Aspects of 3D Shape and Topology Optimization with Multiple Load Cases

In the present study we concentrate on the possibilities for three dimensional design of density distributions. Simple recursive optimizations are performed and the optimal solutions are normally obtained within 10 iterations. For single load cases, being static or dynamic, the necessary optimality criterion for compliance or frequency optimization is loosely stated uniform energy density. For static problems uniform elastic energy and for dynamic problems uniform difference between elastic and kinetic energy, in the active design domains. For size optimization in all non-restricted design domains and for the shape optimization along the non-restricted boundaries. For these single load cases we study the difference between optimizing for compliance (stiffness), and optimizing for von Mises stress or alternative reference stress (strength).

For multiple load cases the formulation with linear combination of the individual cases is in realty just as simple as for the single load cases, but the design solution naturally depends on the chosen linear combination factors. We illustrate this by examples. Linear combinations of static and dynamic load cases are treated in the same manner. By adjusting the linear combination factors this may lead to extremize eigenfrequencies or eigenfrequency gaps with constraint on compliance, and we can still obtain solutions by optimality criterion iterations.

An alternative to the linear combination of different load cases is to optimize with focus on fully stressed design, that is, each part of the non-restricted design domain should be used to the same extent in at least one of the load cases. Examples based on this strategy are compared to the alternative optimizations, and in general the goal of the study is to gain more knowledge related to the effect of the chosen optimality objective.

Even when an actual problem is so complex that mathematical programming must be involved, it is often a good strategy to use optimality criterion optimization for simplified versions of the problem in order to get a good starting design for the mathematical programming.