Growth of solid bodies in the framework of shape and topology optimization
Jean-François Ganghoffer1
, Jan Sokolowski2
1LEMTA, Nancy Université. 2, avenue de la Forêt de Haye. BP 160. 54504 Vandoeuvre Cedex. France.
2Institut Élie Cartan de Nancy Université Henri Poincaré Nancy 1. B.P. 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France.
DOI:
https://doi.org/10.7494/cmms.2012.1.0380
Abstract:
In the present paper, a model for growth of elastic bodies is proposed for purposes of shape and topology optimization. Growth of solid bodies is herewith considered in the framework of topology and shape optimization, with the goal of mimicking the natural generation of both stiff and light biological structures, consisting of a solid skeleton immersed into a softer phase. The growth process is modeled as the nucleation and subsequent growth of islands of a hard elastic phase within a softer elastic matrix, which plays the role of a reservoir of nutrients for the supply of the newly generated material. Islands of the generated solid skeleton are modeled as balls of small radius, and the position of their center is determined in such a way that the effective compliance, the product of the compliance by the relative density of the hard phase, is minimal for each new generation event. A growth model is set up from the mass balance with a source term involving the growth rate of mass, taken as a constant. The modeling of the growth process relies on the evaluation of the topological derivative of the effective compliance, which allows finding the optimal position of the center of new inclusions of the generated hard phase. This nucleation process is then followed by the shape optimization of the growing solid bodies. A proper mathematical formulation of the topology and shape optimization of growing elastic solid body is provided, relying on a domain decomposition technique allowing to replace the singularly perturbed geometrical domain by a regularly perturbation of Steklov-Poincaré operator. As an enrichment of this model, surface energy is lastly considered in the framework of a linear elastic constitutive model with surface stress.
Cite as:
Ganghoffer, J., & Sokolowski, J. (2012). Growth of solid bodies in the framework of shape and topology optimization. Computer Methods in Materials Science, 12(1), 9 – 19. https://doi.org/10.7494/cmms.2012.1.0380
Article (PDF):
Keywords:
Continuum growth, Solid bodies, Topology optimization, Topological derivative, Effective compliance
References:
Altenbach, H., Emereyev, V.A., Lebedev, L.P., 2010, On the existence of solution in the linear elasticity with surface stresses. ZAMM – J Applied Math. Meek. Zeitschrift fur Angewandte Mathematik und Mechanik, 90, 3, 231-240.
Ambrosi, D., Mollica, F., 2002, On the mechanics of a growing tumor; Int. J. Engng Sci, 40, 1297-1316.
Ateshian, G.A., 2007, On the theory of reactive mixtures for modeling biological growth, Biomechan. Model. Mechanobiol, 6, 423-445.
Bagge, M., 2000, A model of bone adaptation as an optimization process, J. Biomech., 33, 1349-1357.
Bendsoe, M.P., Kiuchi, N., 1989, Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Engng, 71, 197-224.
Bensoe, M.P., 1989, Optimal shape design as a material distribution problem, Struct. Optim., 1, 193-203.
Di Carlo, A., 2005, Surface and bulk growth unified, Mechanics of Material Forces. Advances in Mechanics and Mathematics, 11, eds. P. Steinmann and G. A. Maugin, Springer.
Epstein, M., 2010, Kinetics of boundary growth, Mech. Res. Comm., 37(5), 453-457.
Epstein, M., Maugin, G.A., 2000, Thermomechanics of volumetric growth in uniform bodies, Int. J. Plasticity, 16, 951-978.
Fernandes, P.R., Rodriguez, H.C., 1999, A material optimization model for bone remodeling around cementless hip stems, Proc. 9th European Conf. on Computational Mechanics, ed. W. Wunderlich, Munich, Germany.
Ganghoffer, J.F., Haussy, B., 2005, Mechanical modeling of growth considering domain variation. Part I: constitutive framework, Int. J. Solids Struct., 42 (15), 4311-4337.
Ganghoffer, J.F., 2010a, On Eshelby tensors in the context of open systems: application to volumetric growth, Int. J. Engng Sci., 48 (12), 2081-2098.
Ganghoffer, J.F., 2010b, Mechanical modeling of growth considering domain variation – Part II: volumetric and surface growth involving Eshelby tensors, J. Mech. Phys. Solids, 58, 9, 1434-1459.
Ganghoffer, J.F., 2011a, Mechanics and thermodynamics of surface growth viewed as moving discontinuities, Mech. Res. Comm., 38 (5), 372-377.
Ganghoffer, J.F., 2012, A contribution to the mechanics and thermodynamics of surface growth. Application to bone remodeling, Int. J. Engng Sci., 50, 1, 166-191.
Garikipati, K., Narayanan, H., Arruda, E.M., Grosh, K., Calve, A., 2003, Material forces in the context of biotissue remodeling, Proc. of the EUROMECH Colloquium 445 Mechanics of material forces, ed. P. Steinmann, G.A. Maugin, Kaiserslautern, Germany.
Hollister, S.J., Brennan, J.M., Kikuchi, N., 1994, A homogenization sampling procedure for calculating trabecular bone effective stiffness and tissue level stress, J. Biomech., 27, 433-444.
Huiskes, R., 2000, If bone is the answer, then what is the question?, j. Anat. , 197, 145-156.
Humphrey, J.D., Rajagopal, K.R., 2002, A constrained mixture model for growth and remodeling of soft tissues, Math. Meth. Mod. App. Sci., 12 (3), 407-430.
Khludnev, A.M., Novotny, A.A., Sokolowski, J., Zochowski. 2009, Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions. J. Mech. Phys. Solids., 57 (10), 1718-1732.
Lubarda, V.A., Hoger, A., 2002, On the mechanics of solids with a growing mass, Int. J. Solids Struct., 39, 4627-4664.
Mattheck, C. 1990, Engineering components grow like trees. Mat. Wiss. U. Werk-sto tech., 21, 143-168.
Narayanan, H., Arruda, E.M., Grosh, K. Garikipati, K., 2009. The micromechanics of fluid-solid interactions during growth in porous soft biological tissue, Biomech. Model. Mechanobiol, 8, 167-181.
Nazarov, S., Sokolowski, J., 2003, Asymptotic analysis of shape functional, Journal de Mathématiques Pures et Appliquées, 8 (2), 125-196.
Rajagopal, K.R., 1995, Multiple natural configurations in continuum mechanics, Report Inst. Comput. Appl. Mech.. 6.
Rajagopal, K.R, Srinivasa, A.R., 1998, Mechanics of the inelastic behavior of materials; Part I: Theoretical underpinnings, Int. J Plast., 14, 945-967.
Rodriguez, E.K., Hoger, A., McCullogh, A.D., 1994, Stress-dependent finite growth in soft elastic tissues. J. Biomech., 21 (4), 455-46.
Sokolowski, J., Zochowski, A., 1999, On the topological derivatives in shape optimization, SIAM Journal on Control and Optimization, 37 (4), 1251-1272.
Sokolowski, J., Zochowski, A., 2001, Topological derivatives of shape functional for elasticity systems, Mechanics of Structures and Machines, 29 (3), 333-351.
Sokolowski, J., Zochowski, A., 2005, Modelling of topological derivatives for contact problems, Numerische Mathematik, 102(1), 145-179.
Sokolowski, J., Zochowski, A., 2008, Topological derivatives for optimization in plane elasticity contact problems. Engineering Analysis with Boundary Elements, 32 (11). 900-908.
Taber, L., Humphrey, J.D., 2001, Stress-modulated growth, remodeling and morphogenesis, Appl. Mech. Rev., 48, 487-545.
Xie, Y.M., Steven, G.P., 1993, A simple evolutionary procedure for structural optimization, Comput. Struct., 49, 885-896.