TOPOLOGICAL DERIVATIVE-BASED TOPOLOGY OPTIMIZATION OF STRUCTURES SUBJECT TO MULTIPLE LOAD-CASES

Abstract

THE TOPOLOGICAL DERIVATIVE MEASURES THE SENSITIVITY OF A SHAPE FUNCTIONAL WITH RESPECT TO AN INFINITESIMAL SINGULAR DOMAIN PERTURBATION, SUCH AS THE INSERTION OF HOLES, INCLUSIONS OR SOURCE-TERMS. THE TOPOLOGICAL DERIVATIVE HAS BEEN SUCCESSFULLY APPLIED IN OBTAINING THE OPTIMAL TOPOLOGY FOR A LARGE CLASS OF PHYSICS AND ENGINEERING PROBLEMS. IN THIS PAPER THE TOPOLOGICAL DERIVATIVE IS APPLIED IN THE CONTEXT OF TOPOLOGY OPTIMIZATION OF STRUCTURES SUBJECT TO MULTIPLE LOAD-CASES. IN PARTICULAR, THE STRUCTURAL COMPLIANCE UNDER PLANE STRESS OR PLANE STRAIN ASSUMPTIONS IS MINIMIZED UNDER VOLUME CONSTRAINT. FOR THE SAKE OF COMPLETENESS, THE TOPOLOGICAL ASYMPTOTIC ANALYSIS OF THE TOTAL POTENTIAL ENERGY WITH RESPECT TO THE NUCLEATION OF A SMALL CIRCULAR INCLUSION IS DEVELOPED IN ALL DETAILS. SINCE WE ARE DEALING WITH MULTIPLE LOAD-CASES, A MULTI-OBJECTIVE OPTIMIZATION PROBLEM IS PROPOSED AND THE TOPOLOGICAL SENSITIVITY IS OBTAINED AS A SUM OF THE TOPOLOGICAL DERIVATIVES ASSOCIATED WITH EACH LOAD-CASE. THE VOLUME CONSTRAINT IS IMPOSED THROUGH THE AUGMENTED LAGRANGIAN METHOD. THE OBTAINED RESULT IS USED TO DEVISE A TOPOLOGY OPTIMIZATION ALGORITHM BASED ON THE TOPOLOGICAL DERIVATIVE TOGETHER WITH A LEVEL-SET DOMAIN REPRESENTATION METHOD. FINALLY, SEVERAL FINITE ELEMENT-BASED EXAMPLES OF STRUCTURAL OPTIMIZATION ARE PRESENTED.