Structural Optimization and Engineering Feature Design with Semi-Lagrangian Level Set Method.

摘要

Although the basic theory of level set based structural optimization has been well established and many applications have been reported in the last decade, the realm is still under investigation for a number of practical issues, such as to improve computational efficiency, optimal search effectiveness, design capability and industrial applicability.;Firstly, an efficient and numerically stable semi-Lagrangian level set method is proposed for structural optimization with a line search algorithm and a sensitivity modulation scheme. The semi-Lagrange method has an advantage to allow for a large time step without the limitation of Courant-Friedrichs-Lewy (CFL) condition. The line search attempts to adaptively determine an appropriate time step in each iteration of optimization. With consideration of some practical characteristics during topology optimization process, incorporating the line search into semi-Lagrange optimization method can yield fewer design iterations and thus improve the overall computational efficiency. The sensitivity modulation is inspired from the conjugate gradient method in finite-dimensions, and provides an alternative to the standard steepest descent search in level set based optimization. Two benchmark examples are presented to compare the sensitivity modulation and the steepest descent techniques with and without the line search respectively.;Secondly, a generic method to design engineering features for level set based structural optimization is presented. Engineering features are regular and simple shape units containing specific engineering significance for manufacture and assembly consideration. It is practically useful to combine feature design with structural optimization. In this thesis, a Constructive Solid Geometry (CSG) based Level Sets description is proposed to represent a structure based on two basic entities: a level set model containing either a feature shape or a freeform boundary. By treating both entities implicitly and homogeneously, optimal feature design and freeform boundary design are unified under the level set framework. For feature models, a constrained motion of affine transformations is utilized, where the design velocity is obtained through a least square approximation of continuous shape variation. An accurate particle level set updating scheme is employed for the transformation. Meanwhile, freeform models undergo a standard level set updating process using a semi-Lagrange scheme. With this method, various feature characteristics are identified through carefully constructing a CSG model tree with flexible entities and preserved by imposing motion constraints to different stages of the tree. Moreover, because a free shape and topology optimization is enabled over non-feature regions, a truly optimal structural configuration with engineering features can be designed in a convenient way. Several 2D and 3D generative feature design examples are provided to show the applicability of this approach.;Finally, a 3D implementation using adaptive level set method is discussed. This method utilizes both explicit and implicit geometric representations for computation. An octree grid is adopted to accommodate the free structural interface of an implicit level set model and a corresponding 2-manifold triangle mesh model. Within each iteration of optimization, the interface evolves implicitly using a semi-Lagrange level set method, during which the signed distance field is evaluated directly and accurately from the current surface model other than interpolation. After that, another mesh model is extracted from the updated field and serves as the input of subsequent process. This hybrid and adaptive representation scheme not only achieves "narrow band computation", but also facilitates the structural analysis by using a geometry-aware mesh-free approach. Moreover, a feature preserving and topological errorless mesh simplification algorithm is proposed to enhance the computational efficiency. Remarkably, the adaptive level set scheme opens up a gate to incorporate geometric editing into structural optimization in an effective way, which creates a new dimension of opportunity to further develop level set based structural optimization in this direction. A three dimensional benchmark example and possible extensions are presented to demonstrate the capability and potential of this method. (Abstract shortened by UMI.).