Semi-analytical Sensitivity Analysis Research Articles

A Method of “Exact” Numerical Differentiation for Error Elimination in Finite-Element-Based Semi-Analytical Shape Sensitivity Analyses*

ABSTRACT The traditional, simple numerical differentiation of finite-element stiffness matrices by a forward difference scheme is the source of severe error problems that have been reported recently for certain problems of finite-element-based, semi-analytical shape design sensitivity analysis. In order to develop a method for elimination of such errors, without a sacrifice of the simple numerical differentiation and other main advantages of the semi-analytical method, the common mathematical structure of a broad range of finite-element stiffness matrices is studied in this paper. This study leads to the result that element stiffness matrices can generally be expressed in terms of a class of special scalar functions and a class of matrix functions of shape design variables that are defined such that the members of the classes admit “exact” numerical differentiation (exact up to round-off error) by means of very simple correction factors to upgrade standard computationally inexpensive first-order finite di...

Study of inaccuracy in semi-analytical sensitivity analysis — a model problem

The semi-analytical method of sensitivity analysis (Zienkiewicz and Campbell 1973; Esping 1983; Cheng and Liu 1987) of finite element discretized structures is attractive due to the balance between computational cost and ease of implementation (Cheng and Liu 1987; Haftka and Adelman 1989), but unfortunately the method may exhibit serious inaccuracies when applied in shape optimization of structures modelled by beam, plate, shell and Hermite elements (Cheng and Liu 1987; Haftka and Adelman 1989; Barthelemyet al. 1988; Barthelemy and Haftka 1988; Choi and Twu 1991, Pedersenet al. 1989; Chenget al. 1989). In the present paper, we perform an exact analysis of the error of sensitivity for a simple model problem which has earlier been considered by Barthelemyet al. (1988), Barthelemy and Haftka (1988), Pedersenet al. (1989). The analysis gives a deep insight into the nature of the general inaccuracy problem and enables us to devise methods by which the severe error of the sensitivity can be substantially reduced or removed for the model problem. The results of the paper are illustrated via an example. A method of error elimination for an extended class of semianalytical analysis problems is developed and presented in a companion paper (Olhoff and Rasmussen 1991).

Adaptive grid refinement in a bem-based optimal shape synthesis

The boundary element method is used in conjunction with nonlinear programming methods for optimal structural shape synthesis. An approach for grid refinement and grid adaptation for the boundary element method is developed for application in plane elasticity problems. This technique uses a predefined control function in a variational formulation to obtain an optimal node distribution for the computational domain. To prevent distortion of the original domain, master nodes are introduced to control the geometry of the structure. The paper also describes an efficient implementation of a semi-analytical sensitivity analysis for optimum design. Numerical results are presented for a class of shape synthesis problems.

Accuracy of semi-analytic sensitivity analysis

The integration of FEM, CAD and optimization is becoming an important area of research in the field of computational mechanics. Implementing the integration, one of the main difficulties encountered is to perform a sensitivity analysis for the structural response with respect to any design variable. It has been shown in the literature that the semi-analytic sensitivity analysis method provides an ideal tool to solve this difficulty. Nevertheless, the accuracy of the method remains to be studied. The present paper addresses the accuracy problem. Structures composed of beam elements, 3-node constant plane stress triangular elements, 4-node plane stress isoparametric elements, 8-node plane stress isoparametric elements, 8-node solid elements, and 3-node triangular bending elements are studied. It is found that for beam and plate elements, whose nodal degrees of freedom have non-uniform dimensions, the error caused by the semi-analytic sensitivity analysis method is almost proportional to the square of the number of elements. An alternative forward-backward finite difference scheme solves the problem nicely. For the other elements, the accuracy of the semi-analytic sensitivity analysis is acceptable. The results obtained in the present paper justify the usage of the semi-analytic sensitivity analysis method.