## A multiobjective and fixed elements based modification of the evolutionary structural optimization method.(English)Zbl 1120.74708

Summary: In 1993, Y.M. Xie and G.P. Steven introduced an approach called evolutionary structural optimization (ESO). ESO is based on the simple idea that the optimal structure (maximum stiffness, minimum weight) can be produced by gradually removing the ineffectively used material from the design domain. The design domain is constructed by the FE method, and furthermore, external loads and support conditions are applied to the element model. Considering the engineering aspects, ESO seems to have some attractive features: the ESO method is very simple to program via the FEA packages and requires a relatively small amount of FEA time. On the other hand, different constraints cannot be added into the problem. In the ESO optimization the results supposedly approach truss-like, fully stressed topologies, which have the maximum stiffness with respect to the volume. Generally, these types of structures correspond to least-weight trusses.
Although ESO is not capable of handling general stress or displacement constraints, the design problems are often such that these constraints do not need to be included in the topology optimization, especially if the design optimization task is divided into two stages. In the first stage, only the overall geometry is outlined, and for that reason, the actual constraints do not have to be activated. In the second stage, the sizing optimization is performed. It can be concluded that ESO is well suited to solve the first stage optimization problems.
In some design problems it may occur that the structure cannot attain the fully stressed state because of geometrical constraints. It follows that the topology having the maximum stiffness with respect to the volume does not necessarily produce the least-weight structure when the stress constraints are applied in the second stage optimization. The geometrical constraints may force some structural components to be subject to a understressed state, i.e. to carry some “waste material”. As a consequence, the aim of this paper was to study whether ESO can be modified so that some geometrical constraints can be taken into account already in the first stage topology optimization. The modification was based on the assumption that if the stress level of otherwise understressed structural components can be increased during the $$compliance-volume$$ product minimization, a lighter topology may be obtained. This new approach, the multiobjective and fixed elements based modification of the evolutionary structural optimization (MESO) utilizes a new optimization objective in which the overall stiffness of the structure and the loading of some parts of the structure are increased simultaneously. The gradient vector of the MESO objective function was determined by the FE method. Some of the partial derivatives involved were first presolved and then approximated. This approach was justified by large savings in the analysis time. Yet, MESO cannot take the general constraints into account.
To study the performance of the MESO optimization, two numerical examples were evaluated. The main purpose of these examples was to study whether MESO can produce structures lighter than the ESO results for problems having both stress and geometrical constraints.
These example were based on the two-stage optimization approach. In both example the MESO truss turned out to be lighter than the corresponding ESO truss. However, the ESO truss was the stiffest one also having a smaller overall stress value. In the question of shallow structures the deflection criteria may be predominant, and as a consequence, the ESO optimization may yield lighter structures than MESO.

### MSC:

 74P10 Optimization of other properties in solid mechanics 74P15 Topological methods for optimization problems in solid mechanics 65K10 Numerical optimization and variational techniques

CONMIN; ESOFRAME
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### References:

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