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: