Bridging topology optimization and additive manufacturing

Abstract

Topology optimization is a technique that allows for increasingly efficient designs with minimal a priori decisions. Because of the complexity and intricacy of the solutions obtained, topology optimization was often constrained to research and theoretical studies. Additive manufacturing, a rapidly evolving field, fills the gap between topology optimization and application. Additive manufacturing has minimal limitations on the shape and complexity of the design, and is currently evolving towards new materials, higher precision and larger build sizes. Two topology optimization methods are addressed: the ground structure method and density-based topology optimization. The results obtained from these topology optimization methods require some degree of post-processing before they can be manufactured. A simple procedure is described by which output suitable for additive manufacturing can be generated. In this process, some inherent issues of the optimization technique may be magnified resulting in an unfeasible or bad product. In addition, this work aims to address some of these issues and propose methodologies by which they may be alleviated. The proposed framework has applications in a number of fields, with specific examples given from the fields of health, architecture and engineering. In addition, the generated output allows for simple communication, editing, and combination of the results into more complex designs. For the specific case of three-dimensional density-based topology optimization, a tool suitable for result inspection and generation of additive manufacturing output is also provided.

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Acknowledgments

The authors appreciate constructive comments and insightful suggestions from the anonymous reviewers. We are thankful to the support from the US National Science Foundation under grant CMMI #1335160. We also acknowledge the support from SOM (Skidmore, Owings and Merrill LLP) and from the Donald B. and Elizabeth M. Willett endowment at the University of Illinois at Urbana–Champaign. Any opinion, finding, conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the sponsors.

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Appendix: input for TOPslicer

Appendix: input for TOPslicer

TOPslicer has support for various three-dimensional data array formats. The data in a three-dimensional array (or 3D matrix) can be organized in a several ways depending on how the indices span the data. MATLAB was strongly influenced by Fortran, and both were targeted at numerical analysis involving matrices. As a consequence, both MATLAB and Fortran use column-major ordering: the data’s first index spans the rows of the matrix (vertical direction), and the second index spans the columns (horizontal direction). If this idea is extended to three-dimensional positions, it implies that some data is traversed by its indices first in the y axis, and then in the x axis (e.g the entry A 7,1 is 7 spaces away from the origin on the y axis). Other programming/scripting languages (typically not based on matrices) use row-major ordering, where the more natural xyz data arrangement is used.

Take for example the three-dimensional array shown in Fig. 22a. Using MATLAB’s column-major ordering, the size of this array is 3×4×2, with the indices for some key entries are shown in Fig. 22b. However, using row-major ordering, the size of the array is 4×3×2, with the same indices shown in Fig. 22c.

Fig. 22
figure22

Data ordering in a three-dimensional array ρ: (a) Three-dimensional array of size N x =4, N y =3 and N z =2, where each cell represents a stored value. (b) Example for column-major ordering, with the indices for some key entries shown: the array ρ has size 3×4×2. (c) Example for row-major ordering, with the indices for some key entries shown: the array ρ has size 4×3×2.

The user menu in TOPslicer allows the user to specify what ordering was used in the input data: column-major (MATLAB’s default), row-major (arguably the most intuitive), and additionally, TOPslicer also has support for the (rather unconventional) format used in TOP3D (Liu and Tovar 2014). That said, failure to select the correct ordering will only result in the data being displayed either rotated along some axis, or mirrored along some plane(s).

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Zegard, T., Paulino, G.H. Bridging topology optimization and additive manufacturing. Struct Multidisc Optim 53, 175–192 (2016). https://doi.org/10.1007/s00158-015-1274-4

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Keywords

  • Additive manufacturing
  • Ground structure method
  • Density-based topology optimization
  • Three-dimensional optimal structures
  • Structural manufacturing