Weight Optimisation of Structures under Displacment Limitations by Interpolating Constraints

Structural optimisation problems are very complex to solve in practice. Even in the simplest case where the shape of the structure is given and the optimisation is only about some discrete sizing variables, the associated mathematical program is very intricate to formulate since the problem lacks in general to have a closed analytical form. When an explicit formulation could be derived, the resulting problem is habitually non convex and non linear, some extra difficulties may then attend using the common numerical tools to solve the mathematical program. Weight minimisation of continuous beams under displacement limitations pertains to this last category of structural optimisation problems. In this work, a methodology based on performing interpolations of the intervening problem constraints is developed in order to formulate an approximate mathematical program. Constraint interpolations are obtained by using quadratic polynomials and finite element computations. By considering special embedded domains where these interpolations are sequentially performed, a series of approximate problems is then constructed. If a feasible initial domain is used, this sequence is shown to yield through numerical experiments, after only a few number of iterations, an arbitrary close estimate of the exact original problem solution