**6. Conclusions**

In this paper, a concurrent multiscale simulation method is developed to shed light on the material plasticity at the microscale. In this method, the DDD module looks to the FEM module to solve the boundary conditions and the FEM module turns to DDD for plastic strain, which determines the constitutive law of the FEM.

The key issue in the DDD-FEM coupling scheme is how to reasonably regularize the discrete plastic strain to integration points of continuum mechanics. A novel FE mesh-based regularization method is proposed by using a Burgers vector density function and the characteristics of a finite element mesh in ABAQUS. It shows high accuracy and considerably improves the efficiency by avoiding the tedious calculations.

To achieve the concurrent computational model, the DDD module is programmed into Fortran codes, and then it is incorporated into the Abaqus analysis by calling user subroutines provided by Abaqus to determine the constitutive law. The stress field is stored in COMMON blocks and transferred to DDD codes. The realization of the multiscale model capitalizes on the secondary development of FEM software Abaqus, and the implementation details are presented for the first time.

The developed multiscale framework is able to capture the plastic behavior of microscale crystals. Using this multiscale method, uniaxial compression tests of micropillars are performed to study the size effect on plasticity and its underlying mechanisms. The observed secant strain hardening of the modulus and yield stress might be due to the high probability of dislocations in smaller size to escape from the free surface.

This multiscale framework can be potentially employed in studying the plastic behavior of materials with complex microstructures and deeply revealing its underlying mechanisms. In future work, the developed simulation model will consider a variety of complex dislocation motions, such as dislocation climb, interactions between dislocation and grain boundary, and dislocation junction formation.

**Author Contributions:** Z.Z.: Methodology, investigation, data curation, formal analysis, writing—original draft preparation. Z.T.: Conceptualization, supervision, resources, writing—reviewing and editing, funding acquisition. X.J.: Supervision, project administration, funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** The authors would like to acknowledge the funding support from the UK's EPSRC Future Metrology Hub (Ref: EP/P006930/1), the UK's STFC Innovation Partnership Scheme (STFC-IPS) project under gran<sup>t</sup> agreemen<sup>t</sup> No. ST/V001280/1, and the European Union's Horizon 2020 research and innovation program under gran<sup>t</sup> agreemen<sup>t</sup> No. 767589.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable. **Acknowledgments:** The authors would like to acknowledge the scholarship received from the UK's EPSRC Future Metrology Hub (Ref: EP/P006930/1) and the China Scholarship Council (CSC) on Zhenting Zhang's PhD study.

**Conflicts of Interest:** The authors declare no conflict of interest.
