Structure and appearance optimization for controllable shape design

Abstract

The field of topology optimization seeks to optimize shapes under structural objectives, such as achieving the most rigid shape using a given quantity of material. Besides optimal shape design, these methods are increasingly popular as design tools, since they automatically produce structures having desirable physical properties, a task hard to perform by hand even for skilled designers. However, there is no simple way to control the appearance of the generated objects.

In this paper, we propose to optimize shapes for both their structural properties and their appearance, the latter being controlled by a user-provided pattern example. These two objectives are challenging to combine, as optimal structural properties fully define the shape, leaving no degrees of freedom for appearance. We propose a new formulation where appearance is optimized as an objective while structural properties serve as constraints. This produces shapes with sufficient rigidity while allowing enough freedom for the appearance of the final structure to resemble the input exemplar.

Our approach generates rigid shapes using a specified quantity of material while observing optional constraints such as voids, fills, attachment points, and external forces. The appearance is defined by examples, making our technique accessible to casual users. We demonstrate its use in the context of fabrication using a laser cutter to manufacture real objects from optimized shapes.

References

1. Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B., and Sigmund, O. 2011. Efficient topology optimization in Matlab using 88 lines of code. Struct. Multidiscip. Optim. 43, 1, 1--16.
2. Barnes, C., Shechtman, E., Finkelstein, A., and Goldman, D. B. 2009. PatchMatch: A randomized correspondence algorithm for structural image editing. ACM Trans. Graph. 28, 3, 24:1--24:11.
3. Bendsøe, M. P., and Kikuchi, N. 1988. Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering 71, 2, 197--224.
4. Bendsøe, M. P., and Sigmund, O. 2003. Topology Optimization: Theory, Methods and Applications.
5. Bendsøe, M. P. 1989. Optimal shape design as a material distribution problem. Structural Optimization 1, 192--202.
6. Brackett, D., Ashcroft, I., and Hague, R. 2011. Topology optimization for additive manufacturing. In Proc. of the 24th Solid Freeform Fabrication Symposium, 6--8.
7. Bruyneel, M., and Duysinx, P. 2005. Note on topology optimization of continuum structures including self-weight. Struct. Multidiscip. Optim. 29, 4, 245--256.
8. Busto, P. P., Eisenacher, C., Lefebvre, S., and Stamminger, M. 2010. Instant texture synthesis by numbers. In Proc. of the VMV Workshop, 81--85.
9. Christiansen, A. N., Bærentzen, J. A., Nobel-Jørgensen, M., Aage, N., and Sigmund, O. 2015. Combined shape and topology optimization of 3D structures. Computers & Graphics 46, 25--35.
10. Deaton, J., and Grandhi, R. 2014. A survey of structural and multidisciplinary continuum topology optimization: post 2000. Struct. Multidiscip. Optim. 49, 1, 1--38.
11. Dumas, J., Lu, A., Lefebvre, S., Wu, J., and Dick, C. 2015. By-Example Synthesis of Structurally Sound Patterns. ACM Trans. Graph. 34, 4, 137:1--137:12.
12. Duysinx, P., and Bendsøe, M. P. 1998. Topology optimization of continuum structures with local stress constraints. Int. J. Numer. Methods. Eng. 43, 8, 1453--1478.
13. Efros, A. A., and Leung, T. K. 1999. Texture synthesis by non-parametric sampling. In Proceedings of the International Conference on Computer Vision, 1033--1038.
14. Han, Z., Liu, Z., Han, J., and Bu, S. 2015. 3D shape creation by style transfer. Vis. Comput. 31, 9, 1147--1161.
15. Hertzmann, A., Jacobs, C. E., Oliver, N., Curless, B., and Salesin, D. H. 2001. Image analogies. In Proc. of SIGGRAPH 2001, 327--340.
16. Holmberg, E., Torstenfelt, B., and Klarbring, A. 2013. Stress constrained topology optimization. Struct. Multidiscip. Optim. 48, 1, 33--47.
17. Johnson, S. G., 2007. The NLopt nonlinear-optimization package.
18. Kaspar, A., Neubert, B., Lischinski, D., Pauly, M., and Kopf, J. 2015. Self tuning texture optimization. Computer Graphics Forum 34, 2, 349--359.
19. Kopf, J., Fu, C.-W., Cohen-Or, D., Deussen, O., Lischinski, D., and Wong, T.-T. 2007. Solid texture synthesis from 2D exemplars. ACM Trans. Graph. 26, 3.
20. Kosaka, I., and Swan, C. C. 1999. A symmetry reduction method for continuum structural topology optimization. Computers & Structures 70, 1, 47--61.
21. Kwatra, V., Essa, I., Bobick, A., and Kwatra, N. 2005. Texture optimization for example-based synthesis. ACM Trans. Graph. 24, 3, 795--802.
22. Lee, E., James, K. A., and Martins, J. R. 2012. Stress-constrained topology optimization with design-dependent loading. Struct. Multidiscip. Optim. 46, 5, 647--661.
23. Li, H., Zhang, H., Wang, Y., Cao, J., Shamir, A., and Cohen-Or, D. 2013. Curve style analysis in a set of shapes. Computer Graphics Forum 32, 6, 77--88.
24. Lu, L., Sharf, A., Zhao, H., Wei, Y., Fan, Q., Chen, X., Savoye, Y., Tu, C., Cohen-Or, D., and Chen, B. 2014. Build-to-last: Strength to weight 3D printed objects. ACM Trans. Graph. 33, 4, 97:1--97:10.
25. Ma, C., Huang, H., Sheffer, A., Kalogerakis, E., and Wang, R. 2014. Analogy-driven 3D style transfer. Computer Graphics Forum 33, 2, 175--184.
26. Panetta, J., Zhou, Q., Malomo, L., Pietroni, N., Cignoni, P., and Zorin, D. 2015. Elastic textures for additive fabrication. ACM Trans. Graph. 34, 4, 135:1--135:12.
27. París, J., Muínos, I., Navarrina, F., Colominas, I., and Casteleiro, M. 2005. A minimum weight FEM formulation for structural topological optimization with local stress constraints. In VI World Congress on Structural and Multidisciplinary Optimization.
28. Paulino, G. H., and Gain, A. L. 2015. Bridging art and engineering using Escher-based virtual elements. Struct. Multidiscip. Optim. 51, 4, 867--883.
29. Pedersen, N. 2000. Maximization of eigenvalues using topology optimization. Struct. Mult. Optim. 20, 1, 2--11.
30. Schumacher, C., Bickel, B., Rys, J., Marschner, S., Daraio, C., and Gross, M. 2015. Microstructures to control elasticity in 3D printing. ACM Trans. Graph. 34, 4, 136:1--136:13.
31. Sigmund, O., and Maute, K. 2013. Topology optimization approaches: A comparative review. Struct. Multidiscip. Optim. 48, 6, 1031--1055.
32. Sigmund, O. 1997. On the design of compliant mechanisms using topology optimization. Mechanics of Structures and Machines 25, 4, 493--524.
33. Sigmund, O. 2001. A 99 line topology optimization code written in Matlab. Struct. Mult. Optim. 21, 2, 120--127.
34. Sigmund, O. 2009. Manufacturing tolerant topology optimization. Acta Mechanica Sinica 25, 2, 227--239.
35. Stava, O., Vanek, J., Benes, B., Carr, N., and Měch, R. 2012. Stress relief: Improving structural strength of 3D printable objects. ACM Trans. Graph. 31, 4, 48:1--48:11.
36. Svanberg, K. 1987. The method of moving asymptotes---a new method for structural optimization. Int. J. Numer. Methods. Eng. 24, 2, 359--373.
37. Svanberg, K. 1995. A globally convergent version of MMA without linesearch. In Proc. of the first world congress of structural and multidisciplinary optimization, 9--16.
38. Umetani, N., and Schmidt, R. 2013. Cross-sectional structural analysis for 3D printing optimization. In SIGGRAPH Asia 2013 Technical Briefs, 5:1--5:4.
39. Wang, W., Wang, T. Y., Yang, Z., Liu, L., Tong, X., Tong, W., Deng, J., Chen, F., and Liu, X. 2013. Cost-effective printing of 3D objects with skin-frame structures. ACM Trans. Graph. 32, 6, 177:1--177:10.
40. Wei, L.-Y., and Levoy, M. 2000. Fast texture synthesis using tree-structured vector quantization. In Proc. of SIGGRAPH 2000, 479--488.
41. Wei, L.-Y., Lefebvre, S., Kwatra, V., and Turk, G. 2009. State of the art in example-based texture synthesis. In Eurographics 2009, State of the Art Report.
42. Xu, K., Li, H., Zhang, H., Cohen-Or, D., Xiong, Y., and Cheng, Z. 2010. Style-content separation by anisotropic part scales. ACM Trans. Graph. 29, 6, 184:1--184:10.
43. Xu, K., Zhang, H., Cohen-Or, D., and Chen, B. 2012. Fit and diverse: Set evolution for inspiring 3D shape galleries. ACM Trans. Graph. 31, 4, 57:1--57:10.
44. Zhou, Q., Panetta, J., and Zorin, D. 2013. Worst-case structural analysis. ACM Trans. Graph. 32, 4, 137:1--137:12.
45. Zhou, S., Jiang, C., and Lefebvre, S. 2014. Topology-constrained synthesis of vector patterns. ACM Trans. Graph. 33, 6, 215:1--215:11.
46. Zhou, M., Lazarov, B. S., Wang, F., and Sigmund, O. 2015. Minimum length scale in topology optimization by geometric constraints. Computer Methods in Applied Mechanics and Engineering 293, 266--282.

Index Terms

• Applied computing

  • Operations research

    • Computer-aided manufacturing

• Computing methodologies

  • Computer graphics

    • Animation

      • Physical simulation

    • Image manipulation

      • Texturing

    • Shape modeling

• Mathematics of computing

  • Continuous mathematics

    • Topology

      • Geometric topology