New Development towards Manufacturability-Oriented Topology Optimization


Structural topology optimization (STO) has experienced fast development during the last three decades. Various methods and related mechanical and mathematical theories were proposed. Recent years also witnessed many successful applications in the field of aerospace, aeronautic, automobile and mechanical engineering, where many innovative designs were produced with the help of STO. Dedicated commercial software also played an important role at this splendid stage. The recent integration of additive manufacturing and topology optimization (TO) further adds new impetus to the field of STO.

The Solid Isotropic Material with Penalty (SIMP) approach, the most popular approach for TO, is actually a pixel-based approach and describes structural topology by a 0-1valued density function  characterizing the material distribution at each material point of the design domain. Under this circumstance, the corresponding problem is essentially a NP-hard 0-1 discrete optimization problem. To overcome the huge computational burden for solving the aforementioned large-scale integer programming problem, the popular way is to relax the 0-1 constraints imposed on the design variables, and transform the original 0-1 integer programming problem to a relaxed one with continuous design variables that can be solved with use of gradient-based optimization algorithms. While great success has been achieved, a number of unwanted features of the SIMP method remains to be further addressed. Firstly, the pixel-based design description in SIMP and Evolutionary Structural Optimization (ESO) methods is not consistent with the geometry representation schemes adopted in current computer aided design (CAD) systems. Secondly, optimization results obtained by the pixel-based method often exhibit grey regions where the corresponding element densities are not pure 0/1, blurred structural boundary and uncontrolled structural features. These problems become even more severe in three-dimensional cases. Moreover, additive manufacturing also introduces some special constraints (e.g., overhang angle constraints, structural connectivity constraint and thermally-induced distortion constraint) to ensure the manufacturability of an optimized design. Solving these problems calls for the development of new manufacturability-oriented TO methods, which can have a seamless integration with CAD systems. Recent attempts on this aspect can be reflected from the newly developed moving morphable component (MMC)-based approach for TO as well as the approach developed recently by author and his colleagues in the pixel-based framework to solve the original 0-1 TO problem directly.

This presentation will briefly review the research activities towards manufacturability-oriented TO with use of the MMC approach and the newly developed direct approach which can produce pure 0-1 solution efficiently. The central idea of MMC is a component-based approach, which use a set of structural components as the basic building blocks of TO and finding the optimal structural topology by optimizing the size, shape and layout of these components. MMC has the potential to establish a seamless connection with CAD systems. The latter pixel-based approach developed by author and his colleagues follows the classic idea of sequential approximation programming (SAP) in structural optimization and utilizes the sensitivity information to construct the approximated Sequential Quadratic/Linear Integer Programming (SQIP/SLIP) sub-problems explicitly. These SQIP/SLIP sub-problems can then be solved with the so-called canonical relaxation algorithm constructed from the canonical dual theory. The new method supplemented with two different move limit strategies successfully solves a set of different topology optimization problems. Numerical examples demonstrate that compared to the classical branch and bound approach, large-scale discrete TO problems with a large number of 0-1 design variables can be solved with the proposed approach in a computationally very efficient way, and pure black-and-white design can be generated when combined with the move limit strategy of controlling the volume fraction parameter. Finally, the new method can also efficiently solve the discrete variable topology optimization problems with multiple nonlinear constraints or lots of local linear constraints in a unified and systematic way.