The Ritz method and its application to structural optimization

The Ritz method is presented for minimizing a cost functional of the special form J(u) = S L/0 u(x)dx subject to differential constraints which are non-linear in the control u(x), and which have the form q' (x) = A(u,x)q(x). Through the use of a suitable space of cubic splines on a mesh of norm h on the interval [0,L], the method is used to minimize J(u) and leads to an approximate solution of the constraints q'(x). The application of this method is demonstrated for two problems in structural optimization. The first example deals with minimizing the weight of a column under a critical load, and the second example shows an interesting case of requiring a geometric constraint on the design variable to arrive at a minimum weight of a beam on transverse vibrations for a specified natural frequency.