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Non-linear shape optimization using local subspace projections

Published: 11 July 2016 Publication History

Abstract

In this paper we present a novel method for non-linear shape optimization of 3d objects given by their surface representation. Our method takes advantage of the fact that various shape properties of interest give rise to underdetermined design spaces implying the existence of many good solutions. Our algorithm exploits this by performing iterative projections of the problem to local subspaces where it can be solved much more efficiently using standard numerical routines. We demonstrate how this approach can be utilized for various shape optimization tasks using different shape parameterizations. In particular, we show how to efficiently optimize natural frequencies, mass properties, as well as the structural yield strength of a solid body. Our method is flexible, easy to implement, and very fast.

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References

[1]
Bächer, M., Bickel, B., James, D. L., and Pfister, H. 2012. Fabricating articulated characters from skinned meshes. ACM Transactions on Graphics 31, 4 (jul), 1--9.
[2]
Bächer, M., Whiting, E., Bickel, B., and Sorkine-Hornung, O. 2014. Spin-It: Optimizing Moment of Inertia for Spinnable Objects. ACM Transactions on Graphics 33, 4 (jul), 1--10.
[3]
Bathe, K.-J. 2006. Finite Element Procedures. Prentice Hall.
[4]
Bharaj, G., Levin, D. I. W., Tompkin, J., Fei, Y., Pfister, H., Matusik, W., and Zheng, C. 2015. Computational design of metallophone contact sounds. ACM Transactions on Graphics 34, 6 (oct), 1--13.
[5]
Bickel, B., Bächer, M., Otaduy, M. A., Lee, H. R., Pfister, H., Gross, M., and Matusik, W. 2010. Design and fabrication of materials with desired deformation behavior. ACM Transactions on Graphics 29, 4 (jul), 1.
[6]
Botsch, M., and Kobbelt, L. 2005. Real-Time Shape Editing using Radial Basis Functions. Comput. Graph. Forum 24, 3, 611--621.
[7]
Delfour, M. C., and Zolésio, J. P. 2011. Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition. Advances in Design and Control. Society for Industrial and Applied Mathematics (SIAM, 3600 Market Street, Floor 6, Philadelphia, PA 19104).
[8]
Eberly, D. H. 2010. Game physics, 2. edition ed. Morgan Kaufmann, Burlington, Mass.
[9]
Finsterle, S., and Kowalsky, M. B. 2011. A truncated Levenberg--Marquardt algorithm for the calibration of highly parameterized nonlinear models. Computers & Geosciences 37, 6 (jun), 731--738.
[10]
Hafner, C., Musialski, P., Auzinger, T., Wimmer, M., and Kobbelt, L. 2015. Optimization of natural frequencies for fabrication-aware shape modeling. In ACM SIGGRAPH 2015 Posters on - SIGGRAPH '15, ACM Press, New York, New York, USA, 1--1.
[11]
Ipsen, I. C. F., Kelley, C. T., and Pope, S. R. 2011. Rank-Deficient Nonlinear Least Squares Problems and Subset Selection. SIAM Journal on Numerical Analysis 49, 3 (jan), 1244--1266.
[12]
Kuipers, L., and Niederreiter, H. 2012. Uniform Distribution of Sequences. Dover Books on Mathematics. Dover Publications.
[13]
Lu, L., Chen, B., Sharf, A., Zhao, H., Wei, Y., Fan, Q., Chen, X., Savoye, Y., Tu, C., and Cohen-Or, D. 2014. Build-to-Last: Strength to Weight 3D Printed Objects. ACM Transactions on Graphics 33, 4 (jul), 1--10.
[14]
Mises, R. V. 1986. The mechanics of solids in the plastically-deformable state.
[15]
Musialski, P., Auzinger, T., Birsak, M., Wimmer, M., and Kobbelt, L. 2015. Reduced-Order Shape Optimization Using Offset Surfaces. ACM Transactions on Graphics (Proc. ACM SIGGRAPH 2015) 34, 4 (jul), 102:1--102:9.
[16]
Nocedal, J., and Wright, S. 2006. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer New York.
[17]
Panetta, J., Zhou, Q., Malomo, L., Pietroni, N., Cignoni, P., and Zorin, D. 2015. Elastic textures for additive fabrication. ACM Transactions on Graphics 34, 4 (jul), 135:1--135:12.
[18]
Pérez, J., Thomaszewski, B., Coros, S., Bickel, B., Canabal, J. A., Sumner, R., and Otaduy, M. A. 2015. Design and fabrication of flexible rod meshes. ACM Transactions on Graphics 34, 4 (jul), 138:1--138:12.
[19]
Prévost, R., Whiting, E., Lefebvre, S., and Sorkine-Hornung, O. 2013. Make It Stand: Balancing Shapes for 3D Fabrication. ACM Transactions on Graphics 32, 4 (jul), 1.
[20]
Schumacher, C., Bickel, B., Rys, J., Marschner, S., Daraio, C., and Gross, M. 2015. Microstructures to control elasticity in 3D printing. ACM Transactions on Graphics 34, 4 (jul), 136:1--136:13.
[21]
Skouras, M., Thomaszewski, B., Coros, S., Bickel, B., and Gross, M. 2013. Computational design of actuated deformable characters. ACM Transactions on Graphics 32, 4 (jul), 1.
[22]
Skouras, M., Thomaszewski, B., Kaufmann, P., Garg, A., Bickel, B., Grinspun, E., and Gross, M. 2014. Designing inflatable structures. ACM Transactions on Graphics 33, 4 (jul), 1--10.
[23]
Stava, O., Vanek, J., Benes, B., Carr, N., and Měch, R. 2012. Stress relief: improving structural strength of 3D printable objects. ACM Transactions on Graphics 31, 4 (jul), 1--11.
[24]
Tagliasacchi, A., Alhashim, I., Olson, M., and Zhang, H. 2012. Mean Curvature Skeletons. Computer Graphics Forum 31, 5 (aug), 1735--1744.
[25]
Umetani, N., Mitani, J., and Igarashi, T. 2010. Designing Custom-made Metallophone with Concurrent Eigenanalysis. In Proceedings of the Conference on New Interfaces for Musical Expression (NIME).
[26]
Umetani, N., Kaufman, D. M., Igarashi, T., and Grinspun, E. 2011. Sensitive couture for interactive garment modeling and editing. ACM Transactions on Graphics 30, 4 (jul), 1.
[27]
Umetani, N., Igarashi, T., and Mitra, N. J. 2012. Guided exploration of physically valid shapes for furniture design. ACM Transactions on Graphics 31, 4 (jul), 1--11.
[28]
Vallet, B., and Lévy, B. 2008. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27, 2 (apr), 251--260.
[29]
Wang, L., and Whiting, E. 2016. Buoyancy Optimization for Computational Fabrication. Computer Graphics Forum (Proceedings of Eurographics 2016) 35, 2.
[30]
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 Transactions on Graphics 32, 6 (nov), 1--10.
[31]
Zhou, Q., Panetta, J., and Zorin, D. 2013. Worst-case structural analysis. ACM Transactions on Graphics 32, 4 (jul), 1.
[32]
Zhu, L., Xu, W., Snyder, J., Liu, Y., Wang, G., and Guo, B. 2012. Motion-guided mechanical toy modeling. ACM Transactions on Graphics 31, 6 (nov), 1.

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cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 35, Issue 4
July 2016
1396 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/2897824
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Publication History

Published: 11 July 2016
Published in TOG Volume 35, Issue 4

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Author Tags

  1. digital fabrication
  2. geometric design optimization
  3. geometry processing
  4. non-convex optimization
  5. shape optimization

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  • (2023)Additive manufacturing of non-uniform scaffolds for mass center controlProceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications10.1177/14644207231205281238:2(347-356)Online publication date: 9-Oct-2023
  • (2023)Computational Design of Flexible Planar MicrostructuresACM Transactions on Graphics10.1145/361839642:6(1-16)Online publication date: 5-Dec-2023
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