ICM Method of Topology Optimization for Continuum Structure with Globalization of Stress Constraints

1. Abstract There are three difficulties in the topology optimization problem of continuum structure: (1) The problem under multiple load case is not easy to be approached than one under single load case, because the former becomes a multiple objective problem based on compliance objective function. (2) The problem with local constraints, such as elemental stress limit, is not easy to be solved than one with global constraints, such as displacement or frequency limits, because sensitivity analysis of the former has too expensive computation. (3) The problem with a phenomenon of ill-conditioned load, which is similar with ill-conditioned stiffness matrix in the structural analysis, is not easy to get reasonable final topological structure, because the former is difficult to consider different influences between the loads with small forces and the loads with big forces, and some of topology paths of transferring small forces may disappear during the process of iteration. To overcome above difficulties, five measures are adopted below: (1) Topology optimization model is established by ICM (Independent Continuous Mapping) method. Difficulties caused by multiple objective function of compliance are overcome owing to taking weight as objective function. (2) Based on the von Mises criteria in theory of elastic failure, all element’s stress constraints are transformed into a structural energy constraint, namely, a global constraint substitutes for lots of local constraints. (3) To approach of structural energies under multiple load cases, the process of optimization is divided into three steps: the structural energy under the load case corresponding to maximal structural energy subjected to a weight constraint is minimized to converge a minimum energy; according to the minimum energy, a formula based on numerical experience is obtained to determine the allowable structural energy under each load case; an optimization model with the weight function subjected to all allowable structural energies is established and solved to search the optimum topological structure. (4) The phenomenon of the ill-conditioned load is divided to classify into three cases: ill-condition exists between load cases, but not within each load case; ill-condition exists within some load case; ill-condition exists not only between load cases, but also within some load case. (5) A strategy based on strain energy is proposed to adopt ICM method with stress globalization, and the problems of the above mentioned three cases of ill-conditioned load are solved in term of different complementary approaches one by one. Several numerical examples show that the topology path of transferring forces can be obtained more easily by substituting global strain energy constraints for local stresses constraints, and the problem of ill-conditioned load can be solved well by the weighting method which take structural energy as weighting coefficient. 2.