Abstract We present a novel continuous optimization method to the discrete problem of quadtree optimization. The optimization aims at achieving a quadtree structure with the highest mechanical stiffness, where the edges in the quadtree are interpreted as structural elements carrying mechanical loads. We formulate quadtree optimization as a continuous material distribution problem. The discrete design variables (i.e., to refine or not to refine) are replaced by continuous variables on multiple levels in the quadtree hierarchy. In discrete quadtree optimization, a cell is only eligible for refinement if its parent cell has been refined. We propose a continuous analogue to this dependency for continuous multi-level design variables, and integrate it in the iterative optimization process. Our results show that the continuously optimized quadtree structures perform much stiffer than uniform patterns and the heuristically optimized counterparts. We demonstrate the use of adaptive structures as lightweight infill for 3D printed parts, where uniform geometric patterns have been typically used in practice.
One application of quadtree optimization is to obtain lightweight and stiff infill structures for 3D printing. Topology optimization has been recognized as an important design method for 3D printing, as it fully exploits the manufacturing flexibility to create lightweight and mechanically optimal structures @cite_6 . A recent focus in this area has been on incorporating manufacturability constraints (e.g., overhang angle @cite_20 @cite_27 @cite_39 @cite_25 and length scale @cite_12 ) and thus eliminating post-processing of the numerically optimized structures.
@cite_6 @cite_39 @cite_27 @cite_12 @cite_25 @cite_20
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There is significant interest today in integrating additive manufacturing (AM) and topology optimization (TO). However, TO often leads to designs that are not AM friendly. For example, topologically optimized designs may require significant amount of support structures before they can be additively manufactured, resulting in increased fabrication and clean-up costs.In this paper, we propose a TO methodology that will lead to designs requiring significantly reduced support structures. Towards this end, the concept of support structure topological sensitivity is introduced. This is combined with performance sensitivity to result in a TO framework that maximizes performance, subject to support structure constraints. The robustness and efficiency of the proposed method is demonstrated through numerical experiments, and validated through fused deposition modeling, a popular AM process. Addresses additive manufacturing support structure constraint in topology optimization.Introduces topological sensitivity for support structures.Combines support structure sensitivity with performance sensitivity.Proposes a robust and efficient algorithm for support structure constrained topology optimization. Additively manufactured components often require temporary support material to prevent the component from collapsing or warping during fabrication. Whether these support materials are removed chemically as in the case of many polymer additive manufacturing processes, or mechanically as in the case of (for example) Direct Met al Laser Sintering, the use of sacrificial material increases total material usage, build time, and time required in post-fabrication treatments. The goal of this work is to embed a minimum allowable self-supporting angle within the topology optimization framework such that designed components and structures may be manufactured without the use of support material. This is achieved through a series of projection operations that combine a local projection to enforce minimum length scale requirements and a support region projection to ensure a feature is adequately supported from below. The magnitude of the self-supporting angle is process dependent and is thus an input variable provided by the manufacturing or design engineer. The algorithm is demonstrated on standard minimum compliance topology optimization problems and solutions are shown to satisfy minimum length scale, overhang angle, and volume constraints, and are shown to be dependent on the allowable magnitudes of these constraints. Since its original introduction in structural design, density-based topology optimization has been applied to a number of other fields such as microelectromechanical systems, photonics, acoustics and fluid mechanics. The methodology has been well accepted in industrial design processes where it can provide competitive designs in terms of cost, materials and functionality under a wide set of constraints. However, the optimized topologies are often considered as conceptual due to loosely defined topologies and the need of postprocessing. Subsequent amendments can affect the optimized design performance and in many cases can completely destroy the optimality of the solution. Therefore, the goal of this paper is to review recent advancements in obtaining manufacturable topology-optimized designs. The focus is on methods for imposing minimum and maximum length scales, and ensuring manufacturable, well-defined designs with robust performances. The overview discusses the limitations, the advantages and the associated computational costs. The review is completed with optimized designs for minimum compliance, mechanism design and heat transfer. Summary We present an approach for controlling the undercut and the minimal overhang angle in density based topology optimization, which are useful for reducing support structures in additive manufacturing. We cast both the undercut control and the minimal overhang angle control that are inherently constraints on the boundary shape into a domain integral of Heaviside projected density gradient. Such a Heaviside projection based integral of density gradient leads to a single constraint for controlling the undercut or controlling the overhang angle in the optimization. It effectively corresponds to a constraint on the projected perimeter that has undercut or has slope smaller than the prescribed overhang angle. In order to prevent trivial solutions of intermediate density to satisfy the density gradient constraints, a constraint on density grayness is also incorporated into the formulations. Numerical results on Messerschmitt–Bolkow–Blohm beams, cantilever beams, and 2D and 3D heat conduction problems demonstrate the proposed formulations are effective in controlling the undercut and the minimal overhang angle in the optimized designs. Copyright © 2016 John Wiley & Sons, Ltd. Additive manufacturing (AM) offers exciting opportunities to manufacture parts of unprecedented complexity. Topology optimization is essential to fully exploit this capability. However, AM processes have specific limitations as well. When these are not considered during design optimization, modifications are generally needed in post-processing, which add costs and reduce the optimized performance. This paper presents a filter that incorporates the main characteristics of a generic AM process, and that can easily be included in conventional density-based topology optimization procedures. Use of this filter ensures that optimized designs comply with typical geometrical AM restrictions. Its performance is illustrated on compliance minimization problems, and a 2D Matlab implementation is provided.