Incremental Finite Element Analysis of Nonlinear Structural Design Sensitivity Problems

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.

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 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.

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

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

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.

Design sensitivity analysis of dynamic response of nonviscously damped systems

In this paper, the mean-centered first-order second-moment (FOSM) method is employed to perform probabilistic push-over analysis (POA) of structural and/or soil-structure systems. Approximations of first and second statistical moments (FSSMs) of engineering demand parameters (EDPs) of structural and/or geotechnical systems with random material parameters are computed based on finite-element (FE) response and response sensitivity analysis (RSA) results. The FE RSA is performed accurately and efficiently by using the direct differentiation method (DDM) and is employed to evaluate the relative importance (RI) of the various modeling material parameters in influencing the variability of the EDPs. The proposed approximate methodology is illustrated through probabilistic POA results for nonlinear inelastic FE models of: (1) a three-story reinforced-concrete (RC) frame building and (2) a soil-foundation-structure interaction system consisting of a RC frame structure founded on layered soil. FSSMs of EDPs computed through the FOSM method are compared with the corresponding accurate estimates obtained via Monte Carlo simulation. Results obtained from exact (or local) and averaged (or global) response sensitivities are also compared. The RI of the material parameters describing the systems is studied in both the deterministic and probabilistic sense, and presented in the form of tornado diagrams. Effects of statistical correlation between material parameters are also considered and analyzed by the FOSM method. A simple approximation of the probability density function and cumulative distribution function of EDPs due to a single random parameter at a time (while all the other parameters are fixed to their mean values) is also proposed. Conclusions are drawn on both the appropriateness of using local RSA for simplified probabilistic POA and on the application limits of the FOSM method. It is observed that the FOSM method combined with the DDM provides accurate estimates of FSSMs of EDPs for low-to-moderate level of inelastic structural or system behavior and useful qualitative information on the RI ranking of material parameters on the structural or system response for high level of inelastic behavior.

Forced expression of Hnf4a induces hepatic gene activation through directed differentiation

Direct differentiation formulation for boundary element shape sensitivity analysis of axisymmetric elastic solids

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

Boundary element based formulations for crack shape sensitivity analysis

A new approach toward reliability based design optimization (RBDO) is proposed based on the response surface augmented moment method (RSMM). In RSMM, the reliability analysis procedure based on design of experiments (DOE) is combined with the response surface method (RSM) for numerical efficiency. It utilizes the Pearson system with four statistical moments to calculate the failure probability, and the progressive update of the response surface facilitates the calculation of these four statistical moments. In this study, a semi-analytic design sensitivity analysis is performed in connection with RSMM for an efficient implementation of RSMM in RBDO. The sensitivity of failure probability with respect to the design variables is calculated by direct differentiation and finite difference method with the Pearson system. It is integrated into a mathematical programming for RBDO and applied to several test examples. It was demonstrated that the proposed method of RBDO based on RSMM is efficient and robust.

Shape sensitivity analysis of thin-shell structures

Sensitivity analysis of the dynamic response of a frame. Part I: Direct differentiation method