J. Alexandersen and B. Lazarov, Topology optimisation of manufacturable microstructural details without length scale separation using a spectral coarse basis preconditioner, Computer Methods in Applied Mechanics and Engineering, vol.290, pp.156-182, 2015.
DOI : 10.1016/j.cma.2015.02.028

G. Allaire, Shape Optimization by the Homogenization Method, Applied Mathematical Sciences, vol.24, 2002.

G. Allaire, Conception optimale de structures, 2007.

G. Allaire, Numerical Analysis and Optimization. An Introduction to Mathematical Modelling and Numerical Simulation, Numerical Mathematics and Scientific Computation, vol.87, 2007.
URL : https://hal.archives-ouvertes.fr/hal-01242950

G. Allaire and F. Jouve, Coupling the Level Set Method and the Topological Gradient in Structural Optimization, IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials Solid Mechanics and Its Applications, pp.3-12, 2006.
DOI : 10.1007/1-4020-4752-5_1

G. Allaire, F. Jouve, and A. Toader, Structural optimization using sensitivity analysis and a level-set method, Journal of Computational Physics, vol.194, issue.1, pp.363-393, 2004.
DOI : 10.1016/j.jcp.2003.09.032

G. Allaire, D. Gournay, F. Jouve, F. Toader, and A. , Structural optimization using topological and shape sensitivity via a level set method, Control and cybernetics, vol.34, issue.1, pp.59-80, 2005.

G. Allaire, C. Dapogny, G. Delgado, and G. Michailidis, a level set method, ESAIM: Control, Optimisation and Calculus of Variations, vol.20, issue.2, pp.576-611, 2014.
DOI : 10.1051/cocv/2013076

URL : https://hal.archives-ouvertes.fr/hal-00933545

G. Allaire, F. Jouve, and G. Michailidis, Thickness control in structural optimization via a level set method. HAL preprint: hal-00985000v1, to appear in Structural and Multidisciplinary Optimization, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00985000

L. Ambrosio and G. Buttazzo, An optimal design problem with perimeter penalization, Calculus of Variations and Partial Differential Equations, vol.39, issue.312, pp.55-69, 1993.
DOI : 10.1007/BF02163264

J. Archard, Contact and Rubbing of Flat Surfaces, Journal of Applied Physics, vol.24, issue.8, pp.981-988, 1953.
DOI : 10.1063/1.1721448

J. Archard and W. Hirst, The Wear of Metals under Unlubricated Conditions, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.236, issue.1206, pp.397-410, 1206.
DOI : 10.1098/rspa.1956.0144

N. Axén and S. Jacobson, A model for the abrasive wear resistance of multiphase materials, Wear, vol.174, issue.1-2, pp.187-199, 1994.
DOI : 10.1016/0043-1648(94)90101-5

R. Barron, Cryogenic treatment of metals to improve wear resistance, Cryogenics, vol.22, issue.8, pp.409-413, 1982.
DOI : 10.1016/0011-2275(82)90085-6

M. Bendsøe and N. Kikuchi, Generating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, vol.71, issue.2, pp.197-224, 1988.
DOI : 10.1016/0045-7825(88)90086-2

M. Bendsøe and S. O. , Topology optimization: theory, methods and applications Constrained optimization and Lagrange multiplier methods The interaction of wear and dynamics of a simple mechanism, Berlin Bertsekas DP Journal of Tribology, vol.119, issue.3, p.597, 1996.

P. Blau, Fifty years of research on the wear of metals, Tribology International, vol.30, issue.5, pp.321-331, 1997.
DOI : 10.1016/S0301-679X(96)00062-X

J. Céa, Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. RAIRO-Modélisation mathématique et analyse numérique, pp.371-402, 1986.

J. Céa, S. Garreau, P. Guillaume, and M. Masmoudi, The shape and topological optimizations connection, Computer Methods in Applied Mechanics and Engineering, vol.188, issue.4, pp.713-726, 2000.
DOI : 10.1016/S0045-7825(99)00357-6

V. Challis, A. Roberts, and A. Wilkins, Fracture resistance via topology optimization, Structural and Multidisciplinary Optimization, vol.35, issue.2, pp.263-271, 2008.
DOI : 10.1007/s00158-007-0160-0

R. Chattopadhyay, Surface wear: analysis, treatment, and prevention. ASM international Chavel I (2010) Riemannian Geometry, A Modern Introduction, 2001.

J. Chen, V. Shapiro, K. Suresh, and I. Tsukanov, Shape optimization with topological changes and parametric control, International Journal for Numerical Methods in Engineering, vol.54, issue.3, pp.313-346, 2007.
DOI : 10.1002/nme.1943

H. Choi, D. Lee, and J. Lee, Optimization of a railway wheel profile to minimize flange wear and surface fatigue, Wear, vol.300, issue.1-2, pp.225-233, 2013.
DOI : 10.1016/j.wear.2013.02.009

C. Chongyi, W. Chengguo, and Y. J. , Study on numerical method to predict wheel/rail profile evolution due to wear, Wear, vol.269, issue.3-4, pp.3-4167, 2010.
DOI : 10.1016/j.wear.2009.12.031

M. Ciavarella, D. Hills, and G. Monno, The influence of rounded edges on indentation by a flat punch, Proceedings of the Institution of Mechanical Engineers, pp.319-327, 1998.
DOI : 10.1243/0954406981521259

R. Colbert, B. Krick, A. Dunn, J. Vail, N. Argibay et al., Uncertainty in Pin-on-Disk Wear Volume Measurements Using Surface Scanning Techniques, Tribology Letters, vol.35, issue.1, pp.129-131, 2011.
DOI : 10.1007/s11249-010-9744-8

C. Dapogny, ´. Ecole-doctorale-paris-centre-delfour, M. Zolésio, and J. , Shape optimization, level set methods on unstructured meshes and mesh evolution Shapes and geometries Analysis, differential calculus and optimization Evolution of wear in a two-dimensional bushing, Tribology Transactions, vol.4, issue.472, pp.257-262, 2001.

D. Dickrell, D. Dooner, and W. Sawyer, The Evolution of Geometry for a Wearing Circular Cam: Analytical and Computer Simulation With Comparison to Experiment, Journal of Tribology, vol.125, issue.1, p.187, 2003.
DOI : 10.1115/1.1504092

P. Duhamel and M. Vetterli, Fast Fourier Transforms, Signal Processing, vol.19, issue.4, pp.259-299, 1990.
DOI : 10.1201/9781420046076-c7

G. Erickson, M. Sidebottom, J. Curry, D. Kay, S. Kuhn-hendricks et al., Paleo-tribology: Development of wear measurement techniques and a three-dimensional model revealing how grinding dentitions self-wear to enable functionality . Surface Topography: Metrology and Properties Special Issue Article Eschenauer HA Topology optimization of continuum structures: A review, Olhoff N Applied Mechanics Reviews, vol.54, issue.4, p.331, 2001.

F. Feppon, G. Michailidis, M. Sidebottom, B. Krick, N. Vermaak et al., Efficient steady-state computation for wearing of multi-material composites. Preprint submitted to Computational wear prediction of a total knee replacement from in vivo kinematics, ASME Journal of tribology Journal of Biomechanics, vol.38, issue.2, pp.305-314, 2005.

J. Gomes and O. Faugeras, Reconciling distance functions and level sets. Scale-Space Theories in Computer Vision, pp.70-81, 1999.
URL : https://hal.archives-ouvertes.fr/inria-00073006

J. Guest, J. Prévost, and T. Belytschko, Achieving minimum length scale in topology optimization using nodal design variables and projection functions, International Journal for Numerical Methods in Engineering, vol.61, issue.2, pp.238-254, 2004.
DOI : 10.1002/nme.1064

S. Han and T. Blanchet, Experimental Evaluation of a Steady-State Model for the Wear of Particle-Filled Polymer Composite Materials, Journal of Tribology, vol.119, issue.4, pp.694-699, 1997.
DOI : 10.1115/1.2833871

C. Hatchett, Experiments and Observations on the Various Alloys, on the Specific Gravity, and on the Comparative Wear of Gold. Being the Substance of a Report Made to the Right Honourable the Lords of the Committee of Privy Council, Appointed to Take into Consideration the State of the Coins of This Kingdom, and the Present Establishment and Constitution of His Majesty's Mint, Philosophical Transactions of the Royal Society of London, vol.93, issue.0, pp.43-194, 1803.
DOI : 10.1098/rstl.1803.0005

L. He, C. Kao, and S. Osher, Incorporating topological derivatives into shape derivatives based level set methods, Journal of Computational Physics, vol.225, issue.1, pp.891-909, 2007.
DOI : 10.1016/j.jcp.2007.01.003

F. Hecht, New development in freefem++, Journal of Numerical Mathematics, vol.20, issue.3-4, pp.3-4251, 2012.
DOI : 10.1515/jnum-2012-0013

D. Hilding, A. Klarbring, and J. Petersson, Optimization of Structures in Unilateral Contact, Applied Mechanics Reviews, vol.52, issue.4, 1995.
DOI : 10.1115/1.3098931

D. Hilding, B. Torstenfelt, and A. Klarbring, A computational methodology for shape optimization of structures in frictionless contact, Computer Methods in Applied Mechanics and Engineering, vol.190, issue.31, pp.4043-4060, 2001.
DOI : 10.1016/S0045-7825(00)00310-8

W. Hirst and J. Lancaster, Surface Film Formation and Metallic Wear, Journal of Applied Physics, vol.27, issue.9, pp.1057-1065, 1956.
DOI : 10.1063/1.1722541

S. Hovis, J. Talia, and R. Scattergood, Erosion in multiphase systems, Wear, vol.108, issue.2, pp.139-155, 1986.
DOI : 10.1016/0043-1648(86)90093-1

I. Jang, D. Burris, P. Dickrell, P. Barry, C. Santos et al., Sliding orientation effects on the tribological properties of polytetrafluoroethylene, Journal of Applied Physics, vol.102, issue.12, p.123509, 2007.
DOI : 10.1063/1.2821743

M. Jansen, B. Lazarov, M. Schevenels, and S. O. , On the similarities between micro/nano lithography and topology optimization projection methods, Structural and Multidisciplinary Optimization, vol.89, issue.1???3, pp.717-730, 2013.
DOI : 10.1007/s00158-013-0941-6

L. Johansson, Numerical Simulation of Contact Pressure Evolution in Fretting, Journal of Tribology, vol.116, issue.2, pp.247-254, 1994.
DOI : 10.1115/1.2927205

A. Kerr, Elastic and Viscoelastic Foundation Models, Journal of Applied Mechanics, vol.31, issue.3, p.491, 1964.
DOI : 10.1115/1.3629667

M. Khruschov, Principles of abrasive wear, Wear, vol.28, issue.1, pp.69-88, 1974.
DOI : 10.1016/0043-1648(74)90102-1

N. Kim, D. Won, D. Burris, B. Holtkamp, G. Gessel et al., Finite element analysis and experiments of metal/metal wear in oscillatory contacts, Wear, vol.258, issue.11-12, pp.11-121787, 2005.
DOI : 10.1016/j.wear.2004.12.014

J. Lancaster, Abrasive wear of polymers, Wear, vol.14, issue.4, pp.223-239, 1969.
DOI : 10.1016/0043-1648(69)90047-7

G. Lee, C. Dharan, and R. Ritchie, A physically-based abrasive wear model for composite materials, Wear, vol.252, issue.3-4, pp.3-4322, 2002.
DOI : 10.1016/S0043-1648(01)00896-1

J. Lengiewicz and S. Stupkiewicz, Efficient model of evolution of wear in quasi-steady-state sliding contacts, Wear, vol.303, issue.1-2, pp.611-621, 2013.
DOI : 10.1016/j.wear.2013.03.051

J. Luo, Z. Luo, S. Chen, L. Tong, and M. Wang, A new level set method for systematic design of hinge-free compliant mechanisms, Computer Methods in Applied Mechanics and Engineering, vol.198, issue.2, pp.318-331, 2008.
DOI : 10.1016/j.cma.2008.08.003

V. Markine, I. Shevtsov, and C. Esveld, An inverse shape design method for railway wheel profiles, Structural and Multidisciplinary Optimization, vol.1, issue.3, pp.243-253, 2007.
DOI : 10.1007/s00158-006-0049-3

J. Marzeda, V. Pauk, M. Wo´zniakwo´zniak, N. Meisel, A. Gaynor et al., Contact of a rigid flat punch with a wedge supported by the winkler foundation Multiple-material topology optimization of compliant mechanisms created via polyjet 3d printing In: 24th Annual international solid freeform fabrication symposium an additive manufacturing conference Mukras S Numerical integration schemes and parallel computation for wear prediction using finite element method, Journal of Theoretical and Applied Mathematics Wear, vol.266, pp.7-8822, 2001.

S. Mukras, N. Kim, N. Mauntler, T. Schmitz, and W. Sawyer, Analysis of planar multibody systems with revolute joint wear, Wear, vol.268, issue.5-6, pp.5-6643, 2010.
DOI : 10.1016/j.wear.2009.10.014

F. Murat and L. Tartar, Calcul des variations et homogénéisation Topology optimization of quasistatic contact problems, Coll. Dir. ´ Etudes et Recherches EDF International Journal of Applied Mathematics and Computer Science, vol.57, issue.222, pp.269-280, 1985.

P. Neittaanmäki, J. Sokolowski, and J. Zolesio, Optimization of the domain in elliptic variational inequalities, Applied Mathematics & Optimization, vol.54, issue.1, pp.85-98, 1988.
DOI : 10.1007/BF01443616

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces, Applied Mechanics Reviews, vol.57, issue.3, p.15, 2004.
DOI : 10.1115/1.1760520

S. Osher and J. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, vol.79, issue.1, pp.12-49, 1988.
DOI : 10.1016/0021-9991(88)90002-2

P. Põdra and S. Andersson, Wear simulation with the Winkler surface model, Wear, vol.207, issue.1-2, pp.79-85, 1997.
DOI : 10.1016/S0043-1648(96)07468-6

P. Põdra and S. Andersson, Finite element analysis wear simulation of a conical spinning contact considering surface topography, Wear, vol.224, issue.1, pp.13-21, 1999.
DOI : 10.1016/S0043-1648(98)00318-4

P. Põdra and S. Andersson, Simulating sliding wear with finite element method, Tribology International, vol.32, issue.2, pp.71-81, 1999.
DOI : 10.1016/S0301-679X(99)00012-2

I. Páczelt and Z. Mróz, On optimal contact shapes generated by wear, International Journal for Numerical Methods in Engineering, vol.249, issue.229, pp.1250-1287, 2005.
DOI : 10.1002/nme.1321

P. Pasternak, On a new method of analysis of an elastic foundation by means of two foundation constants. Gosudarstvennoe Izdatel'stvo Litearturi po Stroitel'stvu i Arkhitekture, 1954.

B. Prasad, Effect of microstructure on the sliding wear performance of a Zn???Al???Ni alloy, Wear, vol.240, issue.1-2, pp.100-112, 2000.
DOI : 10.1016/S0043-1648(00)00369-0

E. Rabinowicz and A. Mutis, Effect of abrasive particle size on wear, Wear, vol.8, issue.5, pp.381-390, 1965.
DOI : 10.1016/0043-1648(65)90169-9

E. Rabinowicz, L. Dunn, and P. Russell, A study of abrasive wear under three-body conditions, Wear, vol.4, issue.5, pp.345-355, 1961.
DOI : 10.1016/0043-1648(61)90002-3

K. Rowe, G. Erickson, W. Sawyer, and B. Krick, Evolution in Surfaces: Interaction of Topography with Contact Pressure During Wear of Composites Including Dinosaur Dentition, Tribology Letters, vol.1985, issue.3, pp.249-255, 2014.
DOI : 10.1007/s11249-013-0271-2

W. Sawyer, Wear predictions for a simple-cam including the coupled evolution of wear and load, Lubr Eng, vol.57, issue.9, pp.31-36, 2001.

W. Sawyer, Surface Shape and Contact Pressure Evolution in Two Component Surfaces: Application to Copper Chemical Mechanical Polishing, Tribology Letters, vol.17, issue.2, pp.139-145, 2004.
DOI : 10.1023/B:TRIL.0000032438.56992.be

W. Sawyer, N. Argibay, D. Burris, and B. Krick, Mechanistic Studies in Friction and Wear of Bulk Materials, Annual Review of Materials Research, vol.44, issue.1, pp.395-427, 2014.
DOI : 10.1146/annurev-matsci-070813-113533

M. Schevenels, B. Lazarov, and S. O. , Robust topology optimization accounting for spatially varying manufacturing errors, Computer Methods in Applied Mechanics and Engineering, vol.200, issue.49-52, pp.3613-3627, 2011.
DOI : 10.1016/j.cma.2011.08.006

H. Schmidt and J. Hattel, Thermal modelling of friction stir welding, Scripta Materialia, vol.58, issue.5, pp.332-337, 2008.
DOI : 10.1016/j.scriptamat.2007.10.008

T. Schmitz, J. Action, D. Burris, J. Ziegert, and W. Sawyer, Wear-Rate Uncertainty Analysis, Journal of Tribology, vol.126, issue.4, p.802, 2004.
DOI : 10.1115/1.1792675

F. Schury, M. Stingl, F. Wein, C. Seepersad, J. Allen et al., Efficient Two-Scale Optimization of Manufacturable Graded Structures, SIAM Journal on Scientific Computing, vol.34, issue.6, pp.711-7331285, 2006.
DOI : 10.1137/110850335

M. Sidebottom, F. Feppon, N. Vermaak, and B. Krick, Modelling wear of multi-material composite wear surfaces Manufacturing tolerant topology optimization, ASME Journal of Tribology Submitted Sigmund O Acta Mechanica Sinica, vol.25, issue.2, pp.227-239, 2009.

H. Sin, N. Saka, and N. Suh, Abrasive wear mechanisms and the grit size effect, Wear, vol.55, issue.1, pp.163-190, 1979.
DOI : 10.1016/0043-1648(79)90188-1

J. Sokolowski and A. Zochowski, On the Topological Derivative in Shape Optimization, SIAM Journal on Control and Optimization, vol.37, issue.4, pp.1251-1272, 1999.
DOI : 10.1137/S0363012997323230

N. Strömberg and A. Klarbring, Topology optimization of structures in unilateral contact, Structural and Multidisciplinary Optimization, vol.24, issue.1, pp.57-64, 2010.
DOI : 10.1007/s00158-009-0407-z

K. Tankala and R. Kapoor, Method for improving wear resistance of abrasive tools US Patent 6, p.188, 2001.

L. Tartar, The General Theory of Homogenization: A Personalized Introduction Lecture Notes of the Unione Matematica Italiana Simulation of wear in a rolling-sliding contact by asemi-winkler model and the archard's wear law, Wear, vol.7, pp.7-8817, 2004.
DOI : 10.1007/978-3-642-05195-1

N. Vermaak, G. Michailidis, G. Parry, R. Estevez, G. Allaire et al., Material interface effects on the topology optimizationof multi-phase structures using a level set method, Structural and Multidisciplinary Optimization, vol.33, issue.2, pp.623-644, 2014.
DOI : 10.1007/s00158-014-1074-2

URL : https://hal.archives-ouvertes.fr/hal-01120063

F. Wang, B. Lazarov, and S. O. , On projection methods, convergence and robust formulations in topology optimization, Structural and Multidisciplinary Optimization, vol.197, issue.1???2, pp.767-784, 2011.
DOI : 10.1007/s00158-010-0602-y

M. Wang, X. Wang, and D. Guo, A level set method for structural topology optimization, Computer Methods in Applied Mechanics and Engineering, vol.192, issue.1-2, pp.227-246, 2003.
DOI : 10.1016/S0045-7825(02)00559-5

R. Willing and I. Kim, Three dimensional shape optimization of total knee replacements for reduced wear, Structural and Multidisciplinary Optimization, vol.404, issue.4, pp.405-414, 2009.
DOI : 10.1007/s00158-008-0281-0

T. Yamada, K. Izui, S. Nishiwaki, and A. Takezawa, A topology optimization method based on the level set method incorporating a fictitious interface energy, Computer Methods in Applied Mechanics and Engineering, vol.199, issue.45-48, pp.45-482876, 2010.
DOI : 10.1016/j.cma.2010.05.013

M. Zhou, B. Lazarov, and S. O. , Topology optimization for optical projection lithography with manufacturing uncertainties, Applied Optics, vol.53, issue.12, pp.2720-2729, 2014.
DOI : 10.1364/AO.53.002720

A. Zmitrowicz, Wear patterns and laws of wear -a review, Theoretical And Applied Mechanics, vol.44, pp.219-253, 1803.

Z. Gahr and K. , Microstructure and wear of materials, 1987.