Computational Physics Group

Karel Matous










Homogenization methods and multiscale modeling:
non-linear problems

Marc G.D. Geers1, Varvara G. Kouznetsova1, K. Matous2 and Julien Yvonnet3

1Department of Mechanical Engineering,
Eindhoven University of Technology,
Eindhoven, The Netherlands.

2Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN, 46556, USA.

3Laboratoire Modelisation et Simulation Multi Echelle,
Universite Paris-Est Marne-la-Vallee,


        This chapter focuses on computational multiscale methods for the mechanical response of nonlinear heterogeneous materials. After a short historical note, a brief overview is given of some recent activities in the field, with a particular focus on nonlinear homogenization methods. The two-scale nonlinear computational homogenization scheme for mechanics is presented, along with details on representative unit cell aspects and statistics. Model performance is advocated through a decoupled implementation and multiscale schemes based on the nonuniform transformation field analysis. High-performance parallel multi-scale implementations of the computational homogenization scheme are addressed in more detail.


       In spite of the progress made over the past decades, a lot of work still remains to be done in the broad field of multi-scale computational engineering. Among the ongoing and expected trends and challenges, the following ones are highlighted:
  • 3D, the third dimension (see section 7):
 Most applications focus on two-dimensional descriptions of the considered materials with their governing deformation mechanisms. Without any doubt, more realistic threedimensional computations and experimental analyses will be necessary, e.g. Shan and Gokhale (2001); Mosby and Matous (2015). The progress to be made here is evidently coupled to the increase in computational power in the coming decade(s).
  • Model reduction:
The high computational cost of nonlinear multiscale solution methods calls for novel efficient approaches that adequately balance computational speed and accuracy. The further development of model reduction techniques in combination with nonlinear multiscale schemes will be a necessity to make further progress.
  •   Interaction with materials science, physics and mathematics:
The various cross-sections presented in this overview have clearly illustrated the growing interaction with materials science. The need for more accurate microstructural deformation models goes hand in hand with the need for more physics in the applied models. At the micron scale and even more at smaller scales, interaction with other physical phenomena is of major importance. It is expected that this trend will become more pronounced in the near future. As mentioned in the introduction, multi-scale methods are of general importance and attract attention from all fields in science and engineering. In particular, the developments in the physics community and the computational mathematics community are quite relevant for the nonlinear mechanics of materials. Increasing the interaction with these neighbouring fields, may speed up the developments considerably.

  • Multi-scale versus multiple scales:
There is a growing interest in establishing correct scale transitions for various nonlinear problems in mechanics of materials, where in the mean time some problems are probably best tackled by considering multiple scales in a single domain. Challenges remain in both, where the most challenging example is probably to transition from damage to fracture across all length scales.
  • Temporal scale transitions:
Besides spatial scales, a lot more attention has to be given to temporal scales. Engineering approaches easily resort to accelerated tests to assess the lifetime of a material in a particular application. Many small-scale deformation mechanisms are characterized by typical time scales, which cannot be altered. Accelerated testing is therefore not always an adequate tool, since it may inhibit certain small-scale deformation processes. The proper incorporation of various (extreme) time scales in multi-scale models therefore remains a challenge. Methods like the GENERIC scheme (Ottinger, 2005; Hutter and Tervoort, 2008b), offer clear opportunities in this sense.


    KM was supported in part by the U.S. Department of Energy, National Nuclear Security Administration under the Predictive Science Academic Alliance Program, under contract no. DE-NA0002377. KM would like to thank two of his graduate research assistants, M. Mosby and A. Gillman, for their numerous contributions to the multi-scale modeling work presented in this chapter.

Download the chapter here.

(c) 2015 Notre  Dame  and Dr. Karel Matous