On using a zero lower bound on the physical density in material distribution topology optimization. (English) Zbl 1441.74169

Quoc Khanh Nguyen; Serra-Capizzano, Stefano; Wadbro, Eddie

Comput. Methods Appl. Mech. Eng. 359, Article ID 112669, 17 p. (2020).

Summary: The current paper studies the possibility of allowing a zero lower bound on the physical density in material distribution based topology optimization. We limit our attention to the standard test problem of minimizing the compliance of a linearly elastic structure subject to a constant forcing. First order tensor product Finite Elements discretize the problem. An elementwise constant material indicator function defines the discretized, elementwise constant, physical density by using filtering and penalization. To alleviate the ill-conditioning of the stiffness matrix, due to the variation of the elementwise constant physical density, we precondition the system. We provide a specific spectral analysis for large matrix sizes for the one-dimensional problem with Dirichlet-Neumann conditions in detail, even if most of the mathematical tools apply also in a \(d\)-dimensional setting, \(d \geq 2\). It is easy to find an elementwise constant material indicator function so that the resulting preconditioned system matrix is singular when allowing the vanishing physical densities. However, for a large class of material indicator functions, the corresponding preconditioned system matrix has a condition number of the same order as the system matrix for the case when the physical density is one in all elements. Finally, we critically report and illustrate results from numerical experiments: as a conclusion, it is indeed possible to solve large-scale topology optimization problems, allowing a vanishing physical density, without encountering ill-conditioned system matrices during the optimization.

MSC:

74P15	Topological methods for optimization problems in solid mechanics
65K10	Numerical optimization and variational techniques

Keywords:

topology optimization; conditioning; preconditioning; large-scale problems

Software:

GPUTop

References:

[1]	Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engrg., 71, 197-224 (1988) · Zbl 0671.73065
[2]	Bendsøe, M. P.; Sigmund, O., Topology Optimization. Theory, Methods, and Applications (2003), Springer · Zbl 1059.74001
[3]	Aage, N.; Andreassen, E.; Lazarov, B. S.; Sigmund, O., Giga-voxel computational morphogenesis for structural design, Nature, 550, 84-86 (2017)
[4]	Schmidt, S.; Schulz, V., A 2589 line topology optimization code written for the graphics card, Comput. Vis. Sci., 14, 6, 249-256 (2011) · Zbl 1380.74100
[5]	Wadbro, E.; Berggren, M., Megapixel topology optimization on a graphics processing unit, SIAM Rev., 51, 707-721 (2009) · Zbl 1179.65079
[6]	Zhu, J.-H.; Zhang, W.-H.; Xia, L., Topology optimization in Aircraft and Aerospace structures design, Arch. Comput. Methods Eng., 23, 4, 595-622 (2016) · Zbl 1360.74128
[7]	Berggren, M.; Kasolis, F., Weak material approximation of holes with traction-free boundaries, SIAM J. Numer. Anal., 50, 4, 1827-1848 (2012) · Zbl 1253.35176
[8]	Buhl, T.; Pedersen, C. B.W.; Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidiscip. Optim., 19, 2, 93-104 (2000)
[9]	Bruns, T. E.; Tortorelli, D. A., An element removal and reintroduction strategy for the topology optimization of structures and compliant mechanisms, Internat. J. Numer. Methods Engrg., 57, 10, 1413-1430 (2003) · Zbl 1062.74589
[10]	Bruns, T., Zero density lower bounds in topology optimization, Comput. Methods Appl. Mech. Engrg., 196, 1, 566-578 (2006) · Zbl 1120.74711
[11]	Serra-Capizzano, S., Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations, Linear Algebra Appl., 366, 371-402 (2003) · Zbl 1028.65109
[12]	Serra-Capizzano, S., The GLT class as a generalized Fourier analysis and applications, Linear Algebra Appl., 419, 1, 180-233 (2006) · Zbl 1109.65032
[13]	Garoni, C.; Serra-Capizzano, S., The Theory of Generalized Locally Toeplitz Sequences: Theory and Applications - Vol. I (2017), Springer: Springer Cham · Zbl 1376.15002
[14]	Ekström, S.-E.; Serra-Capizzano, S., Eigenvalue isogeometric approximations based on B-splines: Tools and results, (Giannelli, C.; Speleers, H., Advanced Methods for Geometric Modeling and Numerical Simulation. Advanced Methods for Geometric Modeling and Numerical Simulation, INdAM Series 25 (2019), Springer), 57-76 · Zbl 1429.65167
[15]	Serra-Capizzano, S.; Tablino-Possio, C., Spectral and structural analysis of high precision finite difference matrices for elliptic operators, Linear Algebra Appl., 293, 1, 85-131 (1999) · Zbl 0940.65113
[16]	Bhatia, R., Matrix Analysis (1997), Springer: Springer New York
[17]	Böttcher, A.; Grudsky, S., Spectral Properties of Banded Toeplitz Matrices (2005), SIAM: SIAM Philadelphia · Zbl 1089.47001
[18]	Zygmund, A., Trigonometric Series (1959), Cambridge University Press: Cambridge University Press Cambridge · JFM 58.0296.09
[19]	Serra, S., The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems, Numer. Math., 81, 3, 461-495 (1999) · Zbl 0918.65059
[20]	Bertaccini, D.; Golub, G. H.; Serra-Capizzano, S.; Tablino-Possio, C., Preconditioned HSS methods for the solution of non-Hermitian positive definite linear systems and applications to the discrete convection-diffusion equation, Numer. Math., 99, 3, 441-484 (2005) · Zbl 1068.65041
[21]	Russo, A.; Serra-Capizzano, S.; Tablino-Possio, C., Quasi-optimal preconditioners for finite element approximations of diffusion dominated convection-diffusion equations on (nearly) equilateral triangle meshes, Numer. Linear Algebra Appl., 22, 1, 123-144 (2015) · Zbl 1363.65046
[22]	Hägg, L.; Wadbro, E., On minimum length scale control in topology optimization, Struct. Multidiscip. Optim., 58, 3, 1015-1032 (2018)
[23]	Svanberg, K.; Svärd, H., Density filters for topology optimization based on the pythagorean means, Struct. Multidiscip. Optim., 48, 5, 859-875 (2013)
[24]	Allaire, G.; Jouve, F.; Toader, A.-M., Structural optimization using sensitivity analysis and a level-set method, J. Comput. Phys., 194, 1, 363-393 (2004) · Zbl 1136.74368
[25]	Zhang, W.; Yuan, J.; Zhang, J.; Guo, X., A new topology optimization approach based on Moving Morphable Components (MMC) and the ersatz material model, Struct. Multidiscip. Optim., 53, 6, 1243-1260 (2016)