In this paper we propose an optimization framework for interior carving of 3D fabricated shapes. Interior carving is an important technique widely used in industrial and artistic designs to achieve functional purposes by hollowing interior shapes in objects. We formulate such functional purpose as the objective function of an optimization problem whose solution indicates the optimal interior shape. In contrast to previous volumetric methods, we directly represent the boundary of the interior shape as a triangular mesh. We use Eulerian semiderivative to relate the time derivative of the object function to a virtual velocity field and iteratively evolve the interior shape guided by the velocity field with surface tracking. In each iteration, we compute the velocity field guaranteeing the decrease of objective function by solving a linear programming problem. We demonstrate this general framework in a novel application of designing objects floating in fluid and two previously investigated applications, and print various optimized objects to verify its effectiveness.
Prévost R, Whiting E, Lefebvre S, Sorkine-Hornung O. Make it stand: balancing shapes for 3D fabrication. ACM Transactions on Graphics, 2013, 32(4): 81 https://doi.org/10.1145/2461912.2461957
2
Bächer M, Whiting E, Bickel B, Sorkine-Hornung O. Spin-it: optimizing moment of inertia for spinnable objects. ACM Transactions on Graphics, 2014, 33(4): 96 https://doi.org/10.1145/2601097.2601157
Chen S, Torterelli D. Three-dimensional shape optimization with variational geometry. Structural Optimization, 1997, 13(2): 81–94 https://doi.org/10.1007/BF01199226
5
Braibant V, Fleury C. Shape optimal design using B-splines. Computer Methods in Applied Mechanics and Engineering, 1984, 44(3): 247–267 https://doi.org/10.1016/0045-7825(84)90132-4
6
Xu D, Ananthasuresh G K. Freeform skeletal shape optimization of compliant mechanisms. Journal of Mechanical Design, 2003, 125(2): 253–261 https://doi.org/10.1115/1.1563634
7
Bendsoe M P. Optimal shape design as a material distribution problem. Structural Optimization, 1989, 1(4): 193–202 https://doi.org/10.1007/BF01650949
8
Wang M Y, Wang X M, Guo D M. A level set method for structural topology optimization. Computer Methods in Applied Mechanics and Engineering, 2003, 192(1): 227–246 https://doi.org/10.1016/S0045-7825(02)00559-5
9
Zhou M, Pagaldipti N, Thomas H, Shyy Y. An integrated approach to topology, sizing, and shape optimization. Structural and Multidisciplinary Optimization, 2004, 26(5): 308–317 https://doi.org/10.1007/s00158-003-0351-2
10
Haftka R T, Grandhi R V. Structural shape optimizationa——a survey. Computer Methods in Applied Mechanics and Engineering, 1986, 57(1): 91–106 https://doi.org/10.1016/0045-7825(86)90072-1
11
Saitou K, Izui K, Nishiwaki S, Papalambros P. A survey of structural optimization in mechanical product development. Journal of Computing and Information Science in Engineering, 2005, 5(3): 214–226 https://doi.org/10.1115/1.2013290
12
Luo L, Baran I, Rusinkiewicz S, Matusik W. Chopper: partitioning models into 3D-printable parts. ACM Transactions on Graphics, 2012, 31(6) https://doi.org/10.1145/2366145.2366148
13
Attene M. Shapes in a box: disassembling 3D objects for efficient packing and fabrication. Computer Graphics Forum, 2015
14
Zhou Y B, Sueda S, Matusik W, Shamir A. Boxelization: folding 3D objects into boxes. ACM Transactions on Graphics, 2014, 33(4): 71 https://doi.org/10.1145/2601097.2601173
15
Bickel B, Kaufmann P, Skouras M, Thomaszewski B, Bradley D, Beeler T, Jackson P, Marschner S, Matusik W, Gross M. Physical face cloning. ACM Transactions on Graphics, 2012, 31(4): 118 https://doi.org/10.1145/2185520.2185614
16
Skouras M, Thomaszewski B, Coros S, Bickel B, Gross M. Computational design of actuated deformable characters. ACM Transactions on Graphics, 2013, 32(4): 82 https://doi.org/10.1145/2461912.2461979
17
Chen X, Zheng C, Xu W, Zhou K. An asymptotic numerical method for inverse elastic shape design. ACM Transactions on Graphics, 2014, 33(4): 95 https://doi.org/10.1145/2601097.2601189
18
Bächer M, Bickel B, James D L, Pfister H. Fabricating articulated characters from skinned meshes. ACM Transactions on Graphics, 2012, 31(4): 47 https://doi.org/10.1145/2185520.2185543
19
Calì J, Calian D A, Amati C, Kleinberger R, Steed A, Kautz J, Weyrich T. 3D-printing of non-assembly, articulated models. ACM Transactions on Graphics, 2012, 31(6): 130 https://doi.org/10.1145/2366145.2366149
20
Zhu L, Xu W, Snyder J, Liu Y, Wang G, Guo B. Motion-guided mechanical toy modeling. ACM Transactions on Graphics, 2012, 31(6): 127 https://doi.org/10.1145/2366145.2366146
21
Coros S, Thomaszewski B, Noris G, Sueda S, Forberg M, Sumner R W, Matusik W, Bickel B. Computational design of mechanical characters. ACM Transactions on Graphics, 2013, 32(4): 83 https://doi.org/10.1145/2461912.2461953
22
Umetani N, Igarashi T, Mitra N J. Guided exploration of physically valid shapes for furniture design. ACM Transactions on Graphics, 2012, 31(4): 86 https://doi.org/10.1145/2185520.2185582
23
Vouga E, Höbinger M, Wallner J, Pottmann H. Design of selfsupporting surfaces. ACM Transactions on Graphics, 2012, 31(4): 87 https://doi.org/10.1145/2185520.2185583
24
Panozzo D, Block P, Sorkine-Hornung O. Designing unreinforced masonry models. ACM Transactions on Graphics, 2013, 32(4): 91 https://doi.org/10.1145/2461912.2461958
25
De Goes F, Alliez P, Owhadi H, Desbrun M. On the equilibrium of simplicial masonry structures. ACM Transactions on Graphics, 2013, 32(4): 93 https://doi.org/10.1145/2461912.2461932
26
Wang W, Wang T Y, Yang Z, Liu L, Tong X, Tong W, Deng J, Chen F, Liu X. Cost-effective printing of 3D objects with skin-frame structures. ACM Transactions on Graphics, 2013, 32(6): 177 https://doi.org/10.1145/2508363.2508382
27
Lu L, Sharf A, Zhao H, Wei Y, Fan Q, Chen X, Savoye Y, Tu C, CohenOr D, Chen B. Build-to-last: strength to weight 3D printed objects. ACM Transactions on Graphics, 2014, 33(4): 97 https://doi.org/10.1145/2601097.2601168
28
Stava O, Vanek J, Benes B, Carr N, Mˇech R. Stress relief: improving structural strength of 3D printable objects. ACM Transactions on Graphics, 2012, 31(4): 48 https://doi.org/10.1145/2185520.2185544
Xie Y, Xu W, Yang Y, Guo X, Zhou K. Agile structural analysis for fabrication-aware shape editing. Computer Aided Geometric Design, 2015, 35: 163–179 https://doi.org/10.1016/j.cagd.2015.03.019
31
Musialski P, Auzinger T, Birsak M, Wimmer M, Kobbelt L. Reducedorder shape optimization using offset surfaces. ACM Transactions on Graphics, 2015, 34(4): 102 https://doi.org/10.1145/2766955
32
Delfour M C, Zolésio J P. Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. Philadelphia: Siam, 2011 https://doi.org/10.1137/1.9780898719826
McGrail S. Boats of the World: from the Stone Age to Medieval Times. New York: Oxford University Press, 2004
41
Ascher U M, Chin H, Reich S. Stabilization of DAEs and invariant manifolds. Numerische Mathematik, 1994, 67(2): 131–149 https://doi.org/10.1007/s002110050020
42
Mégel J, Kliava J. Metacenter and ship stability. American Journal of Physics, 2010, 78(7): 738–747 https://doi.org/10.1119/1.3285975
43
Byrd R H, Nocedal J, Waltz R A. Knitro: an integrated package for nonlinear optimization. In: Di Pillo G, Roma M, <Eds/>. Large-Scale Nonlinear Optimization, Vol 83. Springer, 2006, 35–59 https://doi.org/10.1007/0-387-30065-1_4
