Synthesis of shape and topology of multi-material structures with a phase-field method

Abstract

In this paper, we present a phase-field method to the problem of shape and topology synthesis of structures with three materials. A single phase model is developed based on the classical phase-transition theory in the fields of mechanics and material sciences. The multi-material synthesis is formulated as a continuous optimization problem within a fixed reference domain. As a single parameter, the phase-field model represents regions made of any of the three distinct material phases and the interface between the regions. The Van der Waals-Cahn-Hilliard theory is applied to define a dynamic process of phase transition. The -convergence theory is used for an approximate numerical solution to this free-discontinuity problem without any explicit tracking of the interface. Within this variational framework, we show that the phase-transition theory leads to a well-posed problem formulation with the effects of "domain regularization" and "region segmentation" incorporated naturally. The proposed phase-field method is illustrated with several 2D examples that have been extensively used in the recent literature of topology optimization, especially in the homogenization based methods. It is further suggested that such a phase-field approach may represent a promising alternative to the widely-used homogenization models for the design of heterogeneous materials and solids, with a possible extension to a general model of multiple material phases.

A phase-field model for compliance shape optimization in nonlinear elasticity

“…The range of usual objective functionals J [u, O] is relatively diverse. The mechanical work of the load, the so-called compliance C = 1 2 ΓN F · u da, is very popular [2][3][4][5]24,[32][33][34][35] since it equals the energy to be absorbed by the elastic structure. A related choice is the L 2 -norm of the internal stresses [2,4,5], O σ 2 F dx.…”

“…Hence, the approach lends itselfto a perimeter regularization. This technique is employed by Wang and Zhou [32] who minimize the compliance of an elastic structure using a triphasic phase field (with one void and two material phases) for which the potential Ψ is equipped with a periodically repeated sequence of three minima to allow for all three possible types of phase transitions. Furthermore, they replace the term Ω |∇v| 2 dx by an edgepreserving smoothing and perform a multiscale relaxation, starting with large ε and successively decreasing it -remarkably beyond the point up to which the grid can still resolve the diffusive interface in a usual fashion.…”

Optimal Design of Nonlinear Multimaterial Structures for Crashworthiness Using Cluster Analysis

“…Phase field-based methods are free boundary tracking methods that avoid the need for re-initialization [29][30][31][32][33][34]. These methods are capable of handling the motion caused by domain states and the motion caused by the domain shape, e.g., the temperature and the mean curvature motion, respectively, so they are also related to shape optimization methods.…”

Phase-field topology optimization model that removes the curvature effects

“…To the best of our knowledge, the first application of the phase-field method to topology optimization was performed by Bourdin and Chambolle (Bourdin and Chambolle, 2003;Bourdin and Chambolle, 2006). For a while since then, the Cahn-Hilliard (CH) model has been widely applied to topology optimization (Burger and Stainko, 2006;Dedè et al, 2012;Wang and Zhou, 2004a;Wang and Zhou, 2004b;Zhou and Wang, 2007;Zhou and Wang, 2006). The CH model can conserve the volume without any operation.…”