Finite-volume micromechanics of periodic materials: Past, present and future

The finite-volume method is now a well-established tool in the numerical engineering community for simulation of a wide range of problems in fluid and solid mechanics. Its acceptance by the mechanics of heterogeneous media community, however, continues to be slow, often characterized by confusion with the finite-element method or so-called higher-order theories. Herein, we provide a brief historical perspective on the evolution of this important technique in the fluid mechanics community, its transition to the solution of solid mechanics boundary-value problems initiated in Europe in 1988, and the recent developments aimed at the solution of unit cell problems of periodic heterogeneous media. The differences and similarities with the finite-element method are highlighted, and the resulting tangible advantages of the finite-volume technique discussed and illustrated. Finally, our most recent results in this area are presented which demonstrate the method's capability of solving unit cell problems with complex architectures in a variety of settings and applications, while revealing undocumented effects of interest in the development of new material microstructures with targeted response. Recent attempts to develop alternative versions of this technique are also discussed, together with our ongoing work to generalize the finite-volume micromechanics approach in order to further enhance its predictive capabilities and efficiency. (C) 2012 Elsevier Ltd. All rights reserved.