# zbMATH — the first resource for mathematics

“Color” level sets: A multi-phase method for structural topology optimization with multiple materials. (English) Zbl 1060.74585
Summary: In this paper we address the problem of structural shape and topology optimization in a multi-material domain. A level-set method is employed as an alternative approach to the popular homogenization-based methods of rule of mixtures for multi-material modeling. A multi-phase level-set model is adapted for material and topology representation. This model eliminates the need for a material interpolation or phase mixing scheme. It only requires $$m$$ level-set functions to represent a structure of $$n=2^m$$ different material phases, in a principle similar to combining colors from the three primary colors. Therefore, this multi-phase model may be referred to as a ”color” level-set representation which has its unique benefits: it is flexible to handle complex topologies; it substantially reduces the number of model functions when $$n>3$$; it automatically avoids the problem of overlap between material phases of a conventional partitioning approach. We describe numerical techniques for efficient and robust implementation of the method, by embedding a rectilinear grid in a fixed finite element mesh defined on a reference design domain. This would separate the issues of accuracy in numerical calculations of the physical equation and in the level-set model propagation. A gradient projection method is described for incorporating multiple constraints in the problem. Finally, the benefits and the advantages of the developed method are illustrated with several 2D examples of mean compliance minimization of multi-material structures.

##### MSC:
 74P15 Topological methods for optimization problems in solid mechanics 74S30 Other numerical methods in solid mechanics (MSC2010) 65K10 Numerical optimization and variational techniques
Full Text:
##### References:
 [1] Allaire, G.; Jouve, F.; Taoder, A.-M., A level-set method for shape optimization, C.R. acad. sci. Paris ser. I, 334, 1-6, (2002) [2] Ambrosio, L.; Buttazzo, G., An optimal design problem with perimeter penalization, Calc. var. partial differen. equat., 1, 55-59, (1993) · Zbl 0794.49040 [3] Bendsoe, M.P., Optimal shape design as a material distribution problem, Struct. optim., 1, 193-202, (1989) [4] Bendsoe, M.P., Optimization of structural topology, shape and material, (1997), Springer Berlin [5] M.P. Bendsoe, Variable-topology optimization: status and challenges, in: W. Wunderlich (Ed.), Proceedings of the European Conference on Computational Mechanics, September 1999 [6] Bendsoe, M.P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenisation method, Comput. methods appl. mech. engrg., 71, 197-224, (1988) · Zbl 0671.73065 [7] Bendsoe, M.P.; Sigmund, O., Material interpolations in topology optimization, Arch. appl. mech., 69, 635-654, (1999) · Zbl 0957.74037 [8] Bourdin, B., Filters in topology optimization, Int. J. numer. methods engrg., 50, 2143-2158, (2001) · Zbl 0971.74062 [9] Bourdin, B.; Chambolle, A., Implementation of an adaptive finite-element approximation of the mumford – shah functional, Numer. math., 85, 4, 609-646, (2000) · Zbl 0961.65062 [10] Bulman, S.; Sienz, J.; Hinton, E., Comparisons between algorithms for structural topology optimization using a series of benchmark studies, Comput. struct., 79, 1203-1218, (2001) [11] Cheng, G., On non-smoothness in optimal design of solid elastic plates, Int. J. solids struct., 17, 795-810, (1981) · Zbl 0515.73088 [12] Diaz, R.; Sigmund, O., Checkerboards patterns in layout optimization, Struct. optim., 10, 10-45, (1995) [13] Fujii, D.; Chen, B.C.; Kicuchi, N., Composite material design of two-dimensional structures using the homogenization design method, Int. J. numer. methods engrg., 50, 2031-2051, (2001) · Zbl 0994.74055 [14] Gibson, L.J.; Ashby, M.F., Cellular solids: structure and properties, (1997), Cambridge University Press Cambridge, UK [15] Haber, R.B.; Jog, C.S.; Bendsoe, M.P., A new approach to variable-topology shape design using a constraint on perimeter, Struct. optim., 11, 1-12, (1996) [16] Haug, E.J.; Choi, K.K.; Komkov, V., Design sensitivity analysis of structural systems, (1986), Academic Press Orlando · Zbl 0618.73106 [17] Jog, C.S., Topology design of structures using a dual algorithm and a constraint on the perimeter, Int. J. numer. methods engrg., 54, 1007-1019, (2002) · Zbl 1098.74659 [18] Lorensen, W.E.; Cline, H.E., Marching cubes: a high resolution 3D surface construction algorithm, Comput. graph., 21, 4, 163-169, (1987) [19] Osher, S.; Sethian, J.A., Front propagating with curvature-dependent speed: algorithms based on hamilton – jacobi formulations, J. comput. phys., 79, 12-49, (1988) · Zbl 0659.65132 [20] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, (2003), Springer New York · Zbl 1026.76001 [21] Osher, S.; Santosa, F., Level set methods for optimization problems involving geometry and constraints. I. frequencies of a two-density inhomogeneous drum, J. comput. phys., 171, 1, 272-288, (2001) · Zbl 1056.74061 [22] Petersson, J., Some convergence results in perimeter-controlled topology optimization, Comput. methods appl. mech. engrg., 171, 123-140, (1999) · Zbl 0947.74050 [23] Rozvany, G., Structural design via optimality criteria, (1988), Kluwer Dordrecht [24] Rozvany, G., Aims, scope, methods, history and unified terminology of computer aided topology optimization in structural mechanics, Struct. multidiscip. optim., 21, 90-108, (2001) [25] De Ruiter, M.J.; van Keulen, F., Topology optimization: approaching the material distribution problem using a topological function description, (), 111-119 [26] Sethian, J.A.; Wiegmann, A., Structural boundary design via level set and immersed interface methods, J. comput. phys., 163, 2, 489-528, (2000) · Zbl 0994.74082 [27] Sethian, J.A., Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science, (1999), Cambridge University Press · Zbl 0973.76003 [28] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock capture schemes, J. comput. phys., 77, 439-471, (1988) · Zbl 0653.65072 [29] Sigmund, O., Design of multiphysics actuators using topology optimization. part II: two-material structures, Comput. methods appl. mech. engrg., 190, 6605-6627, (2001) · Zbl 1116.74407 [30] Sigmund, O., Topology optimization: a tool for the tailoring of structures and materials, Philos. trans.: math. phys. engrg. sci., 358, 211-228, (2000) · Zbl 0984.74062 [31] Sigmund, O.; Petersson, J., Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Struct. optim., 16, 1, 68-75, (1998) [32] Sokolowski, J.; Zolesio, J.P., Introduction to shape optimization: shape sensitivity analysis, (1992), Springer-Verlag New York · Zbl 0761.73003 [33] S. Suresh, A. Mortensen, Fundamentals of Functionally Graded Materials, IDM Communications Ltd., 1988 [34] Swan, C.C.; Kosaka, I., Voigt – reuss topology optimization for structures with linear elastic material behaviors, Int. J. numer. methods engrg., 40, 3033-3057, (1997) · Zbl 0903.73047 [35] Tsai, A.; Yezzi, A.; Willsky, A.S., Curve evolution implementation of the mumford – shah functional for image segmentation, denoising, interpolation, and magnification, IEEE trans. image process., 10, 8, 1169-1186, (2001) · Zbl 1062.68595 [36] Vese, L.A.; Chan, T.F., A multiphase level set framework for image segmentation using the Mumford and Shah model, Int. J. comput. vision, 50, 3, 271-293, (2002) · Zbl 1012.68782 [37] Wang, M.Y.; Wang, X.M.; Guo, D.M., A level set method for structural topology optimization, Comput. methods appl. mech. engrg., 192, 1-2, 227-246, (2003) · Zbl 1083.74573 [38] X.M. Wang, M.Y. Wang, D.M. Guo, Structural shape and topology optimization in a level-set based framework of region representation, Struct. Multidiscip. Optim., in press [39] M.Y. Wang, X.M. Wang, PDE-driven level sets, velocity fields, and perimeter penalization in structural topology optimization, Int. J. Numer. Methods Engrg., submitted for publication [40] Yin, L.; Ananthasuresh, G.K., Topology optimization of compliant mechanisms with multiple materials using a peak function material interpolation scheme, Struct. multidiscip. optim., 23, 49-62, (2001) [41] Zhao, H.-K.; Merriman, B.; Osher, S.; Wang, L., Capturing the behavior of bubbles and drops using the variational level set approach, J. comput. phys., 143, 495-518, (1998) · Zbl 0936.76065
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.