Figure. Minimization of f with respect to x is a non-flat problem: local solutions are well isolated and differ in quality significantly.
Optimization process behavior and results are heavily influenced by the type of the considered problem. Topology optimization problems of load-carrying structures are in general
A consequence of non-linearity is that the optimization process is iterative and can not be solved within a single cycle. Non-convexity, on the other hand means that the problem exhibits many local minima and that it is very difficult to check whether the obtained result is actually the global minimum (solution). What is causing the most confusion in practice, however, is the extreme flatness of topology optimization problems.
In short, a flat problem is the one that exhibits several solutions of approximately the same quality. Proper understanding of the consequences of flatness is therefore very important for approaching correctly to problem formulation and solution.
A non-flat problem is characterized by exhibiting one or more well isolated local minima (local solutions) which differ significantly in the quality. From the mathematical point of view, quality is measured by the value of the objective function (for example, the strain energy); if the problem is a minimization problem, then lower value of the objective function means better quality.
Figure. Minimization of f with respect to x is a non-flat problem: local solutions are well isolated and differ in quality significantly.
The solution process of a non-flat problem typically exhibits relatively good convergence to a single and well isolated local solution. Forcing additional optimization cycles after convergence to one of the local minima was reached, does not change the solution.
A flat problem is characterized by exhibiting a wide range of poorly isolated (smeared) local minima which differ only negligibly in the quality (from the mathematical point of view). From the engineering point of view, however, the results may differ substantially, for example, because the production cost of one design might be substantially different than that of another design. Topology optimization is especially prone to deliver a myriad of possible 'optimal designs', only by running optimization from different starting points and with different optimization tuning parameters.
Figure. Minimization of f with respect to x is a flat problem: local solutions are smeared over wide regions and differ in quality only negligibly.
The solution process of a flat problem typically exhibits relatively poor convergence to any of the local solutions. Since such a local solution is typically not well isolated, the optimization algorithm may end up with a huge number of cycles in order to 'creep' from one solution to another (only slightly better) one. From the numerical point of view this means that in topology optimization it is actually very difficult to set an adequate convergence criterion. There seems to be no cure to this problem, except, that the engineer monitors the optimization process interactively and sets the tuning parameters adequately to steer the process into the desired direction.
The figures below illustrate two solutions of the same topology optimization problem, obtained by running optimization by two different sets of optimization parameters. Mathematically, the quality of both results (strain energy; SENER) is practically the same although the designs differ substantially.
Figure. Optimal design A with strain energy SENER = 133; faster material redistribution rate.
Figure. Optimal design B with strain energy SENER = 134; slower material redistribution rate.
NOTE. Topology optimization problems of load carrying structures are extremely flat problems. Typically, many solutions are possible that exhibit a similar quality, but may look quite different.