Computational Physics Group

Karel Matous










A Review of Predictive Nonlinear Theories for
Multiscale Modeling of Heterogeneous Materials

K. Matous1, Marc G.D. Geers2, Varvara G. Kouznetsova2, and Andrew Gillman1

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

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


      Since the beginning of the industrial age, material performance and design have been in the midst of innovation of many disruptive technologies. Today's electronics, space, medical, transportation, and other industries are enriched by development, design and deployment of composite, heterogeneous and multifunctional materials. As a result, materials innovation is now considerably outpaced by other aspects from component design to product cycle. In this article, we review predictive nonlinear theories for multiscale modeling of heterogeneous materials. Deeper attention is given to multiscale modeling in space and to computational homogenization in addressing challenging materials science questions. Moreover, we discuss a state-of-the-art platform in predictive image-based, multiscale modeling with co-designed simulations and experiments that executes on the world's largest supercomputers. Such a modeling framework consists of experimental tools, computational methods, and digital data strategies. Once fully completed, this collaborative and interdisciplinary framework can be the basis of Virtual Materials Testing standards and aids in the development of new material formulations. Moreover, it will decrease the time to market of innovative products.


    Although existing computational science and engineering capabilities for modeling nonlinear material behavior are impressive, it is unclear how much impact they have had on materials engineering and design in general. Moreover, an integrated vision and/or framework that would be widely usable in multiscale computational materials engineering is lacking. Therefore, establishing microstructure-statistics-property relations underpinned by Integrated Computational Materials Engineering based on an image-based (datadriven) multiscale modeling framework with co-designed simulations and experiments has the potential to transform the materials science and engineering field. Such a framework would ultimately lead to Virtual Materials Testing standards and aid in the development of new material formulations as well as decrease the time to market of innovative products.
    In spite of the progress made over the past decades, a lot of work still remains to be done, and several technological barriers have to be mitigated to make integrated computational materials engineering a reality for a broad spectrum of materials applications. Among the ongoing challenges and expected trends the following ones are outlined:
  • Fully coupled hierarchical and concurrent multiscale models in 3D
          Without any doubt, three-dimensional multiscale simulations (both hierarchical and concurrent) that resolve all essential length (and time) scales are needed. Fully coupled multiscale simulations of realistic devices/components (test specimen) with complex physics are still uncommon and multiscale modeling is accomplished by the input and output of stand-alone codes mostly in twodimensions only. The progress to be made here is evidently coupled to the next item on Advances in Computing Capabilities.
  •     Advances in computing capabilities
            Exponential growth in computer performance over the last several decades is one of the factors in integrated computational materials engineering feasibility. Moreover, high performance computing has had a dramatic effect on multiscale computational materials engineering. If current computer resources can do X, we envision that exascale machines will provide O(1000 X) improvement in terms of larger multiscale problems being solved, more physics being included, etc. Therefore, development and application of modern parallel computational methods will be paramount in the deployment of integrated computational materials engineering and utilization of the next generation of supercomputers (see Section 6).
  • Intelligent model reduction -- whether applied to computational or experimental data
          Large 3D parallel simulations and experimental/imaging materials characterizations will necessitate the intelligent reduction of information required for the next level of materials integration. Mathematical/Statistical (e.g. machine learning) methods that seek meaningful low-dimensional structures hidden in high-dimensional multiscale data (both computational and experimental) will be important for a variety of tasks, i.e. multiscale solution accelerators, visualization, decision making, uncertainty quantification, reduced order model construction (see Section 7).
  • Co-designed simulations and experiments in 3D
        This review article has clearly illustrated the need for growing interaction with materials science. Materials science provides not only the required insight in the physical and mechanical behavior at small scales, but also the necessary quantitative data (parameters and models) on which any multiscale method relies. If the quality of this data is inadequate, not well understood or physical parameters are not reliable, then multiscale methods cannot possibly yield predictive results. Proper characterization of materials at small scales relies on strong collaboration between mechanics of materials experts with materials scientists. Cutting-edge research areas in 3D nondestructive experimental techniques with in situ measurements are not mature yet, but show a promising path for validation. Experimental measurements co-designed with simulations will have a profound impact on filling gaps in theoretical understanding and calibrating and validating computational methods. This can only be achieved by joining the expertise in experimental mechanics, computational mechanics and materials science.
  • Verification, validation and uncertainty quantification
        Verification and Validation in computational science and engineering have been accepted as a key part of modern physics simulation codes and this trend must continue. Verification in a multiscale setting (i.e. how to efficiently construct verified solutions) is still an open area of work, and validation is linked to the discussion above on Co-designed Simulations and Experiments. Uncertainty is a complex issue with pervasive effects that deserves its own attention. Although UQ is a large and richly developed area of research in the context of computer modeling, applications of multiscale UQ techniques in materials science are still quite limited. Epistemic uncertainty in parameters of macroscopic models (i.e. global models) can be properly linked though multiscale simulations to aleatoric uncertainty of the microscopic (i.e. fine scale) models. Therefore, uncertainty quantification, propagation, mitigation and management, which are all relatively recent research areas in multiscale computational science and engineering, have to be further investigated (see Section 8).
  • Multiscale versus multi-level description
        Damage and fracture is one of the challenging problems that hierarchically cascades across all length scales, for example. Although multiscale modeling and multi-level materials design are related, these two endeavors are distinct. Multiscale techniques pertain to data and systems representing two or more distinct spatial and/or temporal scales whereas multi-level approaches aim at data and systems representing multiple levels of material arrangements. In the perspective of integrated computational materials engineering, multiscale modeling (both hierarchical and concurrent) is contributing to progress in multi-level materials design.
  • Multiple temporal scales
        A lot more attention has to be given to temporal scales and to spatial-temporal coupling. Many physical time scales in diffusion and small-scale deformation mechanisms cannot be accelerated. Likewise several small-scale processes happen very quickly, e.g. in many chemical reactions. Scale separation limitations and micro-inertia effects are also areas of interest. Therefore, capturing fully transient regimes and extreme spectra of temporal processes still remains a challenge.
  • Cultural barriers, education and workforce readiness
        If integrated computational materials engineering is to become part of the industrial design chain, a cultural change in the engineering and science disciplines is needed and will require the integration of computational science and engineering into curriculum at all levels (from baccalaureate to doctoral). Skills from statistics, mechanics, physics, chemistry, applied mathematics, scientific computing, data science, materials science, are all needed and will have to enrich traditional single-discipline curricula.


    KM and AG were supported in part by the Department of Energy, National Nuclear Security Administration, under Award Number DE-NA0002377 as part of the Predictive Science Academic Alliance Program II. We would also like to acknowledge computational resources from the 2015 ASCR Leadership Computing Challenge (ALCC) under project number CSC188. KM would like to thank his former graduate research assistant, M. Mosby, for his numerous contributions to the multiscale modeling work presented in this review article. The research of MG has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no [339392].

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2016 Notre  Dame  and Dr. Karel Matous