“Non-linear shape optimization using local subspace projections”

  • ©Przemyslaw Musialski, Christian Hafner, Florian Rist, Michael Birsak, Michael Wimmer, and Leif Kobbelt

Conference:


Type:


Title:

    Non-linear shape optimization using local subspace projections

Session/Category Title:   COMPUTATIONAL DESIGN OF STRUCTURES, SHAPES, AND SOUND


Presenter(s)/Author(s):


Moderator(s):



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.

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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    3. Bathe, K.-J. 2006. Finite Element Procedures. Prentice Hall.Google Scholar
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    6. Botsch, M., and Kobbelt, L. 2005. Real-Time Shape Editing using Radial Basis Functions. Comput. Graph. Forum 24, 3, 611–621.Google ScholarCross Ref
    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). Google ScholarDigital Library
    8. Eberly, D. H. 2010. Game physics, 2. edition ed. Morgan Kaufmann, Burlington, Mass.Google Scholar
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    12. Kuipers, L., and Niederreiter, H. 2012. Uniform Distribution of Sequences. Dover Books on Mathematics. Dover Publications.Google Scholar
    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. Google ScholarDigital Library
    14. Mises, R. V. 1986. The mechanics of solids in the plastically-deformable state.Google Scholar
    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. Google ScholarDigital Library
    16. Nocedal, J., and Wright, S. 2006. Numerical Optimization. Springer Series in Operations Research and Financial Engineering. Springer New York.Google Scholar
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    24. Tagliasacchi, A., Alhashim, I., Olson, M., and Zhang, H. 2012. Mean Curvature Skeletons. Computer Graphics Forum 31, 5 (aug), 1735–1744. Google ScholarDigital Library
    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).Google Scholar
    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. Google ScholarDigital Library
    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. Google ScholarDigital Library
    28. Vallet, B., and Lévy, B. 2008. Spectral Geometry Processing with Manifold Harmonics. Computer Graphics Forum 27, 2 (apr), 251–260.Google ScholarCross Ref
    29. Wang, L., and Whiting, E. 2016. Buoyancy Optimization for Computational Fabrication. Computer Graphics Forum (Proceedings of Eurographics 2016) 35, 2.Google Scholar
    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. Google ScholarDigital Library
    31. Zhou, Q., Panetta, J., and Zorin, D. 2013. Worst-case structural analysis. ACM Transactions on Graphics 32, 4 (jul), 1. Google ScholarDigital Library
    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. Google ScholarDigital Library


ACM Digital Library Publication:



Overview Page: