Optimal geometry design of truss structure (Application of nodal positional differentiation of stiffness matrix in general form)

The geometry of truss structure is expressed in terms of its nodal positions; that is, the nodal positions vector is the fundamental design variable for the optimal geometry design of a truss. In order to solve an optimal design problem, there are various approaches that uses differentiation of the objective function in terms of the design variable, such as the steepest descent method and the gradient projection method. Many of the objective functions of structural optimal design problems are formulated in terms of the stiffness matrix; the gradient or differentiation approach requires differentiation of the stiffness matrix in terms of the nodal positions vector. It is, however, not impossible but still impractical in many cases since we have to deal with the third-order differential coefficient tensor. In this paper, we deal with the differentiation of the product of the stiffness matrix and a vector in terms of the nodal positions vector. We develop a formulation in general form that can be applied to trusses of any configuration. Examples of optimal geometry designs of 2D and 3D trusses are demonstrated.