# Micro-texture design and optimization in hydrodynamic lubrication via two-scale analysis

- 395 Downloads

## Abstract

A novel computational surface engineering framework is developed to design micro-textures which can optimize the macroscopic response of hydrodynamically lubricated interfaces. All macroscopic objectives are formulated and analyzed within a homogenization-based two-scale setting and the micro-texture design is achieved through topology optimization schemes. Two non-standard aspects of this multiscale optimization problem, namely the temporal and spatial variations in the homogenized response of the micro-texture, are individually addressed. Extensive numerical investigations demonstrate the ability of the framework to deliver optimal micro-texture designs as well as the influence of major problem parameters.

### Keywords

Optimization Homogenization Two-scale analysis Texture design Lubrication## Notes

### Acknowledgements

The second author acknowledges support by the Scientific and Technological Research Council of Turkey (TÜBİTAK) under the 1001 Programme (Grant No. 114M406).

### References

- Balzani D, Scheunemann L, Brands D, Schröder J (2014) Construction of two- and three-dimensional statistically similar RVEs for coupled micro-macro simulations. Comput Mech 54:1269–1284CrossRefMATHGoogle Scholar
- Bayada G, Chambat M (1988) New models in the theory of the hydrodynamic lubrication of rough surfaces. J Tribol 110:402–407CrossRefGoogle Scholar
- Bayada G, Ciuperca I, Jai M (2006) Homogenized elliptic equations and variational inequalities with oscillating parameters. Application to the study of thin flow behavior with rough surfaces. Nonlinear Anal Real World Appl 7:950–966MathSciNetCrossRefMATHGoogle Scholar
- Bendsøe MP (1989) Optimal shape design as a material distribution problem. Structural Optimization 1(4):193–202CrossRefGoogle Scholar
- Bendsøe MP, Sigmund O (2004) Topology optimization: theory, methods and applications, 2nd edn. Springer, BerlinCrossRefMATHGoogle Scholar
- Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158MathSciNetCrossRefMATHGoogle Scholar
- Buscaglia G, Jai M (2000) Sensitivity analysis and Taylor expansions in numerical homogenization problems. Numer Math 85:49–75MathSciNetCrossRefMATHGoogle Scholar
- Buscaglia GC, Ausas RF, Jai M (2006) Optimization tools in the analysis of micro-textured lubricated devices. Inverse Prob Sci Eng 14:365–378CrossRefMATHGoogle Scholar
- Chen B-C, Silva ECN, Kikuchi N (2001) Advances in computational design and optimization with application to MEMS. Int J Numer Methods Eng 52(1-2):23–62MathSciNetCrossRefGoogle Scholar
- Christensen PW, Klarbring A (2010) An introduction to structural optimization. Springer, BerlinMATHGoogle Scholar
- Coelho PG, Fernandes PR, Guedes JM, Rodrigues H (2008) A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Struct Multidisc Optim 35:107–115CrossRefGoogle Scholar
- Costa HL, Hutchings IM (2015) Some innovative surface texturing techniques for tribological purposes. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 229:429–448CrossRefGoogle Scholar
- Diaz A, Sigmund O (1995) Checkerboard patterns in layout optimization. Structural Optimization 10(1):40–45CrossRefGoogle Scholar
- Dobrica MB, Fillon M, Pascovici MD, Cicone T (2010) Optimizing surface texture for hydrodynamic lubricated contacts using a mass-conserving numerical approach. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology 224(8):737–750CrossRefGoogle Scholar
- Fabricius J, Tsandzana A, Perez-Rafols F, Wall P (2017) A comparison of the roughness regimes in hydrodynamic lubrication. J Tribol, (in press)Google Scholar
- Fesanghary M, Khonsari MM (2011) On the shape optimization of self-adaptive grooves. Tribol Trans 54(2):256–264CrossRefGoogle Scholar
- Fujii D, Chen BC, Kikuchi N (2001) Composite material design of two-dimensional structures using the homogenization design method. Int J Numer Methods Eng 50(9):2031–2051MathSciNetCrossRefMATHGoogle Scholar
- Guest JK, Prévost JH, Belytschko T (2004) Achieving minimum length scale in topology optimization using nodal design variables and projection functions. Int J Numer Methods Eng 61:238–254MathSciNetCrossRefMATHGoogle Scholar
- Guzek A, Podsiadlo P, Stachowiak GW (2013) Optimization of textured surface in 2D parallel bearings governed by the Reynolds equation including cavitation and temperature. Tribology Online 8(1):7–21CrossRefGoogle Scholar
- Hamrock B, Schmid S, Jacobson B (2004) Fundamentals of fluid film lubrication. CRC Press, Boca RatonCrossRefGoogle Scholar
- Huang X, Zhou S, Sun G, Li G, Xie YM (2015) Topology optimization for microstructures of viscoelastic composite materials. Comput Methods Appl Mech Eng 283:503–516CrossRefGoogle Scholar
- Kato J, Yachi D, Terada K, Kyoya T (2014) Topology optimization of micro-structure for composites applying a decoupling multi-scale analysis. Struct Multidisc Optim 49:595–608MathSciNetCrossRefGoogle Scholar
- Lee J-H, Singer JP, Thomas EL (2012) Micro-/nanostructured mechanical metamaterials. Adv Mater 24:4782–4810CrossRefGoogle Scholar
- Matsui K, Terada K (2004) Continuous approximation of material distribution for topology optimization. Int J Numer Methods Eng 59:1925–1944MathSciNetCrossRefMATHGoogle Scholar
- Michelaris P, Tortorelli DA, Vidal CA (1994) Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity. Int J Numer Methods Eng 37:2471–2499CrossRefMATHGoogle Scholar
- Mitsuya Y, Fukui S (1986) Stokes roughness effects on hydrodynamic lubrication. Part I — comparison between incompressible and compressible lubricating films. J Tribol 108:151–158CrossRefGoogle Scholar
- Nakshatrala PB, Tortorelli DA, Nakshatrala KB (2013) Nonlinear structural design using multiscale topology optimization. Part i: static formulation. Comput Methods Appl Mech Eng 261–262:167–176MathSciNetCrossRefMATHGoogle Scholar
- Neves MM, Rodrigues H, Guedes JM (2000) Optimal design of periodic linear elastic microstructures. Comput Struct 76:421–429CrossRefGoogle Scholar
- Niu B, Yan J, Cheng G (2009) Optimum structure with homogeneous optimum cellular material with maximum fundamental frequency. Struct Multidisc Optim 39:115–132CrossRefGoogle Scholar
- Noël L, Duysinx P (2016) Shape optimization of microstructural designs subject to local stress constraints within an XFEM-level set framework. Struct Multidisc Optim 10.1007/s00158-016-1642-8
- Park S, Cheng X, Böker A, Tsarkova L (2016) Hierarchical manipulation of block copolymer patterns on 3D topographic substrates: beyond graphoepitaxy. Adv Mater 28:6900–6905CrossRefGoogle Scholar
- Rodrigues H, Guedes JM, Bendsøe MP (2002) Hierarchical optimization of material and structure. Struct Multidisc Optim 24:1–10CrossRefGoogle Scholar
- Rozvany G, Zhou M, Birker T (1992) Generalized shape optimization without homogenization. Structural Optimization 4:250–252CrossRefGoogle Scholar
- Sanchez-Palencia E (1980) Non-homogeneous media and vibration theory. Springer, BerlinMATHGoogle Scholar
- Scaraggi M (2014) Optimal textures for increasing the load support in a thrust bearing pad geometry. Tribol Lett 53(1):127–143CrossRefGoogle Scholar
- Shen C, Khonsari MM (2015) Numerical optimization of texture shape for parallel surfaces under unidirectional and bidirectional sliding. Tribol Int 82:1–11CrossRefGoogle Scholar
- Sigmund O (1994) Materials with prescribed constitutive parameters: an inverse homogenization problem. Int J Solids Struct 31:2313–2329MathSciNetCrossRefMATHGoogle Scholar
- Sigmund O (2000) A new class of extremal composites. J Mech Phys Solids 48:397–428MathSciNetCrossRefMATHGoogle Scholar
- Sigmund O (2007) Morphology-based black and white filters for topology optimization. Struct Multidisc Optim 33:401–424CrossRefGoogle Scholar
- Sigmund O, Petersson J (1998) Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima. Structural Optimization 16(1):68–75CrossRefGoogle Scholar
- Sigmund O, Torquato S (1996) Composites with extremal thermal expansion coefficients. Appl Phys Lett 69:3203–3205CrossRefGoogle Scholar
- Sivapuram R, Dunning PD, Kim HA (2016) Simultaneous material and structural optimization by multiscale topology optimization. Struct Multidisc Optim 54:1267–1281MathSciNetCrossRefGoogle Scholar
- Svanberg K (1987) The method of moving asymptotes — a new method for structural optimization. Int J Numer Methods Eng 24:359–373MathSciNetCrossRefMATHGoogle Scholar
- Svanberg K, Svärd H (2013) Density filters for topology optimization based on the Pythagorean means. Struct Multidisc Optim 48: 859–875MathSciNetCrossRefGoogle Scholar
- Szeri AZ (2011) Fluid film lubrication. Cambridge University Press, CambridgeMATHGoogle Scholar
- Torquato S (2002) Random heterogeneous materials: microstructure and macroscopic properties. Springer, BerlinCrossRefMATHGoogle Scholar
- Waseem A, Temizer İ, Kato J, Terada K (2016) Homogenization-based design of surface textures in hydrodynamic lubrication. Int J Numer Methods Eng 108:1427–1450MathSciNetCrossRefGoogle Scholar
- Xia L, Breitkopf P (2014) Concurrent topology optimization design of material and structure within nonlinear multiscale analysis framework. Comput Methods Appl Mech Eng 278:524–542MathSciNetCrossRefGoogle Scholar
- Xia L, Breitkopf P (2015) Multiscale structural topology optimization with an approximate constitutive model for local material microstructure. Comput Methods Appl Mech Eng 286:147–167MathSciNetCrossRefGoogle Scholar
- Yan X, Huang X, Zha Y, Xie YM (2014) Concurrent topology optimization of structures and their composite microstructures. Comput Struct 133:103–110CrossRefGoogle Scholar
- Yıldıran İN, Temizer İ, Çetin B (2017) Homogenization in hydrodynamic lubrication: microscopic regimes and re-entrant textures. J Tribol, (in press)Google Scholar
- Zhang H, Dong G-N, Hua M, Chin K-S (2017) Improvement of tribological behaviors by optimizing concave texture shape under reciprocating sliding motion. J Tribol 139:011702Google Scholar
- Zhang W, Sun S (2006) Scale-related topology optimization of cellular materials and structures. Int J Numer Methods Eng 68:993–1011CrossRefMATHGoogle Scholar
- Zhou S, Li Q (2008) Design of graded two-phase microstructures for tailored elasticity gradients. J Mater Sci 43:5157–5167CrossRefGoogle Scholar