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Topology Optimization
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Tags: descent_methodoptimal_designsensitivity_analysistopology_optimization
Added by Martin on 20100722 11:29
Topology Optimization In lightweight design saving of material has to be done in such a way that the material is not overloaded at any position of the construction. Otherwise a failure of the structure would be possible. Mathematical models are available for the simulation of the local stress in the material if the operating loads are applied. This simulation can also be used to detect regions of the structures, which are not necessary for carrying the loads. The material can be removed from the detected regions, which have only small stresses, in the computer, i.e. the design of the structure can be improved automatically. The development of such algorithms and methods is a recent field of research. Software packages are already available for the automated optimal design of components, which have to carry static or dynamical mechanical loads and which have to fulfill also other functions. In the socalled shape optimization only the boundary of the structure is moved iteratively in such a way that e.g. notch stresses are reduced. On the other hand topology optimization algorithms are iterative methods, where the topology of the structure is changing during the iterations. E.g. new holes are created in unloaded regions of the components. Often the topology optimization is the first step and shape optimization the second step in the design of lightweight structures. Automatic structural optimization method are most efficient, if the are used at the beginning of design of a new product. Castings, which have a very flexible and filigree geometry, can be produced in foundries. Therefore structural optimization methods are often used for the design of castings. A standard problem is the minimization of the compliance of structures under given mechanical loads. This problem can be formulated as an optimization problem with constraints. In this case the constraints contain an elliptic boundary value problem for the equations of linear elasticity. For the solution of this problem many iterative algorithms are proposed in literature. In each iteration of these algorithms the structure is analyzed at first, then the sensitivity with respect to changes of the structures is computed, and finally the geometry of the structure is updated. The structural analysis is often performed by the finiteelement method (FEM) . Generalized derivatives (shape derivative, topological gradient) are used for the sensitivity analysis. The levelset method is a modern highly efficient mathematical tool for the representation of the boundary of the structure and also for the change of this boundary.