Rigorous improvement of semi-analytical design sensitivities by exact differentiation of rigid body motions

In the recent past inaccuracy problems have been reported that arise when computing shape design sensitiv- ities by the semi-analytical method. Since both the analytical and the global nite-di erence method do not show these severe inaccuracies, it has been concluded that these errors are due to the numerical di erenti- ation of the nite-element sti ness matrices, which is inherent in the semi-analytical method. Moreover, it has also been observed that these inaccuracies become especially dominant when relatively large rigid body motions can be identi ed for individual elements. So far, improvements to the semi-analytical method are focusing on the numerical di erentiation of the nite-element sti ness matrices. It is shown in the present paper that the contribution to the design sensitivities corresponding to the rigid body motions can be evalu- ated by exact di erentiation of the rigid body modes. This approach requires only minor programming e ort and the additional computing time is very small. As shown by numerical examples, the proposed method eliminates the problem of abnormal errors occurring in the semi-analytical method.