Multiscale computational modeling has seen tremendous progress inthe past couple of decades as evidenced by an ever-increasingnumber of journal publications, thematic conferences, andworkshops on the topic. One key bottleneck that has not yet beenadequately addressed is that despite continuous developments incomputational modeling and simulation, the computational costsaccrued in multiscale simulations remain significant, and have sofar limited the potential impact of multiscale models and methodsin engineering industry and applications beyond academia.Reduced order modeling ameliorates this tremendous computationalburden by eliminating degrees of freedom from the computationalproblem, as appropriate, to attain computational efficienciesrivaling traditional single scale computations, and has beenidentified as a key area of further research in recent technicalworkshops and conferences. The purpose of this special issue isto provide the state of the art in reduced order modelingapproaches for evaluation of mechanics problems that involvemultiple scales. The contributions in this special issue focuson modeling the response of heterogeneous materials within bothstochastic and deterministic settings. With this special issue,we also attempted to identify those areas that need furtherresearch in addition to establishing the current state of progressin reduced order modeling approaches.

The manuscripts in this special issue address a range of solidmechanics problems concerning model reduction in the context ofmultiscale modeling. The variety of the model reduction methodsadvocated by the manuscripts indicate the multitude of approachesone can take to reduce the computational burden depending on theunderlying physical characteristics of the problem. This varietyalso points to the significant body of research that remains to beundertaken to understand and form the balance betweencomputational tractability and model fidelity in multiscalecomputational modeling of heterogeneous materials.

Sparks and Oskay propose a method to identify the optimal reducedorder models that can best capture the failure response ofheterogeneous materials at a given model order. They employ theEigendeformation-based reduced order homogenization method as themodel reduction approach, which is based on concurrent nonlinearcomputational homogenization. Yvonnet et al. propose anonconcurrent reduced order approach for modeling deformationresponse of heterogeneous materials with hyperelasticconstituents. The primary idea is the numerical construction ofthe constitutive response (via defining a numerical hyperelasticpotential) for the homogenized domain based on off-linecomputations defined over the representative volume element(RVE). The six-dimensional (for 3D problems) strain space isinterrogated using the reduced database model to construct thehomogenized hyperelastic potential. Deng and Chen propose theatomistic field theory (AFT), a new coarse graining method at theatomistic scale that has the capability to accurately predictdynamic fracture. They show that the atomistic features of thecrack propagation can be captured by the AFT method, whileproviding nearly two orders of magnitude computationalefficiency. Gal et al. provide an extended finite elementmodel for inclusion-reinforced matrices that develop a distinctinterphase region. Their model eliminates the need to resolvethe interphase region, the thickness of which may be orders ofmagnitude smaller than the characteristic size of the RVE. Thisapproach is important in modeling the behavior of concrete andother cement based-composites, which are well known to exhibit aninterfacial transition zone. The accuracy characteristics of thereduced order multiscale models are often degraded around fractureprocess zones and high stress gradients. Kerfriden et al.describe a new computational methodology to objectivelydifferentiate between the problem subdomains at which modelreduction approaches can be applied, and the fracture processzones that may require full resolution. They investigatestochastic fracture problems in materials with randominclusions. Bogdanor et al. propose a stochastic calibrationand uncertainty quantification methodology to describe damageevolution in heterogeneous materials. The proposed approachaddresses two important research questions relevant to the reducedorder modeling of heterogeneous materials. They providepromising approaches for calibration of the multiscale modelparameters, and for quantification of various sources and types ofuncertainty such as material parameter variability, as well as themodeling errors due to discretization, model reduction, andothers.

Despite the increasing research activity in the past few years,model reduction methods for multiscale modeling of heterogeneousmaterials continues to be a young field and is likely to remain afertile field of research for quite some time in the future.This special issue addresses some of the critical questionssurrounding reduced order modeling of heterogeneous materialsystems, and poses additional questions that need to be answeredin the near future. The guest editor would like to thank theauthors for their valuable contributions, all the reviewers fortheir assessments of the manuscripts, and Professor Jacob Fish,the editor in chief, for allowing the publication of this specialissue in the International Journal for MultiscaleComputational Engineering.