Geometrische Restriktionen bei der geometriebasierten Strukturoptimierung von Maschinenbauteilen mit Freiformgeometrien

摘要

In this thesis a geometry-based optimization method for machine components by means of the finite element analysis (FEA) is built up which is capable of satisfying geometrical constraints in the context of spline representation. FE-based optimization methods are already supported by the consideration of some selected geometric criteria. In contrast to FE-based approaches a new mesh is generated for each structural variation of a geometry-based optimization. This guarantees a high mesh quality, even if there are big changes in the CAD-model. Moreover, the results of a geometry-based optimization are directly applicable in the engineering design process by a parameterized CAD-model. However, an approach for the geometry-based optimization is still missing which is able to return a directly applicable CAD-model of a geometric feasible optimum back to the engineering design process. The basis of a geometry-based shape optimization is the mathematical description of the component boundaries. As a suitable mathematical representation Non-Uniform Rational B-Splines (NURBS) have been chosen which are industrial standard in technical applications and capable of representing conic sections and other free-form geometries, such as B-splines and Bézier curves exactly. By the consideration of geometric constraints geometric instabilities of the CAD-system and intersections of the available installation space are excluded. Furthermore, in evolution strategies (ES) the established algorithms offer a rapid and robust generation of NURBS which fulfill the defined mass and geometrically describable manufacturing constraints. For gradient based approaches a linearization method was investigated which permits a consideration of geometric constraints on a finite number of points located on the NURBS curve. In contrast to gradient based methods, an ES facilitates a search for the global extremum, even for multimodal optimization tasks. Besides a self-adaptive adjustment of the strategy parameters to the conditions of the current optimization cycle, different modifications and extensions of the ES were investigated which led to an improvement of the convergence speed and search ability. Moreover, metamodels were developed which, through an approximation of already calculated solutions, offer a prediction of the structural properties. Hereby, a metamodel-assisted evolution strategy (MAES) was built up using the metamodels for a local optimization of solutions recommended by the ES. A sequential quadratic programming (SQP) as gradient based approach for the local optimization was chosen. Only in conjunction with the research of new heuristics for the control of the MAES and an adaption of the active set method was a serious acceleration of the convergence speed achieved. By the implementation of the established optimization system a completely automated structural optimization program for machine components was realized. The integration of parameter optimization, parallel to shape optimization with free-form geometries, permits an optimization of dimensions, such as wall thicknesses, which have a significant influence on structural behavior. Thereby, an integration of discrete parameters, such as the number of ribs, enabled a limited change of the topology of a machine component. The combination of the different optimizations methods necessitated the formulation of parameter dependencies for the mathematical representation of geometric relations to avoid problematical intersections in the CAD-model and to include physical interrelationships. Since only geometric permissible NURBS are transferred to the CAE-system, only geometric feasible structural variants are evaluated by the time-consuming FE-analysis. By means of practical examples taken from power train and machine tool engineering, the mechanical properties of the components were improved significantly without violating of the defined geometric constraints. Besides a performance analysis of the developed MAES, the impact of different spline representations and number of interpolation points was investigated in the example of the inner plates of a roller chain. As a second example taken from power train engineering the stress in the tooth root of a pinion was reduced by more than 5% with approximately 30% lower material usage. Furthermore, two typical components derived from machine tool engineering were optimized. The increase of the first eigenfrequency of a boring bar by more than 14% demonstrates that the computational effort for the generation of geometrically feasible solutions is reduced significantly by the established methods. Finally, a cross beam of a portal milling machine was optimized by varying its dimensions, shapes and topology. Thereby, the initial mass was reduced up to 18.5% while retaining the structural stiffness. Through the MAES, the number of FE-calculations was reduced to less than 94% for a comparable solution of the developed ES.