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; Barthelemy et al. 1988; Barthelemy and Haftka 1988; Choi and Twu 1991, Pedersen et al. 1989; Cheng et 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 Barthelemy et al. (1988), Barthelemy and Haftka (1988), Pedersen et 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).