Parametric and Topological Control in Shape Optimization

Overview

It is of great importance to find the best possible material layout or shape for given design objectives and constraints in a product design process. In the last two decades, substantial research efforts have been devoted to develop efficient and robust methods in the field of shape/topology optimization. Three types of shape optimization problems are considered in literatures: parametric shape optimization, traditional (boundary variation) shape optimization, and topology optimization, of which each focuses on different aspect of the design process and has its own advantages and disadvantages. These three shape optimization problems also correspond to the three stages of a design process (in reverse order): conceptual design stage, preliminary design stage and detailed design stage. While shape optimization is an iterative multi-objective process, the boundaries among these three aspects should be relaxed as opposed to the current state in shape optimization in order to facilitate design automation. Our current research is focusing on combining parametric shape optimization and free-form shape optimization (which we refer as the combination of boundary variation based shape optimization and topology optimization).   Parametric, shape and topology optimization Parametric shape optimization deals with shape optimization problems in a particular design space in which the shape is parameterized by a finite set of geometric parameters (dimensions). This set of meaningful geometric parameters are used as design variables in a parametric shape optimization problem, which essentially transfer a shape optimization problem into a easy-solving "sizing" problem. The limitation of parametric shape optimization is that usually the topology of the shape is fixed or the re-parametrization may be required during an optimization process. Parametric shapes are manufacturing friendly and parametric shape optimization method can be integrated into CAD system easily. Traditional parametric optimization In boundary variation shape optimization problems, there is no such high-level geometric parameters, the motion of the shape boundary can vary in any fashion. Typically some parametrization or discretization techniques (for example, B-spline curve fitting or polygonization) are applied to the shape boundary, which generate a set of design variables (such as control points or finite element nodes) for the optimization process. Tracking intersected boundaries is very difficult for parameterized curves/surfaces. So in general, boundary variation based shape optimization methods do not allow topological changes. Boundary variation based shape optimization For the last decade, much attention has been paid to the topology optimization problems, which focus on how to allow topological changes in the shape optimization process. The importance of topology optimization lies in the fact that the choice of appropriate topology of a structure at the initial design stage is in general the most decisive fact for the efficiency of a product. Among all three optimization problems, topology optimization is the most challenge one mainly due to the lacking of theoretic support. Existing methods can be divided into two categories. One is based on material distribution, such as homogenization method and SIMP (Solid Isotropic Microstructure with Penalty) method. Another class of methods are geometry-based in the sense that they focus on how to move the boundary and where to put the holes, which will be our focus. Geometry based topology optimization