Abstract
In this work, we develop a topology optimization method for linear elastic contact problems with maximum contact pressure constraint. First, the Kreisselmeier–Steinhauser (KS) function is adopted as an aggregated measure of the maximum contact pressure over specific contact regions. Then, the maximum contact pressure constraint is introduced into the standard volume-constrained compliance minimization problem and formulated in the framework of B-spline parameterization method. Two geometric constraints are further extended to suppress the intermediate densities and control minimum length scale. The adjoint method is employed for deriving design sensitivities analytically. Finally, both frictionless and frictional problems are tested to demonstrate the effectiveness of the proposed method. It is shown that the maximum contact pressure can be effectively controlled using the contact pressure constraint and thus avoiding the concentration of contact pressure. The influence of maximum contact pressure constraint on the optimization result is discussed in comparison with the standard maximum stiffness design. Effects of friction behavior upon optimized results and contact pressure are also highlighted. It concludes that the maximum contact pressure can be reduced at the cost of the structural stiffness.
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References
Behrou R, Lawry M, Maute K (2017) Level set topology optimization of structural problems with interface cohesion. Int J Numer Meth Eng 112(8):990–1016
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224
Bendsøe MP, Sigmund O (2013) Topology optimization: theory, methods, and applications. Springer, New York
Clausen A, Andreassen E (2017) On filter boundary conditions in topology optimization. Struct Multidisc Optim 56(5):1147–1155
Collins JA, Busby HR, Staab GH (2009) Mechanical design of machine elements and machines: a failure prevention perspective. Wiley, New York
Desmorat B (2007) Structural rigidity optimization with frictionless unilateral contact. Int J Solids Struct 44(3–4):1132–1144
Fancello EA, Feijóo RA (1994) Shape optimization in frictionless contact problems. Int J Numer Meth Eng 37(13):2311–2335
Fernandez F, Puso MA, Solberg J, Tortorelli DA (2020) Topology optimization of multiple deformable bodies in contact with large deformations. Comput Methods Appl Mech Eng 371:113288
Hilding D (2000) The equilibrium state of a structure subject to frictional contact. Eur J Mech A Solids 19(6):1029–1040
Hilding D, Klarbring A (2012) Optimization of structures in frictional contact. Comput Methods Appl Mech Eng 205–208:83–90
Hilding D, Klarbring A, Petersson J (1999) Optimization of structures in unilateral contact. Appl Mech Rev 52(4):139–160
Jeong GE, Youn SK, Park K (2018) Topology optimization of deformable bodies with dissimilar interfaces. Comput Struct 198:1–11
Klarbring A (1992) On the problem of optimizing contact force distributions. J Optim Theory Appl 74(1):131–150
Kreisselmeier G, Steinhauser R (1980) Systematic control design by optimizing a vector performance index. Computer aided design of control systems. Elsevier, New York, pp 113–117
Kristiansen H, Poulios K, Aage N (2020) Topology optimization for compliance and contact pressure distribution in structural problems with friction. Comput Methods Appl Mech Eng 364:112915
Lawry M, Maute K (2018) Level set shape and topology optimization of finite strain bilateral contact problems. Int J Numer Meth Eng 113(8):1340–1369
Li W, Li Q, Steven GP, Xie Y (2003) An evolutionary approach to elastic contact optimization of frame structures. Finite Elem Anal Des 40(1):61–81
Li H, Yamada T, Jolivet P, Furuta K, Kondoh T, Izui K, Nishiwaki S (2021a) Full-scale 3d structural topology optimization using adaptive mesh refinement based on the level-set method. Finite Elem Anal Des 194:103561
Li J, Zhang W, Niu C, Gao T (2021b) Topology optimization of elastic contact problems using b-spline parameterization. Struct Multidisc Optim 63(4):1669–1686
Lohan DJ, Allison JT (2017) Temperature constraint formulations for heat conduction topology optimization. In: Proceedings of the 12th world congress on structural and multidisciplinary optimization, Braunschweig, Germany
Luo Y, Li M, Kang Z (2016) Topology optimization of hyperelastic structures with frictionless contact supports. Int J Solids Struct 81:373–382
Mulaik SA (2009) Foundations of factor analysis. CRC Press, Boca Raton
Myśliński A (2008) Level set method for optimization of contact problems. Eng Anal Boundary Elem 32(11):986–994
Myśliński A (2015) Piecewise constant level set method for topology optimization of unilateral contact problems. Adv Eng Softw 80:25–32
Niu C, Zhang W, Gao T (2019) Topology optimization of continuum structures for the uniformity of contact pressures. Struct Multidisc Optim 60(1):185–210
Niu C, Zhang W, Gao T (2020) Topology optimization of elastic contact problems with friction using efficient adjoint sensitivity analysis with load increment reduction. Comput Struct 238:106296
Ou H, Lu B, Cui Z, Lin C (2013) A direct shape optimization approach for contact problems with boundary stress concentration. J Mech Sci Technol 27(9):2751–2759
Popov VL (2010) Contact mechanics and friction. Springer, New York
Qian X (2013) Topology optimization in b-spline space. Comput Methods Appl Mech Eng 265:15–35
Qian X, Sigmund O (2013) Topological design of electromechanical actuators with robustness toward over-and under-etching. Comput Methods Appl Mech Eng 253:237–251
Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33(4–5):401–424
Strömberg N (2013) The influence of sliding friction on optimal topologies. Recent advances in contact mechanics. Springer, New York, pp 327–336
Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12(2):555–573
Wang M, Qian X (2015) Efficient filtering in topology optimization via b-splines. J Mech Des 137(3):031402
Wang F, Lazarov BS, Sigmund O (2011) On projection methods, convergence and robust formulations in topology optimization. Struct Multidisc Optim 43(6):767–784
Wriggers P (2006) Computational contact mechanics. Springer, New York
Yang K, Fernandez E, Niu C, Duysinx P, Zhu J, Zhang W (2019) Note on spatial gradient operators and gradient-based minimum length constraints in simp topology optimization. Struct Multidisc Optim 60(1):393–400
Zhang W, Niu C (2018) A linear relaxation model for shape optimization of constrained contact force problem. Comput Struct 200:53–67
Zhou M, Lazarov BS, Wang F, Sigmund O (2015) Minimum length scale in topology optimization by geometric constraints. Comput Methods Appl Mech Eng 293:266–282
Zhou Y, Lin Q, Hong J, Yang N (2020) Combined interface shape and material stiffness optimization for uniform distribution of contact stress. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1860086
Zhu J, Zhou H, Wang C, Zhou L, Yuan S, Zhang W (2020) A review of topology optimization for additive manufacturing: status and challenges. Chin J Aeronaut 34(1):91–110
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This work is supported by the National Natural Science Foundation of China (12032018, 11620101002)
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Zhang, W., Li, J. & Gao, T. Topology optimization of elastic contact problems with maximum contact pressure constraint. Struct Multidisc Optim 65, 106 (2022). https://doi.org/10.1007/s00158-022-03195-z
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DOI: https://doi.org/10.1007/s00158-022-03195-z