Sensitivities of displacements and stresses with respect to changes in shape and topology

1. Abstract This presentation deals with sensitivity calculations for load-carrying linearly elastic continuum structures which are discretized by a finite element model. The topology and shape of the considered structure are represented by binary design variables indicating material or void in the various finite elements. Elements containing material are called “black” while elements without material are called “white”. Assume that a certain design, defined by the values of the binary design variables, has been analyzed. This means, in particular, that the corresponding stiffness matrix has been factorized and that the displacements as well as the stresses under the given load conditions have been calculated. Assume further that one design variables should be changed to its opposite value, which means that the corresponding element should be changed from white to black or from black to white. The question addressed here is how to calculate the new displacements and stresses in the structure by exploiting the fact that the new design is very similar to the old one. We answer that question by presenting a fast method which does the required calculations without any need for a refactorization of the stiffness matrix, without any approximations, and without any need for artificial lower bounds on the ”densities” in void elements, which means that it is perfectly valid that the number of degrees of freedom (dofs) for the structure is changed when a design variable is changed!