Abstract
This paper concerns the heuristic-based material stiffness optimization of frictional linear elastic contact problems for having control over the contact stress distribution, aiming to extend the material stiffness optimization to multiple loading conditions, in which each of the loadings acts solely on the structures. A decrease level of the variance of the contact stress is introduced and a weighted sum of the decrease levels under all load cases is constructed as the objective function. The individual criterion for contact problems with multiple contact regions is addressed. The worst case design is adopted for multiple load cases, and an extreme reference stress, which is the highest stress level of the subdomain under all load cases, is defined to control the Young’s modulus modification process in a finite element framework. Through three numerical examples, it is demonstrated how an even distribution of the contact stress can be obtained for contact problems subjected to multiple load cases with single or multiple contact regions. Some new features of the material stiffness optimization with multiple loading conditions are also illustrated.
Similar content being viewed by others
References
Bendsøe, M.P., Díaz, A.R., Lipton, R., Taylor, J.E.: Optimal design of material properties and material distribution for multiple loading conditions. Int. J. Numer. Methods Eng. 38(7), 1149–1170 (1995). https://doi.org/10.1002/nme.1620380705
Ben-Tal, A., Kovara, M., Nemirovski, A., Zowe, J.: Free material design via semidefinite programming: The multiload case with contact conditions. SIAM Rev. 42(4), 695–715 (2000). https://doi.org/10.1137/S0036144500372081
Cai, K., Gao, Z., Shi, J.: Compliance optimization of a continuum with bimodulus material under multiple load cases. Comput. Aided Des. 45(2), 195–203 (2013). https://doi.org/10.1016/j.cad.2012.07.009
Chen, X., Jin, X., Shang, K., Zhang, Z.: Entropy-based method to evaluate contact-pressure distribution for assembly-accuracy stability prediction. Entropy 21(3), 322 (2019). https://doi.org/10.3390/e21030322
Collins, J.A., Staab, G.H.: Mechanical Design of Machine Elements and Machines: A Failure Prevention Perspective, 2nd edn. Wiley, New York (2010)
Conry, T.F., Seireg, A.: A mathematical programming method for design of elastic bodies in contact. J. Appl. Mech. 38, 387–392 (1971). https://doi.org/10.1115/1.3408787
Cramer, A.D., Challis, V.J., Roberts, A.P.: Microstructure interpolation for macroscopic design. Struct. Multidiscip. Optim. 53(3), 489–500 (2015). https://doi.org/10.1007/s00158-015-1344-7
Czarnecki, S., Lewiński, T.: A stress-based formulation of the free material design problem with the trace constraint and multiple load conditions. Struct. Multidiscip. Optim. 49(5), 707–731 (2014). https://doi.org/10.1007/s00158-013-1023-5
Czarnecki, S., Lewiński, T.: On material design by the optimal choice of Young’s modulus distribution. Int. J. Solids. Struct. 110–111, 315–331 (2017a). https://doi.org/10.1016/j.ijsolstr.2016.11.021
Czarnecki, S., Lewiński, T.: Pareto optimal design of non-homogeneous isotropic material properties for the multiple loading conditions. Phys. Status Solidi B 254(12), 1600821 (2017b). https://doi.org/10.1002/pssb.201600821
Diaz, A.R., Bendsøe, M.: Shape optimization of structures for multiple loading conditions using a homogenization method. Struct. Optim. 4(1), 17–22 (1992). https://doi.org/10.1007/BF01894077
Fernandez, F., Puso, M.A., Solberg, J., Tortorelli, D.A.: Topology optimization of multiple deformable bodies in contact with large deformations. Comput. Methods Appl. Mech. Eng. 371, 113288 (2020). https://doi.org/10.1016/j.cma.2020.113288
Goyat, V., Verma, S., Garg, R.K.: On the reduction of stress concentration factor in an infinite panel using different radial functionally graded materials. Int. J. Mater. Prod. Technol. 57(1–3), 109–131 (2018). https://doi.org/10.1504/Ijmpt.2018.092937
Goyat, V., Verma, S., Garg, R.K.: Stress concentration reduction using different functionally graded materials layer around the hole in an infinite panel. Strength Fract Complexity. 12(1), 31–45 (2019). https://doi.org/10.3233/sfc-190232
Haug, E.J., Kwak, B.M.: Contcat stress minimization by contour design. Int. J. Numer. Methods Eng. 12(6), 917–930 (1978). https://doi.org/10.1002/nme.1620120604
Hilding, D., Klarbring, A.: Optimization of structures in frictional contact. Comput. Methods Appl. Mech. Eng. 205–208, 83–90 (2012). https://doi.org/10.1016/j.cma.2011.02.014
Hilding, D., Klarbring, A., Pang, J.-S.: Minimization of maximum unilateral force. Comput. Methods Appl. Mech. Eng. 177(3), 215–234 (1999). https://doi.org/10.1016/S0045-7825(98)00382-X
Hilding, D., Torstenfelt, B., Klarbring, A.: A computational methodology for shape optimization of structures in frictionless contact. Comput. Methods Appl. Mech. Eng. 190(31), 4043–4060 (2001). https://doi.org/10.1016/S0045-7825(00)00310-8
Jeong, G.E., Youn, S.K., Park, K.C.: Topology optimization of deformable bodies with dissimilar interfaces. Comput. Struct. 198, 1–11 (2018). https://doi.org/10.1016/j.compstruc.2018.01.001
Johnson, K.L.: Contact Mechanics. Cambridge University Press, Cambridge (1987)
Klarbring, A.: On the problem of optimizing contact force distributions. J. Optim. Theory Appl. 74(1), 131–150 (1992). https://doi.org/10.1007/Bf00939896
Kočvara, M., Zibulevsky, M., Zowe, J.: Mechanical design problems with unilateral contact. Math. Model. Numer. Anal.. 32(3), 255–281 (1998). https://doi.org/10.1051/m2an/1998320302551
Kristiansen, H., Poulios, K., Aage, N.: Topology optimization for compliance and contact pressure distribution in structural problems with friction. Comput. Methods Appl. Mech. Eng. 364, 112915 (2020). https://doi.org/10.1016/j.cma.2020.112915
Lawry, M., Maute, K.: Level set topology optimization of problems with sliding contact interfaces. Struct. Multidiscip. Optim. 52(6), 1107–1119 (2015). https://doi.org/10.1007/s00158-015-1301-5
Lawry, M., Maute, K.: Level set shape and topology optimization of finite strain bilateral contact problems. Int. J. Numer. Methods Eng. 113(8), 1340–1369 (2018). https://doi.org/10.1002/nme.5582
Li, W., Li, Q., Steven, G.P., Xie, Y.M.: An evolutionary shape optimization procedure for contact problems in mechanical designs. Proc. IMechE Part C: J Mech. Eng. Sci. 217(4), 435–446 (2003). https://doi.org/10.1243/095440603321509711
Li, W., Li, Q., Steven, G.P., Xie, Y.M.: An evolutionary shape optimization for elastic contact problems subject to multiple load cases. Comput. Methods Appl. Mech. Eng. 194(30–33), 3394–3415 (2005). https://doi.org/10.1016/j.cma.2004.12.024
Li, J., Guan, Y., Wang, G., Wang, G., Zhang, H., Lin, J.: A meshless method for topology optimization of structures under multiple load cases. Structures 25, 173–179 (2020). https://doi.org/10.1016/j.istruc.2020.03.005
Marler, R.T., Arora, J.S.: The weighted sum method for multi-objective optimization: new insights. Struct. Multidiscip. Optim. 41(6), 853–862 (2009). https://doi.org/10.1007/s00158-009-0460-7
Myslinski, A.: Level set method for optimization of contact problems. Eng. Anal. Bound. Elem. 32(11), 986–994 (2008). https://doi.org/10.1016/j.enganabound.2007.12.008
Myslinski, A.: Piecewise constant level set method for topology optimization of unilateral contact problems. Adv. Eng. Softw. 80, 25–32 (2015). https://doi.org/10.1016/j.advengsoft.2014.09.020
Myśliński, A., Wróblewski, M.: Structural optimization of contact problems using Cahn–Hilliard model. Comput. Struct. 180, 52–59 (2017). https://doi.org/10.1016/j.compstruc.2016.03.013
Nakazawa, K., Maruyama, N., Hanawa, T.: Effect of contact pressure on fretting fatigue of austenitic stainless steel. Tribol. Int. 36(2), 79–85 (2003). https://doi.org/10.1016/S0301-679x(02)00135-4
Niu, C., Zhang, W.H., Gao, T.: Topology optimization of continuum structures for the uniformity of contact pressures. Struct. Multidiscip. Optim. 60(1), 185–210 (2019). https://doi.org/10.1007/s00158-019-02208-8
Niu, C., Zhang, W., Gao, T.: Topology optimization of elastic contact problems with friction using efficient adjoint sensitivity analysis with load increment reduction. Comput. Struct. 238, 106296 (2020). https://doi.org/10.1016/j.compstruc.2020.106296
Ou, H., Lu, B., Cui, Z.S., Lin, C.: A direct shape optimization approach for contact problems with boundary stress concentration. J. Mech. Sci. Technol. 27(9), 2751–2759 (2013). https://doi.org/10.1007/s12206-013-0721-7
Smyl, D.: An inverse method for optimizing elastic properties considering multiple loading conditions and displacement criteria. J. Mech. Des. (2018). https://doi.org/10.1115/1.4040788
Strömberg, N., Klarbring, A.: Topology optimization of structures in unilateral contact. Struct. Multidiscip. Optim. 41(1), 57–64 (2009). https://doi.org/10.1007/s00158-009-0407-z
Young, V., Querin, O.M., Steven, G.P., Xie, Y.M.: 3D and multiple load case bi-directional evolutionary structural optimization (BESO). Struct. Optim. 18(2–3), 183–192 (1999). https://doi.org/10.1007/BF01195993
Zhang, W.H., Niu, C.: A linear relaxation model for shape optimization of constrained contact force problem. Comput. Struct. 200, 53–67 (2018). https://doi.org/10.1016/j.compstruc.2018.02.005
Zhou, Y.C., Lin, Q.Y., Hong, J., Yang, N.: Combined interface shape and material stiffness optimization for uniform distribution of contact stress. Mech. Based Des. Struct. Mach. (2020). https://doi.org/10.1080/15397734.2020.1860086
Acknowledgements
The work is financially supported by the National Natural Science Foundation of China (No. 51975457 and No. 51635010), and the Foundation of State Key Laboratory of Smart Manufacturing for Special Vehicles and Transmission System (No. GZ2019KF005).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhou, Y., Lin, Q., Yang, X. et al. Material stiffness optimization for contact stress distribution in frictional elastic contact problems with multiple load cases. Int J Mech Mater Des 17, 503–519 (2021). https://doi.org/10.1007/s10999-021-09544-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10999-021-09544-y