Structural Shape and Topology Optimization with Implicit and Parametric Representations

Engineers have utilized CAE technique as an analysis tool to refine the engineering design over decades. However, CAE alone is not the key to open the door for the final goal. In order to achieve the practical solution to the real-time engineering problem, we need to integrate CAD, CAE and optimization techniques into a single framework.;In the problem of the structural optimizations, three categories of the approaches can be identified: size, shape and topology optimizations. For size optimization, explicit dimensions are usually chosen as the design variables, for example, the thickness of a beam or the diameter of a cylinder. For shape optimization, the shape related parameters of the geometrical boundary are always considered to be the design variables, like the positions of the control points for a Bezier curve. However, these two methods are lack of the capability to handle the topological changes of the geometry. On the contrary, topology optimization is the generalization of size and shape optimizations, which offers a more flexible and powerful tool to determine the best layout of the materials and the topology to the design problem, and it is becoming increasingly important in the conceptual design phase. In other words, topology optimization gives one the inspiration for the locations where we put holes to reach the best design.;The material density method and the boundary-variation method are the popular methods adopted in both academia and industrial community. Even though the former method is dominant in industry, the latter method is more preferable these years owing to its boundary description nature. Undoubtedly, the level set based method is the most promising technique of the boundary-variation type. Scientists successfully developed the optimization algorithms based on the level set method (LSM) in the past few years. With the implicit representation of the LSM, topological changes of the design can be handled easily and the geometrical complexity is then reserved.;In this thesis, we put forward the algebraic level set (ALS) model with the consideration of the constructive solid geometry (CSG) model so that it is consistent with half-space primitive concept in CSG. Based on general shape derivative, we propose the general shape design sensitivity analysis (SDSA) formulations for general geometric primitives that are represented implicitly, such as line and circle primitives in two-dimensional space and plane primitive in three-dimensional space. We then extend the relevant formulations into corresponding parametrically represented primitives as they are widely used in today's mainstream CAD systems.;In the optimization algorithm part, apart from the general parametric steepest descent (ST) algorithm, we also study the least square (LSQ) based optimization algorithm. As a result, we can solve the problem arisen from the variant dimensional sizes of the different design variables by using the weighted sensitivity information.;The optimal result given by conventional topology optimization usually involves tedious post-processing to form CAD geometry. Using our parameterizations with basic primitives and the proposed optimization algorithms, we can deliver comparatively complicated shapes with rich topological information. Therefore, the detail design could be conducted directly later.;The numerical examples for the design optimization problem are successfully implemented with both the implicit geometric representation (2D cases) and the parametric geometric representation (3D cases), which proves the feasibility of the proposed framework. The results show that both shape and topology optimizations of a design could be accomplished in a natural way.