# zbMATH — the first resource for mathematics

Topology optimization for compliance and contact pressure distribution in structural problems with friction. (English) Zbl 1442.74171
Summary: This paper concerns density-based topology optimization of linear elastic contact problems, aiming to present robust and practically realizable designs for different objective functions. First we revisit a compliance minimization with frictionless contact problem from the literature and present crisp solid-void designs, based on the so-called modified robust topology optimization formulation. An adaptation of this problem to frictional contact is then solved for various friction coefficients and it is checked that the optimization algorithm indeed exploits the presence of friction for lowering the objective further. Secondly, we propose and demonstrate the use of a $$p$$-norm based objective function to control the distribution and variation of contact pressure, on an a priori unknown area of contact, between a body of unknown topology and an obstacle. To have control over the contact pressure, a Lagrange multiplier based contact formulation is used within a coupled Newton solution, for imposing impenetrability, friction, and the corresponding complementarity conditions. The adjoint method is employed for deriving consistent design sensitivities for the mixed formulation involving both displacements and contact Lagrange multipliers. Through a series of numerical examples, it is demonstrated how an even distribution of contact pressure and crisp solid-void designs can be obtained for problems with and without friction.
##### MSC:
 74P15 Topological methods for optimization problems in solid mechanics 74M10 Friction in solid mechanics
top.m
Full Text:
##### References:
 [1] Bendsøe, M. P., Optimal shape design as a material distribution problem, Struct. Optim., 1, 4, 193-202 (1989) [2] Aage, N.; Andreassen, E.; Lazarov, B. S.; Sigmund, O., Giga-voxel computational morphogenesis for structural design, Nature, 550, 7674, 84 (2017) [3] Andreasen, C. S.; Gersborg, A. R.; Sigmund, O., Topology optimization of microfluidic mixers, Internat. J. Numer. Methods Fluids, 61, 5, 498-513 (2009) · Zbl 1172.76014 [4] Aage, N.; Poulsen, T. H.; Gersborg-Hansen, A.; Sigmund, O., Topology optimization of large scale stokes flow problems, Struct. Multidiscip. Optim., 35, 2, 175-180 (2008) · Zbl 1273.76094 [5] Christiansen, R. E.; Sigmund, O., Designing meta material slabs exhibiting negative refraction using topology optimization, Struct. Multidiscip. Optim., 54, 3, 469-482 (2016) [6] Dühring, M. B.; Jensen, J. S.; Sigmund, O., Acoustic design by topology optimization, J. Sound Vib., 317, 3-5, 557-575 (2008) [7] Hilding, D.; Klarbring, A.; Pang, J.-S., Minimization of maximum unilateral force, Comput. Methods Appl. Mech. Engrg., 177, 3-4, 215-234 (1999) · Zbl 0943.74049 [8] N. Strömberg, A. Klarbring, Minimization of compliance of a linear elastic structure with contact constraints by using sequential linear programming and Newton’s method, in: The Proceedings of the 7th International ASMOUK/ISSMO International Conference on Engineering Design Optimization, Bath, UK, 2008. [9] Strömberg, N.; Klarbring, A., Topology optimization of structures with contact constraints by using a smooth formulation and a nested approach, (8th World Congr. Struct. Multidiscip. Optim., Vol. 1 (2009)), 1-8 [10] Hilding, D., A heuristic smoothing procedure for avoiding local optima in optimization of structures subject to unilateral constraints, Struct. Multidiscip. Optim., 20, 1, 29-36 (2000) [11] Strömberg, N., Topology optimization of structures with manufacturing and unilateral contact constraints by minimizing an adjustable compliance-volume product, Struct. Multidiscip. Optim., 42, 3, 341-350 (2010) · Zbl 1274.74397 [12] Strömberg, N.; Klarbring, A., Topology optimization of structures in unilateral contact, Struct. Multidiscip. Optim., 41, 1, 57-64 (2010) · Zbl 1274.74398 [13] Strömberg, N., The influence of sliding friction on optimal topologies, (Recent Advances in Contact Mechanics (2013), Springer), 327-336 [14] Wang, F.; Lazarov, B. S.; Sigmund, O., On projection methods, convergence and robust formulations in topology optimization, Struct. Multidiscip. Optim., 43, 6, 767-784 (2011) · Zbl 1274.74409 [15] Borrvall, T.; Petersson, J., Topology optimization using regularized intermediate density control, Comput. Methods Appl. Mech. Engrg., 190, 37-38, 4911-4928 (2001) · Zbl 1022.74035 [16] Achieving minimum length scale in topology optimization using nodal design variables and projection functions, Internat. J. Numer. Methods Engrg., 61, September 2003, 238-254 (2004) · Zbl 1079.74599 [17] Klarbring, A., On the problem of optimizing contact force distributions, J. Optim. Theory Appl., 74, 1, 131-150 (1992) · Zbl 0795.49026 [18] Zowe, J.; Kočvara, M.; Bendsøe, M. P., Free material optimization via mathematical programming, Math. Program., 79, 1-3, 445-466 (1997) · Zbl 0886.90145 [19] Patriksson, M.; Petersson, J., A subgradient method for contact structural optimization, (Ferris, M. C.; Pang, J.-S., Complementarity and Variational Problems— State of the Art, Proceedings of the International Conference on Complementarity Problems (ICCP-95) (1997), SIAM: SIAM Philadelphia, PA), 295-314 · Zbl 0886.90182 [20] Petersson, J.; Patriksson, M., Topology optimization of sheets in contact by a subgradient method, Internat. J. Numer. Methods Engrg., 40, 7, 1295-1321 (1997) · Zbl 0890.73046 [21] Kočvara, M.; Zibulevsky, M.; Zowe, J., Mechanical design problems with unilateral contact, ESAIM Math. Model. Numer. Anal., 32, 3, 255-281 (1998) · Zbl 0901.73055 [22] Hilding, D.; Klarbring, A.; Petersson, J., Optimization of structures in unilateral contact, Appl. Mech. Rev., 52, 4, 139-160 (1999) [23] Desmorat, B., Structural rigidity optimization with frictionless unilateral contact, Int. J. Solids Struct., 44, 3, 1132-1144 (2007) · Zbl 1137.74048 [24] Andrade-Campos, A.; Ramos, A.; Simões, J. A., A model of bone adaptation as a topology optimization process with contact, J. Biomed. Sci. Eng., 5, 05, 229 (2012) [25] Luo, Y.; Ll, M.; Kang, Z., Topology optimization of hyperelastic structures with frictionless contact supports, Int. J. Solids Struct., 81, 373-382 (2016) [26] Hilding, D.; Klarbring, A., Optimization of structures in frictional contact, Comput. Methods Appl. Mech. Engrg., 205, 83-90 (2012) · Zbl 1239.74078 [27] Lawry, M.; Maute, K., Level set topology optimization of problems with sliding contact interfaces, Struct. Multidiscip. Optim., 52, 6, 1107-1119 (2015) [28] Klarbring, A.; Petersson, J.; Rönnqvist, M., Truss topology optimization including unilateral contact, J. Optim. Theory Appl., 87, 1, 1-31 (1995) · Zbl 0841.73046 [29] Alart, P.; Curnier, A., A mixed formulation for frictional contact problems prone to Newton like solution methods, Comput. Methods Appl. Mech. Engrg., 92, 3, 353-375 (1991) · Zbl 0825.76353 [30] Poulios, K.; Renard, Y., An unconstrained integral approximation of large sliding frictional contact between deformable solids, Comput. Struct., 153, 75-90 (2015) [31] Renard, Y., Generalized Newton’s methods for the approximation and resolution of frictional contact problems in elasticity, Comput. Methods Appl. Mech. Engrg., 256, 38-55 (2013) · Zbl 1352.74194 [32] Sigmund, O., Design of Material Structures Using Topology Optimization (1994), Technical University of Denmark: Technical University of Denmark Denmark, (Ph.D. thesis) · Zbl 1116.74407 [33] Buhl, T.; Pedersen, C. B.; Sigmund, O., Stiffness design of geometrically nonlinear structures using topology optimization, Struct. Multidiscip. Optim., 19, 2, 93-104 (2000) [34] Sigmund, O., A 99 line topology optimization code written in matlab, Struct. Multidiscip. Optim., 21, 2, 120-127 (2001) [35] Groen, J. P.; Sigmund, O., Homogenization-based topology optimization for high-resolution manufacturable microstructures, Int. J. Numer. Methods Eng., 113, 8, 1148-1163 (2018) [36] Bourdin, B., Filters in topology optimization, Internat. J. Numer. Methods Engrg., 50, 9, 2143-2158 (2001) · Zbl 0971.74062 [37] Bendsøe, M. P.; Sigmund, O., Material interpolation schemes in topology optimization, Arch. Appl. Mech., 69, 9-10, 635-654 (1999) · Zbl 0957.74037 [38] Sigmund, O., Manufacturing tolerant topology optimization, Acta Mech. Sinica, 25, 2, 227-239 (2009) · Zbl 1270.74165 [39] Sigmund, O., Morphology-based black and white filters for topology optimization, Struct. Multidiscip. Optim., 33, 4-5, 401-424 (2007) [40] Svanberg, K., The method of moving asymptotes - a new method for structural optimization, Internat. J. Numer. Methods Engrg., 24, 2, 359-373 (1987) · Zbl 0602.73091 [41] Petersson, J., Optimization of Structures in Unilateral Contact (1995), Univ. [42] Guest, J. K.; Asadpoure, A.; Ha, S.-H., Eliminating beta-continuation from heaviside projection and density filter algorithms, Struct. Multidiscip. Optim., 44, 4, 443-453 (2011) · Zbl 1274.74337 [43] Zhou, M.; Lazarov, B. S.; Wang, F.; Sigmund, O., Minimum length scale in topology optimization by geometric constraints, Comput. Methods Appl. Mech. Engrg., 293, 266-282 (2015) · Zbl 1423.74778
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.