solids subjected to dynamic loading. Sample numerical results are presented and some of the future directions for research on sensitivity analysis of large-scale nonlinear structures are outlined.

Design sensitivity analysis of critical load factor for nonlinear structural systems

In the stochastic sensitivity analysis, a large number of simulation models lead to low computational e‐ciency. With the dimension increasing in assigned problems, the accurate is di‐cult to achieve by popular regression methodologies. Without considering the correlations of parameters, inaccurate sensitivity coe‐cients would be calculated by analyzing the efiect of parametric variables on structures, moreover, the existing methods only calculate the local gradient as sensitivity. According to these problems, approximate model and global sensitivity method are employed for design sensitivity analysis of structures. The approximate model is constructed by the hybrid neural network which possesses signiflcant learning capacity and generalization capability with a small amount of information. The improved chaotic particle swarm is applied to optimize the neural network so as to improve the computational e‐ciency and accuracy. The proposed sensitivity analysis method values the global response of the outputs by varying all the input parameters at a time with the correlations of parameters. Uniform design and Latin hypercube sampling are used to sample points. Numerical analysis show that the proposed method can successfully measure the actual sensitivity.

An incremental finite element formulation for structural sensitivity analysis or nonlinear (possibly path dependent) systems with fixed overall shape is proposed. Both the direct differentiation and adjoint variable methods, as well as a mixed differentiation-adjoint technique are employed to evaluate 1st- and 2nd-order sensitivity increments during the load step. Numerical illustration is given.

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

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